% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_608__5373964_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:30.801

% Could-be-implicit typings (6)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    poly_poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_a : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (38)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    one_on1411366565ly_nat : poly_poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    one_one_poly_poly_a : poly_poly_a).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J, type,
    plus_plus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    power_power_poly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    power_1336127338ly_nat : poly_poly_nat > nat > poly_poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    power_276493840poly_a : poly_poly_a > nat > poly_poly_a).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_Itf__a_J, type,
    power_power_poly_a : poly_a > nat > poly_a).
thf(sy_c_Power_Opower__class_Opower_001tf__a, type,
    power_power_a : a > nat > a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (245)
thf(fact_0_nc, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ p)))))). % nc
thf(fact_1_poly__offset, axiom,
    ((![P : poly_a, A : a]: (?[Q : poly_a]: (((fundam247907092size_a @ Q) = (fundam247907092size_a @ P)) & (![X : a]: ((poly_a2 @ Q @ X) = (poly_a2 @ P @ (plus_plus_a @ A @ X))))))))). % poly_offset
thf(fact_2_poly__pCons, axiom,
    ((![A : a, P : poly_a, X2 : a]: ((poly_a2 @ (pCons_a @ A @ P) @ X2) = (plus_plus_a @ A @ (times_times_a @ X2 @ (poly_a2 @ P @ X2))))))). % poly_pCons
thf(fact_3_poly__pCons, axiom,
    ((![A : nat, P : poly_nat, X2 : nat]: ((poly_nat2 @ (pCons_nat @ A @ P) @ X2) = (plus_plus_nat @ A @ (times_times_nat @ X2 @ (poly_nat2 @ P @ X2))))))). % poly_pCons
thf(fact_4_mult__cancel__left1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_left1
thf(fact_5_mult__cancel__left1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_left1
thf(fact_6_mult__cancel__left2, axiom,
    ((![C : poly_a, A : poly_a]: (((times_times_poly_a @ C @ A) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_left2
thf(fact_7_mult__cancel__left2, axiom,
    ((![C : a, A : a]: (((times_times_a @ C @ A) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_left2
thf(fact_8_mult__cancel__right1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_right1
thf(fact_9_mult__cancel__right1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_right1
thf(fact_10_mult__cancel__right2, axiom,
    ((![A : poly_a, C : poly_a]: (((times_times_poly_a @ A @ C) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_right2
thf(fact_11_mult__cancel__right2, axiom,
    ((![A : a, C : a]: (((times_times_a @ A @ C) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_right2
thf(fact_12_poly__power, axiom,
    ((![P : poly_poly_nat, N : nat, X2 : poly_nat]: ((poly_poly_nat2 @ (power_1336127338ly_nat @ P @ N) @ X2) = (power_power_poly_nat @ (poly_poly_nat2 @ P @ X2) @ N))))). % poly_power
thf(fact_13_poly__power, axiom,
    ((![P : poly_poly_a, N : nat, X2 : poly_a]: ((poly_poly_a2 @ (power_276493840poly_a @ P @ N) @ X2) = (power_power_poly_a @ (poly_poly_a2 @ P @ X2) @ N))))). % poly_power
thf(fact_14_poly__power, axiom,
    ((![P : poly_a, N : nat, X2 : a]: ((poly_a2 @ (power_power_poly_a @ P @ N) @ X2) = (power_power_a @ (poly_a2 @ P @ X2) @ N))))). % poly_power
thf(fact_15_poly__power, axiom,
    ((![P : poly_nat, N : nat, X2 : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X2) = (power_power_nat @ (poly_nat2 @ P @ X2) @ N))))). % poly_power
thf(fact_16_poly__1, axiom,
    ((![X2 : poly_nat]: ((poly_poly_nat2 @ one_on1411366565ly_nat @ X2) = one_one_poly_nat)))). % poly_1
thf(fact_17_poly__1, axiom,
    ((![X2 : poly_a]: ((poly_poly_a2 @ one_one_poly_poly_a @ X2) = one_one_poly_a)))). % poly_1
thf(fact_18_poly__1, axiom,
    ((![X2 : a]: ((poly_a2 @ one_one_poly_a @ X2) = one_one_a)))). % poly_1
thf(fact_19_poly__1, axiom,
    ((![X2 : nat]: ((poly_nat2 @ one_one_poly_nat @ X2) = one_one_nat)))). % poly_1
thf(fact_20_poly__add, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X2 : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q2) @ X2) = (plus_plus_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q2 @ X2)))))). % poly_add
thf(fact_21_poly__add, axiom,
    ((![P : poly_a, Q2 : poly_a, X2 : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q2) @ X2) = (plus_plus_a @ (poly_a2 @ P @ X2) @ (poly_a2 @ Q2 @ X2)))))). % poly_add
thf(fact_22_poly__mult, axiom,
    ((![P : poly_a, Q2 : poly_a, X2 : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q2) @ X2) = (times_times_a @ (poly_a2 @ P @ X2) @ (poly_a2 @ Q2 @ X2)))))). % poly_mult
thf(fact_23_poly__mult, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X2 : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q2) @ X2) = (times_times_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q2 @ X2)))))). % poly_mult
thf(fact_24_poly__0, axiom,
