let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (2 * n) + 1 ) implies for n being Nat holds (Partial_Sums s) . n = (n + 1) |^ 2 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ($1 + 1) |^ 2;
assume A1: for n being Nat holds s . n = (2 * n) + 1 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (n + 1) |^ 2
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 = (n + 1) |^ 2 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((n + 1) |^ 2) + (s . (n + 1)) by SERIES_1:def 1
.= ((n + 1) |^ 2) + ((2 * (n + 1)) + 1) by A1
.= (((n + 1) |^ 2) + (2 * n)) + 3
.= ((((n |^ 2) + ((2 * n) * 1)) + (1 |^ 2)) + (2 * n)) + 3 by Lm3
.= ((((n |^ 2) + (2 * n)) + 1) + (2 * n)) + 3
.= ((n |^ 2) + ((2 * n) * 2)) + (2 |^ 2) by Lm3
.= (n + 2) |^ 2 by Lm3 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (2 * 0) + 1 by A1
.= (0 + 1) |^ 2 ;
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 = (n + 1) |^ 2 ; :: thesis: verum