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
  for m,n being Nat holds r > 1 & m > n implies r|^m > r|^n
proof
  let m,n be Nat;
  assume that
A1: r > 1 and
A2: m > n;
  reconsider m,n as Element of NAT by ORDINAL1:def 12;
  (m-'n)+n = m by A2,XREAL_1:235;
  then
A3: r|^m = r|^(m-'n)*r|^n by NEWTON:8;
  m-'n <> 0 by A2,NAT_D:36;
  then r|^(m-'n) > 1 by A1,Th65;
  hence thesis by A1,A3,NEWTON:83,XREAL_1:155;
end;
