Proofgold Address

Param explicit_Ring_with_id^{explicit_Ring} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known explicit_Ring_with_id_I^{explicit_Ring_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) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x5 x2 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7)) ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4
Known explicit_Ring_with_id_E^{explicit_Ring_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Ring_with_id 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) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x6 x2 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 (x3 x6 x7) x8 = x3 (x4 x6 x8) (x4 x7 x8)) ⟶ x5) ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ x5
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition explicit_Ring_with_id_exp_nat^{explicit_Ring_exp_nat} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 . nat_primrec x2 (λ x6 . x4 x5)
Param ap^ap : ι → ι → ι
Definition explicit_Ring_with_id_eval_poly^{explicit_Ring_eval_poly} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 x6 x7 . nat_primrec x1 (λ x8 . x3 (x4 (ap x6 x8) (explicit_Ring_with_id_exp_nat x0 x1 x2 x3 x4 x7 x8))) x5
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 745b1.. : ∀ 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_Ring_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring_with_id x0 x1 x2 x5 x6 (proof)
Param iff^iff : ο → ο → ο
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem explicit_Ring_with_id_repindep^{explicit_Ring_repindep} : ∀ 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) ⟶ iff (explicit_Ring_with_id x0 x1 x2 x3 x4) (explicit_Ring_with_id x0 x1 x2 x5 x6) (proof)
Param explicit_CRing_with_id^{explicit_CRing} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Known explicit_CRing_with_id_I^{explicit_CRing_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 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4
Known explicit_CRing_with_id_E^{explicit_CRing_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_CRing_with_id 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 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ x5
Theorem 7d9f7.. : ∀ 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_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_CRing_with_id x0 x1 x2 x5 x6 (proof)
Theorem explicit_CRing_with_id_repindep^{explicit_CRing_repindep} : ∀ 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) ⟶ iff (explicit_CRing_with_id x0 x1 x2 x3 x4) (explicit_CRing_with_id x0 x1 x2 x5 x6) (proof)
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
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
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
Theorem 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 (proof)
Theorem explicit_Field_repindep^{explicit_Field_repindep} : ∀ 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) ⟶ iff (explicit_Field x0 x1 x2 x3 x4) (explicit_Field x0 x1 x2 x5 x6) (proof)
Theorem 17e44.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 (proof)
Param pack_b_b_e_e^pack_b_b_e_e : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι
Definition struct_b_b_e_e^{struct_b_b_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 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x1 (pack_b_b_e_e x2 x3 x4 x5 x6)) ⟶ x1 x0
Param unpack_b_b_e_e_o^{unpack_b_b_e_e_o} : ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο) → ο
Known unpack_b_b_e_e_o_eq^{unpack_b_b_e_e_o_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_b_e_e_o (pack_b_b_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition Ring_with_id^Ring := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Ring_with_id x1 x4 x5 x2 x3))
Known prop_ext^prop_ext : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 = x1
Theorem Ring_with_id_unpack_eq^{Ring_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . unpack_b_b_e_e_o (pack_b_b_e_e x0 x1 x2 x3 x4) (λ x6 . λ x7 x8 : ι → ι → ι . λ x9 x10 . explicit_Ring_with_id x6 x9 x10 x7 x8) = explicit_Ring_with_id x0 x3 x4 x1 x2 (proof)
Definition CRing_with_id^CRing := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_CRing_with_id x1 x4 x5 x2 x3))
Theorem CRing_with_id_unpack_eq^{CRing_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . unpack_b_b_e_e_o (pack_b_b_e_e x0 x1 x2 x3 x4) (λ x6 . λ x7 x8 : ι → ι → ι . λ x9 x10 . explicit_CRing_with_id x6 x9 x10 x7 x8) = explicit_CRing_with_id x0 x3 x4 x1 x2 (proof)
Definition Field^Field := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Field x1 x4 x5 x2 x3))
Theorem Field_unpack_eq^{Field_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . unpack_b_b_e_e_o (pack_b_b_e_e x0 x1 x2 x3 x4) (λ x6 . λ x7 x8 : ι → ι → ι . λ x9 x10 . explicit_Field x6 x9 x10 x7 x8) = explicit_Field x0 x3 x4 x1 x2 (proof)
Known explicit_CRing_with_id_Ring_with_id^{explicit_CRing_Ring} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4
Theorem CRing_with_id_is_Ring_with_id^{CRing_is_Ring} : ∀ x0 . CRing_with_id x0 ⟶ Ring_with_id x0 (proof)
Theorem Field_is_CRing_with_id^{Field_is_CRing} : ∀ x0 . Field x0 ⟶ CRing_with_id x0 (proof)