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;
