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 Th9:
  for n being natural Number st 0 < a & a <= b holds a |^ n <= b |^ n
proof
  let n be natural Number;
A1: n is Nat by TARSKI:1;
  assume that
A2: 0 < a and
A3: a <= b;
  defpred P[Nat] means a |^ $1 <= b |^ $1;
A4: for m1 be Nat st P[m1] holds P[m1+1]
  proof
    let m1 be Nat such that
A5: a |^ m1 <= b |^ m1;
    a |^ m1 > 0 by A2,Th6;
    then a |^ m1 * a <= b |^ m1 * b by A2,A3,A5,XREAL_1:66;
    then a |^ (m1+1) <= b |^ m1 * b by NEWTON:6;
    hence thesis by NEWTON:6;
  end;
A6: b |^ 0 = b GeoSeq.0 by Def1
    .= 1 by Th3;
  a |^ 0 = a GeoSeq.0 by Def1
    .= 1 by Th3;
  then
A7: P[0] by A6;
  for m1 be Nat holds P[m1] from NAT_1:sch 2(A7,A4);
  hence thesis by A1;
end;
