<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>consider </b><font color="Maroon">Ss</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>, <font color="Maroon">Ts</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>, <font color="Maroon">Sl</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>, <font color="Maroon">Tl</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E2:55"/>
( <font color="Maroon">S</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">T</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E3:55"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">S</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">Ss</font>,<font color="Maroon">Sl</font> &amp; <font color="Maroon">T</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">Ts</font>,<font color="Maroon">Tl</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T29">WAYBEL19:29</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">T</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">Ts</font> &amp; <font color="Maroon">S</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">Ss</font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T45">YELLOW_9:45</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green">E31</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ss</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Ss</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">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Sl</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Sl</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">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ts</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Ts</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">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Tl</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Tl</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">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">f</font> be  <a href="waybel_0.html#V19">meet-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Sl</font>,<font color="Maroon">Tl</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Ss</font>,<font color="Maroon">Ts</font> <b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Ts</font>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Ts</font> ;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E3:55_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E4:55_1_1"/><b>then </b>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, <a class="ref" href="waybel19.html#T37">WAYBEL19:37</a></i>;<br/><a NAME="E5:55_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:55_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E6:55_1_1"/>
( <font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">h</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i>, <a class="txt" href="waybel21.html#E3:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a>, <a class="ref" href="waybel_9.html#T1">WAYBEL_9:1</a></i>;<br/><a NAME="E7:55_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T2">WAYBEL17:2</a></i>;<br/><a NAME="E8:55_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">C</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/><b>hence </b><a NAME="E9:55_1_1"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, , <a class="ref" href="waybel19.html#T41">WAYBEL19:41</a></i>;<br/></div><b>end;</b></div><a NAME="E3:55_1"/><b>then </b>
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E4:55_1"/><b>then </b>
<font color="Maroon">h</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/><b>hence </b><a NAME="E5:55_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i>, </i>;<br/><a NAME="E6:55_1"/>
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:55_1_2"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">F</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:55_1_2"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:55_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel19.html#T43">WAYBEL19:43</a></i>;<br/>
<a NAME="E4:55_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> &amp; <font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:55_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E5:55_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T13">SETWISEO:13</a></i>;<br/>
<a NAME="E6:55_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel20.html#T25">WAYBEL20:25</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel19.html#T27">WAYBEL19:27</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T94">RELAT_1:94</a></i>
<br/>.= 
<font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
;<br/>
 <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_9.html#D2">WAYBEL_9:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">G</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E10:55_1_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">F</font>,<font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:55_1"/>
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E9:55"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E10:55"/><b>then </b>
<font color="Maroon">h</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:55"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<b>assume </b><a NAME="E12:55"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Sl</font>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E2:55_2"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:55_2"/>
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T65">WAYBEL_0:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:55"/><b>then </b>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T8">WAYBEL19:8</a></i>;<br/>
<b>hence </b><a NAME="E15:55"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , <a class="ref" href="waybel19.html#T24">WAYBEL19:24</a></i>;<br/>


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