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<b>let </b><font color="Maroon">W</font> be  <a href="setfam_1.html#V2">with_non-empty_element</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>



<a NAME="E1:72"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E2:72"/><b>then </b>
( <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> implies <font color="Maroon">L</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span> )
 
;<br/>
<a NAME="E3:72"/><b>then </b>
( <font color="Maroon">L</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> iff <font color="Maroon">L</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:72"/>
( <font color="Maroon">L</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> iff ( <font color="Maroon">L</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) )
 <b>by </b><i/>;<br/>


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