Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_2d7c7a9916fa2967cfb4d546f4e37c43b64368ed4a60618379328e066e9b7e0e with
x0,
λ x2 . IrreflexiveTransitiveReln (BinReln_product x2 x1) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Let x3 of type ι → ι → ο be given.
Assume H2:
∀ x4 . x4 ∈ x2 ⟶ not (x3 x4 x4).
Assume H3: ∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6.
Apply unknownprop_2d7c7a9916fa2967cfb4d546f4e37c43b64368ed4a60618379328e066e9b7e0e with
x1,
λ x4 . IrreflexiveTransitiveReln (BinReln_product (pack_r x2 x3) x4) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Let x5 of type ι → ι → ο be given.
Assume H4:
∀ x6 . x6 ∈ x4 ⟶ not (x5 x6 x6).
Assume H5: ∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ ∀ x8 . x8 ∈ x4 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8.
Apply unknownprop_efcdca50692a8e5dea3b2dabd19b7c98b28ec0cd127c886a2f9539f6c2a2ba01 with
x2,
x3,
x4,
x5,
λ x6 x7 . IrreflexiveTransitiveReln x7.
Apply unknownprop_dbb6377af3127d2bf8cd888143d856b4a86f0ec975822a440e0313d91ee07474 with
setprod x2 x4,
λ x6 x7 . and (x3 (ap x6 0) (ap x7 0)) (x5 (ap x6 1) (ap x7 1)) leaving 2 subgoals.
Let x6 of type ι be given.
Assume H7:
and (x3 (ap x6 0) (ap x6 0)) (x5 (ap x6 1) (ap x6 1)).
Apply H7 with
False.
Assume H8:
x3 (ap x6 0) (ap x6 0).
Apply H4 with
ap x6 1.
Apply ap1_Sigma with
x2,
λ x7 . x4,
x6.
The subproof is completed by applying H6.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H9:
(λ x9 x10 . and (x3 (ap x9 0) (ap x10 0)) (x5 (ap x9 1) (ap x10 1))) x6 x7.
Assume H10:
(λ x9 x10 . and (x3 (ap x9 0) (ap x10 0)) (x5 (ap x9 1) (ap x10 1))) x7 x8.
Apply H9 with
(λ x9 x10 . and (x3 (ap x9 0) (ap x10 0)) (x5 (ap x9 1) (ap x10 1))) x6 x8.
Assume H11:
x3 (ap x6 0) (ap x7 0).
Assume H12:
x5 (ap x6 1) (ap x7 1).
Apply H10 with
(λ x9 x10 . and (x3 (ap x9 0) (ap x10 0)) (x5 (ap x9 1) (ap x10 1))) x6 x8.
Assume H13:
x3 (ap x7 0) (ap x8 0).