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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> holds <br/> <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">p</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> implies  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> )</span><br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">Lk</font> = <span class="p1">{<span class="default"> <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="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  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">L</font> <b>by </b><i><a class="ref" href="waybel_1.html#T46" title="WAYBEL_1:th.46">WAYBEL_1:46</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T53" title="WAYBEL_1:th.53">WAYBEL_1:53</a></i>;<br/>
<a NAME="E4:18"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="waybel_1.html#T55" title="WAYBEL_1:th.55">WAYBEL_1:55</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="waybel_5.html#T28" title="WAYBEL_5:th.28">WAYBEL_5:28</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 
<b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="yellow_2.html#T32" title="YELLOW_2:th.32">YELLOW_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">pk</font> = <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Lk</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_1.html#T49" title="WAYBEL_1:th.49">WAYBEL_1:49</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c4">pk</font> is <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T51" title="WAYBEL_1:th.51">WAYBEL_1:51</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E21">A6</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">pk</font></span>)</span> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel15.html#T1" target="_self" title="WAYBEL15:th.1">Th1</a></i>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>
<br/>.= 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">pk</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T47" title="WAYBEL_1:th.47">WAYBEL_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">pk</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>
;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">pk</font>
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="yellow_0.html#T58" title="YELLOW_0:th.58">YELLOW_0:58</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c5">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font> )</span><br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E23">A8</font></i>: 
( not <font color="Maroon" title="c5">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">X'</font> = <font color="Maroon" title="c5">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3" title="WAYBEL13:th.3">WAYBEL13:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">X'</font> = <font color="Maroon" title="c6">X'</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">L</font> <b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="yellow_2.html#T7" title="YELLOW_2:th.7">YELLOW_2:7</a></i>;<br/><a NAME="E4:18_2"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c2">p</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="#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2: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="c5">X</font></span>)</span> )</span><br/><a NAME="E2:18_2_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T53" title="WAYBEL_1:th.53">WAYBEL_1:53</a></i>;<br/><a NAME="E3:18_2_1"/><b>then </b>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="ref" href="yellow_2.html#T32" title="YELLOW_2:th.32">YELLOW_2:32</a></i>;<br/><b>hence </b><a NAME="E4:18_2_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2: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="c5">X</font></span>)</span></span><br/><a NAME="E5:18_2_1"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:18_2_1"/><i><font color="Green" title="E26">A11</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</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="E7:18_2_1"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</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="E8:18_2_1"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font> is  <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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="yellow_2.html#T17" title="YELLOW_2:th.17">YELLOW_2:17</a></i>;<br/><a NAME="E9:18_2_1"/><i><font color="Green" title="E29">A15</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="waybel_1.html#T55" title="WAYBEL_1:th.55">WAYBEL_1:55</a></i>;<br/><a NAME="E10:18_2_1"/><b>then </b><i><font color="Green" title="E30">A16</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="#E6:18_2_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="waybel_0.html#T7" title="WAYBEL_0:th.7">WAYBEL_0:7</a></i>;<br/><a NAME="E11:18_2_1"/>
<font color="Maroon" title="c5">X</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 ;<br/><a NAME="E12:18_2_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Lk</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E13:18_2_1"/><b>then </b>
<font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/><b>hence </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">X</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:18_2_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E8:18_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:18_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="#E9:18_2_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="waybel_0.html#T7" title="WAYBEL_0:th.7">WAYBEL_0:7</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2: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="txt" href="#E4:18_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="#E6:18_2_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
<br/>.= 
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2: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="c5">X</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:18_2_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E10:18_2_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_2"/>
<font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E13:18"/><b>then </b>
<font color="Maroon" title="c4">pk</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="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<b>hence </b><a NAME="E14:18"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="txt" href="#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E6:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="#E11:18"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="waybel15.html#T16" target="_self" title="WAYBEL15:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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