reserve n,n1,n2,k,D for Nat,
        r,r1,r2 for Real,
        x,y for Integer;

theorem Th16:
  D is non square implies
    ex p be Pell's_solution of D st p is positive
  proof
    assume D is non square;
    then consider x,y be Nat such that
    A1: x^2 - D * y^2 = 1 & y <> 0 by Th14;
    x in INT & y in INT by INT_1:def 2;
    then reconsider ab=[x,y] as Element of [:INT,INT:] by ZFMISC_1:87;
    x = ab`1 & y = ab`2;then
    reconsider ab as Pell's_solution of D by A1,Def1;
    take ab;
    thus thesis by A1;
  end;
