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 Th11:
  Px(a,n) + Py(a,n) * sqrt (a^2-'1)  = (a + sqrt (a^2-'1)) |^ n &
  Px(a,n) + (- Py(a,n)) * sqrt (a^2-'1)  = (a - sqrt (a^2-'1)) |^ n
proof
  set A=a^2-'1,M=min_Pell's_solution_of A;
  set n1=n+1;
  A1: M = [a,1] by Th8;
  A2:Px(a,n) + Py(a,n) * sqrt A = ( M`1 + M`2 * sqrt A) |^n by Def2
  .= (a+ 1*sqrt A) |^n by A1;
  then Px(a,n) - Py(a,n) * sqrt A  = (a- 1*sqrt A) |^n by PELLS_EQ:6;
  hence thesis by A2;
end;
