reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th54:
  L is distributive implies latt F is distributive
proof
  assume
A1: for p,q,r holds p"/\"(q"\/"r) = (p"/\"q)"\/"(p"/\"r);
  let p9,q9,r9 be Element of latt F;
  reconsider p = p9, q = q9, r = r9, qr = q9"\/"r9 as Element of F by Th49;
A2: p9"/\"q9 = p"/\"q & p9"/\"r9 = p"/\"r by Th50;
  thus p9"/\"(q9"\/"r9) = p"/\"qr by Th50
    .= p"/\"(q"\/"r) by Th50
    .= (p"/\"q)"\/"(p"/\"r) by A1
    .= (p9"/\"q9)"\/"(p9"/\"r9) by A2,Th50;
end;
