-- Module: Mathlib.Order.RelSeries
Goal:
α : Type u_3
inst : PartialOrder α
inst_1 : BoundedOrder α
inst_2 : WellFoundedLT α
inst_3 : WellFoundedGT α
s : LTSeries α
this :
  ∀ {α : Type u_3} [inst : PartialOrder α] [inst_4 : BoundedOrder α] [WellFoundedLT α] [WellFoundedGT α]
    (s : LTSeries α)
    (h₁ :
      RelSeries.head (r :=
          {x : α × α |
            match (motive := α × α → Prop) x with
            | (a, b) => a < b})
          s =
        ⊥),
    ∃ (t :
      RelSeries (α := α)
        {x : α × α |
          match (motive := α × α → Prop) x with
          | (a, b) => a ⋖ b})
      (i :
      Fin
          (s.length (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a < b}) +
            1) ↪
        Fin
          (t.length (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) +
            1)),
      t.toFun (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) ∘
            (⇑i : (a : Fin (s.length + 1)) → Fin (t.length + 1)) =
          s.toFun (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a < b}) ∧
        t.head (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) =
            ⊥ ∧
          t.last (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) =
            ⊤
h₁ :
  ¬RelSeries.head (r :=
        {x : α × α |
          match (motive := α × α → Prop) x with
          | (a, b) => a < b})
        s =
      ⊥
inr :
  ∀
    (t :
      RelSeries (α := α)
        {x : α × α |
          match (motive := α × α → Prop) x with
          | (a, b) => a ⋖ b})
    (i :
      Fin
          ((RelSeries.cons (r :=
                  {x : α × α |
                    match (motive := α × α → Prop) x with
                    | (a, b) => a < b})
                  s ⊥ ⋯).length
              (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a < b}) +
            1) ↪
        Fin
          (t.length (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) +
            1))
    (hi :
      t.toFun (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) ∘
          (⇑i : (a : Fin ((RelSeries.cons s ⊥ ⋯).length + 1)) → Fin (t.length + 1)) =
        (RelSeries.cons (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a < b})
              s ⊥ ⋯).toFun
          (r :=
          {x : α × α |
            match (motive := α × α → Prop) x with
            | (a, b) => a < b}))
    (ht :
      t.head (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) =
          ⊥ ∧
        t.last (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) =
          ⊤),
    ∃ (t :
      RelSeries (α := α)
        {x : α × α |
          match (motive := α × α → Prop) x with
          | (a, b) => a ⋖ b})
      (i :
      Fin
          (s.length (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a < b}) +
            1) ↪
        Fin
          (t.length (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) +
            1)),
      t.toFun (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) ∘
            (⇑i : (a : Fin (s.length + 1)) → Fin (t.length + 1)) =
          s.toFun (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a < b}) ∧
        t.head (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) =
            ⊥ ∧
          t.last (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a ⋖ b}) =
            ⊤
⊢ ∃ (t :
    RelSeries (α := α)
      {x : α × α |
        match (motive := α × α → Prop) x with
        | (a, b) => a ⋖ b})
    (i :
    Fin
        (s.length (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a < b}) +
          1) ↪
      Fin
        (t.length (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) +
          1)),
    t.toFun (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) ∘
          (⇑i : (a : Fin (s.length + 1)) → Fin (t.length + 1)) =
        s.toFun (r :=
          {x : α × α |
            match (motive := α × α → Prop) x with
            | (a, b) => a < b}) ∧
      t.head (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) =
          ⊥ ∧
        t.last (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) =
          ⊤

Tactic:
obtain ⟨t, i, hi, ht⟩ := this (s.cons ⊥ (bot_lt_iff_ne_bot.mpr h₁)) rfl

Premises used: Iff.mpr, bot_lt_iff_ne_bot, Bot.bot, RelSeries.cons, rfl