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
  L is modular implies latt F is modular
proof
  assume
A1: for a,b,c being Element of L st a [= c holds a"\/"(b"/\"c) = (a"\/"b
  )"/\"c;
  let a,b,c be Element of latt F such that
A2: a [= c;
  reconsider p = a, q = b, r = c as Element of F by Th49;
  b"/\"c = q"/\"r by Th50;
  then
A3: a"\/"(b"/\"c) = p"\/"(q"/\"r) by Th50;
  a"\/"b = p"\/"q by Th50;
  then
A4: (a"\/"b)"/\"c = (p"\/"q)"/\" r by Th50;
  p [= r by A2,Th51;
  hence a"\/"(b"/\"c) = (a"\/"b)"/\"c by A1,A3,A4;
end;
