Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0.
Assume H3: ∀ x3 . x3 ∈ x0 ⟶ x2 (x2 x3) = x3.
Apply dneg with
∃ x3 . and (x3 ∈ x0) (x2 x3 = x3).
Assume H4:
not (∃ x3 . and (x3 ∈ x0) (x2 x3 = x3)).
Apply H0 with
False.
Assume H6:
∀ x3 . x3 ∈ omega ⟶ x1 = mul_nat 2 x3 ⟶ ∀ x4 : ο . x4.
Apply even_nat_not_odd_nat with
x1 leaving 2 subgoals.
Apply unknownprop_5596b16e05d109cf37a32604ad2d1fa564d64f6c432201fa79c05f594462ca59 with
x1,
x0,
x2 leaving 5 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H5.
Apply equip_sym with
x0,
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H7: x3 ∈ x0.
Assume H8: x2 x3 = x3.
Apply H4.
Let x4 of type ο be given.
Assume H9:
∀ x5 . and (x5 ∈ x0) (x2 x5 = x5) ⟶ x4.
Apply H9 with
x3.
Apply andI with
x3 ∈ x0,
x2 x3 = x3 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H0.