let F, G be sequence of ExtREAL; :: thesis: for n being Nat st ( for m being Nat st m <= n holds

F . m <= G . m ) holds

(Ser F) . n <= (Ser G) . n

let n be Nat; :: thesis: ( ( for m being Nat st m <= n holds

F . m <= G . m ) implies (Ser F) . n <= (Ser G) . n )

assume A1: for m being Nat st m <= n holds

F . m <= G . m ; :: thesis: (Ser F) . n <= (Ser G) . n

defpred S_{1}[ Nat] means ( ( for m being Nat st m <= $1 holds

F . m <= G . m ) implies (Ser F) . $1 <= (Ser G) . $1 );

A2: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A7, A2);

hence (Ser F) . n <= (Ser G) . n by A1; :: thesis: verum

F . m <= G . m ) holds

(Ser F) . n <= (Ser G) . n

let n be Nat; :: thesis: ( ( for m being Nat st m <= n holds

F . m <= G . m ) implies (Ser F) . n <= (Ser G) . n )

assume A1: for m being Nat st m <= n holds

F . m <= G . m ; :: thesis: (Ser F) . n <= (Ser G) . n

defpred S

F . m <= G . m ) implies (Ser F) . $1 <= (Ser G) . $1 );

A2: for k being Nat st S

S

proof

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A3: S_{1}[k]
; :: thesis: S_{1}[k + 1]

_{1}[k + 1]
; :: thesis: verum

end;assume A3: S

now :: thesis: ( ( for m being Nat st m <= k + 1 holds

F . m <= G . m ) implies (Ser F) . (k + 1) <= (Ser G) . (k + 1) )

hence
SF . m <= G . m ) implies (Ser F) . (k + 1) <= (Ser G) . (k + 1) )

assume A4:
for m being Nat st m <= k + 1 holds

F . m <= G . m ; :: thesis: (Ser F) . (k + 1) <= (Ser G) . (k + 1)

then ((Ser F) . k) + (F . (k + 1)) <= ((Ser G) . k) + (G . (k + 1)) by A3, A5, XXREAL_3:36;

then (Ser F) . (k + 1) <= ((Ser G) . k) + (G . (k + 1)) by SUPINF_2:def 11;

hence (Ser F) . (k + 1) <= (Ser G) . (k + 1) by SUPINF_2:def 11; :: thesis: verum

end;F . m <= G . m ; :: thesis: (Ser F) . (k + 1) <= (Ser G) . (k + 1)

A5: now :: thesis: for m being Nat st m <= k holds

F . m <= G . m

F . (k + 1) <= G . (k + 1)
by A4;F . m <= G . m

let m be Nat; :: thesis: ( m <= k implies F . m <= G . m )

assume m <= k ; :: thesis: F . m <= G . m

then m < k + 1 by NAT_1:13;

hence F . m <= G . m by A4; :: thesis: verum

end;assume m <= k ; :: thesis: F . m <= G . m

then m < k + 1 by NAT_1:13;

hence F . m <= G . m by A4; :: thesis: verum

then ((Ser F) . k) + (F . (k + 1)) <= ((Ser G) . k) + (G . (k + 1)) by A3, A5, XXREAL_3:36;

then (Ser F) . (k + 1) <= ((Ser G) . k) + (G . (k + 1)) by SUPINF_2:def 11;

hence (Ser F) . (k + 1) <= (Ser G) . (k + 1) by SUPINF_2:def 11; :: thesis: verum

now :: thesis: ( ( for m being Nat st m <= 0 holds

F . m <= G . m ) implies (Ser F) . 0 <= (Ser G) . 0 )

then A7:
SF . m <= G . m ) implies (Ser F) . 0 <= (Ser G) . 0 )

A6:
( (Ser F) . 0 = F . 0 & (Ser G) . 0 = G . 0 )
by SUPINF_2:def 11;

assume for m being Nat st m <= 0 holds

F . m <= G . m ; :: thesis: (Ser F) . 0 <= (Ser G) . 0

hence (Ser F) . 0 <= (Ser G) . 0 by A6; :: thesis: verum

end;assume for m being Nat st m <= 0 holds

F . m <= G . m ; :: thesis: (Ser F) . 0 <= (Ser G) . 0

hence (Ser F) . 0 <= (Ser G) . 0 by A6; :: thesis: verum

for k being Nat holds S

hence (Ser F) . n <= (Ser G) . n by A1; :: thesis: verum