let F1, F2 be sequence of ExtREAL; :: thesis: ( ( for n being Element of NAT holds F1 . n <= F2 . n ) implies for n being Element of NAT holds (Ser F1) . n <= (Ser F2) . n )
defpred S1[ Nat] means (Ser F1) . $1 <= (Ser F2) . $1;
assume A1: for n being Element of NAT holds F1 . n <= F2 . n ; :: thesis: for n being Element of NAT holds (Ser F1) . n <= (Ser F2) . n
let n be Element of NAT ; :: thesis: (Ser F1) . n <= (Ser F2) . n
A3: (Ser F2) . 0 = F2 . 0 by Def11;
A5: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A6: (Ser F1) . k <= (Ser F2) . k ; :: thesis: S1[k + 1]
A7: F1 . (k + 1) <= F2 . (k + 1) by A1;
A8: (Ser F2) . (k + 1) = ((Ser F2) . k) + (F2 . (k + 1)) by Def11;
(Ser F1) . (k + 1) = ((Ser F1) . k) + (F1 . (k + 1)) by Def11;
hence S1[k + 1] by A6, A8, A7, XXREAL_3:36; :: thesis: verum
end;
(Ser F1) . 0 = F1 . 0 by Def11;
then A9: S1[ 0 ] by A1, A3;
for n being Nat holds S1[n] from NAT_1:sch 2(A9, A5);
 hence (Ser F1) . n <= (Ser F2) . n ; :: thesis: verum