reserve n,k for Element of NAT;

theorem
  for L be finite LATTICE holds L is distributive iff ex X be non empty
  Ring_of_sets st L, InclPoset X are_isomorphic
proof
  let L be finite LATTICE;
  thus L is distributive implies ex X be non empty Ring_of_sets st L,
  InclPoset X are_isomorphic
  proof
    consider X be set such that
A1: X = LOWER(subrelstr Join-IRR L);
A2: X is Ring_of_sets by A1,Th15;
    assume L is distributive;
    then L,InclPoset X are_isomorphic by A1,Th14;
    hence thesis by A1,A2;
  end;
  given X be non empty Ring_of_sets such that
A3: L, InclPoset X are_isomorphic;
  consider f be Function of L, InclPoset X such that
A4: f is isomorphic by A3;
A5: f is one-to-one by A4,WAYBEL_0:66;
  f is infs-preserving & f is join-preserving by A4,Lm3,WAYBEL13:20;
  hence thesis by A5,Lm2,WAYBEL_6:3;
end;
