let F be Real_Sequence; :: thesis: for n being Nat
for a being Real st ( for k being Nat holds F . k = a ) holds
(Partial_Sums F) . n = a * (n + 1)
let n be Nat; :: thesis: for a being Real st ( for k being Nat holds F . k = a ) holds
(Partial_Sums F) . n = a * (n + 1)
let a be Real; :: thesis: ( ( for k being Nat holds F . k = a ) implies (Partial_Sums F) . n = a * (n + 1) )
assume A1: for k being Nat holds F . k = a ; :: thesis: (Partial_Sums F) . n = a * (n + 1)
defpred S₁[ Nat] means (Partial_Sums F) . $1 = a * ($1 + 1);
(Partial_Sums F) . 0 = F . 0 by SERIES_1:def 1;
then A2: S₁[ 0 ] by A1;
A3: for i being Nat st S₁[i] holds
S₁[i + 1]

proof

let i be Nat; :: thesis: ( S₁[i] implies S₁[i + 1] )
assume A4: S₁[i] ; :: thesis: S₁[i + 1]
reconsider i1 = i + 1, One = 1 as Real ;
(Partial_Sums F) . (i + 1) = ((Partial_Sums F) . i) + (F . (i + 1)) by SERIES_1:def 1;
then (Partial_Sums F) . (i + 1) = (a * (i + 1)) + a by A1, A4;
hence S₁[i + 1] ; :: thesis: verum

end;

for i being Nat holds S₁[i] from NAT_1:sch 2(A2, A3);
hence (Partial_Sums F) . n = a * (n + 1) ; :: thesis: verum