Apply explicit_Field_I with real, 0, 1, add_SNo, mul_SNo leaving 14 subgoals.

The subproof is completed by applying real_add_SNo.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ι be given.

Assume H1: x1 ∈ real.

Let x2 of type ι be given.

Assume H2: x2 ∈ real.

Apply add_SNo_assoc with x0, x1, x2 leaving 3 subgoals.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Apply real_SNo with x1.

The subproof is completed by applying H1.

Apply real_SNo with x2.

The subproof is completed by applying H2.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ι be given.

Assume H1: x1 ∈ real.

Apply add_SNo_com with x0, x1 leaving 2 subgoals.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Apply real_SNo with x1.

The subproof is completed by applying H1.

The subproof is completed by applying real_0.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Apply add_SNo_0L with x0.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ο be given.

Assume H1: ∀ x2 . and (x2 ∈ real) (add_SNo x0 x2 = 0) ⟶ x1.

Apply H1 with minus_SNo x0.

Apply andI with minus_SNo x0 ∈ real, add_SNo x0 (minus_SNo x0) = 0 leaving 2 subgoals.

Apply real_minus_SNo with x0.

The subproof is completed by applying H0.

Apply add_SNo_minus_SNo_rinv with x0.

Apply real_SNo with x0.

The subproof is completed by applying H0.

The subproof is completed by applying real_mul_SNo.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ι be given.

Assume H1: x1 ∈ real.

Let x2 of type ι be given.

Assume H2: x2 ∈ real.

Apply mul_SNo_assoc with x0, x1, x2 leaving 3 subgoals.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Apply real_SNo with x1.

The subproof is completed by applying H1.

Apply real_SNo with x2.

The subproof is completed by applying H2.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ι be given.

Assume H1: x1 ∈ real.

Apply mul_SNo_com with x0, x1 leaving 2 subgoals.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Apply real_SNo with x1.

The subproof is completed by applying H1.

The subproof is completed by applying real_1.

The subproof is completed by applying neq_1_0.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Apply mul_SNo_oneL with x0.

Apply real_SNo with x0.

The subproof is completed by applying H0.

The subproof is completed by applying nonzero_real_recip_ex.

Let x0 of type ι be given.

Assume H0: x0 ∈ real.

Let x1 of type ι be given.

Assume H1: x1 ∈ real.

Let x2 of type ι be given.

Assume H2: x2 ∈ real.

Apply mul_SNo_distrL with x0, x1, x2 leaving 3 subgoals.

Apply real_SNo with x0.

The subproof is completed by applying H0.

Apply real_SNo with x1.

The subproof is completed by applying H1.

Apply real_SNo with x2.

The subproof is completed by applying H2.

■

Proofgold Proof