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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι be given.
Assume H0: Group (pack_b x0 x1).
Assume H1: explicit_subgroup x0 x1 x2.
Claim L2: explicit_Group x0 x1
Apply GroupE with x0, x1.
The subproof is completed by applying H0.
Claim L3: explicit_Group x2 x1
Apply H1 with explicit_Group x2 x1.
Assume H3: Group (pack_b 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 H4: Group (pack_b x2 x1).
Assume H5: x2 ⊆ x0.
The subproof is completed by applying H5.
Claim L5: explicit_Group_identity x0 x1 ∈ x0
Apply explicit_Group_identity_in with x0, x1.
The subproof is completed by applying L2.
Claim L6: explicit_Group_identity x2 x1 ∈ x2
Apply explicit_Group_identity_in with x2, x1.
The subproof is completed by applying L3.
Claim L7: explicit_Group_identity x2 x1 ∈ x0
Apply L4 with explicit_Group_identity x2 x1.
The subproof is completed by applying L6.
Apply explicit_Group_rcancel with x0, x1, explicit_Group_identity x0 x1, explicit_Group_identity x2 x1, explicit_Group_identity x2 x1 leaving 5 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L5.
The subproof is completed by applying L7.
The subproof is completed by applying L7.
Apply explicit_Group_identity_rid with x2, x1, explicit_Group_identity x2 x1, λ x3 x4 . x1 (explicit_Group_identity x0 x1) (explicit_Group_identity x2 x1) = x4 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L6.
Apply explicit_Group_identity_lid with x0, x1, explicit_Group_identity x2 x1 leaving 2 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L7.
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