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