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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> implies <font color="Maroon" title="c1">L</font> is  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> )</span><br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  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></span><br/>

<a NAME="E2:34"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font></span><br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E1:34"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow_0.html#T43" title="YELLOW_0:th.43">YELLOW_0:43</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font></span><br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_2"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E1:34"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="waybel_0.html#T76" title="WAYBEL_0:th.76">WAYBEL_0:76</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
<font color="Maroon" title="c1">L</font> is  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>
 <b>by </b><i><a class="ref" href="yellow_2.html#T30" target="_self" title="YELLOW_2:th.30">Th30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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