-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
⊢ Eq (α := (a : Composition n) × Composition a.length)
    ((fun (i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))) ↦
        (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length)
            {
              blocks :=
                flatten
                  (ofFn (n :=
                    i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                    fun
                      (j :
                        Fin
                          i.fst (β := fun (c : Composition n) ↦
                              (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                    (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).blocks),
              blocks_pos := ⋯, blocks_sum := ⋯ }
            {
              blocks :=
                ofFn (n :=
                  i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                  fun
                    (j :
                      Fin
                        i.fst (β := fun (c : Composition n) ↦
                            (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                  (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).length,
              blocks_pos := ⋯, blocks_sum := ⋯ :
              Composition
                {
                    blocks :=
                      flatten
                        (ofFn (n :=
                          i.fst (β := fun (c : Composition n) ↦
                              (i : Fin c.length) → Composition (c.blocksFun i)).length)
                          fun
                            (j :
                              Fin
                                i.fst (β := fun (c : Composition n) ↦
                                    (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                          (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                              j).blocks),
                    blocks_pos := ⋯, blocks_sum := ⋯ : Composition n }.length } :
          (a : Composition n) × Composition a.length))
      ((fun (i : (a : Composition n) × Composition a.length) ↦
          (Sigma.mk (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
              (i.fst (β := fun (a : Composition n) ↦ Composition a.length).gather
                (i.snd (β := fun (a : Composition n) ↦ Composition a.length)))
              (i.fst (β := fun (a : Composition n) ↦ Composition a.length).sigmaCompositionAux
                (i.snd (β := fun (a : Composition n) ↦ Composition a.length))) :
            (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))))
        (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)))
    (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)
have rw : blocks (n := n)
      (Sigma.fst (β := fun (a : Composition n) ↦ Composition a.length)
        ((fun (i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))) ↦
            (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length)
                {
                  blocks :=
                    flatten
                      (ofFn (n :=
                        i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                        fun
                          (j :
                            Fin
                              i.fst (β := fun (c : Composition n) ↦
                                  (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                        (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                            j).blocks),
                  blocks_pos := sorry, blocks_sum := sorry : Composition n }
                {
                  blocks :=
                    ofFn (n :=
                      i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                      fun
                        (j :
                          Fin
                            i.fst (β := fun (c : Composition n) ↦
                                (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                      (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                          j).length,
                  blocks_pos := sorry, blocks_sum := sorry :
                  Composition
                    {
                        blocks :=
                          flatten
                            (ofFn (n :=
                              i.fst (β := fun (c : Composition n) ↦
                                  (i : Fin c.length) → Composition (c.blocksFun i)).length)
                              fun
                                (j :
                                  Fin
                                    i.fst (β := fun (c : Composition n) ↦
                                        (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                              (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                                  j).blocks),
                        blocks_pos := sorry, blocks_sum := sorry : Composition n }.length } :
              (a : Composition n) × Composition a.length))
          ((fun (i : (a : Composition n) × Composition a.length) ↦
              (Sigma.mk (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                  (i.fst (β := fun (a : Composition n) ↦ Composition a.length).gather
                    (i.snd (β := fun (a : Composition n) ↦ Composition a.length)))
                  (i.fst (β := fun (a : Composition n) ↦ Composition a.length).sigmaCompositionAux
                    (i.snd (β := fun (a : Composition n) ↦ Composition a.length))) :
                (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))))
            (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)))) =
    (Sigma.fst (β := fun (a : Composition n) ↦ Composition a.length)
        (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)).blocks ∧
  blocks (n :=
      length (n := n)
        (Sigma.fst (β := fun (a : Composition n) ↦ Composition a.length)
          ((fun (i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))) ↦
              (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length)
                  {
                    blocks :=
                      flatten
                        (ofFn (n :=
                          i.fst (β := fun (c : Composition n) ↦
                              (i : Fin c.length) → Composition (c.blocksFun i)).length)
                          fun
                            (j :
                              Fin
                                i.fst (β := fun (c : Composition n) ↦
                                    (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                          (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                              j).blocks),
                    blocks_pos := sorry, blocks_sum := sorry : Composition n }
                  {
                    blocks :=
                      ofFn (n :=
                        i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                        fun
                          (j :
                            Fin
                              i.fst (β := fun (c : Composition n) ↦
                                  (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                        (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                            j).length,
                    blocks_pos := sorry, blocks_sum := sorry :
                    Composition
                      {
                          blocks :=
                            flatten
                              (ofFn (n :=
                                i.fst (β := fun (c : Composition n) ↦
                                    (i : Fin c.length) → Composition (c.blocksFun i)).length)
                                fun
                                  (j :
                                    Fin
                                      i.fst (β := fun (c : Composition n) ↦
