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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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Olive" title="b1">x</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> ) )</span><br/><b>let </b><font color="Maroon" title="c2">x</font> be    <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"> ( <font color="Maroon" title="c2">x</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">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Maroon" title="c2">x</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> ) )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Maroon" title="c2">x</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> implies <font color="Maroon" title="c2">x</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> )</span>
<div class="add"><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c2">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Maroon" title="c2">x</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> )</span><br/><b>then reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">x</font> as   <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> ;<br/><a NAME="E3:50_1"/>
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 ;<br/><b>hence </b><a NAME="E4:50_1"/>
<font color="Maroon" title="c2">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <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"> <font color="Maroon" title="c2">x</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></span><br/><a NAME="E5:50_1"/>
( <font color="Maroon" title="c3">a</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 <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> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow21.html#K2" title="YELLOW21:func.2">POSETS</a> <font color="Maroon" title="c1">W</font> )
 <b>by </b><i><a class="ref" href="yellow21.html#T12" target="_self" title="YELLOW21:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E6:50_1"/>
<font color="Maroon" title="c2">x</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>
 ; <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>

<b>assume </b><a NAME="E2:50"/>
<font color="Maroon" title="c2">x</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <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"> (  not <font color="Maroon" title="c2">x</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> or <font color="Maroon" title="c2">x</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> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c2">x</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<b>assume </b><a NAME="E4:50"/>
<font color="Maroon" title="c2">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</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></span><br/>

<a NAME="E5:50"/><b>then </b>
( <font color="Maroon" title="c3">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="c3">L</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> )
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a>, <a class="ref" href="yellow21.html#D2" target="_self" title="YELLOW21:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:50"/>
<font color="Maroon" title="c2">x</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>
 <b>by </b><i><a class="ref" href="yellow21.html#D10" target="_self" title="YELLOW21:def.10">Def10</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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