reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;

theorem
  Product (n|->a) = (power K).(a,n)
proof
  defpred P[Nat] means Product ($1|->a) = (power K).(a,$1);
A1: for k st P[k] holds P[k+1]
  proof
    let k such that
A2: P[k];
    thus Product ((k+1) |->a) = (Product (k|->a)) * Product (1|->a) by
FVSUM_1:83
      .= Product (k|->a)*Product<*a*> by FINSEQ_2:59
      .= (power K).(a,k)*a by A2,GROUP_4:9
      .= (power K).(a,k+1) by GROUP_1:def 7;
  end;
  0|->a=<*>(the carrier of K);
  then Product (0|->a) = 1_K by GROUP_4:8
    .= (power K).(a,0) by GROUP_1:def 7;
  then
A3: P[0];
  for k holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
