let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((n * (n + 1)) * (n + 2)) * (n + 3) ) implies for n being Nat holds (Partial_Sums s) . n = ((((n * (n + 1)) * (n + 2)) * (n + 3)) * (n + 4)) / 5 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (((($1 * ($1 + 1)) * ($1 + 2)) * ($1 + 3)) * ($1 + 4)) / 5;
assume A1: for n being Nat holds s . n = ((n * (n + 1)) * (n + 2)) * (n + 3) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((((n * (n + 1)) * (n + 2)) * (n + 3)) * (n + 4)) / 5
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)) * (n + 2)) * (n + 3)) * (n + 4)) / 5 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((((n * (n + 1)) * (n + 2)) * (n + 3)) * (n + 4)) / 5) + (s . (n + 1)) by SERIES_1:def 1
.= (((((n * (n + 1)) * (n + 2)) * (n + 3)) * (n + 4)) / 5) + ((((n + 1) * ((n + 1) + 1)) * ((n + 1) + 2)) * ((n + 1) + 3)) by A1
.= (((((n + 1) * (n + 2)) * (n + 3)) * (n + 4)) * (n + 5)) / 5 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= ((((0 * (0 + 1)) * (0 + 2)) * (0 + 3)) * (0 + 4)) / 4 by A1 ;
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)) * (n + 2)) * (n + 3)) * (n + 4)) / 5 ; :: thesis: verum