Proofgold Address

Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition explicit_Ring := λ x0 x1 . λ x2 x3 : ι → ι → ι . and (and (and (and (and (and (and (and (and (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 (x2 x5 x6) = x2 (x2 x4 x5) x6)) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = x2 x5 x4)) (prim1 x1 x0)) (∀ x4 . prim1 x4 x0 ⟶ x2 x1 x4 = x4)) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (x2 x4 x6 = x1) ⟶ x5) ⟶ x5)) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0)) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x3 x5 x6) = x3 (x3 x4 x5) x6)) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x2 x5 x6) = x2 (x3 x4 x5) (x3 x4 x6))) (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 (x2 x4 x5) x6 = x2 (x3 x4 x6) (x3 x5 x6))
Known and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
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_Ring_I : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 (x2 x5 x6) = x2 (x2 x4 x5) x6) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = x2 x5 x4) ⟶ prim1 x1 x0 ⟶ (∀ x4 . prim1 x4 x0 ⟶ x2 x1 x4 = x4) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (x2 x4 x6 = x1) ⟶ x5) ⟶ x5) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x3 x5 x6) = x3 (x3 x4 x5) x6) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x2 x5 x6) = x2 (x3 x4 x5) (x3 x4 x6)) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 (x2 x4 x5) x6 = x2 (x3 x4 x6) (x3 x5 x6)) ⟶ explicit_Ring x0 x1 x2 x3 (proof)
Known and4E : ∀ x0 x1 x2 x3 : ο . and (and (and x0 x1) x2) x3 ⟶ ∀ x4 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4) ⟶ x4
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_Ring_E : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ο . (explicit_Ring x0 x1 x2 x3 ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = x2 x6 x5) ⟶ prim1 x1 x0 ⟶ (∀ x5 . prim1 x5 x0 ⟶ x2 x1 x5 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (x2 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ 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 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x2 x6 x7) = x2 (x3 x5 x6) (x3 x5 x7)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 (x2 x5 x6) x7 = x2 (x3 x5 x7) (x3 x6 x7)) ⟶ x4) ⟶ explicit_Ring x0 x1 x2 x3 ⟶ x4 (proof)
Definition explicit_Ring_minus := λ x0 x1 . λ x2 x3 : ι → ι → ι . λ x4 . prim0 (λ x5 . and (prim1 x5 x0) (x2 x4 x5 = x1))
Known Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Theorem explicit_Ring_minus_prop : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ and (prim1 (explicit_Ring_minus x0 x1 x2 x3 x4) x0) (x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1) (proof)
Theorem explicit_Ring_minus_clos : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (explicit_Ring_minus x0 x1 x2 x3 x4) x0 (proof)
Theorem explicit_Ring_minus_R : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1 (proof)
Theorem explicit_Ring_minus_L : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 (explicit_Ring_minus x0 x1 x2 x3 x4) x4 = x1 (proof)
Theorem explicit_Ring_plus_cancelL : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 x5 = x2 x4 x6 ⟶ x5 = x6 (proof)
Theorem explicit_Ring_plus_cancelR : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 x6 = x2 x5 x6 ⟶ x4 = x5 (proof)
Theorem explicit_Ring_minus_invol : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x0 ⟶ explicit_Ring_minus x0 x1 x2 x3 (explicit_Ring_minus x0 x1 x2 x3 x4) = x4 (proof)
Definition explicit_Ring_with_id := 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)) (prim1 x2 x0)) (x2 = x1 ⟶ ∀ x5 : ο . x5)) (∀ x5 . prim1 x5 x0 ⟶ x4 x2 x5 = x5)) (∀ x5 . prim1 x5 x0 ⟶ x4 x5 x2 = x5)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7))
Known andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_Ring_with_id_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) ⟶ prim1 x2 x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ x4 x5 x2 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7)) ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4 (proof)
Theorem explicit_Ring_with_id_E : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Ring_with_id 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) ⟶ prim1 x2 x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x4 x6 x2 = x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 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 (proof)
Theorem explicit_Ring_with_id_Ring : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring x0 x1 x3 x4 (proof)
Theorem explicit_Ring_with_id_minus_clos : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (explicit_Ring_minus x0 x1 x3 x4 x5) x0 (proof)
Theorem explicit_Ring_with_id_minus_R : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x5 (explicit_Ring_minus x0 x1 x3 x4 x5) = x1 (proof)
Theorem explicit_Ring_with_id_minus_L : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 (explicit_Ring_minus x0 x1 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_Ring_with_id_plus_cancelL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id 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_Ring_with_id_plus_cancelR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id 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_Ring_with_id_minus_invol : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 (explicit_Ring_minus x0 x1 x3 x4 x5) = x5 (proof)
Theorem explicit_Ring_with_id_minus_one_In : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ prim1 (explicit_Ring_minus x0 x1 x3 x4 x2) x0 (proof)
Theorem explicit_Ring_with_id_zero_multR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_Ring_with_id_zero_multL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_Ring_with_id_minus_mult : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 (explicit_Ring_minus x0 x1 x3 x4 x2) x5 (proof)
Theorem explicit_Ring_with_id_mult_minus : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 x5 (explicit_Ring_minus x0 x1 x3 x4 x2) (proof)
Theorem explicit_Ring_with_id_minus_one_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x2) (explicit_Ring_minus x0 x1 x3 x4 x2) = x2 (proof)
Theorem explicit_Ring_with_id_minus_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x5) (explicit_Ring_minus x0 x1 x3 x4 x5) = x4 x5 x5 (proof)
Param nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition explicit_Ring_with_id_exp_nat := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 . nat_primrec x2 (λ x6 . x4 x5)
Definition explicit_CRing_with_id := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . 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 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))
Theorem explicit_CRing_with_id_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 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 (proof)
Theorem explicit_CRing_with_id_E : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_CRing_with_id 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 ⟶ ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 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 (proof)
