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' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)
proof
  thus (a 'nand' b) 'nand' (a 'nand' c) = 'not' ((a 'nand' b) '&' (a 'nand' c)
  ) by th1
    .= 'not' ('not' (a '&' b) '&' (a 'nand' c)) by th1
    .= 'not' ('not' (a '&' b) '&' 'not' (a '&' c)) by th1
    .= 'not' 'not' (a '&' b) 'or' 'not' 'not' (a '&' c) by BVFUNC_1:14
    .= a '&' (b 'or' c) by BVFUNC_1:12;
end;
