reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th3:
  for S,T being LATTICE, f being Function of S,T st T is
  distributive & f is meet-preserving join-preserving one-to-one holds S is
  distributive
proof
  let S,T be LATTICE, f be Function of S,T;
  assume that
A1: T is distributive and
A2: f is meet-preserving join-preserving one-to-one;
  let x,y,z be Element of S;
  f.( x "/\" (y "\/" z)) = f. x "/\" f.(y "\/" z) by A2,Th1
    .= f. x "/\" (f.y "\/" f.z) by A2,Th2
    .= (f.x "/\" f.y) "\/" (f.x "/\" f.z) by A1
    .= (f.x "/\" f.y) "\/" f.(x "/\" z) by A2,Th1
    .= f.(x "/\" y) "\/" f.(x "/\" z) by A2,Th1
    .= f.((x "/\" y) "\/" (x "/\" z))by A2,Th2;
  hence x "/\" (y "\/" z) = (x "/\" y) "\/" (x "/\" z) by A2;
end;
