Proofgold Address

Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition explicit_Nats := λ x0 x1 . λ x2 : ι → ι . and (and (and (and (prim1 x1 x0) (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0)) (∀ x3 . prim1 x3 x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4)) (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . prim1 x4 x0 ⟶ x3 x4)
Known and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Theorem explicit_Nats_I : ∀ x0 x1 . ∀ x2 : ι → ι . prim1 x1 x0 ⟶ (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ (∀ x3 . prim1 x3 x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4) ⟶ (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . prim1 x4 x0 ⟶ x3 x4) ⟶ explicit_Nats x0 x1 x2 (proof)
Known and5E : ∀ x0 x1 x2 x3 x4 : ο . and (and (and (and x0 x1) x2) x3) x4 ⟶ ∀ x5 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5) ⟶ x5
Theorem explicit_Nats_E : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ο . (explicit_Nats x0 x1 x2 ⟶ prim1 x1 x0 ⟶ (∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0) ⟶ (∀ x4 . prim1 x4 x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5) ⟶ x3) ⟶ explicit_Nats x0 x1 x2 ⟶ x3 (proof)
Known andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_Nats_ind : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . prim1 x4 x0 ⟶ x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . prim1 x4 x0 ⟶ x3 x4 (proof)
Param 48ef8.. : ι
Param 4a7ef.. : ι
Param 4ae4a.. : ι → ι
Known 8ee89.. : prim1 4a7ef.. 48ef8..
Known 98ac3.. : ∀ x0 . prim1 x0 48ef8.. ⟶ prim1 (4ae4a.. x0) 48ef8..
Known 1b1d1.. : ∀ x0 . 4ae4a.. x0 = 4a7ef.. ⟶ ∀ x1 : ο . x1
Known 054cd.. : ∀ x0 x1 . 4ae4a.. x0 = 4ae4a.. x1 ⟶ x0 = x1
Definition ba9d8.. := λ x0 . ∀ x1 : ι → ο . x1 4a7ef.. ⟶ (∀ x2 . x1 x2 ⟶ x1 (4ae4a.. x2)) ⟶ x1 x0
Known c2711.. : ∀ x0 . prim1 x0 48ef8.. ⟶ ba9d8.. x0
Theorem dd179.. : explicit_Nats 48ef8.. 4a7ef.. 4ae4a.. (proof)
Definition bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x1) (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . prim1 x3 x1 ⟶ ∀ x4 : ο . (∀ x5 . and (prim1 x5 x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Theorem explicit_Nats_transfer : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 x4 . ∀ x5 x6 : ι → ι . explicit_Nats x0 x1 x2 ⟶ bij x0 x3 x6 ⟶ x6 x1 = x4 ⟶ (∀ x7 . prim1 x7 x0 ⟶ x6 (x2 x7) = x5 (x6 x7)) ⟶ explicit_Nats x3 x4 x5 (proof)
Definition explicit_Field := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . and (and (and (and (and (and (and (and (and (and (and (and (and (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = x3 x6 x5)) (prim1 x1 x0)) (∀ x5 . prim1 x5 x0 ⟶ x3 x1 x5 = x5)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x4 x5 x6) x0)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = x4 x6 x5)) (prim1 x2 x0)) (x2 = x1 ⟶ ∀ x5 : ο . x5)) (∀ x5 . prim1 x5 x0 ⟶ x4 x2 x5 = x5)) (∀ x5 . prim1 x5 x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))
Known and7I : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6
Theorem explicit_Field_I : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ prim1 x1 x0 ⟶ (∀ x5 . prim1 x5 x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x4 x5 x6) x0) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = x4 x6 x5) ⟶ prim1 x2 x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_Field x0 x1 x2 x3 x4 (proof)
Known and7E : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6 ⟶ ∀ x7 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ x7) ⟶ x7
Theorem explicit_Field_E : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ prim1 (x3 x6 x7) x0) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ prim1 x1 x0 ⟶ (∀ x6 . prim1 x6 x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 : ο . (∀ x8 . and (prim1 x8 x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ prim1 (x4 x6 x7) x0) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ prim1 x2 x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ (x6 = x1 ⟶ ∀ x7 : ο . x7) ⟶ ∀ x7 : ο . (∀ x8 . and (prim1 x8 x0) (x4 x6 x8 = x2) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ x5 (proof)
Definition explicit_Field_minus := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 . prim0 (λ x6 . and (prim1 x6 x0) (x3 x5 x6 = x1))
Known Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Theorem explicit_Field_minus_prop : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ and (prim1 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x0) (x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1) (proof)
