intros x y.
rewrite <- (Qplus_zero_left (Qplus (Qopp x) (Qopp y))).
rewrite <- (Q_opp_def (Qplus x y)).
rewrite (Qplus_sym (Qplus x y)).
pattern (Qplus x y) at 3 in |- *; rewrite Qplus_sym.
rewrite Qplus_assoc.
rewrite Qplus_assoc.
rewrite <- (fun a b : Q => Qplus_assoc a b (Qopp x)).
rewrite Q_opp_def.
rewrite <- (Qplus_sym Zero); rewrite Qplus_zero_left.
rewrite <- Qplus_assoc; rewrite Q_opp_def.
rewrite <- (Qplus_sym Zero); rewrite Qplus_zero_left.
auto.
Qed.

Theorem Q_distr_left :
forall x y z : Q, Qmult (Qplus x y) z = Qplus (Qmult x z) (Qmult y z).