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 Th19:
   a = 2 & k < n implies (3 + 2*sqrt 3) * Py(a,k) < Px(a,n)
proof
A1:(sqrt 3)^2 = 3 & sqrt 3 >= 0 by SQUARE_1:def 2;
  set A=a^2-'1,S = sqrt A;
  assume
A2: a = 2 & k <n;
  then
A3: k+1 <= n by NAT_1:13;
  Py(a,k+1) = Px(a,k) + Py(a,k) * a by Th9;
  then
A4: (sqrt 3) *(Px(a,k) + Py(a,k) * 2) < Px(a,n) by A3,A2,Th18;
  (sqrt 3) * Py(a,k) <= Px(a,k) by A2,Th18;
  then
A5: (sqrt 3) * Py(a,k)*(sqrt 3) <= Px(a,k)* (sqrt 3) by A1,XREAL_1:64;
A6: (sqrt 3) * Py(a,k)*(sqrt 3) = (sqrt 3) *(sqrt 3) * Py(a,k)
     .= 3*Py(a,k) by A1;
  (sqrt 3) *(Px(a,k) + Py(a,k) * 2) = Py(a,k) * (2*sqrt 3) + Px(a,k)* (sqrt 3);
  then (sqrt 3) *(Px(a,k) + Py(a,k) * 2) >= Py(a,k) * (2*sqrt 3) + 3*Py(a,k)
    by A5,A6,XREAL_1:7;
  hence thesis by A4,XXREAL_0:2;
end;
