reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;
reserve r for Real;

theorem Th65:
  for n being Nat holds 0 < n & 1 < r implies 1 < r|^n
proof
  let n be Nat;
  assume that
A1: 0 < n and
A2: r > 1;
  defpred P[Nat] means 0 < $1 implies 1 < r|^$1;
A3: for k being Nat holds P[k] implies P[k+1]
  proof
    let k be Nat;
    assume that
A4: P[k] and
    0 < k+1;
    per cases;
    suppose
      k > 0;
      then
      r|^(k+1) = (r|^k)*r & 1 * r <= (r|^k)*r by A2,A4,NEWTON:6,XREAL_1:64;
      hence thesis by A2,XXREAL_0:2;
    end;
    suppose
      k = 0;
      hence thesis by A2;
    end;
  end;
A5: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A5,A3);
  hence thesis by A1;
end;
