let F1, F2 be Function of NAT , 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 )
assume A2: 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
reconsider N1 = F1 as Num of rng F1 by Def16;
A3: Ser F1 = Ser (rng F1),N1 by Def21;
reconsider N2 = F2 as Num of rng F2 by Def16;
A4: Ser F2 = Ser (rng F2),N2 by Def21;
let n be Element of NAT ; :: thesis: (Ser F1) . n <= (Ser F2) . n
defpred S₁[ Element of NAT ] means (Ser F1) . $1 <= (Ser F2) . $1;
A5: S₁[ 0 ]

proof

A6: (Ser F1) . 0 = F1 . 0 by A3, Def17;
(Ser F2) . 0 = F2 . 0 by A4, Def17;
hence S₁[ 0 ] by A2, A6; :: thesis: verum

end;

A7: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A8: (Ser F1) . k <= (Ser F2) . k ; :: thesis: S₁[k + 1]
A9: (Ser F1) . (k + 1) = ((Ser F1) . k) + (F1 . (k + 1)) by A3, Def17;
A10: (Ser F2) . (k + 1) = ((Ser F2) . k) + (F2 . (k + 1)) by A4, Def17;
F1 . (k + 1) <= F2 . (k + 1) by A2;
hence S₁[k + 1] by A8, A9, A10, Th14; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A5, A7);
hence (Ser F1) . n <= (Ser F2) . n ; :: thesis: verum