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