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 Th26:
  Px(a,n),Py(a,n) are_coprime
proof
  defpred P[Nat] means Px(a,$1) gcd Py(a,$1)=1;
  Px(a,0) =1 & Py(a,0)=0 by Th6;
  then
A1: P[0];
A2: for n st P[n] holds P[n+1]
  proof
    set A=a^2-'1;
A3:   A = a^2 -1 by NAT_1:14,XREAL_1:233;
    let n such that
A4:   P[n];
A5:   Px(a,n+1) = Px(a,n)*a + Py(a,n)*A&
      Py(a,n+1) = Px(a,n) + Py(a,n) * a by Th9;
    thus 1 = (Px(a,n) + Py(a,n) * a) gcd Py(a,n) by A4,EULER_1:8
      .= Py(a,n+1) gcd -Py(a,n) by A5, NEWTON02:1
      .= (-Py(a,n) + a* Py(a,n+1)) gcd Py(a,n+1) by NEWTON02:5
      .= Px(a,n+1) gcd Py(a,n+1) by A3,A5;
  end;
  for n holds P[n] from NAT_1:sch 2(A1,A2);
  then Px(a,n) gcd Py(a,n)=1;
  hence thesis by INT_2:def 3;
end;
