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 → ι be given.
Let x6 of type ι → ι → ι be given.
Assume H0: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x5 x7 x8.
Assume H1: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 = x6 x7 x8.
Apply iffI with
explicit_Field x0 x1 x2 x3 x4,
explicit_Field x0 x1 x2 x5 x6 leaving 2 subgoals.
Apply unknownprop_020ff441acf63956db89023138d2e3bf192b2578e8bfb819c9422a66b469b129 with
x0,
x1,
x2,
x3,
x4,
x5,
x6 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_020ff441acf63956db89023138d2e3bf192b2578e8bfb819c9422a66b469b129 with
x0,
x1,
x2,
x5,
x6,
x3,
x4 leaving 2 subgoals.
Let x7 of type ι be given.
Assume H2: x7 ∈ x0.
Let x8 of type ι be given.
Assume H3: x8 ∈ x0.
Let x9 of type ι → ι → ο be given.
Apply H0 with
x7,
x8,
λ x10 x11 . x9 x11 x10 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x7 of type ι be given.
Assume H2: x7 ∈ x0.
Let x8 of type ι be given.
Assume H3: x8 ∈ x0.
Let x9 of type ι → ι → ο be given.
Apply H1 with
x7,
x8,
λ x10 x11 . x9 x11 x10 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.