let p, q, r be boolean object ; :: thesis: p 'nand' (q 'nand' r) = (('not' p) 'or' q) '&' (('not' p) 'or' r)
( p = FALSE or p = TRUE ) by Def3;
hence p 'nand' (q 'nand' r) = (('not' p) 'or' q) '&' (('not' p) 'or' r) ; :: thesis: verum