
theorem Th27:
  for S,T being complete LATTICE for A being Subset of UPS(S, T)
  holds sup A = "\/"(A, T|^the carrier of S)
proof
  let S,T be complete LATTICE;
  let A be Subset of UPS(S, T);
A1: UPS(S, T) is sups-inheriting full SubRelStr of T|^the carrier of S by Def4
,Th26;
  ex_sup_of A,T|^the carrier of S by YELLOW_0:17;
  then "\/"(A,T|^the carrier of S) in the carrier of UPS(S,T) by A1,
YELLOW_0:def 19;
  hence thesis by A1,YELLOW_0:68;
end;
