let r be Real; :: according to PARTFUN3:def 4 :: thesis: ( not r in rng (F | Y) or 0 <= r )
assume A1: r in rng (F | Y) ; :: thesis: 0 <= r
rng (F | Y) c= rng F by RELAT_1:70;
hence 0 <= r by A1, PARTFUN3:def 4; :: thesis: verum