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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0 ∈ omega.
Let x1 of type ι be given.
Assume H1: x1 ∈ omega.
Assume H2: divides_int_alt1 x0 x1.
Apply H2 with divides_nat x0 x1.
Assume H3: and (x0 ∈ int_alt1) (x1 ∈ int_alt1).
Assume H4: ∃ x2 . and (x2 ∈ int_alt1) (mul_SNo x0 x2 = x1).
Apply and3I with x0 ∈ omega, x1 ∈ omega, ∃ x2 . and (x2 ∈ omega) (mul_nat x0 x2 = x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply H4 with ∃ x2 . and (x2 ∈ omega) (mul_nat x0 x2 = x1).
Let x2 of type ι be given.
Assume H5: (λ x3 . and (x3 ∈ int_alt1) (mul_SNo x0 x3 = x1)) x2.
Apply H5 with ∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1).
Assume H6: x2 ∈ int_alt1.
Assume H7: mul_SNo x0 x2 = x1.
Apply binunionE with omega, {minus_CSNo x3|x3 ∈ omega}, x2, ∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1) leaving 3 subgoals.
The subproof is completed by applying H6.
Assume H8: x2 ∈ omega.
Let x3 of type ο be given.
Assume H9: ∀ x4 . and (x4 ∈ omega) (mul_nat x0 x4 = x1) ⟶ x3.
Apply H9 with x2.
Apply andI with x2 ∈ omega, mul_nat x0 x2 = x1 leaving 2 subgoals.
The subproof is completed by applying H8.
Apply mul_nat_mul_SNo with x0, x2, λ x4 x5 . x5 = x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H8.
The subproof is completed by applying H7.
Assume H8: x2 ∈ {minus_CSNo x3|x3 ∈ omega}.
Apply xm with x1 = 0, ∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1) leaving 2 subgoals.
Assume H9: x1 = 0.
Claim L10: 0 ∈ omega
Apply nat_p_omega with 0.
The subproof is completed by applying nat_0.
Let x3 of type ο be given.
Assume H11: ∀ x4 . and (x4 ∈ omega) (mul_nat x0 x4 = x1) ⟶ x3.
Apply H11 with 0.
Apply andI with 0 ∈ omega, mul_nat x0 0 = x1 leaving 2 subgoals.
The subproof is completed by applying L10.
Apply mul_nat_mul_SNo with x0, 0, λ x4 x5 . x5 = x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L10.
Apply H9 with λ x4 x5 . mul_SNo x0 0 = x5.
Apply mul_nat_mul_SNo with x0, 0, λ x4 x5 . x4 = 0 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply nat_p_omega with 0.
The subproof is completed by applying nat_0.
The subproof is completed by applying mul_nat_0R with x0.
Assume H9: x1 = 0 ⟶ ∀ x3 : ο . x3.
Apply FalseE with ∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1).
Apply ReplE_impred with omega, minus_CSNo, x2, False leaving 2 subgoals.
The subproof is completed by applying H8.
Let x3 of type ι be given.
Assume H10: x3 ∈ omega.
Apply minus_SNo_minus_CSNo with x3, λ x4 x5 . x2 = x4 ⟶ ∀ x6 : ο . x6 leaving 2 subgoals.
Apply omega_SNo with x3.
The subproof is completed by applying H10.
Assume H11: x2 = minus_SNo x3.
Claim L12: ...
...
Claim L13: x1 = minus_SNo (mul_SNo x0 x3)
...
Apply H9.
Claim L14: mul_SNo x0 x3 ∈ omega
Apply mul_SNo_In_omega with x0, x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H10.
Apply nonpos_nonneg_0 with x1, mul_SNo x0 x3, x1 = 0 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L14.
The subproof is completed by applying L13.
Assume H15: x1 = 0.
Assume H16: mul_SNo x0 x3 = 0.
The subproof is completed by applying H15.
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