                                          (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                                (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                                    j).blocks),
                          blocks_pos := sorry, blocks_sum := sorry : Composition n }.length } :
                (a : Composition n) × Composition a.length))
            ((fun (i : (a : Composition n) × Composition a.length) ↦
                (Sigma.mk (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                    (i.fst (β := fun (a : Composition n) ↦ Composition a.length).gather
                      (i.snd (β := fun (a : Composition n) ↦ Composition a.length)))
                    (i.fst (β := fun (a : Composition n) ↦ Composition a.length).sigmaCompositionAux
                      (i.snd (β := fun (a : Composition n) ↦ Composition a.length))) :
                  (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))))
              (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)))))
      (Sigma.snd (β := fun (a : Composition n) ↦ Composition a.length)
          ((fun (i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))) ↦
              (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length)
                  {
                    blocks :=
                      flatten
                        (ofFn (n :=
                          i.fst (β := fun (c : Composition n) ↦
                              (i : Fin c.length) → Composition (c.blocksFun i)).length)
                          fun
                            (j :
                              Fin
                                i.fst (β := fun (c : Composition n) ↦
                                    (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                          (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                              j).blocks),
                    blocks_pos := sorry, blocks_sum := sorry : Composition n }
                  {
                    blocks :=
                      ofFn (n :=
                        i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
                        fun
                          (j :
                            Fin
                              i.fst (β := fun (c : Composition n) ↦
                                  (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                        (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                            j).length,
                    blocks_pos := sorry, blocks_sum := sorry :
                    Composition
                      {
                          blocks :=
                            flatten
                              (ofFn (n :=
                                i.fst (β := fun (c : Composition n) ↦
                                    (i : Fin c.length) → Composition (c.blocksFun i)).length)
                                fun
                                  (j :
                                    Fin
                                      i.fst (β := fun (c : Composition n) ↦
                                          (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                                (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                                    j).blocks),
                          blocks_pos := sorry, blocks_sum := sorry : Composition n }.length } :
                (a : Composition n) × Composition a.length))
            ((fun (i : (a : Composition n) × Composition a.length) ↦
                (Sigma.mk (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                    (i.fst (β := fun (a : Composition n) ↦ Composition a.length).gather
                      (i.snd (β := fun (a : Composition n) ↦ Composition a.length)))
                    (i.fst (β := fun (a : Composition n) ↦ Composition a.length).sigmaCompositionAux
                      (i.snd (β := fun (a : Composition n) ↦ Composition a.length))) :
                  (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))))
              (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b))) :
        Composition
          (length (n := n)
            (Sigma.fst (β := fun (a : Composition n) ↦ Composition a.length)
              ((fun (i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))) ↦
                  (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length)
                      {
                        blocks :=
                          flatten
                            (ofFn (n :=
                              i.fst (β := fun (c : Composition n) ↦
                                  (i : Fin c.length) → Composition (c.blocksFun i)).length)
                              fun
                                (j :
                                  Fin
                                    i.fst (β := fun (c : Composition n) ↦
                                        (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                              (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                                  j).blocks),
                        blocks_pos := sorry, blocks_sum := sorry : Composition n }
                      {
                        blocks :=
                          ofFn (n :=
                            i.fst (β := fun (c : Composition n) ↦
                                (i : Fin c.length) → Composition (c.blocksFun i)).length)
                            fun
                              (j :
                                Fin
                                  i.fst (β := fun (c : Composition n) ↦
                                      (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                            (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                                j).length,
                        blocks_pos := sorry, blocks_sum := sorry :
                        Composition
                          {
                              blocks :=
                                flatten
                                  (ofFn (n :=
                                    i.fst (β := fun (c : Composition n) ↦
                                        (i : Fin c.length) → Composition (c.blocksFun i)).length)
                                    fun
                                      (j :
                                        Fin
                                          i.fst (β := fun (c : Composition n) ↦
                                              (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
                                    (i.snd (β := fun (c : Composition n) ↦
                                        (i : Fin c.length) → Composition (c.blocksFun i)) j).blocks),
                              blocks_pos := sorry, blocks_sum := sorry : Composition n }.length } :
                    (a : Composition n) × Composition a.length))
                ((fun (i : (a : Composition n) × Composition a.length) ↦
                    (Sigma.mk (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i))
                        (i.fst (β := fun (a : Composition n) ↦ Composition a.length).gather
                          (i.snd (β := fun (a : Composition n) ↦ Composition a.length)))
                        (i.fst (β := fun (a : Composition n) ↦ Composition a.length).sigmaCompositionAux
                          (i.snd (β := fun (a : Composition n) ↦ Composition a.length))) :
                      (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))))
                  (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)))))) =
    blocks (n :=
      (Sigma.fst (β := fun (a : Composition n) ↦ Composition a.length)
          (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b)).length)
      (Sigma.snd (β := fun (a : Composition n) ↦ Composition a.length)
        (Sigma.mk (β := fun (a : Composition n) ↦ Composition a.length) a b))
Tactic: rw [sigma_composition_eq_iff]