unfold addfraction_fun in |- *; simpl in |- *.
unfold eqfraction in |- *; simpl in |- *.
intros x x' y H'; try assumption.
apply
Trans
with
(ring_mult
(ring_mult
(sgroup_law R (ring_mult (num x) (den y))
(ring_mult (num y) (den x))) (den x'))
(den y)).
auto with algebra.
apply
Trans
with
(ring_mult
(ring_mult
(sgroup_law R (ring_mult (num x') (den y))
(ring_mult (num y) (den x'))) (den x))
(den y)).
apply RING_comp.
apply
Trans
with
(sgroup_law R (ring_mult (ring_mult (num x) (den y)) (den x'))
(ring_mult (ring_mult (num y) (den x)) (den x'))).
auto with algebra.
apply
Trans
with
(sgroup_law R (ring_mult (ring_mult (num x') (den y)) (den x))
(ring_mult (ring_mult (num y) (den x')) (den x))).
apply SGROUP_comp.
apply Trans with (ring_mult (ring_mult (den y) (num x)) (den x')).
auto with algebra.
apply Trans with (ring_mult (ring_mult (den y) (num x')) (den x)).
apply Trans with (ring_mult (den y) (ring_mult (num x) (den x'))).
auto with algebra.
apply Trans with (ring_mult (den y) (ring_mult (num x') (den x))).
auto with algebra.
auto with algebra.
auto with algebra.
apply Trans with (ring_mult (num y) (ring_mult (den x) (den x'))).
auto with algebra.
apply Trans with (ring_mult (num y) (ring_mult (den x') (den x))).
auto with algebra.
auto with algebra.
auto with algebra.
auto with algebra.
auto with algebra.
Qed.

Lemma addfraction_fun_com :
forall x y : fraction_set, Equal (addfraction_fun x y) (addfraction_fun y x).