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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> iff <font color="Olive" title="b1">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> )</span><br/><b>let </b><font color="Maroon" title="c2">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> iff <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> )</span><br/>



<b>thus </b><a NAME="E1:46"/>
( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> implies <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> )
 <b>by </b><i><a class="ref" href="waybel_6.html#T24" target="_self" title="WAYBEL_6:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> implies <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> )</span><br/>

<b>thus </b><a NAME="E2:46"/>
( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> implies <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6" target="_self" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/>

<b>assume </b><a NAME="E2:46_1"/>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/>

<b>then </b><font color="Maroon" title="c2">l</font> = 
<font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6" title="WAYBEL_1:th.6">WAYBEL_1:6</a></i>
;<br/>
<a NAME="E4:46_1"/><b>then </b>
( <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E1:46_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="waybel_6.html#D2" target="_self" title="WAYBEL_6:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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