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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:27"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E1:27_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</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="E3:27_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a></i>;<br/><a NAME="E4:27_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:27_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel_6.html#E3:27_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E5:27_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E3:27_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><b>hence </b><a NAME="E6:27_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:27"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:27_1"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:27_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:27_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E5:27_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:27_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E6:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel_6.html#E3:27_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
<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#D2" target="_self">Def2</a></i>;<br/>


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