∀ {X : Type u} {Y : Type v} {inst : TopologicalSpace X} {inst_1 : TopologicalSpace Y} {f₀ f₁ : C(X, Y)}
  {toContinuousMap : C(↑unitInterval × X, Y)} {map_zero_left : ∀ (x : X), toContinuousMap.toFun (0, x) = f₀ x}
  {map_one_left : ∀ (x : X), toContinuousMap.toFun (1, x) = f₁ x} {toContinuousMap_1 : C(↑unitInterval × X, Y)}
  {map_zero_left_1 : ∀ (x : X), toContinuousMap_1.toFun (0, x) = f₀ x}
  {map_one_left_1 : ∀ (x : X), toContinuousMap_1.toFun (1, x) = f₁ x},
  { toContinuousMap := toContinuousMap, map_zero_left := map_zero_left, map_one_left := map_one_left } =
      { toContinuousMap := toContinuousMap_1, map_zero_left := map_zero_left_1, map_one_left := map_one_left_1 } →
    toContinuousMap = toContinuousMap_1