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 Th12:
  for n being natural Number st 1 <= a & 1 <= n holds a <= a |^ n
proof
  let n be natural Number;
  assume that
A1: 1 <= a and
A2: 1 <= n;
  consider m being Nat such that
A3: n=m+1 by A2,NAT_1:6;
  defpred P[Nat] means a <= a |^ ($1+1);
A4: for m1 st P[m1] holds P[m1+1]
  proof
    let m1;
    assume a <= a |^ (m1+1);
    then a*1 <= a |^ (m1+1) * a by A1,XREAL_1:66;
    hence thesis by NEWTON:6;
  end;
A5: P[0];
A6: for m1 holds P[m1] from NAT_1:sch 2(A5,A4);
  thus thesis by A3,A6;
end;
