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