
theorem Th14:
  for F be Real_Sequence, n be Nat, a be Real st (for k be Nat holds F.k = a)
  holds (Partial_Sums F).n = a * (n+1)
  proof
    let F be Real_Sequence, n be Nat, a be Real;
    assume
A1: for k be Nat holds F.k = a;
    defpred P[Nat] means (Partial_Sums F).$1 = a*($1 + 1);
    (Partial_Sums F).0 = F.0 by SERIES_1:def 1; then
A2: P[0] by A1;
A3: for i be Nat st P[i] holds P[i+1]
    proof
      let i be Nat;
      assume
A4:   P[i];
      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 P[i+1];
    end;
    for i be Nat holds P[i] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
