let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = ((2 * n) - 1) |^ 2 & s . 0 = 0 ) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (n * ((4 * (n |^ 2)) - 1)) / 3 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ($1 * ((4 * ($1 |^ 2)) - 1)) / 3;
assume A1: for n being Nat st n >= 1 holds
( s . n = ((2 * n) - 1) |^ 2 & s . 0 = 0 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (n * ((4 * (n |^ 2)) - 1)) / 3
A2: for n being Nat st n >= 1 & S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( n >= 1 & S₁[n] implies S₁[n + 1] )
assume that
n >= 1 and
A3: (Partial_Sums s) . n = (n * ((4 * (n |^ 2)) - 1)) / 3 ; :: thesis: S₁[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
(Partial_Sums s) . (n + 1) = ((n * ((4 * (n |^ 2)) - 1)) / 3) + (s . (n + 1)) by A3, SERIES_1:def 1
.= ((n * ((4 * (n |^ 2)) - 1)) / 3) + (((2 * (n + 1)) - 1) |^ 2) by A1, A4
.= ((n * ((4 * (n |^ 2)) - 1)) + ((((2 * n) + 1) |^ 2) * 3)) / 3
.= ((n * ((4 * (n |^ 2)) - 1)) + (((((2 * n) |^ 2) + ((2 * (2 * n)) * 1)) + (1 |^ 2)) * 3)) / 3 by Lm3
.= ((((n * 4) * (n |^ 2)) - (n * 1)) + (((((2 * n) |^ 2) * 3) + ((2 * (2 * n)) * 3)) + ((1 |^ 2) * 3))) / 3
.= (((n * (4 * (n |^ 2))) - n) + (((((2 |^ 2) * (n |^ 2)) * 3) + (((2 * 2) * n) * 3)) + ((1 |^ 2) * 3))) / 3 by NEWTON:7
.= (((n * (4 * (n |^ 2))) - n) + (((((2 |^ 2) * (n |^ 2)) * 3) + ((4 * n) * 3)) + (1 * 3))) / 3
.= (((n * (4 * (n |^ 2))) - n) + (((((2 * 2) * (n |^ 2)) * 3) + (12 * n)) + 3)) / 3 by WSIERP_1:1
.= ((((n * (4 * (n |^ 2))) + ((12 - 1) * n)) + (12 * (n |^ 2))) + 3) / 3
.= ((n + 1) * ((4 * ((n + 1) |^ 2)) - 1)) / 3 by Lm11 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . (0 + 1) = ((Partial_Sums s) . 0) + (s . (0 + 1)) by SERIES_1:def 1
.= (s . 0) + (s . 1) by SERIES_1:def 1
.= 0 + (s . 1) by A1
.= 0 + (((2 * 1) - 1) |^ 2) by A1
.= (1 * ((4 * (1 * 1)) - 1)) / 3
.= (1 * ((4 * (1 |^ 2)) - 1)) / 3 ;
then A5: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A5, A2);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (n * ((4 * (n |^ 2)) - 1)) / 3 ; :: thesis: verum