-- Module: Mathlib.Geometry.Convex.Cone.TensorProduct
Goal:
R : Type u_1
inst : CommRing R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
G : Type u_2
inst_3 : AddCommGroup G
inst_4 : Module R G
H : Type u_3
inst_5 : AddCommGroup H
inst_6 : Module R H
C₁ : PointedCone R G
C₂ : PointedCone R H
z : H ⊗[R] G
h :
  (∃ (x : G ⊗[R] H),
      (∀ (φ : Dual R G),
          Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) φ →
            ∀ (ψ : Dual R H),
              Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) ψ →
                LE.le.{u_1} (α := R) (0 : R)
                  ((((dualDistrib R G H : (a : Dual R G ⊗[R] Dual R H) → Dual R (G ⊗[R] H)) (φ ⊗ₜ[R] ψ) :
                          Dual R (G ⊗[R] H)) :
                        (a : G ⊗[R] H) → R)
                      x :
                    R)) ∧
        ((↑(TensorProduct.comm R G H) : G ⊗[R] H →ₗ[R] H ⊗[R] G) : (a : G ⊗[R] H) → H ⊗[R] G) x = z) →
    ∀ (φ : Dual R H),
      Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) φ →
        ∀ (ψ : Dual R G),
          Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) ψ →
            LE.le.{u_1} (α := R) (0 : R)
              ((((dualDistrib R H G : (a : Dual R H ⊗[R] Dual R G) → Dual R (H ⊗[R] G)) (φ ⊗ₜ[R] ψ) :
                      Dual R (H ⊗[R] G)) :
                    (a : H ⊗[R] G) → R)
                  z :
                R)
h₁ :
  ∀
    (hz :
      ∀ (φ : Dual R H),
        Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) φ →
          ∀ (ψ : Dual R G),
            Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) ψ →
              LE.le.{u_1} (α := R) (0 : R)
                ((((dualDistrib R H G : (a : Dual R H ⊗[R] Dual R G) → Dual R (H ⊗[R] G)) (φ ⊗ₜ[R] ψ) :
                        Dual R (H ⊗[R] G)) :
                      (a : H ⊗[R] G) → R)
                    z :
                  R))
    (φ : Dual R G),
    Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) φ →
      ∀ (ψ : Dual R H),
        Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) ψ →
          LE.le.{u_1} (α := R) (0 : R)
            ((((dualDistrib R G H : (a : Dual R G ⊗[R] Dual R H) → Dual R (G ⊗[R] H)) (φ ⊗ₜ[R] ψ) : Dual R (G ⊗[R] H)) :
                  (a : G ⊗[R] H) → R)
                ((TensorProduct.comm R H G : (a : H ⊗[R] G) → G ⊗[R] H) z : G ⊗[R] H) :
              R)
⊢ (∃ (x : G ⊗[R] H),
      (∀ (φ : Dual R G),
          Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) φ →
            ∀ (ψ : Dual R H),
              Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) ψ →
                LE.le.{u_1} (α := R) (0 : R)
                  ((((dualDistrib R G H : (a : Dual R G ⊗[R] Dual R H) → Dual R (G ⊗[R] H)) (φ ⊗ₜ[R] ψ) :
                          Dual R (G ⊗[R] H)) :
                        (a : G ⊗[R] H) → R)
                      x :
                    R)) ∧
        ((↑(TensorProduct.comm R G H) : G ⊗[R] H →ₗ[R] H ⊗[R] G) : (a : G ⊗[R] H) → H ⊗[R] G) x = z) ↔
    ∀ (φ : Dual R H),
      Membership.mem (γ := PointedCone R (Dual R H)) (dual (Dual.eval R H) (↑C₂ : Set H)) φ →
        ∀ (ψ : Dual R G),
          Membership.mem (γ := PointedCone R (Dual R G)) (dual (Dual.eval R G) (↑C₁ : Set G)) ψ →
            LE.le.{u_1} (α := R) (0 : R)
              ((((dualDistrib R H G : (a : Dual R H ⊗[R] Dual R G) → Dual R (H ⊗[R] G)) (φ ⊗ₜ[R] ψ) :
                      Dual R (H ⊗[R] G)) :
                    (a : H ⊗[R] G) → R)
                  z :
                R)

Tactic:
refine ⟨?_, fun hz ↦
    ⟨(TensorProduct.comm R H G) z, ?_, (TensorProduct.comm R H G).symm_apply_apply z⟩⟩