Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι be given.
Apply GroupE with
x0,
x1.
The subproof is completed by applying H0.
Apply H1 with
explicit_Group x2 x1.
Assume H4: x2 ⊆ x0.
Apply GroupE with
x2,
x1.
The subproof is completed by applying H3.
Claim L4: x2 ⊆ x0
Apply H1 with
x2 ⊆ x0.
Assume H5: x2 ⊆ x0.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Assume H5: x3 ∈ x2.
Claim L6: x3 ∈ x0
Apply L4 with
x3.
The subproof is completed by applying H5.
Apply L4 with
explicit_Group_inverse x2 x1 x3.
Apply explicit_Group_inverse_in with
x2,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H5.
Apply explicit_Group_inverse_in with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L6.
Apply explicit_Group_lcancel with
x0,
x1,
x3,
explicit_Group_inverse x0 x1 x3,
explicit_Group_inverse x2 x1 x3 leaving 5 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L6.
The subproof is completed by applying L8.
The subproof is completed by applying L7.
Apply explicit_Group_inverse_rinv with
x0,
x1,
x3,
λ x4 x5 . x5 = x1 x3 (explicit_Group_inverse x2 x1 x3) leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L6.
Apply explicit_Group_inverse_rinv with
x2,
x1,
x3,
λ x4 x5 . explicit_Group_identity x0 x1 = x5 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H5.
Apply explicit_subgroup_identity_eq with
x0,
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.