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 'nand' b) = 'not' a 'or' b & a 'nand' (a 'nand' b) = a 'imp' b
proof
  a 'nand' (a 'nand' b) = 'not' (a '&' (a 'nand' b)) by th1
    .= 'not' (a '&' 'not' (a '&' b)) by th1
    .= ('not' a) 'or' ('not' 'not' (a '&' b)) by BVFUNC_1:14
    .= ('not' a 'or' a) '&' ('not' a 'or' b) by BVFUNC_1:11
    .= I_el(Y) '&' ('not' a 'or' b) by BVFUNC_4:6
    .= 'not' a 'or' b by BVFUNC_1:6;
  hence thesis by BVFUNC_4:8;
end;
