On elements of large order on elliptic curves and multiplicative dependent images of rational functions over finite fields

Let E1 and E2 be elliptic curves in Legendre form with integer parameters. We show there exists a constant C such that for almost all primes, but at most pairs of points on the reduction E1×E2 modulo p having equal x coordinate, least one among P1 P2 has large group order. also similar abundance over finite fields elements whose images under set rational functions have multiplicative orders.