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 Th59:
  L is C_Lattice & L is M_Lattice implies latt <.p.) is C_Lattice
proof
  assume that
A1: L is C_Lattice and
A2: L is M_Lattice;
  reconsider B = L as C_Lattice by A1;
  reconsider M = latt <.p.) as 01_Lattice by A1,Th58;
  M is complemented
  proof
    let r9 be Element of M;
    reconsider r = r9 as Element of <.p.) by Th49;
    reconsider p1 = p as Element of B;
    consider q such that
A3: q is_a_complement_of r by A1,LATTICES:def 19;
    the carrier of latt <.p.) = <.p.) by Th49;
    then reconsider q9 = p"\/"q as Element of M by Th16;
    take q9;
    thus q9"\/"r9 = p"\/"q"\/"r by Th50
      .= p"\/"(q"\/"r) by LATTICES:def 5
      .= p1"\/"Top B by A3
      .= Top L
      .= Top M by A1,Th57;
    hence r9"\/"q9= Top M;
    p [= r by Th15;
    then (p"\/"q)"/\"r = p"\/"(q"/\"r) by A2,LATTICES:def 12;
    hence q9"/\"r9 = p"\/"(q"/\"r) by Th50
      .= p1"\/"Bottom B by A3
      .= p
      .= Bottom M by Th56;
    hence thesis;
  end;
  hence thesis;
end;
