<?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="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</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> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green">E20</font></i>: 
( <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> is <a href="waybel_0.html#V24">up-complete</a> &amp; <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> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a><br/>

<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</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>;<br/><a NAME="E1:79_1"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:79_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</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/></div><b>end;</b></div>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a>
<div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><a NAME="E2:79_2_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E2:79"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E3:79_2_1"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_3.html#K4">proj1</a> <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>;<br/><a NAME="E4:79_2_1"/>
 <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:79_2_1"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:79_2"/>
<font color="Maroon">S</font> is <a href="waybel_0.html#V24">up-complete</a>
 ;<br/><b>thus </b><a NAME="E3:79_2"/>
<font color="Maroon">S</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a><br/>

<b>consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E2:79_2_2"/>
 <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:79_2_2"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</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/>
<a NAME="E4:79_2_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <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> <span class="p4">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p5"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</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">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p5"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">s</font> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i>, <a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow10.html#E4:79_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


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

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a>
<div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><a NAME="E2:79_3"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E2:79"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E3:79_3"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_3.html#K5">proj2</a> <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>;<br/><a NAME="E4:79_3"/>
 <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:79_3"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:79"/>
<font color="Maroon">T</font> is <a href="waybel_0.html#V24">up-complete</a>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a><br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E8:79"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</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/>;<br/>
<a NAME="E9:79"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <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> <span class="p4">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p5"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</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">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p5"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">t</font> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, <a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div>
