Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Apply unknownprop_bd805e8576afa7aae52c5521a42c8fd7e79d093900ca62c8a59403964f3b7a9e with
λ x5 . λ x6 x7 : ι → ι → ι . λ x8 x9 . explicit_Ring_with_id x5 x8 x9 x6 x7,
x0,
x1,
x2,
x3,
x4.
Let x5 of type ι → ι → ι be given.
Assume H0:
∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x5 x6 x7.
Let x6 of type ι → ι → ι be given.
Assume H1:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x2 x7 x8 = x6 x7 x8.
Let x7 of type ο → ο → ο be given.
Apply prop_ext with
explicit_Ring_with_id x0 x3 x4 x1 x2,
explicit_Ring_with_id x0 x3 x4 x5 x6,
λ x8 x9 : ο . x7 x9 x8.
Apply explicit_Ring_with_id_repindep with
x0,
x3,
x4,
x1,
x2,
x5,
x6 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.