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

<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E17">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="latsum_1.html#R1" title="LATSUM_1:pred.1">tolerates</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<a NAME="E2:23"/><b>then </b>
( <font color="Maroon" title="c5">a</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">R</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">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">R</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">xy</font> = <font color="Maroon" title="c5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="latsum_1.html#T13" target="_self" title="LATSUM_1:th.13">Th13</a>, <a class="ref" href="latsum_1.html#T16" target="_self" title="LATSUM_1:th.16">Th16</a></i>;<br/>
<a NAME="E4:23"/>
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<a NAME="E5:23"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c7">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self" title="LATSUM_1:th.15">Th15</a></i>;<br/>
<a NAME="E6:23"/>
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c7">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self" title="LATSUM_1:th.15">Th15</a></i>;<br/>
<a NAME="E8:23"/>
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font> holds <br/><font color="Maroon" title="c7">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c8">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c8">d</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="c1">R</font> <a href="latsum_1.html#K1" title="LATSUM_1:func.1">[*]</a> <font color="Maroon" title="c2">S</font></span>)</span> <b>by </b><i><a class="ref" href="latsum_1.html#T10" target="_self" title="LATSUM_1:th.10">Th10</a></i>;<br/>
<a NAME="E3:23_1"/>
<font color="Maroon" title="c9">X</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">D</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="latsum_1.html#T8" target="_self" title="LATSUM_1:th.8">Th8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">dd</font> = <font color="Maroon" title="c8">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="latsum_1.html#T18" target="_self" title="LATSUM_1:th.18">Th18</a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c11">dd</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self" title="LATSUM_1:th.15">Th15</a></i>;<br/>
<a NAME="E6:23_1"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">dd</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self" title="LATSUM_1:th.15">Th15</a></i>;<br/>
<a NAME="E7:23_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">dd</font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E5:23_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="yellow_5.html#T9" title="YELLOW_5:th.9">YELLOW_5:9</a></i>;<br/>
<b>hence </b><a NAME="E8:23_1"/>
<font color="Maroon" title="c7">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self" title="LATSUM_1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:23"/>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="latsum_1.html#E5:23"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="latsum_1.html#E7:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>


</div>
