have: concat a₁  b = a₁ ++ (concat (nil _) b) := by 
  rw [append_concat, append_nil]
V : Type u
E : Type v
G✝ : Graph V E
u✝ v✝ w✝ : V
e : EdgeBetween G✝ v✝ w✝
G : Graph V E
v w u : V
a₁ a₂ : EdgePath G v w
b : EdgeBetween G w u
rel : [[a₁]] = [[a₂]]
⊢ [[concat a₁ b]] = [[concat a₂ b]]
V : Type u
E : Type v
G✝ : Graph V E
u✝ v✝ w✝ : V
e : EdgeBetween G✝ v✝ w✝
G : Graph V E
v w u : V
a₁ a₂ : EdgePath G v w
b : EdgeBetween G w u
rel : [[a₁]] = [[a₂]]
this : concat a₁ b = a₁ ++ concat (nil w) b
⊢ [[concat a₁ b]] = [[concat a₂ b]]