% 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_1040__5377712_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:29.453

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (40)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
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__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
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__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
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_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    times_1460995011omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
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__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
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__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Ois__zero_001t__Complex__Ocomplex, type,
    is_zero_complex : poly_complex > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Complex__Ocomplex, type,
    poly_cutoff_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Complex__Ocomplex, type,
    synthe151143547omplex : poly_complex > complex > poly_complex).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
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__Complex__Ocomplex_J, type,
    power_184595776omplex : poly_complex > nat > poly_complex).
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__Complex__Ocomplex_J_J, type,
    power_432682568omplex : poly_poly_complex > nat > poly_poly_complex).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex, type,
    dvd_dvd_complex : complex > complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    dvd_dvd_poly_complex : poly_complex > poly_complex > $o).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).
thf(sy_v_u____, type,
    u : poly_complex).
thf(sy_v_x____, type,
    x : complex).

% Relevant facts (221)
thf(fact_0_that_I3_J, axiom,
    ((~ (((poly_complex2 @ q @ x) = zero_zero_complex))))). % that(3)
thf(fact_1_u, axiom,
    (((power_184595776omplex @ q @ (suc @ n)) = (times_1246143675omplex @ p @ u)))). % u
thf(fact_2__092_060open_062poly_A_Iq_A_094_ASuc_An_J_Ax_A_092_060noteq_062_A0_092_060close_062, axiom,
    ((~ (((poly_complex2 @ (power_184595776omplex @ q @ (suc @ n)) @ x) = zero_zero_complex))))). % \<open>poly (q ^ Suc n) x \<noteq> 0\<close>
thf(fact_3_dp_I1_J, axiom,
    ((~ ((p = zero_z1746442943omplex))))). % dp(1)
thf(fact_4_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X : complex]: (![Y : complex]: ((F @ X) = (F @ Y)))))))). % constant_def
thf(fact_5_dp_I2_J, axiom,
    (((degree_complex @ p) = (suc @ n)))). % dp(2)
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_Aq_A_094_ASuc_An_A_061_Ap_A_K_Au_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U : poly_complex]: (~ (((power_184595776omplex @ q @ (suc @ n)) = (times_1246143675omplex @ p @ U))))))))). % \<open>\<And>thesis. (\<And>u. q ^ Suc n = p * u \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_that_I2_J, axiom,
    (((poly_complex2 @ p @ x) = zero_zero_complex))). % that(2)
thf(fact_8_dvd, axiom,
    ((dvd_dvd_poly_complex @ p @ (power_184595776omplex @ q @ (suc @ n))))). % dvd
thf(fact_9_fundamental__theorem__of__algebra, axiom,
    ((![P : poly_complex]: ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra
thf(fact_10_poly__power, axiom,
    ((![P : poly_poly_complex, N : nat, X2 : poly_complex]: ((poly_poly_complex2 @ (power_432682568omplex @ P @ N) @ X2) = (power_184595776omplex @ (poly_poly_complex2 @ P @ X2) @ N))))). % poly_power
thf(fact_11_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_12_poly__power, axiom,
    ((![P : poly_complex, N : nat, X2 : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X2) = (power_power_complex @ (poly_complex2 @ P @ X2) @ N))))). % poly_power
thf(fact_13_poly__mult, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex, X2 : poly_complex]: ((poly_poly_complex2 @ (times_1460995011omplex @ P @ Q) @ X2) = (times_1246143675omplex @ (poly_poly_complex2 @ P @ X2) @ (poly_poly_complex2 @ Q @ X2)))))). % poly_mult
thf(fact_14_poly__mult, axiom,
    ((![P : poly_nat, Q : poly_nat, X2 : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q) @ X2) = (times_times_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q @ X2)))))). % poly_mult
thf(fact_15_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X2) = (times_times_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_mult
thf(fact_16_poly__0, axiom,
    ((![X2 : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X2) = zero_z1746442943omplex)))). % poly_0
thf(fact_17_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_18_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_19_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N)) = zero_zero_complex)))). % power_0_Suc
thf(fact_20_power__0__Suc, axiom,
    ((![N : nat]: ((power_184595776omplex @ zero_z1746442943omplex @ (suc @ N)) = zero_z1746442943omplex)))). % power_0_Suc
