consider N being Nat such that
A1: for n being Nat st n >= N holds
g . n > 0 by Def4;
take N ; :: according to ASYMPT_0:def 4 :: thesis: for n being Nat st n >= N holds
(max (f,g)) . n > 0
let n be Nat; :: thesis: ( n >= N implies (max (f,g)) . n > 0 )
assume n >= N ; :: thesis: (max (f,g)) . n > 0
then g . n > 0 by A1;
then max ((f . n),(g . n)) > 0 by XXREAL_0:25;
hence (max (f,g)) . n > 0 by Def7; :: thesis: verum