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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: prime_nat x0.
Let x1 of type ι be given.
Assume H1: x1 ∈ x0.
Let x2 of type ι be given.
Assume H2: x2 ∈ x0.
Apply H0 with divides_int x0 (add_SNo ((λ x3 . mul_SNo x3 x3) x1) (minus_SNo ((λ x3 . mul_SNo x3 x3) x2))) ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)).
Assume H3: and (x0 ∈ omega) (1 ∈ x0).
Apply H3 with (∀ x3 . x3 ∈ omega ⟶ divides_nat x3 x0 ⟶ or (x3 = 1) (x3 = x0)) ⟶ divides_int x0 (add_SNo ((λ x3 . mul_SNo x3 x3) x1) (minus_SNo ((λ x3 . mul_SNo x3 x3) x2))) ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)).
Assume H4: x0 ∈ omega.
Assume H5: 1 ∈ x0.
Assume H6: ∀ x3 . x3 ∈ omega ⟶ divides_nat x3 x0 ⟶ or (x3 = 1) (x3 = x0).
Claim L7: ...
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Claim L8: ...
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Claim L9: ...
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Claim L10: ...
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Claim L11: ...
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Claim L12: ...
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Claim L13: ...
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Claim L14: ...
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Apply unknownprop_4a0a3eff54752556487021177bda70d187b173aa30e36acd65a7d5c9c0f8dfae with x1, x2, λ x3 x4 . divides_int x0 x3 ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)) leaving 3 subgoals.
The subproof is completed by applying L13.
The subproof is completed by applying L14.
Assume H15: divides_int x0 (mul_SNo (add_SNo x1 x2) (add_SNo x1 (minus_SNo x2))).
Claim L16: ...
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Claim L17: ...
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Claim L18: ...
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Claim L19: ...
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Apply Euclid_lemma with x0, add_SNo x1 x2, add_SNo x1 (minus_SNo x2), or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)) leaving 6 subgoals.
The subproof is completed by applying H0.
Apply int_add_SNo with x1, x2 leaving 2 subgoals.
Apply nat_p_int with x1.
The subproof is completed by applying L11.
Apply nat_p_int with x2.
The subproof is completed by applying L12.
Apply int_add_SNo with x1, minus_SNo x2 leaving 2 subgoals.
Apply nat_p_int with x1.
The subproof is completed by applying L11.
Apply int_minus_SNo with x2.
Apply nat_p_int with x2.
The subproof is completed by applying L12.
The subproof is completed by applying H15.
Assume H20: divides_int x0 (add_SNo x1 x2).
Apply H20 with or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)).
Assume H21: and (x0 ∈ int) (add_SNo x1 x2 ∈ int).
Assume H22: ∃ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = add_SNo x1 x2).
Apply H22 with or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)).
Let x3 of type ι be given.
Assume H23: (λ x4 . and (x4 ∈ int) (mul_SNo x0 x4 = add_SNo x1 x2)) x3.
Apply H23 with or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)).
Apply int_SNo_cases with λ x4 . mul_SNo x0 x4 = add_SNo x1 x2 ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)), x3 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H24: x4 ∈ omega.
Apply unknownprop_b35032c81ea06ad673f8a0490d5be4e7b984453ec9378fed4adde429c2b88d75 with λ x5 . mul_SNo x0 x5 = add_SNo x1 x2 ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)), x4 leaving 4 subgoals.
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