let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL

for n being Nat holds (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat holds (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

let n be Nat; :: thesis: (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

defpred S_{1}[ Nat] means (Partial_Sums (- F)) . $1 = (- (Partial_Sums F)) . $1;

(Partial_Sums (- F)) . 0 = (- F) . 0 by MESFUNC9:def 4

.= - (F . 0) by Th37

.= - ((Partial_Sums F) . 0) by MESFUNC9:def 4 ;

then A1: S_{1}[ 0 ]
by Th37;

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

S_{1}[k + 1]
_{1}[k]
from NAT_1:sch 2(A1, A2);

hence (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n ; :: thesis: verum

for n being Nat holds (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat holds (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

let n be Nat; :: thesis: (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n

defpred S

(Partial_Sums (- F)) . 0 = (- F) . 0 by MESFUNC9:def 4

.= - (F . 0) by Th37

.= - ((Partial_Sums F) . 0) by MESFUNC9:def 4 ;

then A1: S

A2: for k being Nat st S

S

proof

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

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

(Partial_Sums (- F)) . (k + 1) = ((Partial_Sums (- F)) . k) + ((- F) . (k + 1)) by MESFUNC9:def 4

.= ((- (Partial_Sums F)) . k) + (- (F . (k + 1))) by A3, Th37

.= (- ((Partial_Sums F) . k)) + (- (F . (k + 1))) by Th37

.= - (((Partial_Sums F) . k) + (F . (k + 1))) by MEASUR11:64

.= - ((Partial_Sums F) . (k + 1)) by MESFUNC9:def 4 ;

hence S_{1}[k + 1]
by Th37; :: thesis: verum

end;assume A3: S

(Partial_Sums (- F)) . (k + 1) = ((Partial_Sums (- F)) . k) + ((- F) . (k + 1)) by MESFUNC9:def 4

.= ((- (Partial_Sums F)) . k) + (- (F . (k + 1))) by A3, Th37

.= (- ((Partial_Sums F) . k)) + (- (F . (k + 1))) by Th37

.= - (((Partial_Sums F) . k) + (F . (k + 1))) by MEASUR11:64

.= - ((Partial_Sums F) . (k + 1)) by MESFUNC9:def 4 ;

hence S

hence (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n ; :: thesis: verum