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

<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>;<br/>



<b>reconsider </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>, <font color="Maroon">C2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:80"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
;<br/>
<a NAME="E3:80"/><b>then </b>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E5:80"/>
not  <a href="funct_5.html#K1">proj1</a> <font color="Maroon">D'</font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">D1</font> =  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> ;<br/>
<a NAME="E7:80"/>
not  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">D'</font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">D2</font> =  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font> ;<br/>
<b>assume </b><a NAME="E9:80"/>
( <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> is <a href="lattice3.html#V3">complete</a> or  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> )
 ;<br/>

<a NAME="E10:80"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:80"/><b>then </b><i><font color="Green">E20</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D1</font>,<font color="Maroon">L1</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D2</font>,<font color="Maroon">L2</font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E11:80"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E12:80"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E10:80"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E13:80"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">d1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:80"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">d2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/>
<a NAME="E18:80"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">D1</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">d1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><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">L1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E18:80"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_5.html#T4">FUNCT_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> ;<br/>
<a NAME="E6:80_1"/>
<font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E7:80_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E18:80"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:80_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:80"/>
<font color="Maroon">D2</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">d2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_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">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:80_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E18:80"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_5.html#T4">FUNCT_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ;<br/>
<a NAME="E6:80_2"/>
<font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E7:80_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E18:80"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:80_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E20:80"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d1</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d2</font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E12:80"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow_3.html#E15:80"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E21:80"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:80"/>
<font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:80"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E24:80"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E12:80"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E25:80"/>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E18:80"><i><font color="Green">E29</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
