theorem Th45:
  L1 is C_Lattice & L2 is C_Lattice iff [:L1,L2:] is C_Lattice
proof
  thus L1 is C_Lattice & L2 is C_Lattice implies [:L1,L2:] is C_Lattice
  proof
    assume that
A1: L1 is C_Lattice and
A2: L2 is C_Lattice;
    reconsider L = [:L1,L2:] as 01_Lattice by A1,A2,Th41;
    L is complemented
    proof
      let a be Element of L;
      consider p1,p2 such that
A3:   a = [p1,p2] by DOMAIN_1:1;
      consider q1 such that
A4:   q1 is_a_complement_of p1 by A1,LATTICES:def 19;
      consider q2 such that
A5:   q2 is_a_complement_of p2 by A2,LATTICES:def 19;
      reconsider b = [q1,q2] as Element of L;
      take b;
      thus thesis by A1,A2,A3,A4,A5,Th44;
    end;
    hence thesis;
  end;
  assume
A6: [:L1,L2:] is C_Lattice;
  then reconsider C1 = L1, C2 = L2 as 01_Lattice by Th41;
  C1 is complemented
  proof
    set p29 = the Element of C2;
    let p19 be Element of C1;
    reconsider p1 = p19 as Element of L1;
    reconsider p2 = p29 as Element of L2;
    consider b being Element of [:L1,L2:] such that
A7: b is_a_complement_of [p1,p2] by A6,LATTICES:def 19;
    consider q1,q2 such that
A8: b = [q1,q2] by DOMAIN_1:1;
    reconsider q19 = q1 as Element of C1;
    take q19;
    thus thesis by A7,A8,Th44;
  end;
  hence L1 is C_Lattice;
  C2 is complemented
  proof
    set p19 = the Element of C1;
    let p29 be Element of C2;
    reconsider p1 = p19 as Element of L1;
    reconsider p2 = p29 as Element of L2;
    consider b being Element of [:L1,L2:] such that
A9: b is_a_complement_of [p1,p2] by A6,LATTICES:def 19;
    consider q1,q2 such that
A10: b = [q1,q2] by DOMAIN_1:1;
    reconsider q29 = q2 as Element of C2;
    take q29;
    thus thesis by A9,A10,Th44;
  end;
  hence thesis;
end;
