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 Th16:
  for n being natural Number holds -1 < a implies (1 + a) |^ n >= 1 + n * a
proof
  let n be natural Number;
A1: n is Nat by TARSKI:1;
  defpred P[Nat] means (1 + a) |^ $1 >= 1 + $1 * a;
  assume
A2: -1 < a;
A3: for m be Nat st P[m] holds P[m+1]
  proof
A4: -1+1<1+a by A2,XREAL_1:6;
    let m be Nat;
    assume (1+a) |^ m >= 1 + m * a;
    then (1+a) |^ m * (1+a) >= (1 + m*a)*(1+a) by A4,XREAL_1:64;
    then
A5: (1+a) |^ (m+1) >= 1 + 1*a + m*a + (m*a)*a by NEWTON:6;
    0<=a*a by XREAL_1:63;
    then 1 + (m+1)*a + 0 <= 1 + (m+1)*a + m*(a*a) by XREAL_1:7;
    hence thesis by A5,XXREAL_0:2;
  end;
  (1 + a) |^ 0 = (1+a) GeoSeq.0 by Def1
    .= 1 by Th3;
  then
A6: P[0];
  for m being Nat holds P[m] from NAT_1:sch 2(A6,A3);
  hence thesis by A1;
end;
