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 Th14:
  for n being natural Number st 0 < a & a <= 1 & 1 <= n holds a |^ n <= a
proof
  let n be natural Number;
  assume that
A1: 0 < a and
A2: a <= 1 and
A3: 1 <= n;
  consider m being Nat such that
A4: n = 1+m by A3,NAT_1:10;
  defpred P[Nat] means a |^ (1+$1) <= a;
A5: a*a <= a*1 by A1,A2,XREAL_1:64;
A6: for m1 st P[m1] holds P[m1+1]
  proof
    let m1;
    assume a |^ (1+m1) <= a;
    then a |^ (1+m1) * a <= a*a by A1,XREAL_1:64;
    then a |^ (1+(m1+1)) <= a*a by NEWTON:6;
    hence thesis by A5,XXREAL_0:2;
  end;
A7: P[0];
A8: for m1 holds P[m1] from NAT_1:sch 2(A7,A6);
  thus thesis by A4,A8;
end;