Theorem explicit_CRing_with_id_Ring_with_id : ∀ 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 (proof)
Theorem explicit_CRing_with_id_Ring : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring x0 x1 x3 x4 (proof)
Theorem explicit_CRing_with_id_minus_clos : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (explicit_Ring_minus x0 x1 x3 x4 x5) x0 (proof)
Theorem explicit_CRing_with_id_minus_R : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x5 (explicit_Ring_minus x0 x1 x3 x4 x5) = x1 (proof)
Theorem explicit_CRing_with_id_minus_L : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 (explicit_Ring_minus x0 x1 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_CRing_with_id_plus_cancelL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id 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_CRing_with_id_plus_cancelR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id 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_CRing_with_id_minus_invol : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 (explicit_Ring_minus x0 x1 x3 x4 x5) = x5 (proof)
Theorem explicit_CRing_with_id_minus_one_In : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ prim1 (explicit_Ring_minus x0 x1 x3 x4 x2) x0 (proof)
Theorem explicit_CRing_with_id_zero_multR : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_CRing_with_id_zero_multL : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_CRing_with_id_minus_mult : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 (explicit_Ring_minus x0 x1 x3 x4 x2) x5 (proof)
Theorem explicit_CRing_with_id_mult_minus : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 x5 (explicit_Ring_minus x0 x1 x3 x4 x2) (proof)
Theorem explicit_CRing_with_id_minus_one_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x2) (explicit_Ring_minus x0 x1 x3 x4 x2) = x2 (proof)
Theorem explicit_CRing_with_id_minus_square : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x5) (explicit_Ring_minus x0 x1 x3 x4 x5) = x4 x5 x5 (proof)
Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param eb53d.. : ι → CT2 ι
Definition e707a.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) x3)))
Param f482f.. : ι → ι → ι
Known dcdae.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef..
Known 53a20.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x0 = f482f.. (e707a.. x0 x1 x2 x3) 4a7ef..
Param e3162.. : ι → ι → ι → ι
Known edc55.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6
Known a698e.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (e707a.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5
Known 0a774.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6
Known 7bf04.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (e707a.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5
Known f3f77.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))
Known 54060.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x3 = f482f.. (e707a.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))
Known 63f04.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 . e707a.. x0 x2 x4 x6 = e707a.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (x6 = x7)
Known abad8.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x6 x7 = x4 x6 x7) ⟶ e707a.. x0 x1 x3 x5 = e707a.. x0 x2 x4 x5
Definition 06179.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 . prim1 x5 x2 ⟶ x1 (e707a.. x2 x3 x4 x5)) ⟶ x1 x0
Known 0cbd8.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ 06179.. (e707a.. x0 x1 x2 x3)
Known d484b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0
Known e4f10.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0
Known 02d3f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ prim1 x3 x0
Known b91ee.. : ∀ x0 . 06179.. x0 ⟶ x0 = e707a.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Definition 677e4.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Known cff9f.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 677e4.. (e707a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4
Definition 2f4b2.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Known ad938.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 2f4b2.. (e707a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4
Definition fa6a5.. := λ x0 . and (06179.. x0) (2f4b2.. x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 . explicit_Ring x1 x4 x2 x3))
Definition c77b5.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Known 90132.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef..
Known cb757.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x0 = f482f.. (c77b5.. x0 x1 x2 x3 x4) 4a7ef..
Known 1b277.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7
Known bdaba.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6
Known 45d05.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7
Known 74356.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6
Known 8307f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))
Known 73020.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x3 = f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))
Known 85b9b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Known e32d8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 = f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Known d0777.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 . c77b5.. x0 x2 x4 x6 x8 = c77b5.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11)) (x6 = x7)) (x8 = x9)
Known 836a9.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ c77b5.. x0 x1 x3 x5 x6 = c77b5.. x0 x2 x4 x5 x6
Definition 3f0d0.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (c77b5.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Known 9b39d.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 3f0d0.. (c77b5.. x0 x1 x2 x3 x4)
Known 38e2b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0
Known 2ca4b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0
Known f7980.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0
Known 619d5.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0
Known a296b.. : ∀ x0 . 3f0d0.. x0 ⟶ x0 = c77b5.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Definition 92512.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Known 242ff.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 92512.. (c77b5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition c3510.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Known 24f4f.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ c3510.. (c77b5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition d7e73.. := λ x0 . and (3f0d0.. x0) (c3510.. x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Ring_with_id x1 x4 x5 x2 x3))
Definition dac20.. := λ x0 . and (3f0d0.. x0) (c3510.. x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_CRing_with_id x1 x4 x5 x2 x3))
Definition 2f4c2.. := λ x0 x1 . 677e4.. x0 (λ x2 . λ x3 x4 : ι → ι → ι . λ x5 . explicit_Ring_minus x2 x5 x3 x4 x1)
Definition e33f8.. := λ x0 . 92512.. x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . e707a.. x1 x2 x3 x4)