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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_1.html#V2">distributive</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<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>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<b>thus </b><a NAME="E2:49"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_6.html#V1" target="_self">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a>
 ;<br/>

<a NAME="E2:49_1"/><i><font color="Green">E32</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">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">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> ) ) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</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="E1:49_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E2:49_1_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</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="E5:49_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>4</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>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:49_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/><a NAME="E6:49_1_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E5:49_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/><a NAME="E7:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1"/>
ex <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E5:49_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  as  <a href="waybel_6.html#V1" target="_self">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_6.html#E1:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_6.html#E1:49_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_6.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel_6.html#T26" target="_self">Th26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E5:49_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E6:49_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:49_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E2:49_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:49_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/><b>hence </b><a NAME="E3:49_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:49_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:49_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>


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

<b>thus </b><a NAME="E3:49"/>
( ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_6.html#V1" target="_self">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_6.html#V1" target="_self">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:49_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V2" target="_self">meet-irreducible</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T13" target="_self">Th13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <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>;<br/><b>assume </b><a NAME="E1:49_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>assume </b><a NAME="E2:49_2_1"/><i><font color="Green">E33</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&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>2</sub></font> )
 ;<br/><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_6.html#E1:49_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6">WAYBEL_1:6</a></i>
;<br/><a NAME="E4:49_2_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:49_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_6.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:49_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:49_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D6" target="_self">Def6</a></i>;<br/>


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


</div>
