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 Th18:
  a = 2 & k <= n implies (sqrt 3) * 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
  A*(Py(a,k))^2+1 = Px(a,k)^2;
  then
A3:  3*(Py(a,k))^2 < Px(a,k)^2 by A1,A2,NAT_1:13;
  3* Py(a,k)^2 = (sqrt 3)^2 * Py(a,k)^2 by SQUARE_1:def 2
        .= ((sqrt 3)* Py(a,k))^2;
  then
A4: (sqrt 3)* Py(a,k) < Px(a,k) by A3,SQUARE_1:15;
  Px(a,k)<=Px(a,n) by A2,Th13;
  hence thesis by A4,XXREAL_0:2;
end;
