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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>;<br/><a NAME="E1:21_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E2:21_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span> st <br/>( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>  st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">d</font> holds <br/><font color="Olive">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c</font> = <font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font>;<br/>

<a NAME="E1:21_1_1"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:21_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:21_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:21_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8">YELLOW_0:8</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E5:21_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E6:21_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:21_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:21_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:21_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E10:21_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:21_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E12:21_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:21_1_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:21_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>, <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
 <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V1">with_suprema</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a></i>;<br/>


</div>
