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

theorem Th10:
  D is non square & n <> 0 & |. y .| <= n &
  0 < x - y * sqrt D < 1/n
        implies |. x^2 - D * y^2.| <= 2 * sqrt D + 1 / (n^2)
proof
   assume that A1:D is non square & n <>0 and
   A2: |. y .| <= n & 0 < x - y * (sqrt D) < 1/n;
   A3: sqrt D >0 by A1,SQUARE_1:25;
   then A4: |. sqrt D .| = sqrt D by ABSVALUE:def 1;
A5: -n <= y <=n by A2,ABSVALUE:5;
A6: y * (sqrt D) < x < 1/n +y * (sqrt D) by A2,XREAL_1:19,XREAL_1:47;
A7: (-n) *sqrt D <= y * (sqrt D) <= n * (sqrt D) by XREAL_1:64,A5,A3;
   then A8:y * (sqrt D)+1/n <= n * (sqrt D)+1/n by XREAL_1:6;
   (-n) *sqrt D-(1/n) <= (-n) *sqrt D by XREAL_1:51;
   then (-n) *sqrt D-(1/n) <= y * (sqrt D) by A7,XXREAL_0:2;
   then -(n * (sqrt D)+1/n ) < x < n * (sqrt D)+1/n by A6,A8,XXREAL_0:2;
   then A9: |. x .| <= n * (sqrt D) + 1/n by ABSVALUE:5;
   A10: |.  y .| * |. (sqrt D) .| <= n * |. sqrt D .|
      by A2, XREAL_1:64,A3;
   |. x + y * (sqrt D) .| <= |. x .| + |. y * ( sqrt D) .| by COMPLEX1:56;
   then A11: |. x + y * (sqrt D) .| <= |. x .| + |. y .| * (sqrt D)
      by A4,COMPLEX1:65;
   |. x .| + |. y .| * (sqrt D) <= n * (sqrt D) + 1/n + n* (sqrt D)
      by A4, A9, A10, XREAL_1:7;
   then A12: 0<=|. x + y * (sqrt D) .| <= 2*n * (sqrt D) + 1/n
     by COMPLEX1:46,A11,XXREAL_0:2;
   - 1/n <= 0 <= x - y * (sqrt D) <= 1/n by A2;
   then A13: 0<= |. x - y * (sqrt D) .| <= 1/n by ABSVALUE:5;
   A14: (2*n) / n =2 by A1,XCMPLX_1:89;
   |. x + y * (sqrt D) .| * |. x - y * (sqrt D) .|
       <= (2 *n* (sqrt D) + (1/n) ) * (1/n) by XREAL_1:66, A12, A13;
   then |. x + y * (sqrt D) .| * |. x - y * (sqrt D) .| <=
     (2*n) *( 1/n) * (sqrt D)+ ( 1/n) *( 1/n);
   then  |. x + y * (sqrt D) .| * |. x - y * (sqrt D) .| <=
     ( (2*n) / n) *  (sqrt D) +  ( 1/n) *( 1/n) by XCMPLX_1:99;
   then  |. x + y * (sqrt D) .| * |. x - y * (sqrt D) .| <=
     2 * 1 * (sqrt D) + (1 * 1 ) / (n * n) by XCMPLX_1:76, A14;
   then |.( x + y * (sqrt D)) * ( x - y * (sqrt D)) .|
     <= 2* (sqrt D) + 1 / (n*n) by COMPLEX1:65;
   then |. x^2 - y^2 * (sqrt D) ^2 .| <= 2* (sqrt D) + 1 / (n*n);
   then |. x^2 - y^2 * sqrt (D^2) .| <= 2* (sqrt D) + 1 / (n*n)
     by SQUARE_1:29;
   hence thesis by SQUARE_1:22;
end;
