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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b2">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/><font color="Olive" title="b1">p</font> is  <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">p</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/><font color="Maroon" title="c2">p</font> is  <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <font color="Maroon" title="c2">p</font> is  <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> is  <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">F</font>;<br/>
<a NAME="E2:27"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="waybel_4.html#T56" title="WAYBEL_4:th.56">WAYBEL_4:56</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> implies  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:27_1"/>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E5:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:27_1_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_1"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></span>;<br/><a NAME="E5:27_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:27_1_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E6:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></span>;<br/><span class="kw">hence </span><a NAME="E7:27_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:27_1"/>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E9:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:27_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="waybel_4.html#T56" title="WAYBEL_4:th.56">WAYBEL_4:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:27"/>
<font color="Maroon" title="c2">p</font> is  <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_6.html#D2" target="_self" title="WAYBEL_6:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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