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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 ;<br/>

<b>thus </b><a NAME="E2:15"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">X</font> be  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">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D39" title="WAYBEL_0:def.39">WAYBEL_0:def 39</a><br/>

<b>consider </b><font color="Maroon" title="c4">D</font> being  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="c2">T</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span><b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  st <font color="Olive" title="b1">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span> holds <br/><font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel_0.html#D39" title="WAYBEL_0:def.39">WAYBEL_0:def 39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
;<br/>

<a NAME="E5:15_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1"/>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:15_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="yellow_3.html#T29" title="YELLOW_3:th.29">YELLOW_3:29</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E8:15_1"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E9:15_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E10:15_1"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T39" title="YELLOW_3:th.39">YELLOW_3:39</a></i>;<br/>
<a NAME="E11:15_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E12:15_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E8:15_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="yellow_3.html#T30" title="YELLOW_3:th.30">YELLOW_3:30</a></i>;<br/>
<a NAME="E13:15_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:15_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:15_1"/>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E6:15_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">X</font> be  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="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D39" title="WAYBEL_0:def.39">WAYBEL_0:def 39</a><br/>

<b>consider </b><font color="Maroon" title="c4">D</font> being  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">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span><b> such that </b><br/><a NAME="E5:15"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">X</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green" title="E12">A7</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  st <font color="Olive" title="b1">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">X</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span> holds <br/><font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel_0.html#D39" title="WAYBEL_0:def.39">WAYBEL_0:def 39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
;<br/>

<a NAME="E7:15"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E8:15"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E9:15"/>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E5:15"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="yellow_3.html#T29" title="YELLOW_3:th.29">YELLOW_3:29</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/>

<b>assume </b><a NAME="E10:15"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E11:15"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">X</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E12:15"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T39" title="YELLOW_3:th.39">YELLOW_3:39</a></i>;<br/>
<a NAME="E13:15"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E14:15"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c3">X</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E10:15"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="yellow_3.html#T30" title="YELLOW_3:th.30">YELLOW_3:30</a></i>;<br/>
<a NAME="E15:15"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E6:15"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:15"/>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E8:15"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>


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