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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) implies  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> )</span><br/>

<b>set </b><font color="Maroon" title="c2">L</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a></span><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" title="c3">a</font>, <font color="Maroon" title="c4">b</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> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font></span><br/><a NAME="E1:21_1"/>
 ex <font color="Olive" title="b1">c</font> being  <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> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <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> <font color="Maroon" title="c1">X</font></span>)</span>  st <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b1">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">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" title="c5">c</font> = <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">c</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <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> <font color="Maroon" title="c1">X</font></span>)</span>  st <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> ) )</span><br/>

<a NAME="E1:21_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E1:21"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="yellow_1.html#T8" target="_self" title="YELLOW_1:th.8">Th8</a></i>;<br/>
<a NAME="E2:21_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:21_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" target="_self" title="YELLOW_1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" title="YELLOW_0:th.8">YELLOW_0:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <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> <font color="Maroon" title="c1">X</font></span>)</span>  st <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font></span><br/>

<b>let </b><font color="Maroon" title="c6">d</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> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font> )</span><br/>

<b>assume </b><a NAME="E5:21_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font></span><br/>

<a NAME="E6:21_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:21_1_1"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E5:21_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:21_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:21_1_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E5:21_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E10:21_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">d</font> )
 
<b>by </b><i><a class="txt" href="yellow_1.html#E7:21_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="yellow_1.html#T3" target="_self" title="YELLOW_1:th.3">Th3</a></i>;<br/>
<a NAME="E11:21_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E12:21_1_1"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E1:21"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="yellow_1.html#T8" target="_self" title="YELLOW_1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E13:21_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" target="_self" title="YELLOW_1:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:21_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c1">X</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T20" title="YELLOW_0:th.20">YELLOW_0:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
