Proofgold Asset

Param explicit_Nats^{explicit_Nats} : ι → ι → (ι → ι) → ο
Param explicit_Nats_primrec^{explicit_Nats_primrec} : ι → ι → (ι → ι) → ι → (ι → ι → ι) → ι → ι
Known explicit_Nats_E^{explicit_Nats_E} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ο . (explicit_Nats x0 x1 x2 ⟶ x1 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5) ⟶ x3) ⟶ explicit_Nats x0 x1 x2 ⟶ x3
Known explicit_Nats_ind^{explicit_Nats_ind} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4
Known explicit_Nats_primrec_base^{explicit_Nats_primrec_base} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . ∀ x4 : ι → ι → ι . explicit_Nats x0 x1 x2 ⟶ explicit_Nats_primrec x0 x1 x2 x3 x4 x1 = x3
Known explicit_Nats_primrec_S^{explicit_Nats_primrec_S} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . ∀ x4 : ι → ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Nats_primrec x0 x1 x2 x3 x4 (x2 x5) = x4 x5 (explicit_Nats_primrec x0 x1 x2 x3 x4 x5)
Theorem explicit_Nats_primrec_P^{explicit_Nats_primrec_P} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 : ι → ο . ∀ x4 . x3 x4 ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x3 x7 ⟶ x3 (x5 x6 x7)) ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 (explicit_Nats_primrec x0 x1 x2 x4 x5 x6) (proof)
Definition explicit_Nats_zero_plus^{explicit_Nats_zero_plus} := λ x0 x1 . λ x2 : ι → ι . λ x3 x4 . explicit_Nats_primrec x0 x1 x2 x4 (λ x5 . x2) x3
Definition explicit_Nats_zero_mult^{explicit_Nats_zero_mult} := λ x0 x1 . λ x2 : ι → ι . λ x3 x4 . explicit_Nats_primrec x0 x1 x2 x1 (λ x5 . explicit_Nats_zero_plus x0 x1 x2 x4) x3
Theorem explicit_Nats_zero_plus_N^{explicit_Nats_zero_plus_N} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_zero_plus x0 x1 x2 x3 x4 ∈ x0 (proof)
Theorem explicit_Nats_zero_mult_N^{explicit_Nats_zero_mult_N} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_zero_mult x0 x1 x2 x3 x4 ∈ x0 (proof)
Known explicit_Nats_zero_plus_0L^{explicit_Nats_zero_plus_0L} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Nats_zero_plus x0 x1 x2 x1 x3 = x3
Known explicit_Nats_zero_plus_SL^{explicit_Nats_zero_plus_SL} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_zero_plus x0 x1 x2 (x2 x3) x4 = x2 (explicit_Nats_zero_plus x0 x1 x2 x3 x4)
Known explicit_Nats_zero_mult_0L^{explicit_Nats_zero_mult_0L} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Nats_zero_mult x0 x1 x2 x1 x3 = x1
Known explicit_Nats_zero_mult_SL^{explicit_Nats_zero_mult_SL} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_zero_mult x0 x1 x2 (x2 x3) x4 = explicit_Nats_zero_plus x0 x1 x2 x4 (explicit_Nats_zero_mult x0 x1 x2 x3 x4)
Definition explicit_Nats_one_plus^{explicit_Nats_one_plus} := λ x0 x1 . λ x2 : ι → ι . λ x3 x4 . explicit_Nats_primrec x0 x1 x2 (x2 x4) (λ x5 . x2) x3
Theorem explicit_Nats_one_plus_N^{explicit_Nats_one_plus_N} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_plus x0 x1 x2 x3 x4 ∈ x0 (proof)
Definition explicit_Nats_one_mult^{explicit_Nats_one_mult} := λ x0 x1 . λ x2 : ι → ι . λ x3 x4 . explicit_Nats_primrec x0 x1 x2 x4 (λ x5 . explicit_Nats_one_plus x0 x1 x2 x4) x3
Theorem explicit_Nats_one_mult_N^{explicit_Nats_one_mult_N} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_mult x0 x1 x2 x3 x4 ∈ x0 (proof)
Definition explicit_Nats_one_exp^{explicit_Nats_one_exp} := λ x0 x1 . λ x2 : ι → ι . λ x3 . explicit_Nats_primrec x0 x1 x2 x3 (λ x4 . explicit_Nats_one_mult x0 x1 x2 x3)
Theorem explicit_Nats_one_exp_N^{explicit_Nats_one_exp_N} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_exp x0 x1 x2 x3 x4 ∈ x0 (proof)
Known explicit_Nats_one_plus_1L^{explicit_Nats_one_plus_1L} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Nats_one_plus x0 x1 x2 x1 x3 = x2 x3
Known explicit_Nats_one_plus_SL^{explicit_Nats_one_plus_SL} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_plus x0 x1 x2 (x2 x3) x4 = x2 (explicit_Nats_one_plus x0 x1 x2 x3 x4)
Known explicit_Nats_one_mult_1L^{explicit_Nats_one_mult_1L} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Nats_one_mult x0 x1 x2 x1 x3 = x3
Known explicit_Nats_one_mult_SL^{explicit_Nats_one_mult_SL} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_mult x0 x1 x2 (x2 x3) x4 = explicit_Nats_one_plus x0 x1 x2 x4 (explicit_Nats_one_mult x0 x1 x2 x3 x4)
Known explicit_Nats_one_exp_1L^{explicit_Nats_one_exp_1L} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Nats_one_exp x0 x1 x2 x3 x1 = x3
