-- Module: Mathlib.RingTheory.LocalRing.Quotient
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : IsLocalRing R
inst_4 : Module.Finite R S
s : Set S
rw :
  Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (span R
      ((⇑(↑(IsScalarTower.toAlgHom R S
                  (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p))) :
              [inst_5 : FunLike (S →ₐ[R] S ⧸ pS) S (S ⧸ pS)] →
                [SemilinearMapClass (S →ₐ[R] S ⧸ pS) (RingHom.id R) S (S ⧸ pS)] → S →ₗ[R] S ⧸ pS) :
          (a : S) → S ⧸ pS) ''
        s))
rw₁ :
  Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(↑(IsScalarTower.toAlgHom R S
                    (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p))) :
                [inst_5 : FunLike (S →ₐ[R] S ⧸ pS) S (S ⧸ pS)] →
                  [SemilinearMapClass (S →ₐ[R] S ⧸ pS) (RingHom.id R) S (S ⧸ pS)] → S →ₗ[R] S ⧸ pS) :
            (a : S) → S ⧸ pS) ''
          s)))
rw₂ :
  Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(IsScalarTower.toAlgHom R S
                (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p))) :
            (a : S) → S ⧸ pS) ''
          s)))
rw₃ :
  Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(algebraMap S (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p))) :
            (a : S) → S ⧸ pS) ''
          s)))
rw₄ rw₅ :
  Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
⊢ Eq (α := Submodule R (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
    (Submodule.restrictScalars R (R := HasQuotient.Quotient R (B := Ideal R) p)
      (span (HasQuotient.Quotient R (B := Ideal R) p)
        ((⇑(Ideal.Quotient.mk (Ideal.map (F := R →+* S) (algebraMap R S) p)) : (a : S) → S ⧸ pS) '' s)))
    (map (σ₁₂ := RingHom.id R)
      (↑(IsScalarTower.toAlgHom R S
            (HasQuotient.Quotient S (B := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p))) :
        [inst_5 : FunLike (S →ₐ[R] S ⧸ pS) S (S ⧸ pS)] →
          [SemilinearMapClass (S →ₐ[R] S ⧸ pS) (RingHom.id R) S (S ⧸ pS)] → S →ₗ[R] S ⧸ pS)
      (span R s))

Tactic:
rw [map_span, ← restrictScalars_span R (R ⧸ p) Ideal.Quotient.mk_surjective,
      LinearMap.coe_coe, IsScalarTower.coe_toAlgHom', Ideal.Quotient.algebraMap_eq]

Premises used: IsScalarTower.coe_toAlgHom', Ideal.Quotient.algebraMap_eq, Submodule.map_span, LinearMap.coe_coe