-- Module: Mathlib.LinearAlgebra.AffineSpace.Midpoint
Goal:
R : Type u_1
V : Type u_2
P : Type u_4
inst : Ring R
inst_1 : Invertible (α := R) 2
inst_2 : AddCommGroup V
inst_3 : Module R V
inst_4 : AddTorsor V P
p₁ p₂ p : P
rw :
  -(midpoint R p₁ p₂ -ᵥ p) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₁ :
  -(HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p)) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₂ :
  -HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₁ -ᵥ p) +
      -HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₃ :
  HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (-(p₁ -ᵥ p)) +
      -HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p₂ -ᵥ p) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₄ :
  HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (-(p₁ -ᵥ p)) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (-(p₂ -ᵥ p)) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₅ :
  HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (-(p₂ -ᵥ p)) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
rw₆ rw₇ :
  HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂) =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)
⊢ p -ᵥ midpoint R p₁ p₂ =
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₁) +
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne 2 _) (p -ᵥ p₂)

Tactic:
rw [← neg_vsub_eq_vsub_rev, midpoint_vsub, neg_add, ← smul_neg, ← smul_neg, neg_vsub_eq_vsub_rev,
    neg_vsub_eq_vsub_rev]

Premises used: neg_add, smul_neg, neg_vsub_eq_vsub_rev, midpoint_vsub