Known explicit_Nats_one_exp_SL^{explicit_Nats_one_exp_SL} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_exp x0 x1 x2 x3 (x2 x4) = explicit_Nats_one_mult x0 x1 x2 x3 (explicit_Nats_one_exp x0 x1 x2 x3 x4)
Theorem AssocComm_identities^{AssocComm_identities} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 (x1 x3 x4) = x1 (x1 x2 x3) x4) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = x1 x3 x2) ⟶ ∀ x2 : ο . ((∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5)) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x3 (x1 x4 x5) = x1 x5 (x1 x3 x4)) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 (x1 x3 x4) (x1 x5 x6) = x1 (x1 x3 x5) (x1 x4 x6)) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x3 (x1 x4 (x1 x5 x6)) = x1 x6 (x1 x3 (x1 x4 x5))) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x3 (x1 x4 (x1 x5 x6)) = x1 x5 (x1 x6 (x1 x3 x4))) ⟶ x2) ⟶ x2 (proof)
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known explicit_Field_E^{explicit_Field_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ (x6 = x1 ⟶ ∀ x7 : ο . x7) ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x4 x6 x8 = x2) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ x5
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known explicit_Field_zero_multL^{explicit_Field_zero_multL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x1 x5 = x1
Theorem explicit_Field_mult_zero_inv^{explicit_Field_mult_zero_inv} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x1 ⟶ or (x5 = x1) (x6 = x1) (proof)
Param explicit_OrderedField^{explicit_OrderedField} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Known explicit_OrderedField_E^{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 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 x8 x9 ⟶ x5 x7 x9) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (and (x5 x7 x8) (x5 x8 x7)) (x7 = x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ or (x5 x7 x8) (x5 x8 x7)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 (x3 x7 x9) (x3 x8 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x1 x7 ⟶ x5 x1 x8 ⟶ x5 x1 (x4 x7 x8)) ⟶ x6) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6
Theorem explicit_OrderedField_leq_refl^{explicit_OrderedField_leq_refl} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x5 x6 x6 (proof)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_OrderedField_leq_antisym^{explicit_OrderedField_leq_antisym} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x6 ⟶ x6 = x7 (proof)
Theorem explicit_OrderedField_leq_tra^{explicit_OrderedField_leq_tra} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8 (proof)
Known explicit_OrderedField_square_nonneg^{explicit_OrderedField_square_nonneg} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x5 x1 (x4 x6 x6)
Theorem explicit_OrderedField_leq_zero_one^{explicit_OrderedField_leq_zero_one} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x5 x1 x2 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Definition natOfOrderedField_p^{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
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known explicit_Nats_I^{explicit_Nats_I} : ∀ x0 x1 . ∀ x2 : ι → ι . x1 ∈ x0 ⟶ (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ (∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4) ⟶ explicit_Nats x0 x1 x2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known explicit_Field_plus_cancelR^{explicit_Field_plus_cancelR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known explicit_Nats_natOfOrderedField^{explicit_Nats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) x1 (λ x6 . x3 x6 x2)
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem explicit_PosNats_natOfOrderedField^{explicit_PosNats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats {x6 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x6 = x1 ⟶ ∀ x7 : ο . x7} x2 (λ x6 . x3 x6 x2) (proof)
Param explicit_Field_minus^{explicit_Field_minus} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι
Definition explicit_OrderedField_rationalp^{explicit_OrderedField_rationalp} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 . ∀ x7 : ο . (∀ x8 . and (x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) (∀ x9 : ο . (∀ x10 . and (x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12}) (x4 x10 x6 = x8) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7
