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 Th9:
  Px(a,n+1) = Px(a,n)*a + Py(a,n)*(a^2-'1) &
  Py(a,n+1) = Px(a,n) + Py(a,n)*a
proof
  set A=a^2-'1,M=min_Pell's_solution_of A;
  set n1=n+1;
A1:(sqrt A)^2 =A by SQUARE_1:def 2;
A2: M = [a,1] by Th8;
  Px(a,n1) + Py(a,n1) * sqrt A = ( M`1 + M`2 * sqrt A) |^n1 by Def2
    .= (M`1 + M`2 * sqrt A) |^n * (M`1 + M`2 * sqrt A) by NEWTON:6
    .= (Px(a,n) + Py(a,n) * sqrt A) * (M`1 + M`2 * sqrt A) by Def2
    .= (Px(a,n) * a + Py(a,n)* A) + (Px(a,n) + Py(a,n) * a) * sqrt A by A2,A1;
  hence thesis by PELLS_EQ:3;
end;
