Let x0 of type ι → ι be given.
Let x1 of type ι → ι be given.
Assume H0: ∀ x2 . x2 ∈ 0 ⟶ x0 x2 = x1 x2.
Apply unknownprop_4376a6c44e07ee63cfd63de35739aa1967e60551544bcab6d1b8284e5b2ad2ba with
x1,
λ x2 x3 . finite_add_SNo 0 x0 = x3.
The subproof is completed by applying unknownprop_4376a6c44e07ee63cfd63de35739aa1967e60551544bcab6d1b8284e5b2ad2ba with x0.
Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H2:
∀ x3 . x3 ∈ ordsucc x0 ⟶ x1 x3 = x2 x3.
Apply unknownprop_69faeba9a75f2f8d58865148c9f0f7e35d3ac66d15111e3cc3404d6a2eed4dcd with
x1,
x0,
λ x3 x4 . x4 = finite_add_SNo (ordsucc x0) x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_69faeba9a75f2f8d58865148c9f0f7e35d3ac66d15111e3cc3404d6a2eed4dcd with
x2,
x0,
λ x3 x4 . add_SNo (finite_add_SNo x0 x1) (x1 x0) = x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Claim L3: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Apply H2 with
x3.
Apply ordsuccI1 with
x0,
x3.
The subproof is completed by applying H3.
Apply H1 with
x1,
x2,
λ x3 x4 . add_SNo x4 (x1 x0) = add_SNo (finite_add_SNo x0 x2) (x2 x0) leaving 2 subgoals.
The subproof is completed by applying L3.
Claim L4: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
set y6 to be λ x6 . x5
Apply H2 with
x2,
λ x7 x8 . y6 (add_SNo (finite_add_SNo x2 y4) x7) (add_SNo (finite_add_SNo x2 y4) x8) leaving 2 subgoals.
The subproof is completed by applying ordsuccI2 with x2.
The subproof is completed by applying H4.
Let x5 of type ι → ι → ο be given.
Apply L4 with
λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H5: x5 y4 y4.
The subproof is completed by applying H5.