Known explicit_Field_dist_R^{explicit_Field_dist_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7)
Theorem explicit_OrderedField_Npos_props^{explicit_OrderedField_Npos_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . ({x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⊆ x0 ⟶ explicit_Nats {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} x2 (λ x7 . x3 x7 x2) ⟶ x2 ∈ {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ x3 x7 x2 = x2 ⟶ ∀ x8 : ο . x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 : ι → ο . x8 x2 ⟶ (∀ x9 . x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} ⟶ x8 (x3 x9 x2)) ⟶ x8 x7) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ explicit_Nats_one_plus {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} x2 (λ x10 . x3 x10 x2) x7 x8 = x3 x7 x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ explicit_Nats_one_mult {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} x2 (λ x10 . x3 x10 x2) x7 x8 = x4 x7 x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x3 x7 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x4 x7 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) ⟶ x6) ⟶ x6 (proof)
Known explicit_Field_minus_invol^{explicit_Field_minus_invol} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x5
Known explicit_Field_minus_mult_R^{explicit_Field_minus_mult_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x6) = explicit_Field_minus x0 x1 x2 x3 x4 (x4 x5 x6)
Known explicit_Field_zero_multR^{explicit_Field_zero_multR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5 x1 = x1
Known explicit_Field_minus_plus_dist^{explicit_Field_minus_plus_dist} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (x3 x5 x6) = x3 (explicit_Field_minus x0 x1 x2 x3 x4 x5) (explicit_Field_minus x0 x1 x2 x3 x4 x6)
Known explicit_Field_minus_clos^{explicit_Field_minus_clos} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x5 ∈ x0
Known explicit_Field_minus_L^{explicit_Field_minus_L} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x5 = x1
Known explicit_Field_minus_zero^{explicit_Field_minus_zero} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x1 = x1
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Theorem explicit_OrderedField_Z_props^{explicit_OrderedField_Z_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . ((∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})}) ⟶ x1 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⊆ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⊆ x0 ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 : ο . (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x8) ⟶ (x7 = x1 ⟶ x8) ⟶ (x7 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x8) ⟶ x8) ⟶ x2 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x2 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ x3 x7 x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ x4 x7 x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) ⟶ x6) ⟶ x6 (proof)
Theorem explicit_OrderedField_Q_props^{explicit_OrderedField_Q_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . (Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) ⊆ x0 ⟶ (∀ x7 . x7 ∈ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) ⟶ ∀ x8 : ο . (x7 ∈ x0 ⟶ ∀ x9 . x9 ∈ {x10 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12}) (x10 = x1)) (x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12})} ⟶ ∀ x10 . x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12} ⟶ x4 x10 x7 = x9 ⟶ x8) ⟶ x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ ∀ x9 . x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} ⟶ x4 x9 x7 = x8 ⟶ x7 ∈ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5)) ⟶ x6) ⟶ x6 (proof)
Known explicit_Field_I^{explicit_Field_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_Field x0 x1 x2 x3 x4
Known explicit_Field_minus_R^{explicit_Field_minus_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1
Known explicit_Field_minus_mult_L^{explicit_Field_minus_mult_L} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x6 = explicit_Field_minus x0 x1 x2 x3 x4 (x4 x5 x6)
Theorem explicit_OrderedField_explicit_Field_Q^{explicit_OrderedField_explicit_Field_Q} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Field (Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5)) x1 x2 x3 x4 (proof)
