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 Th10:
  for n being natural Number st 0 <= a & a < b & 1 <= n holds a |^ n < b |^ n
proof
  let n be natural Number;
  assume that
A1: 0 <= a and
A2: a < b and
A3: 1 <= n;
  per cases by A1;
  suppose
    a>0;
    hence thesis by A2,A3,Lm1;
  end;
  suppose
A4: a=0;
    reconsider k=n, k1=1 as Integer;
    reconsider m=k-k1 as Element of NAT by A3,INT_1:5;
    a |^ n = a |^ (m+1) .= a |^ m * a |^ 1 by NEWTON:8
      .= a |^ m * a GeoSeq.(0+1) by Def1
      .= a |^ m * (a GeoSeq.0 * 0) by A4,Th3
      .= 0;
    hence thesis by A1,A2,Th6;
  end;
end;
