-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : ¬i = j
a_1 : HomotopyGroup N X x
a : (↑(Ω^ N X x) : Type (max u_2 u_1))
b_1 : HomotopyGroup N X x
b : (↑(Ω^ N X x) : Type (max u_2 u_1))
c_1 : HomotopyGroup N X x
c : (↑(Ω^ N X x) : Type (max u_2 u_1))
d_1 : HomotopyGroup N X x
d : (↑(Ω^ N X x) : Type (max u_2 u_1))
⊢ Quotient.mk'
      ((Equiv.symm (toEquiv (loopHomeo i)) :
          (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
        (Path.trans (y := FundamentalGroupoid.as { as := const })
          ((toEquiv (loopHomeo i) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo j)) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo j) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  d)
                ((toEquiv (loopHomeo j) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  c))))
          ((toEquiv (loopHomeo i) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo j)) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo j) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  b)
                ((toEquiv (loopHomeo j) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  a)))))) =
    HMul.hMul (α := HomotopyGroup N X x)
        (Quot.mk
          (⇑(Homotopic.setoid N x) : (↑(Ω^ N X x) : Type (max u_2 u_1)) → (↑(Ω^ N X x) : Type (max u_2 u_1)) → Prop) a)
        (Quot.mk
          (⇑(Homotopic.setoid N x) : (↑(Ω^ N X x) : Type (max u_2 u_1)) → (↑(Ω^ N X x) : Type (max u_2 u_1)) → Prop)
          c) *
      HMul.hMul (α := HomotopyGroup N X x)
        (Quot.mk
          (⇑(Homotopic.setoid N x) : (↑(Ω^ N X x) : Type (max u_2 u_1)) → (↑(Ω^ N X x) : Type (max u_2 u_1)) → Prop) b)
        (Quot.mk
          (⇑(Homotopic.setoid N x) : (↑(Ω^ N X x) : Type (max u_2 u_1)) → (↑(Ω^ N X x) : Type (max u_2 u_1)) → Prop) d)
have neg : Eq (α := (↑(Ω^ N X x) : Type (max u_2 u_1)))
  ((Equiv.symm (toEquiv (loopHomeo i)) :
        (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
      (Path.trans (y := FundamentalGroupoid.as { as := const })
        ((toEquiv (loopHomeo i) : (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
          ((Equiv.symm (toEquiv (loopHomeo j)) :
              (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
            (Path.trans (y := FundamentalGroupoid.as { as := const })
              ((toEquiv (loopHomeo j) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                d)
              ((toEquiv (loopHomeo j) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                c))))
        ((toEquiv (loopHomeo i) : (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
          ((Equiv.symm (toEquiv (loopHomeo j)) :
              (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
            (Path.trans (y := FundamentalGroupoid.as { as := const })
              ((toEquiv (loopHomeo j) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                b)
              ((toEquiv (loopHomeo j) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                a))))) :
    (↑(Ω^ N X x) : Type (max u_2 u_1)))
  ((Equiv.symm (toEquiv (loopHomeo j)) :
        (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
      (Path.trans (y := FundamentalGroupoid.as { as := const })
        ((toEquiv (loopHomeo j) : (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
          ((Equiv.symm (toEquiv (loopHomeo i)) :
              (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
            (Path.trans (y := FundamentalGroupoid.as { as := const })
              ((toEquiv (loopHomeo i) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                d)
              ((toEquiv (loopHomeo i) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                b))))
        ((toEquiv (loopHomeo j) : (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
          ((Equiv.symm (toEquiv (loopHomeo i)) :
              (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
            (Path.trans (y := FundamentalGroupoid.as { as := const })
              ((toEquiv (loopHomeo i) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                c)
              ((toEquiv (loopHomeo i) :
                  (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                a))))) :
    (↑(Ω^ N X x) : Type (max u_2 u_1)))
Tactic: apply congr_arg Quotient.mk'