<?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/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="waybel21.html#M1" title="WAYBEL21:mode.1">SemilatticeHomomorphism</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>thus </b><a NAME="E1:58"/>
( <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> implies <font color="Maroon" title="c3">f</font> is <a href="waybel21.html#V1" title="WAYBEL21:attr.1">lim_infs-preserving</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp; <font color="Maroon" title="c3">f</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="waybel21.html#T46" target="_self" title="WAYBEL21:th.46">Th46</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D4" target="_self" title="WAYBEL21:def.4">WAYBEL21:def 4</a><br/>

<b>set </b><font color="Maroon" title="c5">M</font> = <font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : verum </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">S</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</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="ref" href="waybel21.html#T25" target="_self" title="WAYBEL21:th.25">Th25</a></i>;<br/>
<a NAME="E4:58_1"/><i><font color="Green" title="E37">A2</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E5:58_1"/><i><font color="Green" title="E38">A3</font></i>: 
( 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="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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 <font color="Maroon" title="c4">N</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> #) <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="c4">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</font> #) )
 
<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 NAME="E6:58_1"/><i><font color="Green" title="E39">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>($1),<font color="Maroon" title="c1">S</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>verum;<br/>
<a NAME="E7:58_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="lattice3.html#S2" title="LATTICE3:sch.2">LATTICE3:sch 2</a>(<a class="txt" href="waybel21.html#E6:58_1"><i><font color="Green" title="E39">A4</font></i></a>);<br/></i>
<a NAME="E8:58_1"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E42">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>;<br/><b>let </b><font color="Maroon" title="c9">j'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>;<br/><b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c9">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>] <b>means </b>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">j</font>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>] <b>means </b>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j'</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> = <font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $1</span>)</span>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><a NAME="E2:58_1_1_1"/><i><font color="Green" title="E44">A9</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">v</font>] holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">v</font>)
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c3">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 <font color="Maroon" title="c4">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/><b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $1;<br/><a NAME="E3:58_1_1_1"/><i><font color="Green" title="E45">A10</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">v</font>] holds <br/><font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">v</font>)
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E4:58_1_1_1"/><i><font color="Green" title="E46">A11</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">v</font>] holds <br/><font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b1">v</font>)
 <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j'</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</font>;<br/><a NAME="E5:58_1_1_1"/><i><font color="Green" title="E47">A12</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">v</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">v</font>] )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:58_1_1_1"/><i><font color="Green" title="E48">A13</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> holds <br/> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">v</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">v</font>] )
 <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> = <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $1;<br/><b>thus </b><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>) = 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] </span>}</span> 

<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive" title="b1">k</font>)</span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="lattice3.html#S2" title="LATTICE3:sch.2">LATTICE3:sch 2</a>(<a class="txt" href="waybel21.html#E6:58_1"><i><font color="Green" title="E39">A4</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 

<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E2:58_1_1_1"><i><font color="Green" title="E44">A9</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">o</font>) where <font color="Olive" title="b1">o</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">o</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E3:58_1_1_1"><i><font color="Green" title="E45">A10</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">r</font>) where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">r</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="waybel21.html#E5:58_1_1_1"><i><font color="Green" title="E47">A12</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">m</font>) where <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font>] </span>}</span> 
<b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green" title="E38">A3</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">q</font>) where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">q</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="waybel21.html#E6:58_1_1_1"><i><font color="Green" title="E48">A13</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E4:58_1_1_1"><i><font color="Green" title="E46">A11</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b1">n</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j'</font> </span>}</span> 

;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E43">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>;<br/><a NAME="E1:58_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_1_2_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>)
 ;<br/>

<a NAME="E2:58_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</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="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">j</font> )
 
;<br/>
<b>hence </b><a NAME="E3:58_1_1_2_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E3:58_1_1_2"/>
( <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c9">A</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c1">S</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E2:58_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2"/>
<font color="Maroon" title="c3">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#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>),<font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">j</font>)</span>)</span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:58_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_1_3"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font><b> such that </b><br/><a NAME="E3:58_1_1_3"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font> </span>}</span> ,<font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:58_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>
<a NAME="E4:58_1_1_3"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">j</font>)</span>)</span>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1_1"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="waybel21.html#E3:58_1_1_3"><i><font color="Green" title="E44">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c9">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>
<a NAME="E6:58_1_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b1">n</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">j'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1"><i><font color="Green" title="E42">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1_3"/>
<font color="Maroon" title="c8">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="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel21.html#E4:58_1_1_3"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:58_1_1"/>
<font color="Maroon" title="c8">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="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">j'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span><b> such that </b><br/><a NAME="E6:58_1_1"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span> : <font color="Olive" title="b1">n</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j'</font> </span>}</span> ,<font color="Maroon" title="c2">T</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">j'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>
<font color="Maroon" title="c8">a</font> = 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">j</font>)</span>)</span>,<font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="waybel21.html#E6:58_1_1"><i><font color="Green" title="E44">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">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#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">j</font>),<font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1_1"><i><font color="Green" title="E43">A14</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:58_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:58_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c4">N</font></span>)</span> = 
<font color="Maroon" title="c3">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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D6" title="WAYBEL11:def.6">WAYBEL11:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E4:58_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E8:58_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="waybel11.html#D6" title="WAYBEL11:def.6">WAYBEL11:def 6</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:58"/><i><font color="Green" title="E36">A18</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Olive" title="b1">N</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D4" target="_self" title="WAYBEL21:def.4">WAYBEL21:def 4</a><br/>

<a NAME="E3:58"/><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:58_2"/>
( not <font color="Maroon" title="c4">D</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">D</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">D'</font> = <font color="Maroon" title="c4">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</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="E3:58_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E4:58_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">D</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/>

<b>thus </b><font color="Maroon" title="c3">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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p2">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c5">D'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel17.html#T10" title="WAYBEL17:th.10">WAYBEL17:10</a></i>
<br/>.= 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c5">D'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E2:58"><i><font color="Green" title="E36">A18</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">D</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel21.html#T33" target="_self" title="WAYBEL21:th.33">Th33</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/><i><font color="Green" title="E38">A20</font></i>: 
for <font color="Olive" title="b1">X</font> being <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="c1">S</font> holds  <font color="Maroon" title="c3">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#D1" target="_self" title="WAYBEL21:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">X</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>reconsider </b><font color="Maroon" title="c5">fX</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</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/><b>thus </b><a NAME="E2:58_3"/>
<font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c4">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_3_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">X</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="E2:58_3_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</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/>

<font color="Maroon" title="c3">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="c4">X</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</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="waybel21.html#T28" target="_self" title="WAYBEL21:th.28">Th28</a></i>
<br/>.= 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</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="txt" href="waybel21.html#E2:58"><i><font color="Green" title="E36">A18</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">fX</font>
<b>by </b><i><a class="ref" href="waybel21.html#T29" target="_self" title="WAYBEL21:th.29">Th29</a></i>
<br/>.= 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fX</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="waybel21.html#T28" target="_self" title="WAYBEL21:th.28">Th28</a></i>
;<br/>
<b>hence </b><a NAME="E4:58_3_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c4">X</font>,<font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel21.html#T28" target="_self" title="WAYBEL21:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:58"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E4:58"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="waybel_0.html#T71" title="WAYBEL_0:th.71">WAYBEL_0:71</a></i>;<br/>
<b>hence </b><a NAME="E7:58"/>
<font color="Maroon" title="c3">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#E3:58"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="waybel21.html#T46" target="_self" title="WAYBEL21:th.46">Th46</a></i>;<br/>


</div>
