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

<b>consider </b><font color="Maroon" title="c1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">D</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c3">R</font> being   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>take </b><font color="Maroon" title="c4">A</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">R</font> #);<br/>

<b>thus </b><a NAME="E3:31_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12" target="_self" title="LATTICE3:def.12">LATTICE3:def 12</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c1">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">s</font>;<br/><b>thus </b><a NAME="E2:31_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" target="_self" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font>;<br/><b>assume </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font>
 ;<br/><b>thus </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E5:31_1_1"/>
not <font color="Maroon" title="c4">A</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/>


</div>
