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 S_{1}[ 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 S_{1}[k] holds

S_{1}[k + 1]

then A9: S_{1}[ 0 ]
by A1, A3;

for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A9, A5);

hence (Ser F1) . n <= (Ser F2) . n ; :: thesis: verum

defpred S

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 S

S

proof

(Ser F1) . 0 = F1 . 0
by Def11;
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A6: (Ser F1) . k <= (Ser F2) . k ; :: thesis: S_{1}[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 S_{1}[k + 1]
by A6, A8, A7, XXREAL_3:36; :: thesis: verum

end;assume A6: (Ser F1) . k <= (Ser F2) . k ; :: thesis: S

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 S

then A9: S

for n being Nat holds S

hence (Ser F1) . n <= (Ser F2) . n ; :: thesis: verum