let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n |^ 6 ) implies for n being Nat holds (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((($1 * ($1 + 1)) * ((2 * $1) + 1)) * ((((3 * ($1 |^ 4)) + (6 * ($1 |^ 3))) - (3 * $1)) + 1)) / 42;
assume A1: for n being Nat holds s . n = n |^ 6 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42
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 * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42) + (s . (n + 1)) by SERIES_1:def 1
.= ((((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42) + ((n + 1) |^ 6) by A1
.= ((((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) + (((n + 1) |^ (5 + 1)) * 42)) / 42
.= ((((n * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) + ((((n + 1) |^ 5) * (n + 1)) * 42)) / 42 by NEWTON:6
.= ((n + 1) * ((((2 * (n * n)) + n) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) + (((n + 1) |^ 5) * 42))) / 42
.= ((n + 1) * ((((2 * (n |^ 2)) + n) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) + (((n + 1) |^ 5) * 42))) / 42 by WSIERP_1:1
.= ((n + 1) * (((((((3 * ((n |^ 4) * (n |^ 2))) * 2) + (3 * ((n |^ 4) * n))) + (((6 * ((n |^ 3) * (n |^ 2))) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n * (n |^ 2))) * 2) + (3 * (n * n)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42
.= ((n + 1) * (((((((3 * (n |^ (4 + 2))) * 2) + (3 * ((n |^ 4) * n))) + (((6 * ((n |^ 3) * (n |^ 2))) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n * (n |^ 2))) * 2) + (3 * (n * n)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by NEWTON:8
.= ((n + 1) * (((((((3 * (n |^ (4 + 2))) * 2) + (3 * ((n |^ 4) * n))) + (((6 * (n |^ (3 + 2))) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n * (n |^ 2))) * 2) + (3 * (n * n)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by NEWTON:8
.= ((n + 1) * (((((((3 * (n |^ (4 + 2))) * 2) + (3 * (n |^ (4 + 1)))) + (((6 * (n |^ (3 + 2))) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n * (n |^ 2))) * 2) + (3 * (n * n)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by NEWTON:6
.= ((n + 1) * (((((((3 * (n |^ (4 + 2))) * 2) + (3 * (n |^ (4 + 1)))) + (((6 * (n |^ (3 + 2))) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n |^ (2 + 1))) * 2) + (3 * (n * n)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by NEWTON:6
.= ((n + 1) * (((((((3 * (n |^ 6)) * 2) + (3 * (n |^ 5))) + (((6 * (n |^ 5)) * 2) + (6 * ((n |^ 3) * n)))) - (((3 * (n |^ 3)) * 2) + (3 * (n |^ 2)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by WSIERP_1:1
.= ((n + 1) * ((((((6 * (n |^ 6)) + (3 * (n |^ 5))) + ((12 * (n |^ 5)) + (6 * (n |^ (3 + 1))))) - ((6 * (n |^ 3)) + (3 * (n |^ 2)))) + ((2 * (n |^ 2)) + n)) + (((n + 1) |^ 5) * 42))) / 42 by NEWTON:6
.= ((n + 1) * (((((((6 * (n |^ 6)) + (6 * (n |^ 4))) + (15 * (n |^ 5))) - (6 * (n |^ 3))) - (1 * (n |^ 2))) + n) + (((((((n |^ 5) + (5 * (n |^ 4))) + (10 * (n |^ 3))) + (10 * (n |^ 2))) + (5 * n)) + 1) * 42))) / 42 by Lm6
.= ((n + 1) * (((((((6 * (n |^ 6)) + (57 * (n |^ 5))) + (216 * (n |^ 4))) + (414 * (n |^ 3))) + (419 * (n |^ 2))) + (211 * n)) + 42)) / 42
.= ((n + 1) * (((n + 2) * ((2 * (n + 1)) + 1)) * ((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - (3 * (n + 1))) + 1))) / 42 by Lm16
.= ((((n + 1) * ((n + 1) + 1)) * ((2 * (n + 1)) + 1)) * ((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - (3 * (n + 1))) + 1)) / 42 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 |^ 6 by A1
.= (((0 * (0 + 1)) * ((2 * 0) + 1)) * ((((3 * (0 |^ 4)) + (6 * (0 |^ 3))) - (3 * 0)) + 1)) / 42 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 * (n + 1)) * ((2 * n) + 1)) * ((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (3 * n)) + 1)) / 42 ; :: thesis: verum