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

<b>let </b><font color="Maroon" title="c1">H</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:119"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="waybel_1.html#NR13" title="WAYBEL_1:NR.13">is_a_Heyting_algebra</a> 
 ;<br/>

<a NAME="E2:119"/><b>then </b><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c1">H</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D19" target="_self" title="WAYBEL_1:def.19">Def19</a></i>;<br/>
<a NAME="E3:119"/><i><font color="Green" title="E75">A3</font></i>: 
<font color="Maroon" title="c1">H</font> is <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:119"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="waybel_1.html#T71" target="_self" title="WAYBEL_1:th.71">Th71</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font>;<br/>

<i><font color="Green" title="E76">A4</font></i>: <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E2:119"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="lattice3.html#T15" title="LATTICE3:th.15">LATTICE3:15</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E2:119"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="waybel_1.html#E3:119"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="waybel_1.html#T5" target="_self" title="WAYBEL_1:th.5">Th5</a></i>
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:119_2"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:119_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:119"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/><b>hence </b><a NAME="E3:119_2"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:119"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="waybel_1.html#E4:119"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:119"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E7:119"/><b>then </b><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:119"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="waybel_1.html#E4:119"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</a></i>;<br/>
<a NAME="E8:119"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:119"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="waybel_1.html#E3:119"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/>
<b>hence </b><a NAME="E9:119"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:119"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="waybel_1.html#E7:119"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div>
