Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Assume H0: explicit_Ring x0 x1 x2 x3.

Let x4 of type ι be given.

Assume H1: x4 ∈ x0.

Apply explicit_Ring_minus_prop with x0, x1, x2, x3, x4, x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1 leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Assume H2: explicit_Ring_minus x0 x1 x2 x3 x4 ∈ x0.

Assume H3: x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1.

The subproof is completed by applying H3.

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Proofgold Proof