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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</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>;<br/>

<a NAME="E1:55"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">Ss</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">S</font>, <font color="Maroon" title="c4">Ts</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="c2">T</font>, <font color="Maroon" title="c5">Sl</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c6">Tl</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E3:55"/>
( <font color="Maroon" title="c1">S</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">T</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c1">S</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c3">Ss</font>,<font color="Maroon" title="c5">Sl</font> &amp; <font color="Maroon" title="c2">T</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c4">Ts</font>,<font color="Maroon" title="c6">Tl</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T29" title="WAYBEL19:th.29">WAYBEL19:29</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c2">T</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c4">Ts</font> &amp; <font color="Maroon" title="c1">S</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c3">Ss</font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T45" title="YELLOW_9:th.45">YELLOW_9:45</a></i>;<br/>
<a NAME="E6:55"/><i><font color="Green" title="E38">A4</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="c3">Ss</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">Ss</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">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #) &amp;  <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="c5">Sl</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">Sl</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">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #) &amp;  <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="c4">Ts</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Ts</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="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #) &amp;  <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="c6">Tl</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">Tl</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="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</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/>
<b>let </b><font color="Maroon" title="c7">f</font> be  <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">Sl</font>,<font color="Maroon" title="c6">Tl</font> <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">Ss</font>,<font color="Maroon" title="c4">Ts</font> <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/>
<a NAME="E9:55"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">Ts</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">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" title="c10">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">Ts</font>;<br/><b>reconsider </b><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c10">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">Ts</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">C</font> = <font color="Maroon" title="c11">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/><b>assume </b><a NAME="E3:55_1_1"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><a NAME="E4:55_1_1"/><b>then </b>
<font color="Maroon" title="c12">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E5:55"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="waybel19.html#T37" title="WAYBEL19:th.37">WAYBEL19:37</a></i>;<br/><a NAME="E5:55_1_1"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:55"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="waybel21.html#E1:55_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E6:55_1_1"/>
( <font color="Maroon" title="c11">A</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c9">h</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="waybel21.html#E3:55_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a>, <a class="ref" href="waybel_9.html#T1" title="WAYBEL_9:th.1">WAYBEL_9:1</a></i>;<br/><a NAME="E7:55_1_1"/><b>then </b>
<font color="Maroon" title="c9">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">A</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T2" title="WAYBEL17:th.2">WAYBEL17:2</a></i>;<br/><a NAME="E8:55_1_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">C</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></i>;<br/><b>hence </b><a NAME="E9:55_1_1"/>
<font color="Maroon" title="c9">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E5:55"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="waybel21.html#E5:55_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="waybel19.html#T41" title="WAYBEL19:th.41">WAYBEL19:41</a></i>;<br/></div><b>end;</b></div><a NAME="E3:55_1"/><b>then </b>
<font color="Maroon" title="c9">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E9:55"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E4:55_1"/><b>then </b>
<font color="Maroon" title="c9">h</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T22" title="WAYBEL17:th.22">WAYBEL17:22</a></i>;<br/><b>hence </b><a NAME="E5:55_1"/>
<font color="Maroon" title="c7">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="waybel21.html#T6" target="_self" title="WAYBEL21:th.6">Th6</a></i>;<br/><a NAME="E6:55_1"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c7">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">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" title="c10">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:55_1_2"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c1">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:55_1_2"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:55_1_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel19.html#T43" title="WAYBEL19:th.43">WAYBEL19:43</a></i>;<br/>
<a NAME="E4:55_1_2"/><b>then </b>
(  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:55_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="waybel21.html#T24" target="_self" title="WAYBEL21:th.24">Th24</a></i>;<br/>
<a NAME="E5:55_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">F</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T13" title="SETWISEO:th.13">SETWISEO:13</a></i>;<br/>
<a NAME="E6:55_1_2"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="ref" href="waybel20.html#T25" title="WAYBEL20:th.25">WAYBEL20:25</a></i>;<br/>
<i><font color="Green" title="E42">A10</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_9.html#D8" title="WAYBEL_9:def.8">WAYBEL_9:def 8</a></i>
<br/>.= 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c10">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel19.html#T27" title="WAYBEL19:th.27">WAYBEL19:27</a></i>
<br/>.= 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">F</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>
<br/>.= 
<font color="Maroon" title="c11">G</font>
<b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>
;<br/>
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="waybel21.html#T44" target="_self" title="WAYBEL21:th.44">Th44</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_9.html#D2" title="WAYBEL_9:def.2">WAYBEL_9:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c11">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="waybel21.html#E8:55_1_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E10:55_1_2"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">F</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:55_1_2"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:55_1"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c7">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">X</font>
 <b>by </b><i><a class="ref" href="waybel21.html#T4" target="_self" title="WAYBEL21:th.4">Th4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:55"/>
<font color="Maroon" title="c7">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E12:55"/><b>then </b>
<font color="Maroon" title="c9">h</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="waybel21.html#T6" target="_self" title="WAYBEL21:th.6">Th6</a></i>;<br/>
<a NAME="E13:55"/><b>then </b><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c9">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22" title="WAYBEL17:th.22">WAYBEL17:22</a></i>;<br/>
<b>assume </b><a NAME="E14:55"/><i><font color="Green" title="E41">A12</font></i>: 
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c7">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">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" title="c10">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">Sl</font>;<br/><b>reconsider </b><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c10">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/><a NAME="E2:55_2"/>
<font color="Maroon" title="c7">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c11">Y</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E14:55"><i><font color="Green" title="E41">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_2"/>
<font color="Maroon" title="c8">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c10">X</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:55"/><b>then </b>
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T8" title="WAYBEL19:th.8">WAYBEL19:8</a></i>;<br/>
<b>hence </b><a NAME="E17:55"/>
<font color="Maroon" title="c7">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E4:55"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="waybel21.html#E13:55"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="waybel19.html#T24" title="WAYBEL19:th.24">WAYBEL19:24</a></i>;<br/>


</div>
