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

theorem Th5:
  for c,d be Integer, n be Nat ex a,b be Integer st
     a + b * sqrt D = (c + d *sqrt D) |^ n
  proof
    let c,d be Integer;set cd=c+d *sqrt (D);
    defpred P[Nat] means
    ex a,b be Integer st a +b * sqrt(D) = cd |^ $1;
    A1: P[0]
    proof
      take 1,0;
      thus thesis by NEWTON:4;
    end;
    A2: P[n] implies P[n+1]
    proof
      assume P[n];
      then consider a,b be Integer such that
      A3: a +b * sqrt(D) = cd |^ n;
      A4: D = (sqrt D)^2 by SQUARE_1:def 2;
      cd |^ (n+1) = cd * (a +b * sqrt(D)) by A3, NEWTON:6
      .= c*a + d*b * D + (sqrt D) *(c*b+a*d) by A4;
      hence thesis;
    end;
    P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
