let r be Real; :: according to PARTFUN3:def 4 :: thesis: ( not r in rng (Liouville_seq (a,b)) or 0 <= r )

assume r in rng (Liouville_seq (a,b)) ; :: thesis: 0 <= r

then consider n being object such that

A1: ( n in dom (Liouville_seq (a,b)) & r = (Liouville_seq (a,b)) . n ) by FUNCT_1:def 3;

thus 0 <= r by A1, Th33; :: thesis: verum

assume r in rng (Liouville_seq (a,b)) ; :: thesis: 0 <= r

then consider n being object such that

A1: ( n in dom (Liouville_seq (a,b)) & r = (Liouville_seq (a,b)) . n ) by FUNCT_1:def 3;

thus 0 <= r by A1, Th33; :: thesis: verum