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<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><b>that </b><br/><a NAME="E1:63"/><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 <font color="Maroon">S</font>  st <font color="Olive">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> holds <br/> <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E20</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 <font color="Maroon">T</font>  st <font color="Olive">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</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 <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>; <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="E3:63"/>
<font color="Maroon">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>
 ;<br/>

<a NAME="E4:63"/><b>then </b><i><font color="Green">E21</font></i>: 
(  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_5.html#T13">FUNCT_5:13</a></i>;<br/>
<a NAME="E5:63"/>
( not  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a> &amp; not  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/>
<a NAME="E6:63"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:63"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<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/>
<a NAME="E8:63"/><b>then </b>
 <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="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> <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">proj2</a> <font color="Maroon">D</font></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/>
<b>hence </b><a NAME="E9:63"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">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>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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