thf(fact_21_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062p_A_061_A0_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062p_A_092_060noteq_062_A0_059_Adegree_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_059_A_092_060And_062n_O_A_092_060lbrakk_062p_A_092_060noteq_062_A0_059_Adegree_Ap_A_061_ASuc_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    (((~ ((p = zero_z1746442943omplex))) => (((~ ((p = zero_z1746442943omplex))) => (~ (((degree_complex @ p) = zero_zero_nat)))) => (~ (((~ ((p = zero_z1746442943omplex))) => (![N2 : nat]: (~ (((degree_complex @ p) = (suc @ N2)))))))))))). % \<open>\<And>thesis. \<lbrakk>p = 0 \<Longrightarrow> thesis; \<lbrakk>p \<noteq> 0; degree p = 0\<rbrakk> \<Longrightarrow> thesis; \<And>n. \<lbrakk>p \<noteq> 0; degree p = Suc n\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_23_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_24_mult__zero__left, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_25_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_26_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_27_mult__zero__right, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right
thf(fact_28_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_29_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_30_mult__eq__0__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex) = (((A = zero_z1746442943omplex)) | ((B = zero_z1746442943omplex))))))). % mult_eq_0_iff
thf(fact_31_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_32_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_33_mult__cancel__left, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ C @ A) = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((A = B))))))). % mult_cancel_left
thf(fact_34_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_35_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_36_mult__cancel__right, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ C) = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((A = B))))))). % mult_cancel_right
thf(fact_37_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_38_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X : complex]: ((poly_complex2 @ P @ X) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_39_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X : poly_complex]: ((poly_poly_complex2 @ P @ X) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_40_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_41_dvd__0__left__iff, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) = (A = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_42_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_43_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_44_dvd__0__right, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_45_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_46_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_47_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_48_dvd__times__right__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ B @ A) @ (times_1246143675omplex @ C @ A)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_49_dvd__times__right__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_50_dvd__times__left__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ (times_1246143675omplex @ A @ C)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_51_dvd__times__left__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_52_dvd__mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((dvd_dvd_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((dvd_dvd_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_53_dvd__mult__cancel__right, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ C) @ (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((dvd_dvd_poly_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_54_dvd__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((dvd_dvd_complex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((dvd_dvd_complex @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_55_dvd__mult__cancel__left, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ C @ A) @ (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((dvd_dvd_poly_complex @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_56_power__Suc0__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_57_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_58_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_59_dvd__trans, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((dvd_dvd_poly_complex @ B @ C) => (dvd_dvd_poly_complex @ A @ C)))))). % dvd_trans
thf(fact_60_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_61_dvd__refl, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ A)))). % dvd_refl
thf(fact_62_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_63_dvd__0__left, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) => (A = zero_zero_complex))))). % dvd_0_left
