reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th8:
  [a,1] = min_Pell's_solution_of (a^2-'1)
proof
  set A=a^2-'1,M=min_Pell's_solution_of A;
  assume
A1: [a,1] <> M;
  a^2-'1 = a^2-1 by XREAL_1:233,NAT_1:14;
  then a^2 - (a^2-'1)*1^2 = 1;
  then reconsider a1 = [a,1] as Pell's_solution of A by Lm3;
  M`1 <> a or M`2 <> 1 by A1;
  then
A2: M`1 + M`2 * sqrt A <> a + 1 *sqrt A by PELLS_EQ:3;
A3: sqrt A >=0 by SQUARE_1:def 2;
  a1 is positive;
  then
A4: M`1 <= a1`1 & M`2 <= a1`2 by PELLS_EQ:def 3;
  then M`2 * sqrt A <= a1`2 *sqrt A by A3,XREAL_1:64;
  then M`1 + M`2 * sqrt A <= a1`1 + a1`2 *sqrt A by A4,XREAL_1:7;
  then
A5: M`1 + M`2 * sqrt A < a1`1 + a1`2 *sqrt A by A2,XXREAL_0:1;
  1 < M`1 + M`2 * sqrt A by PELLS_EQ:18;
  hence thesis by NAT_1:14,A5,PELLS_EQ:19;
end;
