Let x0 of type ι be given.
Apply equip_tra with
{x1 ∈ prim4 x0|equip x1 u2},
{x1 ∈ prim4 u3|equip x1 u2},
u3 leaving 2 subgoals.
Apply H0 with
equip {x1 ∈ prim4 x0|equip x1 u2} {x1 ∈ prim4 u3|equip x1 u2}.
Let x1 of type ι → ι be given.
Apply bijE with
x0,
u3,
x1,
equip {x2 ∈ prim4 x0|equip x2 u2} {x2 ∈ prim4 u3|equip x2 u2} leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2:
∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ u3.
Assume H3: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Assume H4:
∀ x2 . x2 ∈ u3 ⟶ ∃ x3 . and (x3 ∈ x0) (x1 x3 = x2).
Apply bijE with
u3,
x0,
inv x0 x1,
equip {x2 ∈ prim4 x0|equip x2 u2} {x2 ∈ prim4 u3|equip x2 u2} leaving 2 subgoals.
The subproof is completed by applying L5.
Assume H6:
∀ x2 . x2 ∈ u3 ⟶ inv x0 x1 x2 ∈ x0.
Assume H7:
∀ x2 . x2 ∈ u3 ⟶ ∀ x3 . x3 ∈ u3 ⟶ inv x0 x1 x2 = inv x0 x1 x3 ⟶ x2 = x3.
Assume H8:
∀ x2 . x2 ∈ x0 ⟶ ∃ x3 . and (x3 ∈ u3) (inv x0 x1 x3 = x2).
Let x2 of type ο be given.
Apply H9 with
λ x3 . {x1 x4|x4 ∈ x3}.
Apply bijI with
{x3 ∈ prim4 x0|equip x3 u2},
{x3 ∈ prim4 u3|equip x3 u2},
λ x3 . {x1 x4|x4 ∈ x3} leaving 3 subgoals.
Let x3 of type ι be given.
Apply SepE with
prim4 x0,
λ x4 . equip x4 u2,
x3,
(λ x4 . {x1 x5|x5 ∈ x4}) x3 ∈ {x4 ∈ prim4 u3|equip x4 u2} leaving 2 subgoals.
The subproof is completed by applying H10.
Assume H11:
x3 ∈ prim4 x0.
Apply SepI with
prim4 u3,
λ x4 . equip x4 u2,
(λ x4 . {x1 x5|x5 ∈ x4}) x3 leaving 2 subgoals.
Apply PowerI with
u3,
{x1 x4|x4 ∈ x3}.
Let x4 of type ι be given.
Assume H13: x4 ∈ {x1 x5|x5 ∈ x3}.
Apply ReplE_impred with
x3,
x1,
x4,
x4 ∈ u3 leaving 2 subgoals.
The subproof is completed by applying H13.
Let x5 of type ι be given.
Assume H14: x5 ∈ x3.
Assume H15: x4 = x1 x5.
Apply H15 with
λ x6 x7 . x7 ∈ u3.
Apply H2 with
x5.
Apply PowerE with
x0,
x3,
x5 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H14.
Apply equip_tra with
{...|x4 ∈ x3},
...,
... leaving 2 subgoals.