reserve a,b,c,d for set;

theorem
  (not AND2(a,b) is non empty iff OR2(NOT1 a, NOT1 b) is non empty)& (
OR2(a,b) is non empty & OR2(c,b) is non empty iff OR2(AND2(a,c),b) is non empty
) & (OR2(a,b) is non empty & OR2(c,b) is non empty & OR2(d,b) is non empty iff
OR2(AND3(a,c,d),b) is non empty)& (OR2(a,b) is non empty & (a is non empty iff
  c is non empty) implies OR2(c,b) is non empty )
proof
A1: OR2(a,b) is non empty & OR2(c,b) is non empty iff (a is non empty or b
  is non empty) & (c is non empty or b is non empty);
A2: OR2(a,b) is non empty & OR2(c,b) is non empty & OR2(d,b) is non empty
iff (a is non empty or b is non empty ) & (c is non empty or b is non empty ) &
  (d is non empty or b is non empty );
  not AND2(a,b) is non empty iff not (a is non empty & b is non empty);
  hence thesis by A1,A2;
end;