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param omega^omega : ι
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition lt^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)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ap^ap : ι → ι → ι
Known explicit_Reals_E^{explicit_Reals_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x8 (x4 x10 x7)) ⟶ x9) ⟶ x9) ⟶ (∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x8 . x8 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap x8 (x3 x9 x2)) (ap x8 x9))) ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ x0) (∀ x11 . x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x7 x11) x10) (x5 x10 (ap x8 x11))) ⟶ x9) ⟶ x9) ⟶ x6) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ x6
Param nat_p^nat_p : ι → ο
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Param ordsucc^ordsucc : ι → ι
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Known nat_primrec_S^{nat_primrec_S} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . nat_p x2 ⟶ nat_primrec x0 x1 (ordsucc x2) = x1 x2 (nat_primrec x0 x1 x2)
Theorem explicit_Reals_characteristic_0^{explicit_Reals_characteristic_0} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ omega ⟶ nat_primrec x2 (λ x8 . x3 x2) x6 = x1 ⟶ ∀ x7 : ο . x7 (proof)
Known explicit_OrderedField_I^{explicit_OrderedField_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7)) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5
Theorem explicit_Reals_sub^{explicit_Reals_sub} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ⊆ x0 ⟶ x1 ∈ x6 ⟶ x2 ∈ x6 ⟶ (∀ x7 . x7 ∈ x6 ⟶ ∀ x8 . x8 ∈ x6 ⟶ x3 x7 x8 ∈ x6) ⟶ (∀ x7 . x7 ∈ x6 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ x6) ⟶ (∀ x7 . x7 ∈ x6 ⟶ ∀ x8 . x8 ∈ x6 ⟶ x4 x7 x8 ∈ x6) ⟶ (∀ x7 . x7 ∈ x6 ⟶ (x7 = x1 ⟶ ∀ x8 : ο . x8) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x6) (x4 x7 x9 = x2) ⟶ x8) ⟶ x8) ⟶ explicit_OrderedField x6 x1 x2 x3 x4 x5 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known explicit_Field_plus_cancelL^{explicit_Field_plus_cancelL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7
Theorem explicit_Reals_Q_min_props^{explicit_Reals_Q_min_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6 ⊆ x0 ⟶ explicit_Field x6 x1 x2 x3 x4 ⟶ ∀ x7 : ο . ((∀ x8 . x8 ∈ x6 ⟶ explicit_Field_minus x6 x1 x2 x3 x4 x8 = explicit_Field_minus x0 x1 x2 x3 x4 x8) ⟶ (∀ x8 . x8 ∈ x6 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ x6) ⟶ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) = Sep x6 (natOfOrderedField_p x6 x1 x2 x3 x4 x5) ⟶ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} = {x9 ∈ Sep x6 (natOfOrderedField_p x6 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} = {x9 ∈ x6|or (or (explicit_Field_minus x6 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x6 (natOfOrderedField_p x6 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x6 (natOfOrderedField_p x6 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) = Sep x6 (explicit_OrderedField_rationalp x6 x1 x2 x3 x4 x5) ⟶ x7) ⟶ x7 (proof)
Theorem explicit_Reals_Q_min^{explicit_Reals_Q_min} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ⊆ x0 ⟶ explicit_Field x6 x1 x2 x3 x4 ⟶ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) ⊆ x6 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param encode_b^encode_b : ι → CT2 ι
Param encode_r^encode_r : ι → (ι → ι → ο) → ι
Definition pack_b_b_r_e_e^{pack_b_b_r_e_e} := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι → ο . λ x4 x5 . lam 6 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) (encode_b x0 x1) (If_i (x6 = 2) (encode_b x0 x2) (If_i (x6 = 3) (encode_r x0 x3) (If_i (x6 = 4) x4 x5)))))
Known tuple_6_0_eq^tuple_6_0_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 0 = x0
Theorem pack_b_b_r_e_e_0_eq^{pack_b_b_r_e_e_0_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ x1 = ap x0 0 (proof)
Theorem pack_b_b_r_e_e_0_eq2^{pack_b_b_r_e_e_0_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x0 = ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 0 (proof)
Param decode_b^decode_b : ι → ι → ι → ι
Known tuple_6_1_eq^tuple_6_1_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 1 = x1
Known decode_encode_b^{decode_encode_b} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_b (encode_b x0 x1) x2 x3 = x1 x2 x3