thf(fact_64_dvd__0__left, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) => (A = zero_z1746442943omplex))))). % dvd_0_left
thf(fact_65_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_66_dvd__triv__right, axiom,
    ((![A : poly_complex, B : poly_complex]: (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ B @ A))))). % dvd_triv_right
thf(fact_67_dvd__triv__right, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ B @ A))))). % dvd_triv_right
thf(fact_68_dvd__mult__right, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ C) => (dvd_dvd_poly_complex @ B @ C))))). % dvd_mult_right
thf(fact_69_dvd__mult__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ B @ C))))). % dvd_mult_right
thf(fact_70_mult__dvd__mono, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex, D : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((dvd_dvd_poly_complex @ C @ D) => (dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ C) @ (times_1246143675omplex @ B @ D))))))). % mult_dvd_mono
thf(fact_71_mult__dvd__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ C @ D) => (dvd_dvd_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))). % mult_dvd_mono
thf(fact_72_dvd__triv__left, axiom,
    ((![A : poly_complex, B : poly_complex]: (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ A @ B))))). % dvd_triv_left
thf(fact_73_dvd__triv__left, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ A @ B))))). % dvd_triv_left
thf(fact_74_dvd__mult__left, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ C) => (dvd_dvd_poly_complex @ A @ C))))). % dvd_mult_left
thf(fact_75_dvd__mult__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ A @ C))))). % dvd_mult_left
thf(fact_76_dvd__mult2, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ B @ C)))))). % dvd_mult2
thf(fact_77_dvd__mult2, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult2
thf(fact_78_dvd__mult, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ C) => (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ B @ C)))))). % dvd_mult
thf(fact_79_dvd__mult, axiom,
    ((![A : nat, C : nat, B : nat]: ((dvd_dvd_nat @ A @ C) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult
thf(fact_80_dvd__def, axiom,
    ((dvd_dvd_poly_complex = (^[B2 : poly_complex]: (^[A2 : poly_complex]: (?[K : poly_complex]: (A2 = (times_1246143675omplex @ B2 @ K)))))))). % dvd_def
thf(fact_81_dvd__def, axiom,
    ((dvd_dvd_nat = (^[B2 : nat]: (^[A2 : nat]: (?[K : nat]: (A2 = (times_times_nat @ B2 @ K)))))))). % dvd_def
thf(fact_82_dvdI, axiom,
    ((![A : poly_complex, B : poly_complex, K2 : poly_complex]: ((A = (times_1246143675omplex @ B @ K2)) => (dvd_dvd_poly_complex @ B @ A))))). % dvdI
thf(fact_83_dvdI, axiom,
    ((![A : nat, B : nat, K2 : nat]: ((A = (times_times_nat @ B @ K2)) => (dvd_dvd_nat @ B @ A))))). % dvdI
thf(fact_84_dvdE, axiom,
    ((![B : poly_complex, A : poly_complex]: ((dvd_dvd_poly_complex @ B @ A) => (~ ((![K3 : poly_complex]: (~ ((A = (times_1246143675omplex @ B @ K3))))))))))). % dvdE
thf(fact_85_dvdE, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ A) => (~ ((![K3 : nat]: (~ ((A = (times_times_nat @ B @ K3))))))))))). % dvdE
thf(fact_86_dvd__power__same, axiom,
    ((![X2 : poly_complex, Y2 : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ X2 @ Y2) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X2 @ N) @ (power_184595776omplex @ Y2 @ N)))))). % dvd_power_same
thf(fact_87_dvd__power__same, axiom,
    ((![X2 : nat, Y2 : nat, N : nat]: ((dvd_dvd_nat @ X2 @ Y2) => (dvd_dvd_nat @ (power_power_nat @ X2 @ N) @ (power_power_nat @ Y2 @ N)))))). % dvd_power_same
thf(fact_88_power__mult, axiom,
    ((![A : poly_complex, M : nat, N : nat]: ((power_184595776omplex @ A @ (times_times_nat @ M @ N)) = (power_184595776omplex @ (power_184595776omplex @ A @ M) @ N))))). % power_mult
thf(fact_89_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_90_degree__power__eq, axiom,
    ((![P : poly_complex, N : nat]: ((~ ((P = zero_z1746442943omplex))) => ((degree_complex @ (power_184595776omplex @ P @ N)) = (times_times_nat @ N @ (degree_complex @ P))))))). % degree_power_eq
thf(fact_91_degree__mult__eq__0, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((degree_complex @ (times_1246143675omplex @ P @ Q)) = zero_zero_nat) = (((P = zero_z1746442943omplex)) | ((((Q = zero_z1746442943omplex)) | ((((~ ((P = zero_z1746442943omplex)))) & ((((~ ((Q = zero_z1746442943omplex)))) & (((((degree_complex @ P) = zero_zero_nat)) & (((degree_complex @ Q) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_92_nullstellensatz__lemma, axiom,
    ((![P : poly_complex, Q : poly_complex, N : nat]: ((![X3 : complex]: (((poly_complex2 @ P @ X3) = zero_zero_complex) => ((poly_complex2 @ Q @ X3) = zero_zero_complex))) => (((degree_complex @ P) = N) => ((~ ((N = zero_zero_nat))) => (dvd_dvd_poly_complex @ P @ (power_184595776omplex @ Q @ N)))))))). % nullstellensatz_lemma
