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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  iff ( <font color="Maroon" title="c1">N</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:48"/>
( <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies ( <font color="Maroon" title="c1">N</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">N</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  implies <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:48"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
( <font color="Maroon" title="c1">N</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:48"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
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="c1">N</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">N</font> is  <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:48"/>
<font color="Maroon" title="c1">N</font> is  <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> </span><br/>

<a NAME="E5:48"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">J</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">O</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b2">F</font></span>)</span></span>)</span> where <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">F</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span>}</span>  as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#T32" title="WAYBEL19:th.32">WAYBEL19:32</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">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="E3:48_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">N</font>  st  ex <font color="Olive" title="b4">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">N</font> st <br/>( <font color="Olive" title="b4">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">x</font> &amp; not <font color="Olive" title="b4">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">y</font> ) holds <br/> ex <font color="Olive" title="b4">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">N</font> st <br/>( <font color="Olive" title="b4">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">x</font> &amp; not <font color="Olive" title="b4">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">N</font> be   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st  ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">y</font> ) holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> ) implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> ) )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c7">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><span class="kw"> such that </span><a NAME="E2:48_1_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E3:48_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
not <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> &amp; not <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:48_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font></span><br/>

<span class="kw">assume </span><a NAME="E5:48_1_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:48_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></span>;<br/>
<a NAME="E7:48_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:48_1"/>
not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<a NAME="E5:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">W</font>, <font color="Maroon" title="c7">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E7:48_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c6">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E8:48_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E9:48_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">and </span><br/><a NAME="E10:48_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">and </span><br/><a NAME="E11:48_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a></span>;<br/>
<a NAME="E12:48_1"/>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="yellow_8.html#T18" title="YELLOW_8:th.18">YELLOW_8:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:48_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">and </span><br/><a NAME="E15:48_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c8">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E17:48_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">V1</font>
 <span class="kw">and </span><br/><a NAME="E18:48_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c9">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">and </span><br/><a NAME="E19:48_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c9">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">U2</font>, <font color="Maroon" title="c11">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E21:48_1"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c9">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">U2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:48_1"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c10">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font>
 <span class="kw">and </span><br/><a NAME="E23:48_1"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c11">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a></span>;<br/>
<a NAME="E24:48_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
 <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="c3">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">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> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> ;<br/>
<a NAME="E26:48_1"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T21" title="YELLOW_0:th.21">YELLOW_0:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">U1</font> = <font color="Maroon" title="c10">U2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">WU1</font> = <font color="Maroon" title="c6">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">U2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">W1</font> = <font color="Maroon" title="c15">WU1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a></span>;<br/>
<a NAME="E30:48_1"/><span class="lab"><font color="Green" title="E45">A22</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c16">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c15">WU1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E31:48_1"/>
<font color="Maroon" title="c10">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E22:48_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></span>;<br/>
<a NAME="E32:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A23</font></span>: 
<font color="Maroon" title="c14">U1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel14.html#T24" title="WAYBEL14:th.24">WAYBEL14:24</a></span>;<br/>
<a NAME="E33:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c14">U1</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></span>;<br/>
<a NAME="E34:48_1"/>
<font color="Maroon" title="c15">WU1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E48">A25</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E19:48_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E2:48_1_2"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
<font color="Maroon" title="c15">WU1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">WU1</font> or <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:48_1_2"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">WU1</font>
 <span class="kw">and </span><br/><a NAME="E4:48_1_2"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
not <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:48_1_2"/><span class="lab"><font color="Green" title="E52">A29</font></span>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:48_1_2"/>
