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.
Let x5 of type ι → ι → ο be given.
Let x6 of type ι → ι → ι be given.
Let x7 of type ι → ι → ι be given.
Let x8 of type ι → ι → ο be given.
Assume H0: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10.
Assume H1: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10.
Assume H2:
∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10).
Apply iffI with
explicit_OrderedField x0 x1 x2 x3 x4 x5,
explicit_OrderedField x0 x1 x2 x6 x7 x8 leaving 2 subgoals.
Apply unknownprop_bae4375e88c2d44b2130484c0c2686703cdc53a204e289a5b7a073f35c0f816b with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply unknownprop_bae4375e88c2d44b2130484c0c2686703cdc53a204e289a5b7a073f35c0f816b with
x0,
x1,
x2,
x6,
x7,
x8,
x3,
x4,
x5 leaving 3 subgoals.
Let x9 of type ι be given.
Assume H3: x9 ∈ x0.
Let x10 of type ι be given.
Assume H4: x10 ∈ x0.
Let x11 of type ι → ι → ο be given.
Apply H0 with
x9,
x10,
λ x12 x13 . x11 x13 x12 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Let x9 of type ι be given.
Assume H3: x9 ∈ x0.
Let x10 of type ι be given.
Assume H4: x10 ∈ x0.
Let x11 of type ι → ι → ο be given.
Apply H1 with
x9,
x10,
λ x12 x13 . x11 x13 x12 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Let x9 of type ι be given.
Assume H3: x9 ∈ x0.
Let x10 of type ι be given.
Assume H4: x10 ∈ x0.
Apply iff_sym with
x5 x9 x10,
x8 x9 x10.
Apply H2 with
x9,
x10 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.