reserve n, k, r, m, i, j for Nat;

theorem Th41:
  for n being non zero Element of NAT holds Fib (n) divides Fib ( n*k)
proof
  let n be non zero Element of NAT;
  defpred P[Nat] means Fib (n) divides Fib (n*$1);
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A2: P[k];
    Fib (n * (k+1)) = Fib ((n*k) + n)
      .= Fib (n) * Fib (n*k + 1) + Fib (n*k) * Fib (n -' 1) by Th40;
    hence thesis by A2,Th12;
  end;
A3: P[0] by NAT_D:6,PRE_FF:1;
  for n being Nat holds P[n] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