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">WU1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_2"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E4:48_1_2"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:48_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_2"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E21:48_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E5:48_1_2"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">WU1</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T20" title="WAYBEL12:th.20">WAYBEL12:20</a></span>;<br/>
<a NAME="E36:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A30</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c15">WU1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow12.html#T28" title="YELLOW12:th.28">YELLOW12:28</a></span>;<br/>
<a NAME="E37:48_1"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> = 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></span>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c3">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E26:48_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="yellow_0.html#T27" title="YELLOW_0:th.27">YELLOW_0:27</a></span>
<br/>.= 
<font color="Maroon" title="c12">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></span>
;<br/>
<a NAME="E39:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E21:48_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E40:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:48_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="txt" href="#E37:48_1"><span class="lab"><font color="Green" title="E49">A31</font></span></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></span>;<br/>
<a NAME="E41:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A32</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E42:48_1"/>
<font color="Maroon" title="c16">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c16">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></span>;<br/>
<a NAME="E43:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c16">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1"><span class="lab"><font color="Green" title="E50">A32</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">F1</font> = <font color="Maroon" title="c11">F</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E23:48_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/>
<a NAME="E45:48_1"/><span class="lab"><font color="Green" title="E52">A34</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c17">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E46:48_1"/>
<font color="Maroon" title="c1">N</font> is   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E47:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">U2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:48_1"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="waybel19.html#T37" title="WAYBEL19:th.37">WAYBEL19:37</a></span>;<br/>
<a NAME="E48:48_1"/><span class="kw">then </span>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c16">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c17">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E36:48_1"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="txt" href="#E30:48_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E45:48_1"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="waybel30.html#T15" target="_self" title="WAYBEL30:th.15">Th15</a>, <a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></span>;<br/>
<a NAME="E49:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A35</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c17">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:48_1"><span class="lab"><font color="Green" title="E51">A33</font></span></a></span>;<br/>
<a NAME="E50:48_1"/>
<font color="Maroon" title="c3">S</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="yellow12.html#T14" title="YELLOW12:th.14">YELLOW12:14</a></span>;<br/>
<a NAME="E51:48_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c17">F1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F1</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel30.html#T29" target="_self" title="WAYBEL30:th.29">Th29</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E53:48_1"/><span class="lab"><font color="Green" title="E54">A36</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E54:48_1"/><span class="lab"><font color="Green" title="E55">A37</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F1</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:48_1"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E56:48_1"/><span class="lab"><font color="Green" title="E56">A38</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">and </span><br/><a NAME="E57:48_1"/><span class="lab"><font color="Green" title="E57">A39</font></span>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:48_1"><span class="lab"><font color="Green" title="E55">A37</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c19">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a></span>;<br/>
<a NAME="E59:48_1"/><span class="lab"><font color="Green" title="E58">A40</font></span>: 
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c20">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="yellow12.html#T13" title="YELLOW12:th.13">YELLOW12:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c21">u1</font> = <font color="Maroon" title="c20">u1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">u1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Maroon" title="c21">u1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> )</span><br/><span class="kw">thus </span><a NAME="E1:48_1_4"/>
<font color="Maroon" title="c21">u1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:48_1"><span class="lab"><font color="Green" title="E54">A36</font></span></a>, <a class="txt" href="#E56:48_1"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="txt" href="#E59:48_1"><span class="lab"><font color="Green" title="E58">A40</font></span></a>, <a class="ref" href="waybel_3.html#T8" title="WAYBEL_3:th.8">WAYBEL_3:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c21">u1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font></span><br/><a NAME="E2:48_1_4"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c21">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:48_1"><span class="lab"><font color="Green" title="E57">A39</font></span></a>, <a class="ref" href="yellow12.html#T30" title="YELLOW12:th.30">YELLOW12:30</a></span>;<br/><a NAME="E3:48_1_4"/><span class="kw">then </span>
not <font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c21">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E21:48_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_4"/>
not <font color="Maroon" title="c21">u1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">u2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E62:48_1"/><span class="lab"><font color="Green" title="E59">A41</font></span>: 
<font color="Maroon" title="c21">u2</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E63:48_1"/><span class="lab"><font color="Green" title="E60">A42</font></span>: 
not <font color="Maroon" title="c21">u2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c21">u2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">u2</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Maroon" title="c21">u2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E64:48_1"/>
( <font color="Maroon" title="c21">u2</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Maroon" title="c21">u2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:48_1"><span class="lab"><font color="Green" title="E59">A41</font></span></a>, <a class="txt" href="#E63:48_1"><span class="lab"><font color="Green" title="E60">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:48"/>
<font color="Maroon" title="c1">N</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="waybel_3.html#T24" title="WAYBEL_3:th.24">WAYBEL_3:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
