simp_rw [← e.init_eq]
case cons
V : Type u_1
E : Type u_2
G : Graph V E
H : Subgraph G
u v v✝ w✝ u✝ : V
e : EdgeBetween G v✝ w✝
a✝ : EdgePath G w✝ u✝
a_ih✝ : contains H a✝ → w✝ ∈ H.verts
⊢ contains H (EdgePath.cons e a✝) → v✝ ∈ H.verts
case cons
V : Type u_1
E : Type u_2
G : Graph V E
H : Subgraph G
u v v✝ w✝ u✝ : V
e : EdgeBetween G v✝ w✝
a✝ : EdgePath G w✝ u✝
a_ih✝ : contains H a✝ → w✝ ∈ H.verts
⊢ contains H (EdgePath.cons e a✝) → G.ι e.edge ∈ H.verts