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 Th10:
  Px(a,n)^2 - (a^2-'1) *Py(a,n)^2 = 1
proof
  set m=min_Pell's_solution_of (a^2-'1);
  per cases;
  suppose n=0;
    then Px(a,n) = 1 & Py(a,n) = 0  by Th6;
    hence thesis;
  end;
  suppose
A1:   n >0;
    Px(a,n) + Py(a,n)*sqrt (a^2-'1) = (m`1 + m`2 *sqrt ((a^2-'1)) ) |^ n
      by Def2;
    then reconsider PP=[Px(a,n),Py(a,n)] as
    positive Pell's_solution of (a^2-'1) by A1,PELLS_EQ:20;
    (PP`1)^2 - (a^2-'1)* (PP`2)^2 =1 by PELLS_EQ:def 1;
    hence thesis;
  end;
end;
