reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem ::Remark 2.4  (Part II)
  for L, M being complete LATTICE for f being Function of L, M st f is
  sups-preserving or f is infs-preserving holds Image f is complete LATTICE
proof
  let L, M be complete LATTICE;
  let f be Function of L, M such that
A1: f is sups-preserving or f is infs-preserving;
  per cases by A1;
  suppose
    f is sups-preserving;
    then Image f is sups-inheriting by Th32;
    hence thesis by Th31;
  end;
  suppose
    f is infs-preserving;
    then Image f is infs-inheriting by Th33;
    hence thesis by Th30;
  end;
end;
