intros w w' w''.
elim (interp_non_zero w); intros p Hex; elim Hex; intros q Heq; clear Hex.
elim (interp_non_zero w'); intros p' Hex; elim Hex; intros q' Heq'; clear Hex.
elim (interp_non_zero w''); intros p'' Hex; elim Hex; intros q'' Heq'';
clear Hex.
unfold Qpositive_mult in |- *.
rewrite Heq; rewrite Heq'; rewrite Heq''.
elim
(construct_correct2' (S p * S p' + S q * S q') (S p * S p') (S q * S q')).
intros d;
elim
(interp_non_zero
(Qpositive_c (S p * S p') (S q * S q') (S p * S p' + S q * S q')));
intros p3 Hex; elim Hex; intros q3 Heq3; rewrite Heq3;
clear Hex.
intros (Heq1, Heq2).
elim
(construct_correct2' (S p' * S p'' + S q' * S q'')
(S p' * S p'') (S q' * S q'')).
intros d';
elim
(interp_non_zero
(Qpositive_c (S p' * S p'') (S q' * S q'') (S p' * S p'' + S q' * S q'')));
intros p4 Hex; elim Hex; intros q4 Heq6; rewrite Heq6;
clear Hex.
intros (Heq4, Heq5).
apply construct_correct4'.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
auto.
unfold fst, snd in Heq1, Heq2, Heq4, Heq5.
apply mult_reg_l with d'.
rewrite (mult_comm (S p * S p4)).
repeat rewrite (mult_comm (S d')); repeat rewrite <- mult_assoc;
rewrite <- Heq4; rewrite <- Heq5.
clear Heq4 Heq5 Heq6.
apply mult_reg_l with d; repeat rewrite (mult_assoc (S d));
repeat rewrite (mult_comm (S d)); rewrite <- Heq1;
rewrite <- Heq2; ring.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
Qed.

Theorem Qpositive_mult_distr :
forall w w' w'' : Qpositive,
Qpositive_mult (Qpositive_plus w w') w'' =
Qpositive_plus (Qpositive_mult w w'') (Qpositive_mult w' w'').