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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> or  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">b</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:37"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T46" title="YELLOW_2:th.46">YELLOW_2:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X1</font> = <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c8">Y1</font> = <font color="Maroon" title="c5">Y</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 <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="yellow_4.html#E1:37"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">d</font> =  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X1</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c8">Y1</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<a NAME="E8:37"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">d</font>
 
;<br/>
<a NAME="E9:37"/>
( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">d</font> )
 
<b>by </b><i><a class="ref" href="yellow_4.html#T29" target="_self" title="YELLOW_4:th.29">Th29</a></i>;<br/>
<a NAME="E10:37"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">d</font> )
 
<b>by </b><i><a class="txt" href="yellow_4.html#E1:37"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E11:37"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/>
<a NAME="E12:37"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E8:37"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>;<br/>
<b>hence </b><a NAME="E13:37"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<a NAME="E14:37"/>
<font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>
<b>thus </b><a NAME="E4:37_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="yellow_4.html#E1:37"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow_4.html#E3:37"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yellow_4.html#E3:37_1"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:37"/>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="yellow_4.html#E5:37"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="yellow_3.html#T6" title="YELLOW_3:th.6">YELLOW_3:6</a></i>;<br/>


</div>
