
theorem

:: 1.4. LEMMA, p. 144:: Refinements
  for T,T2 being lower complete TopLattice st T2 is TopAugmentation of
  [:T, T qua LATTICE:] for f being Function of T2,T st f = inf_op T holds f is
  continuous
proof
  let T,T2 be lower complete TopLattice such that
A1: the RelStr of T2 = the RelStr of [:T, T:];
  let f be Function of T2,T such that
A2: f = inf_op T;
  f is infs-preserving
  proof
    let X be Subset of T2;
    reconsider Y = X as Subset of [:T,T:] by A1;
    assume
A3: ex_inf_of X,T2;
    thus ex_inf_of f.:X,T by YELLOW_0:17;
A4: inf_op T preserves_inf_of Y by WAYBEL_0:def 32;
    ex_inf_of Y, [:T,T:] by A3,A1,YELLOW_0:14;
    hence inf (f.:X) = f.inf X by A1,A2,A4,YELLOW_0:27;
  end;
  hence thesis by Th9;
end;
