reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th2:
  1 < a implies b < a |^ b
proof
  defpred P[Nat] means $1 < a |^ $1;
  assume
A1: 1 < a;
  then reconsider a1 = a-1 as Element of NAT by INT_1:5;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    then
A3: k+1 < a |^ k + 1 by XREAL_1:6;
A4: now
      set x = a |^ k;
      assume x + 1 > x * a;
      then x*a - (x+1) < (x+1) - (x+1) by XREAL_1:14;
      then x*a1 - 1 + 1 < 0+1 by XREAL_1:6;
      then x = 0 or a1 = 0 by NAT_1:13;
      hence contradiction by A1;
    end;
    a |^ (k+1) = a |^ k * a by NEWTON:6;
    hence thesis by A3,A4,XXREAL_0:2;
  end;
A5: P[ 0 ] by A1;
  for k being Nat holds P[k] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
