
theorem Th54:
  for n being Element of NAT, r being Real st r > 0 holds
  Product (n|-> r) = r to_power n
proof
  defpred P[Nat] means for r being Real st r>0 holds Product ($1
  |-> r) = r to_power $1;
A1: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A2: P[n];
    now
      let r be Real;
      assume
A3:   r>0;
      Product ((n+1)|-> r) = Product ((n |-> r) ^ <*r*>) by FINSEQ_2:60
        .= Product (n |-> r) * r by RVSUM_1:96
        .= (r to_power n) * r by A2,A3
        .= (r to_power n) * (r to_power 1) by POWER:25;
      hence Product ((n+1)|-> r) = r to_power (n+1) by A3,POWER:27;
    end;
    hence thesis;
  end;
A4: P[0] by POWER:24,RVSUM_1:94;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
