theorem ofEven_ι (x y : M × R) :
    ofEven Q ((even.ι (Q' Q)).bilin x y) =
      (ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by
  -- porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
  unfold ofEven
  lift_lets
  

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : M × R
⊢ let f :=
    LinearMap.compl₂
      (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
        (LinearMap.comp (ι Q) (LinearMap.fst R M R) +
          LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
      (LinearMap.comp (ι Q) (LinearMap.fst R M R) -
        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R));
  ((even.lift (Q' Q))
        { bilin := f,
          contract :=
            (_ :
              ∀ (m : M × R),
                ((LinearMap.compl₂
                        (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
                          (LinearMap.comp (ι Q) (LinearMap.fst R M R) +
                            LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                        (LinearMap.comp (ι Q) (LinearMap.fst R M R) -
                          LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                      m)
                    m =
                  (algebraMap R (CliffordAlgebra Q)) ((Q' Q) m)),
          contract_mid :=
            (_ :
              ∀ (m₁ m₂ m₃ : M × R),
                ((LinearMap.compl₂
                          (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
                            (LinearMap.comp (ι Q) (LinearMap.fst R M R) +
                              LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                          (LinearMap.comp (ι Q) (LinearMap.fst R M R) -
                            LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                        m₁)
                      m₂ *
                    ((LinearMap.compl₂
                          (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
                            (LinearMap.comp (ι Q) (LinearMap.fst R M R) +
                              LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                          (LinearMap.comp (ι Q) (LinearMap.fst R M R) -
                            LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                        m₂)
                      m₃ =
                  (Q' Q) m₂ •
                    ((LinearMap.compl₂
                          (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
                            (LinearMap.comp (ι Q) (LinearMap.fst R M R) +
                              LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                          (LinearMap.comp (ι Q) (LinearMap.fst R M R) -
                            LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))
                        m₁)
                      m₃) })
      (((even.ι (Q' Q)).bilin x) y) =
    ((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2)

Next tactic:
intro f