Theorem pack_b_b_r_e_e_1_eq^{pack_b_b_r_e_e_1_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ ∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = decode_b (ap x0 1) x7 x8 (proof)
Theorem pack_b_b_r_e_e_1_eq2^{pack_b_b_r_e_e_1_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x6 x7 = decode_b (ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 1) x6 x7 (proof)
Known tuple_6_2_eq^tuple_6_2_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 2 = x2
Theorem pack_b_b_r_e_e_2_eq^{pack_b_b_r_e_e_2_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ ∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x3 x7 x8 = decode_b (ap x0 2) x7 x8 (proof)
Theorem pack_b_b_r_e_e_2_eq2^{pack_b_b_r_e_e_2_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 = decode_b (ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 2) x6 x7 (proof)
Param decode_r^decode_r : ι → ι → ι → ο
Known tuple_6_3_eq^tuple_6_3_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 3 = x3
Known decode_encode_r^{decode_encode_r} : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_r (encode_r x0 x1) x2 x3 = x1 x2 x3
Theorem pack_b_b_r_e_e_3_eq^{pack_b_b_r_e_e_3_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ ∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x4 x7 x8 = decode_r (ap x0 3) x7 x8 (proof)
Theorem pack_b_b_r_e_e_3_eq2^{pack_b_b_r_e_e_3_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = decode_r (ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 3) x6 x7 (proof)
Known tuple_6_4_eq^tuple_6_4_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 4 = x4
Theorem pack_b_b_r_e_e_4_eq^{pack_b_b_r_e_e_4_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ x5 = ap x0 4 (proof)
Theorem pack_b_b_r_e_e_4_eq2^{pack_b_b_r_e_e_4_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x4 = ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 4 (proof)
Known tuple_6_5_eq^tuple_6_5_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 5 = x5
Theorem pack_b_b_r_e_e_5_eq^{pack_b_b_r_e_e_5_eq} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . x0 = pack_b_b_r_e_e x1 x2 x3 x4 x5 x6 ⟶ x6 = ap x0 5 (proof)
Theorem pack_b_b_r_e_e_5_eq2^{pack_b_b_r_e_e_5_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x5 = ap (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) 5 (proof)
Known and6I^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 pack_b_b_r_e_e_inj^{pack_b_b_r_e_e_inj} : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 x10 x11 . pack_b_b_r_e_e x0 x2 x4 x6 x8 x10 = pack_b_b_r_e_e x1 x3 x5 x7 x9 x11 ⟶ and (and (and (and (and (x0 = x1) (∀ x12 . x12 ∈ x0 ⟶ ∀ x13 . x13 ∈ x0 ⟶ x2 x12 x13 = x3 x12 x13)) (∀ x12 . x12 ∈ x0 ⟶ ∀ x13 . x13 ∈ x0 ⟶ x4 x12 x13 = x5 x12 x13)) (∀ x12 . x12 ∈ x0 ⟶ ∀ x13 . x13 ∈ x0 ⟶ x6 x12 x13 = x7 x12 x13)) (x8 = x9)) (x10 = x11) (proof)
Known encode_r_ext^encode_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ encode_r x0 x1 = encode_r x0 x2
Known encode_b_ext^encode_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ encode_b x0 x1 = encode_b x0 x2
Theorem pack_b_b_r_e_e_ext^{pack_b_b_r_e_e_ext} : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 x8 . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x1 x9 x10 = x2 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x4 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ pack_b_b_r_e_e x0 x1 x3 x5 x7 x8 = pack_b_b_r_e_e x0 x2 x4 x6 x7 x8 (proof)
Definition struct_b_b_r_e_e^{struct_b_b_r_e_e} := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x4 x5 x6 ∈ x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 . x6 ∈ x2 ⟶ ∀ x7 . x7 ∈ x2 ⟶ x1 (pack_b_b_r_e_e x2 x3 x4 x5 x6 x7)) ⟶ x1 x0
Theorem pack_struct_b_b_r_e_e_I^{pack_struct_b_b_r_e_e_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) (proof)
Theorem pack_struct_b_b_r_e_e_E1^{pack_struct_b_b_r_e_e_E1} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x6 x7 ∈ x0 (proof)
Theorem pack_struct_b_b_r_e_e_E2^{pack_struct_b_b_r_e_e_E2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ∈ x0 (proof)
Theorem pack_struct_b_b_r_e_e_E4^{pack_struct_b_b_r_e_e_E4} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ x4 ∈ x0 (proof)
Theorem pack_struct_b_b_r_e_e_E5^{pack_struct_b_b_r_e_e_E5} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ x5 ∈ x0 (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_b_b_r_e_e_eta^{struct_b_b_r_e_e_eta} : ∀ x0 . struct_b_b_r_e_e x0 ⟶ x0 = pack_b_b_r_e_e (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4) (ap x0 5) (proof)
