let a, b be Real; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = (a * n) + b ) holds
for n being Nat holds (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b
let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (a * n) + b ) implies for n being Nat holds (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((((a * ($1 + 1)) * $1) / 2) + ($1 * b)) + b;
assume A1: for n being Nat holds s . n = (a * n) + b ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((((a * (n + 1)) * n) / 2) + (n * b)) + b) + (s . (n + 1)) by SERIES_1:def 1
.= (((((a * (n + 1)) * n) / 2) + (n * b)) + b) + ((a * (n + 1)) + b) by A1
.= ((((a * (n + 1)) * (n + 2)) / 2) + ((n + 1) * b)) + b ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= ((((a * (0 + 1)) * 0) / 2) + (0 * b)) + b by A1 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence for n being Nat holds (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b ; :: thesis: verum