    ((![X2 : a]: ((poly_a2 @ zero_zero_poly_a @ X2) = zero_zero_a)))). % poly_0
thf(fact_25_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_26_add__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q2 : poly_nat]: ((plus_plus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q2)) = (pCons_nat @ (plus_plus_nat @ A @ B) @ (plus_plus_poly_nat @ P @ Q2)))))). % add_pCons
thf(fact_27_add__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q2 : poly_a]: ((plus_plus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q2)) = (pCons_a @ (plus_plus_a @ A @ B) @ (plus_plus_poly_a @ P @ Q2)))))). % add_pCons
thf(fact_28_pCons__eq__iff, axiom,
    ((![A : a, P : poly_a, B : a, Q2 : poly_a]: (((pCons_a @ A @ P) = (pCons_a @ B @ Q2)) = (((A = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_29_pCons__eq__iff, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q2 : poly_nat]: (((pCons_nat @ A @ P) = (pCons_nat @ B @ Q2)) = (((A = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_30_mult__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_31_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_32_mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_33_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_34_mult__eq__0__iff, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % mult_eq_0_iff
thf(fact_35_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_36_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_37_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_38_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_39_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_40_power__one, axiom,
    ((![N : nat]: ((power_power_a @ one_one_a @ N) = one_one_a)))). % power_one
thf(fact_41_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_42_power__one, axiom,
    ((![N : nat]: ((power_power_poly_nat @ one_one_poly_nat @ N) = one_one_poly_nat)))). % power_one
thf(fact_43_power__one, axiom,
    ((![N : nat]: ((power_power_poly_a @ one_one_poly_a @ N) = one_one_poly_a)))). % power_one
thf(fact_44_power__one__right, axiom,
    ((![A : a]: ((power_power_a @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_45_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_46_power__one__right, axiom,
    ((![A : poly_nat]: ((power_power_poly_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_47_power__one__right, axiom,
    ((![A : poly_a]: ((power_power_poly_a @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_48_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_49_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_50_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_51_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_52_psize__eq__0__iff, axiom,
    ((![P : poly_a]: (((fundam247907092size_a @ P) = zero_zero_nat) = (P = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_53_one__poly__eq__simps_I1_J, axiom,
    ((one_on1411366565ly_nat = (pCons_poly_nat @ one_one_poly_nat @ zero_z1059985641ly_nat)))). % one_poly_eq_simps(1)
thf(fact_54_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_poly_a = (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)))). % one_poly_eq_simps(1)
thf(fact_55_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_56_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_a = (pCons_a @ one_one_a @ zero_zero_poly_a)))). % one_poly_eq_simps(1)
thf(fact_57_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_58_one__poly__eq__simps_I2_J, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % one_poly_eq_simps(2)
thf(fact_59_one__poly__eq__simps_I2_J, axiom,
    (((pCons_poly_nat @ one_one_poly_nat @ zero_z1059985641ly_nat) = one_on1411366565ly_nat))). % one_poly_eq_simps(2)
thf(fact_60_one__poly__eq__simps_I2_J, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % one_poly_eq_simps(2)
thf(fact_61_power__mult, axiom,
    ((![A : a, M : nat, N : nat]: ((power_power_a @ A @ (times_times_nat @ M @ N)) = (power_power_a @ (power_power_a @ A @ M) @ N))))). % power_mult
thf(fact_62_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_63_power__mult, axiom,
    ((![A : poly_nat, M : nat, N : nat]: ((power_power_poly_nat @ A @ (times_times_nat @ M @ N)) = (power_power_poly_nat @ (power_power_poly_nat @ A @ M) @ N))))). % power_mult
thf(fact_64_power__mult, axiom,
    ((![A : poly_a, M : nat, N : nat]: ((power_power_poly_a @ A @ (times_times_nat @ M @ N)) = (power_power_poly_a @ (power_power_poly_a @ A @ M) @ N))))). % power_mult
thf(fact_65_pderiv_Oinduct, axiom,
    ((![P2 : poly_a > $o, A0 : poly_a]: ((![A2 : a, P3 : poly_a]: (((~ ((P3 = zero_zero_poly_a))) => (P2 @ P3)) => (P2 @ (pCons_a @ A2 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_66_pderiv_Oinduct, axiom,
    ((![P2 : poly_nat > $o, A0 : poly_nat]: ((![A2 : nat, P3 : poly_nat]: (((~ ((P3 = zero_zero_poly_nat))) => (P2 @ P3)) => (P2 @ (pCons_nat @ A2 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_67_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q2 : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a, B2 : a, Q : poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_68_poly__induct2, axiom,