Theorem explicit_Field_minus_clos : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x0 (proof)
Theorem explicit_Field_minus_R : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1 (proof)
Theorem explicit_Field_minus_L : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_Field_plus_cancelL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7 (proof)
Theorem explicit_Field_plus_cancelR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6 (proof)
Theorem explicit_Field_minus_invol : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x5 (proof)
Theorem explicit_Field_minus_one_In : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ prim1 (explicit_Field_minus x0 x1 x2 x3 x4 x2) x0 (proof)
Theorem explicit_Field_zero_multR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_Field_zero_multL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_Field_minus_mult : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x5 = x4 (explicit_Field_minus x0 x1 x2 x3 x4 x2) x5 (proof)
Theorem explicit_Field_minus_one_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x2) (explicit_Field_minus x0 x1 x2 x3 x4 x2) = x2 (proof)
Theorem explicit_Field_minus_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x4 x5 x5 (proof)
Definition iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Definition or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition explicit_OrderedField := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . and (and (and (and (and (explicit_Field x0 x1 x2 x3 x4) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8)) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7))) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ or (x5 x6 x7) (x5 x7 x6))) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8))) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7))
Known and6I : ∀ x0 x1 x2 x3 x4 x5 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ and (and (and (and (and x0 x1) x2) x3) x4) x5
Theorem explicit_OrderedField_I : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ or (x5 x6 x7) (x5 x7 x6)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7)) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 (proof)
Known and6E : ∀ x0 x1 x2 x3 x4 x5 : ο . and (and (and (and (and x0 x1) x2) x3) x4) x5 ⟶ ∀ x6 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6) ⟶ x6
Theorem explicit_OrderedField_E : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x5 x7 x8 ⟶ x5 x8 x9 ⟶ x5 x7 x9) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (and (x5 x7 x8) (x5 x8 x7)) (x7 = x8)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ or (x5 x7 x8) (x5 x8 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x5 x7 x8 ⟶ x5 (x3 x7 x9) (x3 x8 x9)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x1 x7 ⟶ x5 x1 x8 ⟶ x5 x1 (x4 x7 x8)) ⟶ x6) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6 (proof)
Definition lt := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 x7 . and (x5 x6 x7) (x6 = x7 ⟶ ∀ x8 : ο . x8)
Definition natOfOrderedField_p := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 . ∀ x7 : ι → ο . x7 x1 ⟶ (∀ x8 . x7 x8 ⟶ x7 (x3 x8 x2)) ⟶ x7 x6
Param 1216a.. : ι → (ι → ο) → ι
Known b2421.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ x1 x2 ⟶ prim1 x2 (1216a.. x0 x1)
Known 492ff.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 (1216a.. x0 x1) ⟶ ∀ x3 : ο . (prim1 x2 x0 ⟶ x1 x2 ⟶ x3) ⟶ x3
Theorem 801aa.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) x1 (λ x6 . x3 x6 x2) (proof)
Param 3097a.. : ι → (ι → ι) → ι
Definition b5c9f.. := λ x0 x1 . 3097a.. x1 (λ x2 . x0)
Param f482f.. : ι → ι → ι
Definition 62ee1.. := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . and (and (explicit_OrderedField x0 x1 x2 x3 x4 x5) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (prim1 x9 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8)) (∀ x6 . prim1 x6 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ ∀ x7 . prim1 x7 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ (∀ x8 . prim1 x8 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (and (x5 (f482f.. x6 x8) (f482f.. x7 x8)) (x5 (f482f.. x6 x8) (f482f.. x6 (x3 x8 x2)))) (x5 (f482f.. x7 (x3 x8 x2)) (f482f.. x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (prim1 x9 x0) (∀ x10 . prim1 x10 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (x5 (f482f.. x6 x10) x9) (x5 x9 (f482f.. x7 x10))) ⟶ x8) ⟶ x8)
