-- 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
seed : Seed K M R
f : Partial seed
inst_12 : IsOrderedAddMonoid R
inst_13 : Archimedean R
x y : ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain
hnontrivial : Nontrivial M
hnonempty : Nonempty.{u_2 + 1} (FiniteArchimedeanClass M)
c : FiniteArchimedeanClass M
this : Nontrivial R
inr inr₁ :
  Eq (α := ArchimedeanClass (Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          x))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          y)) ↔
    Eq (α := ArchimedeanClass (Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          x))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          y))
⊢ Eq.{u_2 + 1} (α := WithTop (FiniteArchimedeanClass M))
      (((archimedeanClassOrderIsoWithTop (FiniteArchimedeanClass M) R :
              ArchimedeanClass (Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)) ≃o
                WithTop (FiniteArchimedeanClass M)) :
            (a : ArchimedeanClass (Lex R⟦FiniteArchimedeanClass M⟧)) → WithTop (FiniteArchimedeanClass M))
          (ArchimedeanClass.mk
            ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
                ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
              x)) :
        WithTop (FiniteArchimedeanClass M))
      (((archimedeanClassOrderIsoWithTop (FiniteArchimedeanClass M) R :
              ArchimedeanClass (Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)) ≃o
                WithTop (FiniteArchimedeanClass M)) :
            (a : ArchimedeanClass (Lex R⟦FiniteArchimedeanClass M⟧)) → WithTop (FiniteArchimedeanClass M))
          (ArchimedeanClass.mk
            ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
                ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
              y)) :
        WithTop (FiniteArchimedeanClass M)) ↔
    Eq (α := ArchimedeanClass (Lex (HahnSeries.{u_2, u_3} (FiniteArchimedeanClass M) R)))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          x))
      (ArchimedeanClass.mk
        ((↑(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧) :
            ↥(↑f : M →ₗ.[K] Lex R⟦FiniteArchimedeanClass M⟧).domain → Lex R⟦FiniteArchimedeanClass M⟧)
          y))

Tactic:
rw [(HahnSeries.archimedeanClassOrderIsoWithTop (FiniteArchimedeanClass M) R).injective.eq_iff]

Premises used: HahnSeries.archimedeanClassOrderIsoWithTop, Function.Injective.eq_iff, FiniteArchimedeanClass, OrderIso.injective