thf(fact_93_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_94_mult__right__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_95_mult__right__cancel, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ A @ C) = (times_1246143675omplex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_96_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_97_mult__left__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_98_mult__left__cancel, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ C @ A) = (times_1246143675omplex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_99_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_100_no__zero__divisors, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_101_no__zero__divisors, axiom,
    ((![A : poly_complex, B : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((~ ((B = zero_z1746442943omplex))) => (~ (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex)))))))). % no_zero_divisors
thf(fact_102_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_103_divisors__zero, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) => ((A = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_104_divisors__zero, axiom,
    ((![A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex) => ((A = zero_z1746442943omplex) | (B = zero_z1746442943omplex)))))). % divisors_zero
thf(fact_105_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_106_mult__not__zero, axiom,
    ((![A : complex, B : complex]: ((~ (((times_times_complex @ A @ B) = zero_zero_complex))) => ((~ ((A = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_107_mult__not__zero, axiom,
    ((![A : poly_complex, B : poly_complex]: ((~ (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex))) => ((~ ((A = zero_z1746442943omplex))) & (~ ((B = zero_z1746442943omplex)))))))). % mult_not_zero
thf(fact_108_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_109_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_110_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_111_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_112_power__commuting__commutes, axiom,
    ((![X2 : poly_complex, Y2 : poly_complex, N : nat]: (((times_1246143675omplex @ X2 @ Y2) = (times_1246143675omplex @ Y2 @ X2)) => ((times_1246143675omplex @ (power_184595776omplex @ X2 @ N) @ Y2) = (times_1246143675omplex @ Y2 @ (power_184595776omplex @ X2 @ N))))))). % power_commuting_commutes
thf(fact_113_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_114_power__mult__distrib, axiom,
    ((![A : poly_complex, B : poly_complex, N : nat]: ((power_184595776omplex @ (times_1246143675omplex @ A @ B) @ N) = (times_1246143675omplex @ (power_184595776omplex @ A @ N) @ (power_184595776omplex @ 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_complex, N : nat]: ((times_1246143675omplex @ (power_184595776omplex @ A @ N) @ A) = (times_1246143675omplex @ A @ (power_184595776omplex @ A @ N)))))). % power_commutes
thf(fact_117_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_118_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_119_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_120_power__Suc2, axiom,
    ((![A : poly_complex, N : nat]: ((power_184595776omplex @ A @ (suc @ N)) = (times_1246143675omplex @ (power_184595776omplex @ A @ N) @ A))))). % power_Suc2
thf(fact_121_power__Suc2, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ (power_power_nat @ A @ N) @ A))))). % power_Suc2
thf(fact_122_power__Suc, axiom,
    ((![A : poly_complex, N : nat]: ((power_184595776omplex @ A @ (suc @ N)) = (times_1246143675omplex @ A @ (power_184595776omplex @ A @ N)))))). % power_Suc
thf(fact_123_power__Suc, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_Suc
thf(fact_124_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_125_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_126_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_127_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_128_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_129_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_130_mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel1
thf(fact_131_mult__cancel2, axiom,
    ((![M : nat, K2 : nat, N : nat]: (((times_times_nat @ M @ K2) = (times_times_nat @ N @ K2)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel2
thf(fact_132_dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ (suc @ zero_zero_nat)) = (M = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_133_dvd__1__left, axiom,
    ((![K2 : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K2)))). % dvd_1_left
thf(fact_134_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_135_Suc__inject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))). % Suc_inject
