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

<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#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:43"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <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> holds <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>;<br/><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/>



<a NAME="E2:43"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</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">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green">E29</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_4.html#K4">"/\"</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="E5:43"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<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="E6:43"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</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="E3:43_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E2:43"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:43_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:43_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E7:43_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E8:43_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E5:43_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_2.html#E7:43_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E9:43_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:43_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E7:43_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_4.html#T37">YELLOW_4:37</a></i>;<br/>
<a NAME="E11:43_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<a NAME="E12:43_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E13:43_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E5:43_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_2.html#E8:43_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>
<b>hence </b><a NAME="E14:43_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:43_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_2.html#E11:43_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:43"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E5:43"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E8:43"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:43"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:43"/>
 <a href="yellow_0.html#R1">ex_sup_of</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="E10:43"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_4.html#T53">YELLOW_4:53</a></i>;<br/>
<b>hence </b><a NAME="E11:43"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_2.html#E8:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
