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<b>let </b><font color="Maroon" title="c1">W</font> be  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> holds <br/>( <font color="Olive" title="b1">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> iff ( <font color="Olive" title="b1">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Olive" title="b1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies ( <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> iff ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> ) ) )</span><br/>



<a NAME="E1:51"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E2:51"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</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="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> iff ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> ) )</span><br/>

<a NAME="E3:51"/><b>then </b>
( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow21.html#K2" title="YELLOW21:func.2">POSETS</a> <font color="Maroon" title="c1">W</font> iff <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> )
 
<b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="yellow21.html#D2" target="_self" title="YELLOW21:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:51"/>
( <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> iff ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> ) )
 <b>by </b><i><a class="ref" href="yellow21.html#T13" target="_self" title="YELLOW21:th.13">Th13</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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