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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:62"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>  st <font color="Olive">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span> holds <br/> <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E2:62_1"/>
<font color="Maroon">X</font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a><br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t'</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="yellow10.html#E6:62_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E5:62_1_1"/>
<font color="Maroon">t'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="yellow10.html#E6:62_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:62_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E7:62_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>:  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span> = 
<span class="p1"><a href="yellow_3.html#K7">[</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">proj1</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span></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">proj2</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46">YELLOW_3:46</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></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">proj2</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">t'</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow10.html#E6:62_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
;<br/>
<a NAME="E9:62_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">t'</font></span><a href="yellow_3.html#K10">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="yellow10.html#E8:62_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:62_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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

<b>assume </b><a NAME="E3:62"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a><br/>

<b>assume </b><a NAME="E4:62"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:62"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s'</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E8:62"/>
<font color="Maroon">s'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:62"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<b>then </b><i><font color="Green">E58</font></i>:  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span> = 
<span class="p1"><a href="yellow_3.html#K7">[</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">proj1</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span></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">proj2</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46">YELLOW_3:46</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">s'</font></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">proj2</a> <span class="p4"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">s'</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
;<br/>
<a NAME="E12:62"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">s'</font>,<font color="Maroon">D</font></span><a href="yellow_3.html#K10">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:62"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div>
