
theorem Th21:
  for f be Real_Sequence,n be Nat st f.0 = 0
  holds Sum FinSeq (f,n) = (Partial_Sums f).n
  proof
    let f be Real_Sequence, n be Nat;
    assume
A0: f.0 = 0;
    defpred P[Nat] means
    Sum FinSeq (f,$1) = (Partial_Sums f).$1;
    Sum FinSeq (f,0) = 0 .= (Partial_Sums f).0 by A0,SERIES_1:def 1; then
A1: P[0];
A2: for i be Nat st P[i] holds P[i+1]
    proof
      let i be Nat;
      assume
A3:   P[i];
A4:   (FinSeq (f,i))^<*f.(i+1)*> = FinSeq (f,i+1) by Th20;
      (Partial_Sums f).(i+1) = (Partial_Sums f).i + f.(i+1) by SERIES_1:def 1
   .= (addreal "**" (FinSeq (f,i))) + f.(i+1) by RVSUM_1:def 12,A3
   .= addreal.((addreal "**" (FinSeq (f,i))),f.(i+1)) by BINOP_2:def 9
   .= addreal "**" ((FinSeq (f,i)) ^ <*f.(i+1)*>) by FINSOP_1:4
   .= Sum FinSeq (f,i+1) by RVSUM_1:def 12,A4;
      hence thesis;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
