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 Th44:
  L1 is 01_Lattice & L2 is 01_Lattice implies (p1
is_a_complement_of q1 & p2 is_a_complement_of q2 iff [p1,p2] is_a_complement_of
  [q1,q2])
proof
  assume that
A1: L1 is 01_Lattice and
A2: L2 is 01_Lattice;
  thus p1 is_a_complement_of q1 & p2 is_a_complement_of q2 implies [p1,p2]
  is_a_complement_of [q1,q2]
  proof
    assume that
A3: p1 is_a_complement_of q1 and
A4: p2 is_a_complement_of q2;
A5: p2"\/"q2 = Top L2 by A4;
    p1"\/"q1 = Top L1 by A3;
    hence [p1,p2]"\/"[q1,q2] = [Top L1,Top L2] by A5,Th21
      .= Top [:L1,L2:] by A1,A2,Th43;
    hence [q1,q2]"\/"[p1,p2] = Top [:L1,L2:];
A6: p2"/\"q2 = Bottom L2 by A4;
    p1"/\"q1 = Bottom L1 by A3;
    hence [p1,p2]"/\"[q1,q2] = [Bottom L1,Bottom L2] by A6,Th21
      .= Bottom [:L1,L2:] by A1,A2,Th42;
    hence [q1,q2]"/\"[p1,p2] = Bottom [:L1,L2:];
  end;
  assume
A7: [p1,p2] is_a_complement_of [q1,q2];
  then
A8: [p1,p2]"/\"[q1,q2] = Bottom [: L1,L2:];
  [Bottom L1,Bottom L2] = Bottom [:L1,L2:] by A1,A2,Th42;
  then
A9: [p1"/\"q1,p2"/\"q2] = [Bottom L1,Bottom L2] by A8,Th21;
  then
A10: p1"/\"q1 = Bottom L1 by XTUPLE_0:1;
A11: [p1,p2]"\/"[q1,q2] = Top [:L1,L2:] by A7;
A12: p2"/\"q2 = Bottom L2 by A9,XTUPLE_0:1;
A13: p1"\/"q1 = q1"\/"p1 & p1"/\"q1 = q1"/\"p1;
  [Top L1,Top L2] = Top [:L1,L2:] by A1,A2,Th43;
  then
A14: [Top L1,Top L2] = [p1"\/"q1,p2"\/"q2] by A11,Th21;
  then p1"\/"q1 = Top L1 by XTUPLE_0:1;
  hence p1 is_a_complement_of q1 by A10,A13;
A15: p2"\/"q2 = q2"\/"p2 & p2"/\"q2 = q2"/\"p2;
  p2"\/"q2 = Top L2 by A14,XTUPLE_0:1;
  hence thesis by A12,A15;
end;
