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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:40"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E5:40"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font>
 <b>and </b><br/><a NAME="E6:40"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel16.html#E1:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="waybel16.html#T20" target="_self" title="WAYBEL16:th.20">Th20</a></i>;<br/>
<b>assume </b><a NAME="E7:40"/><i><font color="Green" title="E29">A5</font></i>: 
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E8:40"/><i><font color="Green" title="E30">A6</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font> ) )
 
<b>by </b><i><a class="txt" href="waybel16.html#E2:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<a NAME="E9:40"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:40"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E8:40"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><a NAME="E3:40_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E7:40"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="waybel16.html#E1:40_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E5:40"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:40"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:40_2"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font>
 ;<br/><a NAME="E2:40_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E2:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><b>hence </b><a NAME="E3:40_2"/>
<font color="Maroon" title="c4">q</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E9:40"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:40"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel16.html#E2:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="waybel16.html#E4:40"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel16.html#E11:40"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>


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