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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_1.html#V2">distributive</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>thus </b><a NAME="E1:46"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V5" target="_self">prime</a> implies <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#T24" target="_self">Th24</a></i>;<br/>

<b>thus </b><a NAME="E2:46"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V2" target="_self">meet-irreducible</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="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V2" target="_self">meet-irreducible</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6" target="_self">WAYBEL_6:def 6</a><br/>

<b>assume </b><a NAME="E2:46_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then </b><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>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><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>3</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>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6">WAYBEL_1:6</a></i>
;<br/>
<a NAME="E4:46_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>3</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>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E1:46_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_6.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <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="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/>


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


</div>
