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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="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#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:30"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">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="E3:30"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E18">A2</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="c2">x</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="c3">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="E5:30"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">x</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="c2">x</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="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel16.html#T20" target="_self" title="WAYBEL16:th.20">Th20</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">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>assume </b><b>that </b><br/><a NAME="E6:30"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E7:30"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c2">x</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="c4">X</font>
 <b>and </b><br/><a NAME="E8:30"/><i><font color="Green" title="E21">A5</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<a NAME="E9:30"/><i><font color="Green" title="E22">A6</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></i>;<br/>
<a NAME="E10:30"/>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:30_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E6:30"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E4:30_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:30"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="waybel16.html#E1:30_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E8:30"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="waybel16.html#E1:30_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:30"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:30"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E6:30"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="waybel16.html#E9:30"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></i>;<br/>
<a NAME="E12:30"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E7:30"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></i>;<br/>
<b>hence </b><a NAME="E13:30"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel16.html#E3:30"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></i>;<br/>


</div>
