let seq be Real_Sequence; :: thesis: Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq)
defpred S₁[ Element of NAT ] means (Partial_Sums (R_EAL seq)) . $1 = (R_EAL (Partial_Sums seq)) . $1;
A1: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then (Partial_Sums (R_EAL seq)) . (k + 1) = (R_EAL ((Partial_Sums seq) . k)) + (R_EAL (seq . (k + 1))) by MESFUNC9:def 1
.= R_EAL (((Partial_Sums seq) . k) + (seq . (k + 1))) by MESFUN6C:1 ;
hence S₁[k + 1] by SERIES_1:def 1; :: thesis: verum

end;

(Partial_Sums (R_EAL seq)) . 0 = R_EAL (seq . 0 ) by MESFUNC9:def 1;
then A2: S₁[ 0 ] by SERIES_1:def 1;
for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A2, A1);
hence Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq) by FUNCT_2:113; :: thesis: verum