let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT st n >= 1 holds
( s . n = ((2 * n) - 1) |^ 3 & s . 0 = 0 ) ) implies for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (n |^ 2) * ((2 * (n |^ 2)) - 1) )
assume A1: for n being Element of NAT st n >= 1 holds
( s . n = ((2 * n) - 1) |^ 3 & s . 0 = 0 ) ; :: thesis: for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (n |^ 2) * ((2 * (n |^ 2)) - 1)
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ($1 |^ 2) * ((2 * ($1 |^ 2)) - 1);
(Partial_Sums s) . (1 + 0 ) = ((Partial_Sums s) . 0 ) + (s . (1 + 0 )) by SERIES_1:def 1
.= (s . 0 ) + (s . 1) by SERIES_1:def 1
.= (s . 1) + 0 by A1
.= ((2 * 1) - 1) |^ 3 by A1
.= 1 |^ (2 + 1)
.= (1 |^ 2) * ((2 * (1 * 1)) - 1) by NEWTON:11
.= (1 |^ 2) * ((2 * (1 |^ 2)) - 1) by WSIERP_1:2 ;
then A2: S₁[1] ;
A3: 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 A4: ( n >= 1 & (Partial_Sums s) . n = (n |^ 2) * ((2 * (n |^ 2)) - 1) ) ; :: thesis: S₁[n + 1]
A5: n + 1 >= 1 by NAT_1:11;
A6: n in NAT by ORDINAL1:def 13;
then (Partial_Sums s) . (n + 1) = ((n |^ 2) * ((2 * (n |^ 2)) - 1)) + (s . (n + 1)) by A4, SERIES_1:def 1
.= ((n |^ 2) * ((2 * (n |^ 2)) - 1)) + (((2 * (n + 1)) - 1) |^ 3) by A1, A5
.= ((n |^ 2) * ((2 * (n |^ 2)) - 1)) + (((2 * n) + 1) |^ 3)
.= ((n |^ 2) * ((2 * (n |^ 2)) - 1)) + (((((2 * n) |^ 3) + (3 * ((2 * n) |^ 2))) + (3 * (2 * n))) + 1) by Lm4
.= (((((((n |^ 2) * 2) * (n |^ 2)) - (n |^ 2)) + ((2 * n) |^ 3)) + (3 * ((2 * n) |^ 2))) + ((3 * 2) * n)) + 1
.= (((((((n |^ 2) * 2) * (n |^ 2)) - (n |^ 2)) + ((2 * n) |^ 3)) + (3 * ((2 |^ 2) * (n |^ 2)))) + (6 * n)) + 1 by NEWTON:12
.= (((((((n |^ 2) * 2) * (n |^ 2)) - (n |^ 2)) + ((2 |^ 3) * (n |^ 3))) + (3 * ((2 |^ 2) * (n |^ 2)))) + (6 * n)) + 1 by NEWTON:12
.= (((((((n |^ 2) * 2) * (n |^ 2)) - (n |^ 2)) + ((2 |^ 3) * (n |^ 3))) + (3 * ((2 * 2) * (n |^ 2)))) + (6 * n)) + 1 by WSIERP_1:2
.= (((((((n |^ 2) * 2) * (n |^ 2)) - (n |^ 2)) + (((2 * 2) * 2) * (n |^ 3))) + (3 * ((2 * 2) * (n |^ 2)))) + (6 * n)) + 1 by Lm1
.= ((((((n |^ 2) * (n |^ 2)) * 2) + ((12 - 1) * (n |^ 2))) + (8 * (n |^ 3))) + (6 * n)) + 1
.= ((n + 1) |^ 2) * ((2 * ((n + 1) |^ 2)) - 1) by A6, Lm12 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A2, A3);
hence for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (n |^ 2) * ((2 * (n |^ 2)) - 1) ; :: thesis: verum