    ((![P2 : poly_a > poly_nat > $o, P : poly_a, Q2 : poly_nat]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A2 : a, P3 : poly_a, B2 : nat, Q : poly_nat]: ((P2 @ P3 @ Q) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_nat @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_69_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_a > $o, P : poly_nat, Q2 : poly_a]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A2 : nat, P3 : poly_nat, B2 : a, Q : poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_70_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_nat > $o, P : poly_nat, Q2 : poly_nat]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat, B2 : nat, Q : poly_nat]: ((P2 @ P3 @ Q) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_nat @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_71_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_72_pCons__one, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % pCons_one
thf(fact_73_pCons__one, axiom,
    (((pCons_poly_nat @ one_one_poly_nat @ zero_z1059985641ly_nat) = one_on1411366565ly_nat))). % pCons_one
thf(fact_74_pCons__one, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % pCons_one
thf(fact_75_pderiv_Ocases, axiom,
    ((![X2 : poly_a]: (~ ((![A2 : a, P3 : poly_a]: (~ ((X2 = (pCons_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_76_pderiv_Ocases, axiom,
    ((![X2 : poly_nat]: (~ ((![A2 : nat, P3 : poly_nat]: (~ ((X2 = (pCons_nat @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_77_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q : poly_a]: (~ ((P = (pCons_a @ A2 @ Q)))))))))). % pCons_cases
thf(fact_78_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A2 : nat, Q : poly_nat]: (~ ((P = (pCons_nat @ A2 @ Q)))))))))). % pCons_cases
thf(fact_79_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X3 : a]: (![Y : a]: ((F @ X3) = (F @ Y)))))))). % constant_def
thf(fact_80_mult__right__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_81_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_82_mult__left__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_83_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_84_no__zero__divisors, axiom,
    ((![A : a, B : a]: ((~ ((A = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (~ (((times_times_a @ A @ B) = zero_zero_a)))))))). % no_zero_divisors
thf(fact_85_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_86_divisors__zero, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) => ((A = zero_zero_a) | (B = zero_zero_a)))))). % divisors_zero
thf(fact_87_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_88_mult__not__zero, axiom,
    ((![A : a, B : a]: ((~ (((times_times_a @ A @ B) = zero_zero_a))) => ((~ ((A = zero_zero_a))) & (~ ((B = zero_zero_a)))))))). % mult_not_zero
thf(fact_89_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_90_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_nat = one_one_poly_nat))))). % zero_neq_one
thf(fact_91_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_a = one_one_poly_a))))). % zero_neq_one
thf(fact_92_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_93_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_94_combine__common__factor, axiom,
    ((![A : a, E : a, B : a, C : a]: ((plus_plus_a @ (times_times_a @ A @ E) @ (plus_plus_a @ (times_times_a @ B @ E) @ C)) = (plus_plus_a @ (times_times_a @ (plus_plus_a @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_95_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_96_distrib__right, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % distrib_right
thf(fact_97_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_98_distrib__left, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A @ B) @ (times_times_a @ A @ C)))))). % distrib_left
thf(fact_99_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_100_comm__semiring__class_Odistrib, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_101_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_102_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A @ B) @ (times_times_a @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_103_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_104_power__not__zero, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => (~ (((power_power_a @ A @ N) = zero_zero_a))))))). % power_not_zero
thf(fact_105_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_106_power__not__zero, axiom,
    ((![A : poly_nat, N : nat]: ((~ ((A = zero_zero_poly_nat))) => (~ (((power_power_poly_nat @ A @ N) = zero_zero_poly_nat))))))). % power_not_zero
thf(fact_107_power__not__zero, axiom,
    ((![A : poly_a, N : nat]: ((~ ((A = zero_zero_poly_a))) => (~ (((power_power_poly_a @ A @ N) = zero_zero_poly_a))))))). % power_not_zero
thf(fact_108_power__commuting__commutes, axiom,
    ((![X2 : poly_nat, Y2 : poly_nat, N : nat]: (((times_times_poly_nat @ X2 @ Y2) = (times_times_poly_nat @ Y2 @ X2)) => ((times_times_poly_nat @ (power_power_poly_nat @ X2 @ N) @ Y2) = (times_times_poly_nat @ Y2 @ (power_power_poly_nat @ X2 @ N))))))). % power_commuting_commutes
