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 Th20:
  (2*a-1)|^n *(a-1) <= Px(a,n+1) <= a * (2*a) |^ n &
  (2*a-1)|^n        <= Py(a,n+1) <= (2*a) |^ n
proof
  defpred P[Nat] means
    (2*a-1)|^$1        <= Py(a,$1+1) <= (2*a) |^ $1 &
    (2*a-1)|^$1 *(a-1) <= Px(a,$1+1) <= a * (2*a) |^ $1;
A1: a^2-'1 = a^2-1 by XREAL_1:233,NAT_1:14;
A2: (2*a-1)|^0 = 1 & (2*a) |^ 0 =1 by NEWTON:4;
A3:  a-1 <= a-0 by XREAL_1:10;
  Py(a,0) = 0 & Px(a,0)= 1 by Th6;
  then Py(a,1+0) = 1+0*a & Px(a,1+0) = 1*a+ 0*(a^2-'1)  by Th9;
  then
A4: P[0] by A2,A3;
A5: P[k] implies P[k+1]
  proof
    set k1=k+1;
    assume
A6:   P[k];
A7:   Py(a,k1+1) = Px(a,k1) + Py(a,k1)*a &
    Px(a,k1+1) = Px(a,k1)*a + Py(a,k1)*(a^2-'1) by Th9;
A8:   Py(a,k1)*a <= (2*a) |^ k*a by A6,XREAL_1:64;
A9:  (2*a) |^ k*a + (2*a) |^ k*a = (2*a) |^ k*(2*a)
        .= (2*a) |^ k1 by NEWTON:6;
A10:  Py(a,k1)*a >= (2*a-1)|^k*a by A6,XREAL_1:64;
A11:  (2*a-1)|^k *(a-1) + (2*a-1)|^k*a = (2*a-1)|^k *(2*a-1)
        .= (2*a-1)|^k1 by NEWTON:6;
    Px(a,k1)*a <= a * (2*a) |^ k *a & Py(a,k1)*(a^2-'1)
       <= (2*a) |^ k * (a^2-'1) by A6,XREAL_1:64;
    then
A12:  Px(a,k1+1) <= a * (2*a) |^ k *a + (2*a) |^ k * (a^2-'1) by A7,XREAL_1:7;
    a^2 +a^2-1 <= a^2+a^2-0 by XREAL_1:10;
   then
A13:  (2*a) |^ k * (a^2 +a^2-1) <= (2*a) |^ k * (a^2 +a^2) by XREAL_1:64;
   (2*a-1)|^k * (a^2-'1) <= Py(a,k1)*(a^2-'1) &
     (2*a-1)|^k *(a-1)*a <= Px(a,k1)*a by A6,XREAL_1:64;
   then
A14: (2*a-1)|^k * (a^2-'1) + (2*a-1)|^k *(a-1)*a <= Px(a,k1+1)
       by A7,XREAL_1:7;
    2*1 <= 2*a by XREAL_1:64,NAT_1:14;
    then 1--1 <= 2*a;
    then 2*a^2-a+(1-2*a) <= 2*a^2-a+-1 by XREAL_1:6,12;
    then (2*a-1)|^k * ((2*a-1)*(a-1)) <= (2*a-1)|^k * (a^2-1 + (a-1)*a)
      by XREAL_1:64;
    hence thesis by A8,A7,A6,XREAL_1:7,A9,A10,A11,A1,A13,A12,XXREAL_0:2,A14;
  end;
  P[k] from NAT_1:sch 2(A4,A5);
  hence thesis;
end;
