reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem
  for X being set, F,G being Subset-Family of X st COMPLEMENT F =
  COMPLEMENT G holds F = G
proof
  let X be set, F,G be Subset-Family of X;
  assume COMPLEMENT F = COMPLEMENT G;
  hence F = COMPLEMENT COMPLEMENT G .= G;
end;