thf(fact_109_power__commuting__commutes, axiom,
    ((![X2 : poly_a, Y2 : poly_a, N : nat]: (((times_times_poly_a @ X2 @ Y2) = (times_times_poly_a @ Y2 @ X2)) => ((times_times_poly_a @ (power_power_poly_a @ X2 @ N) @ Y2) = (times_times_poly_a @ Y2 @ (power_power_poly_a @ X2 @ N))))))). % power_commuting_commutes
thf(fact_110_power__commuting__commutes, axiom,
    ((![X2 : a, Y2 : a, N : nat]: (((times_times_a @ X2 @ Y2) = (times_times_a @ Y2 @ X2)) => ((times_times_a @ (power_power_a @ X2 @ N) @ Y2) = (times_times_a @ Y2 @ (power_power_a @ X2 @ N))))))). % power_commuting_commutes
thf(fact_111_power__commuting__commutes, axiom,
    ((![X2 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X2 @ Y2) = (times_times_nat @ Y2 @ X2)) => ((times_times_nat @ (power_power_nat @ X2 @ N) @ Y2) = (times_times_nat @ Y2 @ (power_power_nat @ X2 @ N))))))). % power_commuting_commutes
thf(fact_112_power__mult__distrib, axiom,
    ((![A : poly_nat, B : poly_nat, N : nat]: ((power_power_poly_nat @ (times_times_poly_nat @ A @ B) @ N) = (times_times_poly_nat @ (power_power_poly_nat @ A @ N) @ (power_power_poly_nat @ B @ N)))))). % power_mult_distrib
thf(fact_113_power__mult__distrib, axiom,
    ((![A : poly_a, B : poly_a, N : nat]: ((power_power_poly_a @ (times_times_poly_a @ A @ B) @ N) = (times_times_poly_a @ (power_power_poly_a @ A @ N) @ (power_power_poly_a @ B @ N)))))). % power_mult_distrib
thf(fact_114_power__mult__distrib, axiom,
    ((![A : a, B : a, N : nat]: ((power_power_a @ (times_times_a @ A @ B) @ N) = (times_times_a @ (power_power_a @ A @ N) @ (power_power_a @ B @ N)))))). % power_mult_distrib
thf(fact_115_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_116_power__commutes, axiom,
    ((![A : poly_nat, N : nat]: ((times_times_poly_nat @ (power_power_poly_nat @ A @ N) @ A) = (times_times_poly_nat @ A @ (power_power_poly_nat @ A @ N)))))). % power_commutes
thf(fact_117_power__commutes, axiom,
    ((![A : poly_a, N : nat]: ((times_times_poly_a @ (power_power_poly_a @ A @ N) @ A) = (times_times_poly_a @ A @ (power_power_poly_a @ A @ N)))))). % power_commutes
thf(fact_118_power__commutes, axiom,
    ((![A : a, N : nat]: ((times_times_a @ (power_power_a @ A @ N) @ A) = (times_times_a @ A @ (power_power_a @ A @ N)))))). % power_commutes
thf(fact_119_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_120_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_121_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_122_left__right__inverse__power, axiom,
    ((![X2 : poly_nat, Y2 : poly_nat, N : nat]: (((times_times_poly_nat @ X2 @ Y2) = one_one_poly_nat) => ((times_times_poly_nat @ (power_power_poly_nat @ X2 @ N) @ (power_power_poly_nat @ Y2 @ N)) = one_one_poly_nat))))). % left_right_inverse_power
thf(fact_123_left__right__inverse__power, axiom,
    ((![X2 : poly_a, Y2 : poly_a, N : nat]: (((times_times_poly_a @ X2 @ Y2) = one_one_poly_a) => ((times_times_poly_a @ (power_power_poly_a @ X2 @ N) @ (power_power_poly_a @ Y2 @ N)) = one_one_poly_a))))). % left_right_inverse_power
thf(fact_124_left__right__inverse__power, axiom,
    ((![X2 : a, Y2 : a, N : nat]: (((times_times_a @ X2 @ Y2) = one_one_a) => ((times_times_a @ (power_power_a @ X2 @ N) @ (power_power_a @ Y2 @ N)) = one_one_a))))). % left_right_inverse_power
thf(fact_125_left__right__inverse__power, axiom,
    ((![X2 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X2 @ Y2) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X2 @ N) @ (power_power_nat @ Y2 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_126_power__0, axiom,
    ((![A : a]: ((power_power_a @ A @ zero_zero_nat) = one_one_a)))). % power_0
thf(fact_127_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_128_power__0, axiom,
    ((![A : poly_nat]: ((power_power_poly_nat @ A @ zero_zero_nat) = one_one_poly_nat)))). % power_0
thf(fact_129_power__0, axiom,
    ((![A : poly_a]: ((power_power_poly_a @ A @ zero_zero_nat) = one_one_poly_a)))). % power_0
thf(fact_130_power__add, axiom,
    ((![A : poly_nat, M : nat, N : nat]: ((power_power_poly_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_poly_nat @ (power_power_poly_nat @ A @ M) @ (power_power_poly_nat @ A @ N)))))). % power_add
thf(fact_131_power__add, axiom,
    ((![A : poly_a, M : nat, N : nat]: ((power_power_poly_a @ A @ (plus_plus_nat @ M @ N)) = (times_times_poly_a @ (power_power_poly_a @ A @ M) @ (power_power_poly_a @ A @ N)))))). % power_add
thf(fact_132_power__add, axiom,
    ((![A : a, M : nat, N : nat]: ((power_power_a @ A @ (plus_plus_nat @ M @ N)) = (times_times_a @ (power_power_a @ A @ M) @ (power_power_a @ A @ N)))))). % power_add
thf(fact_133_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_134_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_a @ zero_zero_a @ N) = one_one_a)) & ((~ ((N = zero_zero_nat))) => ((power_power_a @ zero_zero_a @ N) = zero_zero_a)))))). % power_0_left
