let seq be Real_Sequence; :: thesis: Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq)

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

A1: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

then A2: S_{1}[ 0 ]
by SERIES_1:def 1;

for i being Nat holds S_{1}[i]
from NAT_1:sch 2(A2, A1);

hence Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq) ; :: thesis: verum

defpred S

A1: for k being Nat st S

S

proof

(Partial_Sums (R_EAL seq)) . 0 = seq . 0
by MESFUNC9:def 1;
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

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

then (Partial_Sums (R_EAL seq)) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by MESFUNC9:def 1

.= ((Partial_Sums seq) . k) + (seq . (k + 1)) ;

hence S_{1}[k + 1]
by SERIES_1:def 1; :: thesis: verum

end;assume S

then (Partial_Sums (R_EAL seq)) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by MESFUNC9:def 1

.= ((Partial_Sums seq) . k) + (seq . (k + 1)) ;

hence S

then A2: S

for i being Nat holds S

hence Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq) ; :: thesis: verum