
theorem Th6:
  for S1,T1,S2,T2 being non empty RelStr
  st the RelStr of S1 = the RelStr of S2 & the RelStr of T1 = the RelStr of T2
  for f1 being Function of S1,T1, f2 being Function of S2,T2 st f1 = f2 holds
  (f1 is infs-preserving implies f2 is infs-preserving) &
  (f1 is directed-sups-preserving implies f2 is directed-sups-preserving)
proof
  let S1,T1,S2,T2 be non empty RelStr such that
A1: the RelStr of S1 = the RelStr of S2 and
A2: the RelStr of T1 = the RelStr of T2;
  let f1 be Function of S1,T1, f2 be Function of S2,T2 such that
A3: f1 = f2;
  thus f1 is infs-preserving implies f2 is infs-preserving
  by A1,A2,A3,WAYBEL_0:65;
  assume
A4: for X being Subset of S1 st X is non empty directed
  holds f1 preserves_sup_of X;
  let X be Subset of S2;
  reconsider Y = X as Subset of S1 by A1;
  assume X is non empty directed;
  then f1 preserves_sup_of Y by A1,A4,WAYBEL_0:3;
  hence thesis by A1,A2,A3,WAYBEL_0:65;
end;
