image_mul_closed

forall (G : Sort u), forall (oneG : G), forall (mulG : forall (a : G), forall (b : G), G), forall (invG : forall (a : G), G), forall (H : Sort v), forall (oneH : H), forall (mulH : forall (a : H), forall (b : H), H), forall (invH : forall (a : H), H), forall (f : forall (x : G), H), forall (hom_args : @GroupHomLawArgs.{u,v} G oneG mulG invG H oneH mulH invH f), forall (x : H), forall (y : H), forall (hx : @ImagePred.{u,v} G H f x), forall (hy : @ImagePred.{u,v} G H f y), @ImagePred.{u,v} G H f (mulH x y)

fun G => fun oneG => fun mulG => fun invG => fun H => fun oneH => fun mulH => fun invH => fun f => fun hom_args => fun x => fun y => fun hx => fun hy => hx (@ImagePred.{u,v} G H f (mulH x y)) (fun (a : G) => fun (hax : @Eq.{v} H (f a) x) => hy (@ImagePred.{u,v} G H f (mulH x y)) (fun (b : G) => fun (hby : @Eq.{v} H (f b) y) => fun (P : Prop) => fun (mk : forall (c : G), forall (h : @Eq.{v} H (f c) (mulH x y)), P) => mk (mulG a b) (@eq_trans.{v} H (f (mulG a b)) (mulH (f a) (f b)) (mulH x y) (@hom_mul.{u,v} G oneG mulG invG H oneH mulH invH f hom_args a b) (@eq_congr2.{v,v,v} H H H mulH (f a) x (f b) y hax hby))))