-- Module: Mathlib.Algebra.Order.Module.HahnEmbedding
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
inst_12 : IsOrderedAddMonoid R
inst_13 : Archimedean R
h : Nonempty (HahnEmbedding.Seed K M R)
f : ↥⊤ →ₗ[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)
hpartial : HahnEmbedding.IsPartial h.some { domain := ⊤, toFun := f }
harch :
  ∀
    (x :
      ↥(↑(Subtype.mk (p := fun (f : M →ₗ.[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)) ↦
                HahnEmbedding.IsPartial h.some f)
                { domain := ⊤, toFun := f : M →ₗ.[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R) }
                hpartial) :
            M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain),
    Eq (α := ArchimedeanClass M)
      ((withTopOrderIso M : (a : WithTop (FiniteArchimedeanClass M)) → ArchimedeanClass M)
          (orderTop.{u_2, u_3} (R := R)
            ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
              ((↑(↑(Subtype.mk (p := fun (f : M →ₗ.[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)) ↦
                          HahnEmbedding.IsPartial h.some f)
                          { domain := ⊤, toFun := f :
                            M →ₗ.[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R) }
                          hpartial) :
                      M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
                  ↥(↑⟨{ domain := ⊤, toFun := f }, hpartial⟩ : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain →
                    Lex R⟦FiniteArchimedeanClass M⟧)
                x))) :
        ArchimedeanClass M)
      (ArchimedeanClass.mk (↑x : M) : ArchimedeanClass M)
refine_1 :
  StrictMono (α := M) (β := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
    (⇑(f ∘ₗ LinearMap.codRestrict ⊤ LinearMap.id ⋯) : (a : M) → Lex R⟦FiniteArchimedeanClass M⟧)
refine_2 :
  ∀ (a : M),
    Eq (α := ArchimedeanClass M) (ArchimedeanClass.mk a)
      ((withTopOrderIso M : (a : WithTop (FiniteArchimedeanClass M)) → ArchimedeanClass M)
        (orderTop.{u_2, u_3} (R := R)
          ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
            ((f ∘ₗ LinearMap.codRestrict ⊤ LinearMap.id ⋯ : (a : M) → Lex R⟦FiniteArchimedeanClass M⟧) a))))
⊢ ∃ (f : M →ₗ[K] Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)),
    StrictMono (α := M) (β := Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R))
        (⇑f : (a : M) → Lex R⟦FiniteArchimedeanClass M⟧) ∧
      ∀ (a : M),
        Eq (α := ArchimedeanClass M) (ArchimedeanClass.mk a)
          ((withTopOrderIso M : (a : WithTop (FiniteArchimedeanClass M)) → ArchimedeanClass M)
            (orderTop.{u_2, u_3} (R := R)
              ((ofLex : (a : Lex R⟦FiniteArchimedeanClass M⟧) → R⟦FiniteArchimedeanClass M⟧)
                ((f : (a : M) → Lex R⟦FiniteArchimedeanClass M⟧) a))))

Tactic:
refine ⟨f ∘ₗ LinearMap.id.codRestrict ⊤ (by simp), ?_, ?_⟩