Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ι → ι be given.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x1 ⟶ or (x5 = x1) (x6 = x1).
Assume H1: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0.
Assume H2: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7.
Assume H3: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5.
Assume H4: x1 ∈ x0.
Assume H5: ∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5.
Assume H6:
∀ x5 . x5 ∈ x0 ⟶ ∃ x6 . and (x6 ∈ x0) (x3 x5 x6 = x1).
Assume H7: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0.
Assume H8: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7.
Assume H9: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5.
Assume H10: x2 ∈ x0.
Assume H11: x2 = x1 ⟶ ∀ x5 : ο . x5.
Assume H12: ∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5.
Assume H13:
∀ x5 . x5 ∈ x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∃ x6 . and (x6 ∈ x0) (x4 x5 x6 = x2).
Assume H14: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7).
Let x5 of type ι be given.
Assume H15: x5 ∈ x0.
Let x6 of type ι be given.
Assume H16: x6 ∈ x0.
Assume H17: x4 x5 x6 = x1.
Apply xm with
x6 = x1,
or (x5 = x1) (x6 = x1) leaving 2 subgoals.
Assume H18: x6 = x1.
Apply orIR with
x5 = x1,
x6 = x1.
The subproof is completed by applying H18.
Assume H18: x6 = x1 ⟶ ∀ x7 : ο . x7.
Apply orIL with
x5 = x1,
x6 = x1.
Apply H13 with
x6,
x5 = x1 leaving 3 subgoals.
The subproof is completed by applying H16.
The subproof is completed by applying H18.
Let x7 of type ι be given.
Assume H19:
(λ x8 . and ... ...) ....