-- Module: Mathlib.LinearAlgebra.Dimension.Finite
Goal:
R : Type u_1
M : Type u_2
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : Module.Finite R M
inst_4 : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun (x : M) ↦ x - x₀, inj' := ⋯ }
let let₁ let₂ :
  ∃ (f : M → R), Eq (α := M) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
let₃ :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t'),
    ∃ (f : M → R), Eq (α := M) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
let₄ let₅ :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    ∃ (f : M → R), Eq (α := M) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
refine_1 :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    let f : M → R := fun (z : M) ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀);
    Eq (α := M) (∑ e ∈ t, f e • e) 0
refine_2 :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    let f : M → R := fun (z : M) ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀);
    Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0
refine_3 :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    let f : M → R := fun (z : M) ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀);
    ∃ x ∈ t, f x ≠ 0
refine_2₁ :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    let f : M → R := fun (z : M) ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀);
    Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0
refine_3₁ refine_3₂ :
  let t' := Finset.map shift (t.erase x₀);
  ∀ (h' : finrank R M < #t') (g : M → R) (gsum : Eq (α := M) (∑ e ∈ t', g e • e) 0) (x₁ : M) (x₁_mem : x₁ ∈ t')
    (nz : g x₁ ≠ 0),
    let f : M → R := fun (z : M) ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀);
    ∃ x ∈ t, f x ≠ 0
⊢ ∃ (f : M → R), Eq (α := M) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := R) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0

Tactic:
classical
  let t' := (t.erase x₀).map shift
  have h' : finrank R M < t'.card := by
    rw [card_map, card_erase_of_mem x₀_mem]
    exact Nat.lt_pred_iff.mpr h
  -- to obtain a function `g`.
  obtain ⟨g, gsum, x₁, x₁_mem, nz⟩ := exists_nontrivial_relation_of_finrank_lt_card h'
  -- Then obtain `f` by translating back by `x₀`,
  -- and setting the value of `f` at `x₀` to ensure `∑ e ∈ t, f e = 0`.
  let f : M → R := fun z ↦ if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀)
  refine ⟨f, ?_, ?_, ?_⟩
  -- After this, it's a matter of verifying the properties,
  -- based on the corresponding properties for `g`.
  · rw [sum_map, Embedding.coeFn_mk] at gsum
    simp_rw [f, ← t.sum_erase_add _ x₀_mem, if_pos, neg_smul, sum_smul,
             ← sub_eq_add_neg, ← sum_sub_distrib, ← gsum, smul_sub]
    refine sum_congr rfl fun x x_mem ↦ ?_
    rw [if_neg (mem_erase.mp x_mem).1]
  · simp_rw [f, ← t.sum_erase_add _ x₀_mem, if_pos, add_neg_eq_zero]
    exact sum_congr rfl fun x x_mem ↦ if_neg (mem_erase.mp x_mem).1
  · obtain ⟨x₁, x₁_mem', rfl⟩ := Finset.mem_map.mp x₁_mem
    have := mem_erase.mp x₁_mem'
    exact ⟨x₁, by
      simpa only [f, Embedding.coeFn_mk, sub_add_cancel, this.2, true_and, if_neg this.1]⟩

Premises used: Finset.sum_smul, neg_smul, add_neg_eq_zero, Finset.sum_map, sub_eq_add_neg, if_pos, Finset.mem_erase, smul_sub, Finset.sum_sub_distrib, And.left, Iff.mp, Function.Embedding.coeFn_mk, if_neg, Finset.sum_congr, rfl