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 Th22:
  for x be positive Nat st a > 2*n*(x|^n) holds
     x|^n - 1/2 < Py(a*x,n+1)/Py(a,n+1) < x|^n + 1/2
proof
  let x be positive Nat;
A1: 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) by Th21;
  assume
A2: a > 2*n*(x|^n);
A3: 1- n*(1/(2*a)) = 1- ((n*1)/(2*a)) by XCMPLX_1:74;
  2*(x|^n)*n >= 1*n by XREAL_1:64,NAT_1:14;
  then a >= n by A2,XXREAL_0:2;
  then a-n >=n-n by XREAL_1:9;
  then
A4: a-n+a >= a+0 by XREAL_1:6;
  then n / (2*a -n)<= n/a by XREAL_1:118;
  then 1 + n/(2*a -n) <= 1+n/a by XREAL_1:6;
  then
A5: (x |^ n) * (1 + n/(2*a -n)) <= (x|^n) * (1+n/a) by XREAL_1:64;
A6: 2*a - n >0 by A4;
A7: 1/1>=1/(2*a) by XREAL_1:118,NAT_1:14;
  2*a >n by A6,XREAL_1:47;
  then n/(2*a)<1 by XREAL_1:191;
  then
A8: 1- (n/(2*a)) >1-1 by XREAL_1:10;
  1-1<=1-1/(2*a)<=1-0 by A7,XREAL_1:10;
  then (1-1/(2*a))|^n >= 1- (n/(2*a)) by Lm5,A3;
  then 1/ ((1-1/(2*a))|^n) <= 1 / (1- (n/(2*a))) by A8,XREAL_1:118;
  then
A9: x|^n * (1/((1-1/(2*a))|^n)) <= x|^n * (1/(1- (n/(2*a)))) by XREAL_1:64;
  1- (n/(2*a)) = (2*a)/(2*a) - (n/(2*a)) by XCMPLX_1:60
    .= (2*a -n)/(2*a) by XCMPLX_1:120; then
A10: 1/(1- (n/(2*a))) = (2*a-n+n) / (2*a -n) by XCMPLX_1:57
    .= (2*a -n)/(2*a -n) + n/(2*a -n) by XCMPLX_1:62
    .= 1 + n/(2*a -n) by XCMPLX_1:60,A4;
A11: x|^n * (1/((1-1/(2*a))|^n)) <= x|^n * (1+n/a) by A5,A9,A10,XXREAL_0:2;
  2*n*(x|^n) = 2*(n*(x|^n));
  then n*(x|^n) < a/2 by A2,XREAL_1:81;
  then n*(x|^n) < 1/2*a;
  then (n * (x|^n))/a < 1/2 by XREAL_1:83;
  then (n/a) * (x|^n) < 1/2 by XCMPLX_1:74;
  then (x|^n) + (x|^n) * (n/a)  < x|^n + 1/2 by XREAL_1:6;
  then
A12: x|^n * (1 / (1-1/(2*a))|^n) < x|^n + 1/2 by A11,XXREAL_0:2;
  1 >= 1/(2*a*x) by XREAL_1:185,NAT_1:14;
  then 1-1<= 1-1/(2*a*x) <=1-0 by XREAL_1:10;
  then
A13: x|^n * (1-1/(2*a*x)) |^ n >= x|^n * (1 - n*(1/(2*a*x)))
  by Lm5,XREAL_1:64;
  (x|^n)*(n*(1/(2*a*x)))*2 = (x|^n)*((n*1)/(2*a*x))*2 by XCMPLX_1:74
  .= ((x|^n)*2)*(n/(2*a*x))
  .= (x|^n *2*n) /(2*a*x) by XCMPLX_1:74;
  then
A14: (x|^n)*(n*(1/(2*a*x)))*2 < a / (2*a*x) by A2,XREAL_1:74;
  2*x*a >= 1*a by XREAL_1:64,NAT_1:14;
  then a / (2*a*x) <= 1 by XREAL_1:183;
  then (x|^n)*(n*(1/(2*a*x)))*2 < 1 by A14,XXREAL_0:2;
  then (x|^n)*(n*(1/(2*a*x))) < 1/2 by XREAL_1:81;
  then x|^n - (x|^n)*(n*(1/(2*a*x))) > x|^n -1/2 by XREAL_1:15;
  then x|^n * (1-1/(2*a*x)) |^ n > (x|^n -1/2) by A13,XXREAL_0:2;
  hence thesis by A12,A1,XXREAL_0:2;
end;