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem 1f659.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (prim1 x9 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8) ⟶ (∀ x6 . prim1 x6 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ ∀ x7 . prim1 x7 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ (∀ x8 . prim1 x8 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (and (x5 (f482f.. x6 x8) (f482f.. x7 x8)) (x5 (f482f.. x6 x8) (f482f.. x6 (x3 x8 x2)))) (x5 (f482f.. x7 (x3 x8 x2)) (f482f.. x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (prim1 x9 x0) (∀ x10 . prim1 x10 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (x5 (f482f.. x6 x10) x9) (x5 x9 (f482f.. x7 x10))) ⟶ x8) ⟶ x8) ⟶ 62ee1.. x0 x1 x2 x3 x4 x5 (proof)
Known and3E : ∀ x0 x1 x2 : ο . and (and x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3) ⟶ x3
Theorem f2fa8.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (62ee1.. x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∀ x9 : ο . (∀ x10 . and (prim1 x10 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) (x5 x8 (x4 x10 x7)) ⟶ x9) ⟶ x9) ⟶ (∀ x7 . prim1 x7 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ ∀ x8 . prim1 x8 (b5c9f.. x0 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5))) ⟶ (∀ x9 . prim1 x9 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (and (x5 (f482f.. x7 x9) (f482f.. x8 x9)) (x5 (f482f.. x7 x9) (f482f.. x7 (x3 x9 x2)))) (x5 (f482f.. x8 (x3 x9 x2)) (f482f.. x8 x9))) ⟶ ∀ x9 : ο . (∀ x10 . and (prim1 x10 x0) (∀ x11 . prim1 x11 (1216a.. x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ and (x5 (f482f.. x7 x11) x10) (x5 x10 (f482f.. x8 x11))) ⟶ x9) ⟶ x9) ⟶ x6) ⟶ 62ee1.. x0 x1 x2 x3 x4 x5 ⟶ x6 (proof)
Theorem explicit_Field_transfer : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 x7 . ∀ x8 x9 : ι → ι → ι . ∀ x10 : ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ bij x0 x5 x10 ⟶ x10 x1 = x6 ⟶ x10 x2 = x7 ⟶ (∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x10 (x3 x11 x12) = x8 (x10 x11) (x10 x12)) ⟶ (∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x10 (x4 x11 x12) = x9 (x10 x11) (x10 x12)) ⟶ explicit_Field x5 x6 x7 x8 x9 (proof)
Known iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem explicit_OrderedField_transfer : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 x8 . ∀ x9 x10 : ι → ι → ι . ∀ x11 : ι → ι → ο . ∀ x12 : ι → ι . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ explicit_OrderedField x6 x7 x8 x9 x10 x11 (proof)
Param 0fc90.. : ι → (ι → ι) → ι
Known d0a1f.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 (1216a.. x0 x1) ⟶ prim1 x2 x0
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Known d8d74.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . prim1 x2 (3097a.. x0 x1) ⟶ prim1 x3 x0 ⟶ prim1 (f482f.. x2 x3) (x1 x3)
Known 27474.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) (x1 x3)) ⟶ prim1 (0fc90.. x0 x2) (3097a.. x0 x1)
Theorem 5ece9.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 x8 . ∀ x9 x10 : ι → ι → ι . ∀ x11 : ι → ι → ο . ∀ x12 : ι → ι . 62ee1.. x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ 62ee1.. x6 x7 x8 x9 x10 x11 (proof)
Definition 11fac.. := λ x0 . λ x1 x2 : ι → ι . λ x3 x4 x5 . λ x6 x7 : ι → ι → ι . and (and (and (and (and (and (and (and (explicit_Field x0 x3 x4 x6 x7) (∀ x8 : ο . (∀ x9 : ι → ι → ο . 62ee1.. (1216a.. x0 (λ x10 . x1 x10 = x10)) x3 x4 x6 x7 x9 ⟶ x8) ⟶ x8)) (∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) (1216a.. x0 (λ x9 . x1 x9 = x9)))) (prim1 x5 x0)) (∀ x8 . prim1 x8 x0 ⟶ prim1 (x1 x8) x0)) (∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0)) (∀ x8 . prim1 x8 x0 ⟶ x8 = x6 (x1 x8) (x7 x5 (x2 x8)))) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x1 x8 = x1 x9 ⟶ x2 x8 = x2 x9 ⟶ x8 = x9)) (x6 (x7 x5 x5) x4 = x3)
Theorem be4f2.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 . ∀ x6 x7 : ι → ι → ι . explicit_Field x0 x3 x4 x6 x7 ⟶ (∀ x8 : ο . (∀ x9 : ι → ι → ο . 62ee1.. (1216a.. x0 (λ x10 . x1 x10 = x10)) x3 x4 x6 x7 x9 ⟶ x8) ⟶ x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) (1216a.. x0 (λ x9 . x1 x9 = x9))) ⟶ prim1 x5 x0 ⟶ (∀ x8 . prim1 x8 x0 ⟶ prim1 (x1 x8) x0) ⟶ (∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0) ⟶ (∀ x8 . prim1 x8 x0 ⟶ x8 = x6 (x1 x8) (x7 x5 (x2 x8))) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x1 x8 = x1 x9 ⟶ x2 x8 = x2 x9 ⟶ x8 = x9) ⟶ x6 (x7 x5 x5) x4 = x3 ⟶ 11fac.. x0 x1 x2 x3 x4 x5 x6 x7 (proof)