∀ {V : Type u_1} {G : SimpleGraph V} {E : Type u_3} [inst : MetricSpace E] (p : V ↪ E)
  (unit_dist : ∀ {u v : V}, G.Adj u v → dist (p u) (p v) = 1) (p_1 : V ↪ E)
  (unit_dist_1 : ∀ {u v : V}, G.Adj u v → dist (p_1 u) (p_1 v) = 1),
  ({ p := p, unit_dist := unit_dist } = { p := p_1, unit_dist := unit_dist_1 }) = (p = p_1)