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

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">A</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a>  holds <br/> <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a> </span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</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">A</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a>  implies  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a>  )</span><br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">D</font> being 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">N</font>  st  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; ( for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3" title="WAYBEL11:def.3">WAYBEL11:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a> </span><br/>

<b>let </b><font color="Maroon" title="c3">D</font> be  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">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3" title="WAYBEL11:def.3">WAYBEL11:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <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">N</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <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">N</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> ) ) ) )</span><br/>

<b>assume </b><a NAME="E2:26"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <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">N</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <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">N</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> ) ) )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E4:26"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">aa</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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">N</font> <b>by </b><i><a class="ref" href="waybel_0.html#T5" title="WAYBEL_0:th.5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E6:26"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:26"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_2.html#T52" title="WAYBEL_2:th.52">WAYBEL_2:52</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E8:26"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">aa</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D</font> &amp; ( 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">N</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">aa</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) )
 <b>by </b><i><a class="txt" href="waybel30.html#E1:26"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel30.html#E4:26"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<a NAME="E9:26"/>
<font color="Maroon" title="c5">aa</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D</font> <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="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">d</font></span>)</span> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42" title="YELLOW_4:th.42">YELLOW_4:42</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E11:26"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> )
 <b>by </b><i><a class="txt" href="waybel30.html#E8:26"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <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">N</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  not <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> ) ) )</span><br/>

<b>thus </b><a NAME="E12:26"/>
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E11:26"><i><font color="Green" title="E15">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <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">N</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  not <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  not <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E13:26"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c8">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E14:26"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E8:26"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/>
<a NAME="E15:26"/>
<font color="Maroon" title="c7">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E11:26"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E16:26"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E13:26"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E17:26"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E14:26"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</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>
