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

let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat holds (Partial_Sums (- F)) . n = (- ()) . n
let n be Nat; :: thesis: (Partial_Sums (- F)) . n = (- ()) . n
defpred S1[ Nat] means (Partial_Sums (- F)) . \$1 = (- ()) . \$1;
(Partial_Sums (- F)) . 0 = (- F) . 0 by MESFUNC9:def 4
.= - (F . 0) by Th37
.= - (() . 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
.= ((- ()) . k) + (- (F . (k + 1))) by
.= (- (() . k)) + (- (F . (k + 1))) by Th37
.= - ((() . k) + (F . (k + 1))) by MEASUR11:64
.= - (() . (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 = (- ()) . n ; :: thesis: verum