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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> (  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> iff <font color="Olive" title="b1">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> (  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> iff <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )</span><br/>



<b>thus </b><a NAME="E1:18"/>
(  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> implies <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> implies  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D21" title="WAYBEL_0:def.21">WAYBEL_0:def 21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y'</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<a NAME="E6:18_1"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel13.html#T2" target="_self" title="WAYBEL13:th.2">Th2</a></i>;<br/>
<a NAME="E7:18_1"/><b>then </b>
<font color="Maroon" title="c4">y'</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E4:18_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="yellow_0.html#T63" title="YELLOW_0:th.63">YELLOW_0:63</a></i>;<br/>
<a NAME="E8:18_1"/><b>then </b>
<font color="Maroon" title="c4">y'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y'</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>
<a NAME="E10:18_1"/>
( <font color="Maroon" title="c4">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">y'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:18_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>
<b>hence </b><a NAME="E11:18_1"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:18_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</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>

<b>thus </b><a NAME="E2:18"/>
( <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> implies  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<a NAME="E2:18_2"/><i><font color="Green" title="E16">A5</font></i>: 
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  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="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">I</font> =  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="ref" href="waybel13.html#T2" target="_self" title="WAYBEL13:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">I</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel13.html#E2:18_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="waybel10.html#T24" title="WAYBEL10:th.24">WAYBEL10:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c6">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:18_2_1"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c6">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font></span><br/><a NAME="E3:18_2_1"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><a NAME="E4:18_2_1"/>
<font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:18_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/><a NAME="E5:18_2_1"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:18_2_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E6:18_2_1"/>
<font color="Maroon" title="c8">z'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/><b>hence </b><a NAME="E7:18_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:18_2_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</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>
<b>then reconsider </b><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c4">I</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel13.html#E1:18_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E8:18_2"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
;<br/>
<a NAME="E9:18_2"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:18_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>
<a NAME="E10:18_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E11:18_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>
<a NAME="E12:18_2"/><b>then </b>
<font color="Maroon" title="c6">x'</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T62" title="YELLOW_0:th.62">YELLOW_0:62</a></i>;<br/>
<b>hence </b><a NAME="E13:18_2"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V14" title="WAYBEL_0:attr.14">principal</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E9:18_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="waybel_0.html#D21" title="WAYBEL_0:def.21">WAYBEL_0:def 21</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>


</div>
