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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<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#D1">WAYBEL11:def 1</a><br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E32</font></i>: 
 <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">S</font>
 ;<br/>

<a NAME="E2:28"/>
 <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">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E2:28_1"/>
<font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D'</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D'</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:28_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E6:28_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E7:28_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:28_1"/>
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">c</font> ) holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:28_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E3:28_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:28_1"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:28_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">D</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
not <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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