reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

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;
