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 S₁[ 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: S₁[ 0 ] by Th37;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[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 S₁[k + 1] by Th37; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence (Partial_Sums (- F)) . n = (- (Partial_Sums F)) . n ; :: thesis: verum