reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th21:
  for x be positive Nat holds
    x|^n * (1-1/(2*a*x)) |^ n
      <= Py(a*x,n+1)/Py(a,n+1)
        <= (x|^n) * (1 / (1-1/(2*a))|^n)
proof
  let x be positive Nat; set z=n;
  1-1/(2*a*x) = (2*a*x)/(2*a*x)-1/(2*a*x) by XCMPLX_1:60
    .= (2*a*x-1)/(2*a*x) by XCMPLX_1:120;
  then x *(1-1/(2*a*x)) = (2*a*x-1)/(2*a) by XCMPLX_1:92;
  then
A1:   x|^z * (1-1/(2*a*x)) |^ z = ((2*a*x-1)/(2*a)) to_power z by NEWTON:7
    .= ((2*a*x-1) to_power z) / ((2*a) to_power z) by POWER:31;
A2: (2*a-1)|^z <= Py(a,z+1) <= (2*a) |^ z by Th20;
A3:  (2*(a*x)-1)|^z <= Py(a*x,z+1) <= (2*(a*x)) |^ z by Th20;
  ((2*a*x-1)|^z)*Py(a,z+1) <=Py(a*x,z+1)*((2*a) |^ z) by A2,A3,XREAL_1:66;
  hence x|^z * (1-1/(2*a*x)) |^ z <= Py(a*x,z+1)/Py(a,z+1) by A1,XREAL_1:102;
  a>=1 by NAT_1:14;
  then a+a >= 1+1 by XREAL_1:7;
  then 1*(2*a) > 1 by NAT_1:13;
  then 1 > 1/(2*a) by XREAL_1:83;
  then
A4: 1 - 1/(2*a) >1/(2*a)-1/(2*a) by XREAL_1:14;
A5: Py(a*x,z+1)*((2*a-1) |^ z)<= ((2*a*x) |^ z)*Py(a,z+1) by A2,A3,XREAL_1:66;
  (2*a)/(2*a)=1 by XCMPLX_1:60;
  then (2*a-1)/(2*a) = 1 - (1/(2*a)) by XCMPLX_1:120;
  then (2*a)/(2*a-1) = 1 / (1 - (1/(2*a))) by XCMPLX_1:57;
  then (2*a*x)/(2*a-1)= x* (1 / (1 - (1/(2*a)))) by XCMPLX_1:74;
  then (2*a*x)/(2*a-1)= (x*1) / (1 - 1/(2*a)) by XCMPLX_1:74;
  then ((2*a*x) to_power z) / ((2*a-1) to_power z) =
      (x / (1 - 1/(2*a))) to_power z by POWER:31
    .= (1*(x to_power z)) / ((1 - 1/(2*a)) to_power z) by A4,POWER:31
    .= (x|^z) * (1 / (1-1/(2*a))|^z) by XCMPLX_1:74;
  hence thesis by A5,XREAL_1:102;
end;
