reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  Product(n |-> a) = a |^ n
proof
  defpred P[Nat] means Product($1 |-> a) = a |^ $1;
A1: for n being Nat st P[n] holds P[n+1] by Lm2;
A2: P[0] by Lm1;
  for n being Nat holds P[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
