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 Th18:
  D is non square implies
    (p is positive iff p`1 + p`2 * sqrt D > 1)
  proof
    assume A1: D is non square;
    thus p is positive implies p`1 + p`2 * sqrt(D) > 1
    proof
      assume A2: p is positive;
      A3:1^2 = 1;
      A4: sqrt D > 1 by A1,SQUARE_1:27,NAT_1:25, SQUARE_1:18, A3;
      p`2 >= 0 + 1 by A2, INT_1:7;
      then A5: p`2 * sqrt D >= 1 * sqrt D by A4, XREAL_1:64;
      p`2 * sqrt D > 1 by XXREAL_0:2,A5,A4;
      then (p`2 * sqrt D ) + (p`1) > 1 + 0 by XREAL_1:8, A2;
      hence thesis;
    end;
    assume A6:  p`1 + p`2 * sqrt(D) > 1;
    A7: sqrt D >0 by A1,SQUARE_1:25;
    (p`1)^2 - D * (p`2)^2 = (p`1)^2 - (sqrt D)^2 * (p`2)^2 by SQUARE_1:def 2;
    then A8: (p`1 + p`2 * sqrt(D)) * (p`1 - p`2 * sqrt(D)) = 1*1 by Def1;
    then A9: (p`1 - p`2 * sqrt(D)) > 0 & (p`1 - p`2 * sqrt(D)) < 1
      by A6,XREAL_1:98;
    p`1 - p`2 * sqrt(D) + p`2 * sqrt(D) < 1 + p`2 * sqrt(D) by A9, XREAL_1:8;
    then p`1 - p`1 < 1 + p`2 * sqrt(D) - p`1 by XREAL_1:14;
    then 0 - 1 < 1 + p`2 * sqrt(D) - p`1 -1 by XREAL_1:14;
    then (p`2 * sqrt(D) - p`1) + (p`1 + p`2 * sqrt(D)) > -1 + 1
      by A6, XREAL_1:8;
    then 2 * p`2 * sqrt(D) /2 > 0 /2;
    hence p is positive by A7, A6,A8;
  end;
