reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem
  Product (f|^a) = (Product f) |^ a
proof
  defpred P[Nat] means Product (f|^$1) = (Product f) |^ $1;
A1: P[b] implies P[b+1]
  proof
    assume
A2: P[b];
    thus Product (f|^(b+1)) = Product (f|^b) * Product f by Th14
      .= (Product f) |^ (b+1) by A2,NEWTON:6;
  end;
  Product (f|^0) = Product ((len f) |-> 1) by Th9
    .= 1 by RVSUM_1:102
    .= (Product f) |^ 0 by NEWTON:4;
  then
A3: P[ 0 ];
  P[b] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
