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 Th17:
   a <> 2 & k <= n implies 2 * Py(a,k) < Px(a,n)
proof
  set A=a^2-'1,S = sqrt A;
A1: a^2 -'1 = a^2 -1 by NAT_1:14,XREAL_1:233;
  assume
A2: a<>2 & k <=n;
  Px(a,k)^2 - (a^2-'1) *Py(a,k)^2 =1 by Th10;then
A3: A*(Py(a,k))^2+1 = Px(a,k)^2;
  a >=2+1 by A2,NAT_1:22,NAT_2:29;
  then a^2 >= 3*3 by XREAL_1:66;
  then A >= 9-1 by A1,XREAL_1:9;
  then A >= 4 by XXREAL_0:2;
  then 4*(Py(a,k))^2 <= A*(Py(a,k))^2 by XREAL_1:64;
  then (2*Py(a,k))^2 < Px(a,k)^2 by A3,NAT_1:13;
  then
A4:  2 * Py(a,k) < Px(a,k) by SQUARE_1:15;
  Px(a,k)<=Px(a,n) by A2,Th13;
  hence thesis by A4,XXREAL_0:2;
end;
