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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: diadic_rational_alt1_p x0.
Apply H0 with diadic_rational_alt1_p x1diadic_rational_alt1_p (mul_SNo x0 x1).
Let x2 of type ι be given.
Assume H1: (λ x3 . and (x3omega) (∃ x4 . and (x4int_alt1) (x0 = mul_SNo (eps_ x3) x4))) x2.
Apply H1 with diadic_rational_alt1_p x1diadic_rational_alt1_p (mul_SNo x0 x1).
Assume H2: x2omega.
Assume H3: ∃ x3 . and (x3int_alt1) (x0 = mul_SNo (eps_ x2) x3).
Apply H3 with diadic_rational_alt1_p x1diadic_rational_alt1_p (mul_SNo x0 x1).
Let x3 of type ι be given.
Assume H4: (λ x4 . and (x4int_alt1) (x0 = mul_SNo (eps_ x2) x4)) x3.
Apply H4 with diadic_rational_alt1_p x1diadic_rational_alt1_p (mul_SNo x0 x1).
Assume H5: x3int_alt1.
Assume H6: x0 = mul_SNo (eps_ x2) x3.
Assume H7: diadic_rational_alt1_p x1.
Apply H7 with diadic_rational_alt1_p (mul_SNo x0 x1).
Let x4 of type ι be given.
Assume H8: (λ x5 . and (x5omega) (∃ x6 . and (x6int_alt1) (x1 = mul_SNo (eps_ x5) x6))) x4.
Apply H8 with diadic_rational_alt1_p (mul_SNo x0 x1).
Assume H9: x4omega.
Assume H10: ∃ x5 . and (x5int_alt1) (x1 = mul_SNo (eps_ x4) x5).
Apply H10 with diadic_rational_alt1_p (mul_SNo x0 x1).
Let x5 of type ι be given.
Assume H11: (λ x6 . and (x6int_alt1) (x1 = mul_SNo (eps_ x4) x6)) x5.
Apply H11 with diadic_rational_alt1_p (mul_SNo x0 x1).
Assume H12: x5int_alt1.
Assume H13: x1 = mul_SNo (eps_ x4) x5.
Let x6 of type ο be given.
Assume H14: ∀ x7 . and (x7omega) (∃ x8 . and (x8int_alt1) (mul_SNo x0 x1 = mul_SNo (eps_ x7) x8))x6.
Apply H14 with add_SNo x2 x4.
Apply andI with add_SNo x2 x4omega, ∃ x7 . and (x7int_alt1) (mul_SNo x0 x1 = mul_SNo (eps_ (add_SNo x2 x4)) x7) leaving 2 subgoals.
Apply add_SNo_In_omega with x2, x4 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H9.
Let x7 of type ο be given.
Assume H15: ∀ x8 . and (x8int_alt1) (mul_SNo x0 x1 = mul_SNo (eps_ (add_SNo x2 x4)) x8)x7.
Apply H15 with mul_SNo x3 x5.
Apply andI with mul_SNo x3 x5int_alt1, mul_SNo x0 x1 = mul_SNo (eps_ (add_SNo x2 x4)) (mul_SNo x3 x5) leaving 2 subgoals.
Apply unknownprop_3f000c087708670f2a9497d0587231f65beb85593b7ebdaf7909fe5e7d4b8c27 with x3, x5 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H12.
Apply mul_SNo_eps_eps_add_SNo with x2, x4, λ x8 x9 . mul_SNo x0 x1 = mul_SNo x8 (mul_SNo x3 x5) leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H9.
Apply mul_SNo_com_4_inner_mid with eps_ x2, eps_ x4, x3, x5, λ x8 x9 . mul_SNo x0 x1 = x9 leaving 5 subgoals.
Apply SNo_eps_ with x2.
The subproof is completed by applying H2.
Apply SNo_eps_ with ....
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