
theorem Th2:
  for a being non trivial Nat,
      n,m being Nat st n > m holds a|^n > a|^m
proof
let a be non trivial Nat;
let n,m be Nat;
assume A1: n > m;
then consider k being Nat such that
A2: n = m + k by NAT_1:10;
k <> 0 by A1,A2;
then k + 1 > 0 + 1 by XREAL_1:6;
then k >= 1 by NAT_1:13;
then a|^k >= a|^1 by Th1;
then A3: a|^k >= a;
a >= 2 by NAT_2:29;
then a|^k >= 1 + 1 by A3,XXREAL_0:2;
then a|^k > 1 by NAT_1:13;
then 1 * a|^m < a|^k * a|^m by XREAL_1:68;
hence thesis by A2,NEWTON:8;
end;
