let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = n |^ 4 ) implies for n being Element of NAT holds (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30 )
assume A1: for n being Element of NAT holds s . n = n |^ 4 ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 = ((($1 * ($1 + 1)) * ((2 * $1) + 1)) * (((3 * ($1 |^ 2)) + (3 * $1)) - 1)) / 30;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 |^ 4 by A1
.= (((0 * (0 + 1)) * ((2 * 0 ) + 1)) * (((3 * (0 |^ 2)) + (3 * 0 )) - 1)) / 30 by NEWTON:16 ;
then A2: S₁[ 0 ] ;
A3: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30) + (s . (n + 1)) by SERIES_1:def 1
.= ((((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30) + ((n + 1) |^ 4) by A1
.= ((((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) + (((n + 1) |^ (3 + 1)) * 30)) / 30
.= ((((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) + ((((n + 1) |^ 3) * (n + 1)) * 30)) / 30 by NEWTON:11
.= ((n + 1) * (((n * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) + (((n + 1) |^ 3) * 30))) / 30
.= ((n + 1) * (((n + 2) * ((2 * (n + 1)) + 1)) * (((3 * ((n + 1) |^ 2)) + (3 * (n + 1))) - 1))) / 30 by Lm5 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence for n being Element of NAT holds (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) / 30 ; :: thesis: verum