reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem
  L is C_Lattice & L is modular implies latt (L,(.p.>) is C_Lattice
proof
  assume that
A1: L is C_Lattice and
A2: L is modular;
  reconsider K = latt (L,(.p.>) as bounded Lattice by A1,Th81;
  K is complemented
  proof
    let b9 be Element of K;
    reconsider b = b9 as Element of L by Th68;
    consider a being Element of L such that
A3: a is_a_complement_of b by A1,LATTICES:def 19;
A4: a"\/"b = Top L by A3;
A5: carr(K) = (.p.> by Th72;
    then
A6: b [= p by Th28;
    p"/\"a [= p by LATTICES:6;
    then reconsider a9 = p"/\"a as Element of K by A5,Th28;
    take a9;
    thus a9"\/"b9 = (p"/\"a)"\/"b by Th73
      .= (b"\/"a)"/\" p by A2,A6
      .= p by A1,A4
      .= Top K by Th79;
    hence b9"\/"a9 = Top K;
A7: a"/\"b = Bottom L by A3;
    thus a9"/\"b9 = (p"/\"a)"/\"b by Th73
      .= p"/\"Bottom L by A7,LATTICES:def 7
      .= Bottom L by A1
      .= Bottom K by A1,Th80;
    hence thesis;
  end;
  hence thesis;
end;
