Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Assume H1:
∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).
Assume H2:
∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).
Let x1 of type ι be given.
Apply equip_tra with
lam (DirGraphOutNeighbors u18 x0 x1) (λ x2 . setminus (DirGraphOutNeighbors u18 x0 x2) (Sing x1)),
setprod u5 u4,
u20 leaving 2 subgoals.
Apply equip_tra with
lam (DirGraphOutNeighbors u18 x0 x1) (λ x2 . setminus (DirGraphOutNeighbors u18 x0 x2) (Sing x1)),
setprod (DirGraphOutNeighbors u18 x0 x1) u4,
setprod u5 u4 leaving 2 subgoals.
Apply unknownprop_ca35929bb9516758ce3c56c5bf7b9f0713e81693444871247a9d2c8c2835f309 with
DirGraphOutNeighbors u18 x0 x1,
u4,
λ x2 . setminus (DirGraphOutNeighbors u18 x0 x2) (Sing x1).
Let x2 of type ι be given.
Apply unknownprop_52137fbb4a2ce93c26a5099f50d500c9abe51677b43046d7cccb609dc3338329 with
x0,
x1,
x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply unknownprop_114be8822b6413aaa4f094e79f9ed7dccf7f251af45e6784755c2493769eabfa with
DirGraphOutNeighbors u18 x0 x1,
u5,
u4,
u4 leaving 2 subgoals.
Apply unknownprop_942eb02a74c10f16602e1ae1f344c3023e05004e91bcaa34f489f7c49867be93 with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying equip_ref with
u4.
Apply unknownprop_9999811b816ecc962e9d03bd28c7ff2f2b07bac5ef7fb31aad1b993404e01924 with
λ x2 x3 . equip (setprod u5 u4) x2.
Apply equip_sym with
mul_nat u5 u4,
setprod u5 u4.
Apply unknownprop_b67c3f99b7dcb350283e28e0205622aac5c4480ca587d55b1fc9b439a7a164e1 with
u5,
u4,
u5,
u4 leaving 4 subgoals.
The subproof is completed by applying nat_5.
The subproof is completed by applying nat_4.
The subproof is completed by applying equip_ref with
u5.
The subproof is completed by applying equip_ref with
u4.