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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( not  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2" target="_self">satisfying_axiom_of_approximation</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:42_1"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:42_1"/>
( not  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:42"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E4:42_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:42_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_3.html#D5" target="_self">Def5</a>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E5:42_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:42_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:42_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E8:42_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E7:42_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="waybel_3.html#T7" target="_self">Th7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:42"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D5" target="_self">WAYBEL_3:def 5</a><br/>

<a NAME="E4:42"/>
( not  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:42"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E5:42"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E6:42"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T9" target="_self">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:42_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:42"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E4:42_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:42_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:42_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel_3.html#E3:42_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:42"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" target="_self">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E5:42"><i><font color="Green">E20</font></i></a>, <a class="txt" href="waybel_3.html#E6:42"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


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