thf(fact_135_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_136_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_poly_nat @ zero_zero_poly_nat @ N) = one_one_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_poly_nat @ zero_zero_poly_nat @ N) = zero_zero_poly_nat)))))). % power_0_left
thf(fact_137_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_poly_a @ zero_zero_poly_a @ N) = one_one_poly_a)) & ((~ ((N = zero_zero_nat))) => ((power_power_poly_a @ zero_zero_poly_a @ N) = zero_zero_poly_a)))))). % power_0_left
thf(fact_138_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_139_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_140_mult_Oright__neutral, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ A @ one_one_poly_nat) = A)))). % mult.right_neutral
thf(fact_141_mult_Oright__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ one_one_poly_a) = A)))). % mult.right_neutral
thf(fact_142_mult_Oright__neutral, axiom,
    ((![A : a]: ((times_times_a @ A @ one_one_a) = A)))). % mult.right_neutral
thf(fact_143_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_144_mult_Oleft__neutral, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ one_one_poly_nat @ A) = A)))). % mult.left_neutral
thf(fact_145_mult_Oleft__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ one_one_poly_a @ A) = A)))). % mult.left_neutral
thf(fact_146_mult_Oleft__neutral, axiom,
    ((![A : a]: ((times_times_a @ one_one_a @ A) = A)))). % mult.left_neutral
thf(fact_147_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_148_zero__eq__add__iff__both__eq__0, axiom,
    ((![X2 : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X2 @ Y2)) = (((X2 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_149_add__eq__0__iff__both__eq__0, axiom,
    ((![X2 : nat, Y2 : nat]: (((plus_plus_nat @ X2 @ Y2) = zero_zero_nat) = (((X2 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_150_add__cancel__right__right, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ A @ B)) = (B = zero_zero_a))))). % add_cancel_right_right
thf(fact_151_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_152_add__cancel__right__left, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ B @ A)) = (B = zero_zero_a))))). % add_cancel_right_left
thf(fact_153_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_154_add__cancel__left__right, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = A) = (B = zero_zero_a))))). % add_cancel_left_right
thf(fact_155_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_156_add__cancel__left__left, axiom,
    ((![B : a, A : a]: (((plus_plus_a @ B @ A) = A) = (B = zero_zero_a))))). % add_cancel_left_left
thf(fact_157_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_158_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_159_add__left__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_160_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_161_add__right__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_162_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_163_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_164_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_165_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_166_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_167_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_168_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_169_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_170_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_171_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_172_zero__reorient, axiom,
    ((![X2 : a]: ((zero_zero_a = X2) = (X2 = zero_zero_a))))). % zero_reorient
thf(fact_173_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_174_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A @ B) @ C) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_175_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_176_mult_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A @ B) @ C) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % mult.assoc
thf(fact_177_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_178_mult_Ocommute, axiom,
    ((times_times_a = (^[A3 : a]: (^[B3 : a]: (times_times_a @ B3 @ A3)))))). % mult.commute
thf(fact_179_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_180_mult_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((times_times_a @ B @ (times_times_a @ A @ C)) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % mult.left_commute
thf(fact_181_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_182_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_183_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_184_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_185_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_186_group__cancel_Oadd1, axiom,
    ((![A4 : a, K : a, A : a, B : a]: ((A4 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A4 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_187_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_188_group__cancel_Oadd2, axiom,
