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

theorem Th11:
  D is non square implies
    ex x,y be Integer st
        y <> 0 & 0 < x - y * (sqrt D)
        & |. x^2 - D * y^2.| < 2*sqrt D+1
proof
  assume A1:D is non square;
  then consider x,y be Integer such that
  A2: y <> 0 and
  A3: |. y .| <= 2 and
  A4: 0 < x - y * (sqrt D) < 1/2 by Th9;
  A5: |. x^2 - D * y^2.| <= 2 * sqrt D + 1 / (2^2) by A1,A3,A4,Th10;
  take x,y;
  2 * sqrt D + 1 / (2^2) < 2*sqrt D+1 by XREAL_1:6;
  hence thesis by A2,A4,A5,XXREAL_0:2;
end;
