-- Lean 4 proof step from Mathlib.Algebra.Order.Module.HahnEmbedding (goal)
Goal:
K : Type u_1
inst : DivisionRing K
inst_1 : LinearOrder K
inst_2 : IsOrderedRing K
inst_3 : Archimedean K
M : Type u_2
inst_4 : AddCommGroup M
inst_5 : LinearOrder M
inst_6 : IsOrderedAddMonoid M
inst_7 : Module K M
inst_8 : IsOrderedModule K M
R : Type u_3
inst_9 : AddCommGroup R
inst_10 : LinearOrder R
inst_11 : Module K R
seed : Seed K M R
f : Partial seed
inst_12 : IsOrderedAddMonoid R
inst_13 : Archimedean R
x : M
hx : x ∉ (↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain
c : FiniteArchimedeanClass M
y' : M
hy'_1 : y' ∈ (f.extendFun hx).domain
hy' : y' ∈ (↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain ⊔ K ∙ x
a : M
ha : a ∈ (↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain
b : M
hb : Membership.mem (γ := Submodule K M) (K ∙ x) b
hab : a + b = y'
k : K
hk : k • x = b
refine_1 :
  Membership.mem (α := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
    ((↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).supSpanSingleton x (f.eval x) hx).toFun.range
    ((toLex : (a : R⟦FiniteArchimedeanClass M⟧) → Lex R⟦FiniteArchimedeanClass M⟧)
        ((truncLTLinearMap K c : (a : R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
          ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
            ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
                ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
              (Subtype.mk (p := fun (x : M) ↦ x ∈ (↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain) a ha)))) :
      Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
⊢ HAdd.hAdd (α := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
      ((toLex : (a : R⟦FiniteArchimedeanClass M⟧) → Lex R⟦FiniteArchimedeanClass M⟧)
          ((truncLTLinearMap K c : (a : R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
            ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
              ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
                  ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
                (Subtype.mk (p := fun (x : M) ↦ x ∈ (↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain) a ha)))) :
        Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
      (HSMul.hSMul (α := K) (β := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
        ((RingHom.id K : (a : K) → K) k : K)
        ((toLex : (a : R⟦FiniteArchimedeanClass M⟧) → Lex R⟦FiniteArchimedeanClass M⟧)
            ((truncLTLinearMap K c : (a : R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
              ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧) (f.eval x))) :
          Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))) ∈
    ((↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).supSpanSingleton x (f.eval x) hx).toFun.range
have refine_2 : Membership.mem (α := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
  ((↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).supSpanSingleton x (f.eval x) hx).toFun.range
  ((toLex : (a : R⟦FiniteArchimedeanClass M⟧) → Lex R⟦FiniteArchimedeanClass M⟧)
      ((truncLTLinearMap K c : (a : R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
        ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧) (f.eval x))) :
    Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
Tactic: refine Submodule.add_mem _ ?_ (Submodule.smul_mem _ _ ?_)