consider N being Element of NAT such that
A1: for n being Element of NAT st n >= N holds
g . n >= 0 by Def4;
take N ; :: according to ASYMPT_0:def 2 :: thesis: for n being Element of NAT st n >= N holds
(max (f,g)) . n >= 0
let n be Element of NAT ; :: thesis: ( n >= N implies (max (f,g)) . n >= 0 )
assume n >= N ; :: thesis: (max (f,g)) . n >= 0
then A2: g . n >= 0 by A1;
max ((f . n),(g . n)) >= g . n by XXREAL_0:25;
hence (max (f,g)) . n >= 0 by A2, Def10; :: thesis: verum