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

<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E24</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E26</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 ;<br/>

<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E3:4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/><a NAME="E3:4_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/>
<font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/><a NAME="E3:4_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:4"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:4_2"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:4_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font> &amp; <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T9">YELLOW_0:9</a></i>;<br/>
<a NAME="E4:4_2"/><b>then </b>
(  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<b>hence </b><a NAME="E5:4_2"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:4"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>


</div>
