reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th17:
  for n being natural Number st 0 < a & a < 1 holds
  (1 + a) |^ n <= 1 + 3 |^ n * a
proof
  let n be natural Number;
A1: n is Nat by TARSKI:1;
  assume that
A2: 0 < a and
A3: a < 1;
A4: 1 + 0 < 1 + a by A2,XREAL_1:6;
  defpred P[Nat] means (1 + a) |^ $1 <= 1 + 3 |^ $1 * a;
A5: for n be Nat holds P[n] implies P[n+1]
  proof
    let n be Nat;
    assume (1 + a) |^ n <= 1 + 3 |^ n * a;
    then
A6: (1 + a) |^ n * (1 + a) <= (1 + 3 |^ n * a) * (1+a) by A2,XREAL_1:64;
A7: 1 + 3 |^ n * a + (3 |^ n + 3 |^ n) * a = 1 + (3 |^ n * (1 + 2)) * a
      .= 1 + 3 |^ (n+1) * a by NEWTON:6;
A8: 1 <= 3 |^ n
    proof
      per cases;
      suppose
A9:     0 = n;
        3 |^ n = 3 GeoSeq.n by Def1
          .= 1 by A9,Th3;
        hence thesis;
      end;
      suppose
        0 <> n;
        then 0 + 1 <= n by NAT_1:13;
        then 1 |^ n < 3 |^ n by Lm1;
        hence thesis;
      end;
    end;
    then 1 * a <= 3 |^ n * a by A2,XREAL_1:64;
    then
A10: 1 + a <= 1 + 3 |^ n * a by XREAL_1:7;
    a * a < 1 * a by A2,A3,XREAL_1:68;
    then 3 |^ n * (a * a) < 3 |^ n * a by A8,XREAL_1:68;
    then
A11: 3 |^ n * a + 3 |^ n * (a*a) < 3 |^ n * a + 3 |^ n * a by XREAL_1:6;
    (1 + 3 |^ n * a) * (1+a) = 1 + a + (3 |^ n * a + 3 |^ n * (a * a));
    then (1 + 3 |^ n * a) * (1+a) < 1 + 3 |^ (n+1) * a by A10,A7,A11,XREAL_1:8;
    then (1 + a) |^ n * (1 + a) <= 1 + 3 |^ (n + 1) * a by A6,XXREAL_0:2;
    hence thesis by NEWTON:6;
  end;
A12: 1 + 3 |^ 0 * a = 1 + 3 GeoSeq.0 * a by Def1
    .= 1 + 1 * a by Th3
    .= 1 + a;
  (1 + a) |^ 0 = (1 + a) GeoSeq.0 by Def1
    .= 1 by Th3;
  then
A13: P[0] by A12,A4;
  for n be Nat holds P[n] from NAT_1:sch 2(A13,A5);
  hence thesis by A1;
end;
