reserve m,n for Nat;
reserve r for Real;
reserve c for Element of F_Complex;

theorem Th2:
  2 <= m implies for A being Real
  ex n being positive Nat st A <= m|^n
  proof
    assume 2 <= m; then 1 + 1 <= m;
    then
A1: 1 < m by NAT_1:13;
    let A be Real;
    reconsider a=|.[/ A \].| as Nat by TARSKI:1;
    reconsider n=a+1 as positive Nat;
    A <= [/A\] & [/ A \] <= a by ABSVALUE:4,INT_1:def 7;
    then A2: A <= a by XXREAL_0:2;
    a < n & n <= m |^ n by A1,NAT_3:2,NAT_1:13;
    then a <= m |^ n by XXREAL_0:2;
    hence thesis by A2,XXREAL_0:2;
  end;
