Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Assume H1: ∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0.
Assume H2: ∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4.
Apply nat_ind with
λ x4 . 1319b.. x4 x2 x1 = 1319b.. x4 x3 x1 leaving 2 subgoals.
Apply unknownprop_039fc83525f9619f7cfecb750766b6bca3d944a312bec3cbff47462eeab06c10 with
x3,
x1,
λ x4 x5 . 1319b.. 0 x2 x1 = x5.
The subproof is completed by applying unknownprop_039fc83525f9619f7cfecb750766b6bca3d944a312bec3cbff47462eeab06c10 with x2, x1.
Let x4 of type ι be given.
Apply unknownprop_002dbea24d6e2c65c6cefd906b209766da62711ebb920e89995e5f3cbbd95f66 with
x4,
x2,
x1,
λ x5 x6 . x6 = 1319b.. (ordsucc x4) x3 x1 leaving 2 subgoals.
Apply nat_p_omega with
x4.
The subproof is completed by applying H3.
Apply unknownprop_002dbea24d6e2c65c6cefd906b209766da62711ebb920e89995e5f3cbbd95f66 with
x4,
x3,
x1,
λ x5 x6 . x2 (1319b.. x4 x2 x1) = x6 leaving 2 subgoals.
Apply nat_p_omega with
x4.
The subproof is completed by applying H3.
Apply H4 with
λ x5 x6 . x2 (1319b.. x4 x2 x1) = x3 x5.
Apply H2 with
1319b.. x4 x2 x1.
Apply unknownprop_634dfe6db60252fae5fb11b99d8182640cd8d27dd5c6123b0e5c5ff1eb699c89 with
x0,
x1,
x2,
x4 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.