    ((![B4 : a, K : a, B : a, A : a]: ((B4 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B4) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_189_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_190_add_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_191_add_Oleft__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_192_add_Oright__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_193_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_194_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_195_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_196_add_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A @ C)) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_197_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_198_add__left__imp__eq, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_199_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_200_add__right__imp__eq, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_201_one__reorient, axiom,
    ((![X2 : nat]: ((one_one_nat = X2) = (X2 = one_one_nat))))). % one_reorient
thf(fact_202_one__reorient, axiom,
    ((![X2 : a]: ((one_one_a = X2) = (X2 = one_one_a))))). % one_reorient
thf(fact_203_one__reorient, axiom,
    ((![X2 : poly_nat]: ((one_one_poly_nat = X2) = (X2 = one_one_poly_nat))))). % one_reorient
thf(fact_204_one__reorient, axiom,
    ((![X2 : poly_a]: ((one_one_poly_a = X2) = (X2 = one_one_poly_a))))). % one_reorient
thf(fact_205_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_206_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_207_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_208_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_209_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_210_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_211_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_212_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_213_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_214_add_Ogroup__left__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.group_left_neutral
thf(fact_215_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_216_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_217_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_218_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K))))). % left_add_mult_distrib
thf(fact_219_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_220_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_221_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_222_nat__power__eq__Suc__0__iff, axiom,
    ((![X2 : nat, M : nat]: (((power_power_nat @ X2 @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X2 = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_223_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_224_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_225_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_226_mult__Suc__right, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ M @ (suc @ N)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N)))))). % mult_Suc_right
thf(fact_227_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_228_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_229_Suc__inject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))). % Suc_inject
thf(fact_230_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_231_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_232_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_233_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_234_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((P2 @ N2) => (P2 @ (suc @ N2)))) => (P2 @ N)))))). % nat_induct
thf(fact_235_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X4 : nat]: (P2 @ X4 @ zero_zero_nat)) => ((![Y3 : nat]: (P2 @ zero_zero_nat @ (suc @ Y3))) => ((![X4 : nat, Y3 : nat]: ((P2 @ X4 @ Y3) => (P2 @ (suc @ X4) @ (suc @ Y3)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_236_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_237_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_238_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_239_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_240_old_Onat_Oexhaust, axiom,
    ((![Y2 : nat]: ((~ ((Y2 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y2 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_241_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_242_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_243_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_244_nat__arith_Osuc1, axiom,
    ((![A4 : nat, K : nat, A : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((suc @ A4) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[K2 : nat, A5 : a, Q3 : poly_a]: ((~ ((A5 = zero_zero_a))) & ((~ ((K2 = zero_zero_nat))) & (((plus_plus_nat @ (plus_plus_nat @ (fundam247907092size_a @ Q3) @ K2) @ one_one_nat) = (fundam247907092size_a @ p)) & (![Z : a]: ((poly_a2 @ p @ Z) = (plus_plus_a @ (poly_a2 @ p @ zero_zero_a) @ (times_times_a @ (power_power_a @ Z @ K2) @ (poly_a2 @ (pCons_a @ A5 @ Q3) @ Z))))))))))).
