Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply H0 with
x1 (x1 x3 x2) (x1 x3 x2) = explicit_Group_identity x0 x1.
Assume H4:
and (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x4 (x1 x5 x6) = x1 (x1 x4 x5) x6).
Apply H4 with
(∃ x4 . and (prim1 x4 x0) (and (∀ x5 . prim1 x5 x0 ⟶ and (x1 x4 x5 = x5) (x1 x5 x4 = x5)) (∀ x5 . prim1 x5 x0 ⟶ ∃ x6 . and (prim1 x6 x0) (and (x1 x5 x6 = x4) (x1 x6 x5 = x4))))) ⟶ x1 (x1 x3 x2) (x1 x3 x2) = explicit_Group_identity x0 x1.
Assume H6:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x4 (x1 x5 x6) = x1 (x1 x4 x5) x6.
Assume H7:
∃ x4 . and (prim1 x4 x0) (and (∀ x5 . prim1 x5 x0 ⟶ and (x1 x4 x5 = x5) (x1 x5 x4 = x5)) (∀ x5 . prim1 x5 x0 ⟶ ∃ x6 . and (prim1 x6 x0) (and (x1 x5 x6 = x4) (x1 x6 x5 = x4)))).
Apply explicit_Group_rcancel with
x0,
x1,
x1 (x1 x3 x2) (x1 x3 x2),
explicit_Group_identity x0 x1,
x3 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L11.
Apply explicit_Group_identity_in with
x0,
x1.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply explicit_Group_identity_lid with
x0,
x1,
x3,
λ x4 x5 . x1 (x1 (x1 x3 x2) (x1 x3 x2)) x3 = x5 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply H6 with
x1 x3 x2,
x1 x3 x2,
x3,
λ x4 x5 . x4 = x3 leaving 4 subgoals.
The subproof is completed by applying L10.
The subproof is completed by applying L10.
The subproof is completed by applying H2.
Apply H6 with
x3,
x2,
x3,
λ x4 x5 . x1 (x1 x3 x2) x4 = x3 leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply H6 with
x3,
x2,
x1 x3 (x1 ... ...),
... leaving 4 subgoals.