reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;
reserve Y for non empty set,
  a,b,c for Function of Y,BOOLEAN;

theorem
  'not' (a 'nand' b) = a '&' b
proof
  'not' (a 'nand' b) = 'not' 'not' (a '&' b) by th1;
  hence thesis;
end;
