unfold prod_ap. simplify.
intros.
elim H
- [apply (ap_irreflexive A t H1)
- |apply (ap_irreflexive B t1 H1)
+ [apply (ap_irreflexive A a H1)
+ |apply (ap_irreflexive B b H1)
]
|unfold.
intros 2.
unfold prod_ap in H. simplify in H.
unfold prod_ap. simplify.
elim H
- [cut ((t \neq t4) \or (t4 \neq t2))
+ [cut ((a \neq a2) \or (a2 \neq a1))
[elim Hcut
[left. left. assumption
|right. left. assumption
]
|apply (ap_cotransitive A). assumption
]
- |cut ((t1 \neq t5) \or (t5 \neq t3))
+ |cut ((b \neq b2) \or (b2 \neq b1))
[elim Hcut
[left. right. assumption
|right. right. assumption
|intro. unfold. intro.
elim H.
elim H1
- [apply (eq_imp_not_ap A t t2 H2). assumption
- |apply (eq_imp_not_ap B t1 t3 H3). assumption
+ [apply (eq_imp_not_ap A a a1 H2). assumption
+ |apply (eq_imp_not_ap B b b1 H3). assumption
]
]
]