let seq be Complex_Sequence; :: thesis: for z being Complex holds Partial_Sums (z (#) seq) = z (#) (Partial_Sums seq)
let z be Complex; :: thesis: Partial_Sums (z (#) seq) = z (#) (Partial_Sums seq)
defpred S₁[ Nat] means (Partial_Sums (z (#) seq)) . $1 = (z (#) (Partial_Sums seq)) . $1;

A1: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A2: S₁[n] ; :: thesis: S₁[n + 1]
(Partial_Sums (z (#) seq)) . (n + 1) = ((Partial_Sums (z (#) seq)) . n) + ((z (#) seq) . (n + 1)) by SERIES_1:def 1
.= (z * ((Partial_Sums seq) . n)) + ((z (#) seq) . (n + 1)) by A2, VALUED_1:6
.= (z * ((Partial_Sums seq) . n)) + (z * (seq . (n + 1))) by VALUED_1:6
.= z * (((Partial_Sums seq) . n) + (seq . (n + 1)))
.= z * ((Partial_Sums seq) . (n + 1)) by SERIES_1:def 1
.= (z (#) (Partial_Sums seq)) . (n + 1) by VALUED_1:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums (z (#) seq)) . 0 = (z (#) seq) . 0 by SERIES_1:def 1
.= z * (seq . 0) by VALUED_1:6
.= z * ((Partial_Sums seq) . 0) by SERIES_1:def 1
.= (z (#) (Partial_Sums seq)) . 0 by VALUED_1:6 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence Partial_Sums (z (#) seq) = z (#) (Partial_Sums seq) ; :: thesis: verum