Definition unpack_b_b_r_e_e_i^{unpack_b_b_r_e_e_i} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ι . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4) (ap x0 5)
Theorem unpack_b_b_r_e_e_i_eq^{unpack_b_b_r_e_e_i_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . (∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x2 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ x3 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . x10 ∈ x1 ⟶ ∀ x11 . x11 ∈ x1 ⟶ iff (x4 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x7 x8 x9 x5 x6 = x0 x1 x2 x3 x4 x5 x6) ⟶ unpack_b_b_r_e_e_i (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) x0 = x0 x1 x2 x3 x4 x5 x6 (proof)
Definition unpack_b_b_r_e_e_o^{unpack_b_b_r_e_e_o} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4) (ap x0 5)
Theorem unpack_b_b_r_e_e_o_eq^{unpack_b_b_r_e_e_o_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 x6 . (∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x2 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ x3 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . x10 ∈ x1 ⟶ ∀ x11 . x11 ∈ x1 ⟶ iff (x4 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x7 x8 x9 x5 x6 = x0 x1 x2 x3 x4 x5 x6) ⟶ unpack_b_b_r_e_e_o (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) x0 = x0 x1 x2 x3 x4 x5 x6 (proof)
Definition OrderedFieldStruct^{struct_b_b_r_e_e_ordered_field} := λ x0 . and (struct_b_b_r_e_e x0) (unpack_b_b_r_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 : ι → ι → ο . λ x5 x6 . explicit_OrderedField x1 x5 x6 x2 x3 x4))
Known f16ac.. : ∀ x0 x1 x2 . ∀ x3 x4 x5 x6 : ι → ι → ι . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x5 x7 x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 = x6 x7 x8) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ explicit_Field x0 x1 x2 x5 x6
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem a7a4d.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 : ι → ι → ι . ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10)) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x6 x7 x8 (proof)
Known iff_sym^iff_sym : ∀ x0 x1 : ο . iff x0 x1 ⟶ iff x1 x0
Theorem explicit_OrderedField_repindep^{explicit_OrderedField_repindep} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 : ι → ι → ι . ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10)) ⟶ iff (explicit_OrderedField x0 x1 x2 x3 x4 x5) (explicit_OrderedField x0 x1 x2 x6 x7 x8) (proof)
Known prop_ext^prop_ext : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 = x1
Theorem OrderedFieldStruct_unpack_eq^{OrderedFieldStruct_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . unpack_b_b_r_e_e_o (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) (λ x7 . λ x8 x9 : ι → ι → ι . λ x10 : ι → ι → ο . λ x11 x12 . explicit_OrderedField x7 x11 x12 x8 x9 x10) = explicit_OrderedField x0 x4 x5 x1 x2 x3 (proof)
Definition RealsStruct^RealsStruct := λ x0 . and (struct_b_b_r_e_e x0) (unpack_b_b_r_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 : ι → ι → ο . λ x5 x6 . explicit_Reals x1 x5 x6 x2 x3 x4))
Known explicit_Reals_I^{explicit_Reals_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8) ⟶ (∀ x6 . x6 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x8 . x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) (ap x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x10) x9) (x5 x9 (ap x7 x10))) ⟶ x8) ⟶ x8) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Theorem 32ac0.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 : ι → ι → ι . ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10)) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_Reals x0 x1 x2 x6 x7 x8 (proof)
Theorem explicit_Reals_repindep^{explicit_Reals_repindep} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 : ι → ι → ι . ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10)) ⟶ iff (explicit_Reals x0 x1 x2 x3 x4 x5) (explicit_Reals x0 x1 x2 x6 x7 x8) (proof)
Theorem RealsStruct_unpack_eq^{RealsStruct_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . unpack_b_b_r_e_e_o (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) (λ x7 . λ x8 x9 : ι → ι → ι . λ x10 : ι → ι → ο . λ x11 x12 . explicit_Reals x7 x11 x12 x8 x9 x10) = explicit_Reals x0 x4 x5 x1 x2 x3 (proof)