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
  for n being Nat st 0 < a & a < 1 & 2 <= n holds a |^ n < a
proof
  let n be Nat;
  assume that
A1: 0 < a and
A2: a < 1 and
A3: 2 <= n;
  consider m being Nat such that
A4: n = 2+m by A3,NAT_1:10;
  defpred P[Nat] means a |^ (2+$1) < a;
A5: a*a < a*1 by A1,A2,XREAL_1:68;
A6: for m1 st P[m1] holds P[m1+1]
  proof
    let m1;
    assume a |^ (2+m1) < a;
    then a |^ (2+m1) * a < a*a by A1,XREAL_1:68;
    then a |^ (2+m1+1) < a*a by NEWTON:6;
    hence thesis by A5,XXREAL_0:2;
  end;
  a |^ (2+0) = a GeoSeq.(1+1) by Def1
    .= a GeoSeq.(0+1) * a by Th3
    .= a GeoSeq.0 * a * a by Th3
    .= 1*a*a by Th3
    .= a*a;
  then
A7: P[0] by A5;
A8: for m1 holds P[m1] from NAT_1:sch 2(A7,A6);
  thus thesis by A4,A8;
end;
