
theorem Th5:
  for S, T being lower-bounded antisymmetric non empty RelStr, D
being Subset of [:S,T:] st [:S,T:] is complete or ex_sup_of D,[:S,T:] holds sup
  D = [sup proj1 D,sup proj2 D]
proof
  let S, T be lower-bounded antisymmetric non empty RelStr, D be Subset of [:S
  ,T:] such that
A1: [:S,T:] is complete or ex_sup_of D,[:S,T:];
  per cases;
  suppose
    D <> {};
    hence thesis by A1,YELLOW_3:46;
  end;
  suppose
A2: D = {};
    then
A3: sup proj2 D = Bottom T;
    sup D = Bottom [:S,T:] & sup proj1 D = Bottom S by A2;
    hence thesis by A3,Th4;
  end;
end;
