
theorem
  for S,T being complete LATTICE for d being sups-preserving Function of T,S
  for s being Element of S holds (UpperAdj d).s = sup (d"downarrow s)
proof
  let S,T be complete LATTICE;
  let d be sups-preserving Function of T,S;
  let s be Element of S;
  [UpperAdj d, d] is Galois by Def2;
  then (UpperAdj d).s is_maximum_of d"(downarrow s) by WAYBEL_1:11;
  hence thesis;
end;
