have leq :
  liftTerminal (proj G x₀) (basepoint G x₀) rfl p₁ =
  liftTerminal (proj G x₀) (basepoint G x₀) rfl p₂ := by
  simp [liftTerminal]
  apply congrArg
  apply liftClass_eq_of_homotopic
  assumption
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G✝ : Graph V E
x₀✝ : V
G : Graph V E
x₀ τ : V
p₁ p₂ : EdgePath G x₀ τ
hyp₁ : reduced p₁
hyp₂ : reduced p₂
hyp : [[p₁]] = [[p₂]]
⊢ p₁ = p₂
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G✝ : Graph V E
x₀✝ : V
G : Graph V E
x₀ τ : V
p₁ p₂ : EdgePath G x₀ τ
hyp₁ : reduced p₁
hyp₂ : reduced p₂
hyp : [[p₁]] = [[p₂]]
leq :
  liftTerminal (proj G x₀) (basepoint G x₀)
      (_ : (proj G x₀).toFuncV (basepoint G x₀) = (proj G x₀).toFuncV (basepoint G x₀)) p₁ =
    liftTerminal (proj G x₀) (basepoint G x₀)
      (_ : (proj G x₀).toFuncV (basepoint G x₀) = (proj G x₀).toFuncV (basepoint G x₀)) p₂
⊢ p₁ = p₂