simp only [unitInterval, ne_eq, image_subset_iff, preimage_setOf_eq, add_re, mul_re, ofReal_re,
  ofReal_im, mul_zero, sub_zero, re_ofNat, sub_re, one_re, im_ofNat, sub_im, one_im, sub_self,
  f, mem_setOf_eq]
case neg
w : ℂ
hw : ¬w = 1 ∧ 0 < w.re
w_im : ¬w.im = 0
f : ℝ → ℂ := fun t => w * ↑t + 2 * (1 - ↑t)
cont : Continuous f
⊢ f '' unitInterval ⊆ {z | z ≠ 1 ∧ 0 < z.re}
case neg
w : ℂ
hw : ¬w = 1 ∧ 0 < w.re
w_im : ¬w.im = 0
f : ℝ → ℂ := fun t => w * ↑t + 2 * (1 - ↑t)
cont : Continuous f
⊢ Icc 0 1 ⊆ {a | ¬w * ↑a + 2 * (1 - ↑a) = 1 ∧ 0 < w.re * a + 2 * (1 - a)}