let x be set ; :: according to VALUED_0:def 12 :: thesis: ( not x in proj1 (a * b) or (a * b) . x is natural )
assume x in dom (a * b) ; :: thesis: (a * b) . x is natural
(a * b) . x = a * (b . x) by Def2;
hence (a * b) . x is natural ; :: thesis: verum