let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n |^ 3 ) implies for n being Nat holds (Partial_Sums s) . n = ((n |^ 2) * ((n + 1) |^ 2)) / 4 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (($1 |^ 2) * (($1 + 1) |^ 2)) / 4;
assume A1: for n being Nat holds s . n = n |^ 3 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((n |^ 2) * ((n + 1) |^ 2)) / 4
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 |^ 2) * ((n + 1) |^ 2)) / 4 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((n |^ 2) * ((n + 1) |^ 2)) / 4) + (s . (n + 1)) by SERIES_1:def 1
.= (((n |^ 2) * ((n + 1) |^ 2)) / 4) + ((n + 1) |^ 3) by A1
.= (((n |^ 2) * ((n + 1) |^ 2)) + (((n + 1) |^ 3) * 4)) / 4
.= (((n |^ 2) * ((n + 1) |^ 2)) + ((((n + 1) |^ 2) * (n + 1)) * 4)) / 4 by Lm2
.= (((n + 1) |^ 2) * ((n |^ 2) + (((2 * 2) * n) + 4))) / 4
.= (((n + 1) |^ 2) * ((n + 2) |^ 2)) / 4 by Lm3 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 |^ 3 by A1
.= (0 * ((0 + 1) |^ 2)) / 4 by NEWTON:11
.= ((0 |^ 2) * ((0 + 1) |^ 2)) / 4 by NEWTON:11 ;
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 |^ 2) * ((n + 1) |^ 2)) / 4 ; :: thesis: verum