reserve n,n1,n2,k,D for Nat,
        r,r1,r2 for Real,
        x,y for Integer;
reserve p,p1,p2 for Pell's_solution of D;

theorem Th19:
    1 < p1`1 + p1`2 * sqrt D < p2`1 + p2`2 * sqrt D
  &
    D is non square
  implies p1`1 < p2`1 & p1`2 < p2`2
  proof
    assume A1: 1 < p1`1 + p1`2 * sqrt D < p2`1 + p2`2 * sqrt D &
    D is non square &
    (p1`1 >= p2`1 or p1`2 >= p2`2);
    per cases by A1;
    suppose A2: p1`2 >= p2`2;
      A3: sqrt D > 0 by A1,SQUARE_1:25;
      A4: p1 is positive by Th18,A1;
      p2`1 + p2`2 * sqrt D > 1 by A1, XXREAL_0:2;
      then A5: p2 is positive by Th18, A1;
      (p1`1)^2 - (p1`2)^2 * D + (p1`2)^2 * D = 1 + (p1`2)^2 * D by Def1;
      then A6: p1`1 = sqrt (1 + (p1`2)^2 * D) by SQUARE_1:def 2, A4;
      (p2`1)^2 - (p2`2)^2 * D + (p2`2)^2 * D = 1 + (p2`2)^2 * D by Def1;
      then A7: p2`1 = sqrt (1 + (p2`2)^2 * D) by SQUARE_1:def 2, A5;
      (p1`2)^2 >= (p2`2)^2 by SQUARE_1:15, A5, A2;
      then (p1`2)^2 * D >= (p2`2)^2 * D by XREAL_1:64;
      then (p1`2)^2 * D + 1 >= (p2`2)^2 * D + 1 by XREAL_1:6;
      then  A8: p1`1 >= p2`1 by A6,A7, SQUARE_1:26;
      p1`2 * sqrt D >= p2`2 * sqrt D by A2, XREAL_1:64, A3;
      hence contradiction by A1, A8, XREAL_1:7;
    end;
    suppose A9: p1`1 >= p2`1;
      A10: sqrt D >0 by A1,SQUARE_1:25;
      A11: p1 is positive by Th18, A1;
      p2`1 + p2`2 * sqrt D > 1 by A1, XXREAL_0:2;
      then A12: p2 is positive by Th18, A1;
      A13: (p1`1)^2 - (p1`2)^2 * D + (p1`2)^2 * D = 1 + (p1`2)^2 * D by Def1;
      A14:  p1`1 = sqrt (1 + (p1`2)^2 * D) by A13,SQUARE_1:def 2, A11;
      A15: (p2`1)^2 - (p2`2)^2 * D + (p2`2)^2 * D = 1 + (p2`2)^2 * D by Def1;
      A16:  p2`1 = sqrt (1 + (p2`2)^2 * D) by A15,SQUARE_1:def 2, A12;
      sqrt (1 + (p1`2)^2 * D) >=0 & sqrt (1 + (p2`2)^2 * D) >= 0
        by SQUARE_1:25;
      then A17: (sqrt (1 + (p1`2)^2 * D))^2 >=
      (sqrt (1 + (p2`2)^2 * D))^2 by SQUARE_1:15,A9,A14,A16;
      sqrt (1 + (p2`2)^2 * D) * sqrt (1 + (p2`2)^2 * D) =
      sqrt ( (1 + (p2`2)^2 * D) * (1 + (p2`2)^2 * D)) by SQUARE_1:29;
      then A18: sqrt ( (1 + (p1`2)^2 * D)^2) >=
      sqrt ( (1 + (p2`2)^2 * D)^2) by A17, SQUARE_1:29;
      A19: sqrt ( (1 + (p1`2)^2 * D)^2) = (1 + (p1`2)^2 * D)
        by SQUARE_1:22;
      sqrt ( (1 + (p2`2)^2 * D)^2) = (1 + (p2`2)^2 * D) by SQUARE_1:22;
      then 1 + (p1`2)^2 * D -1 >= 1 + (p2`2)^2 * D -1
        by XREAL_1:13, A18, A19;
      then A20: (p1`2)^2 * D / D  >= (p2`2)^2 * D / D by XREAL_1:72;
      A21: (p1`2)^2 * D / D = (p1`2)^2 by XCMPLX_1:89, A1;
      (p2`2)^2 * D / D = (p2`2)^2 by XCMPLX_1:89, A1;
      then A22: sqrt ((p1`2)^2) >= sqrt ((p2`2)^2) by SQUARE_1:26, A20, A21;
      sqrt ( (p2`2)^2 ) = p2`2 by SQUARE_1:22, A12;
      then A23: p1`2 >= p2`2 by A22, SQUARE_1:22, A11;
      p1`2 * sqrt D >= p2`2 * sqrt D by A23, XREAL_1:64, A10;
      hence contradiction by A1,A9,XREAL_1:7;
    end;
  end;