thf(fact_136_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_137_Suc__mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K2) @ M) = (times_times_nat @ (suc @ K2) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_138_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_139_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_140_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_141_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_142_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_143_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P2 @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P2 @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P2 @ X3 @ Y3) => (P2 @ (suc @ X3) @ (suc @ Y3)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_144_zero__induct, axiom,
    ((![P2 : nat > $o, K2 : nat]: ((P2 @ K2) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_145_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_146_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_147_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_148_old_Onat_Oexhaust, axiom,
    ((![Y2 : nat]: ((~ ((Y2 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y2 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_149_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_150_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_151_nat__mult__dvd__cancel__disj, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((dvd_dvd_nat @ (times_times_nat @ K2 @ M) @ (times_times_nat @ K2 @ N)) = (((K2 = zero_zero_nat)) | ((dvd_dvd_nat @ M @ N))))))). % nat_mult_dvd_cancel_disj
thf(fact_152_psize__def, axiom,
    ((fundam1709708056omplex = (^[P3 : poly_complex]: (if_nat @ (P3 = zero_z1746442943omplex) @ zero_zero_nat @ (suc @ (degree_complex @ P3))))))). % psize_def
thf(fact_153_nat__mult__eq__cancel__disj, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N)) = (((K2 = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_154_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_155_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_156_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_157_zero__reorient, axiom,
    ((![X2 : poly_complex]: ((zero_z1746442943omplex = X2) = (X2 = zero_z1746442943omplex))))). % zero_reorient
thf(fact_158_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_159_mult_Oleft__commute, axiom,
    ((![B : poly_complex, A : poly_complex, C : poly_complex]: ((times_1246143675omplex @ B @ (times_1246143675omplex @ A @ C)) = (times_1246143675omplex @ A @ (times_1246143675omplex @ B @ C)))))). % mult.left_commute
thf(fact_160_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_161_mult_Ocommute, axiom,
    ((times_1246143675omplex = (^[A2 : poly_complex]: (^[B2 : poly_complex]: (times_1246143675omplex @ B2 @ A2)))))). % mult.commute
thf(fact_162_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_163_mult_Oassoc, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((times_1246143675omplex @ (times_1246143675omplex @ A @ B) @ C) = (times_1246143675omplex @ A @ (times_1246143675omplex @ B @ C)))))). % mult.assoc
thf(fact_164_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_165_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((times_1246143675omplex @ (times_1246143675omplex @ A @ B) @ C) = (times_1246143675omplex @ A @ (times_1246143675omplex @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_166_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_167_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_168_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_169_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_170_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_171_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_172_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_173_gcd__nat_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A2 @ B2)) & ((dvd_dvd_nat @ B2 @ A2)))))))). % gcd_nat.eq_iff
thf(fact_174_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_175_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_176_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_177_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_178_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_179_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A2 : nat]: (^[B2 : nat]: (((((dvd_dvd_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))) | ((A2 = B2)))))))). % gcd_nat.order_iff_strict
thf(fact_180_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_181_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_182_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_183_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ B) & (~ ((A = B))))))))). % gcd_nat.not_eq_order_implies_strict
thf(fact_184_dvd__productE, axiom,
    ((![P : nat, A : nat, B : nat]: ((dvd_dvd_nat @ P @ (times_times_nat @ A @ B)) => (~ ((![X3 : nat, Y3 : nat]: ((P = (times_times_nat @ X3 @ Y3)) => ((dvd_dvd_nat @ X3 @ A) => (~ ((dvd_dvd_nat @ Y3 @ B)))))))))))). % dvd_productE
thf(fact_185_division__decomp, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ (times_times_nat @ B @ C)) => (?[B3 : nat, C2 : nat]: ((A = (times_times_nat @ B3 @ C2)) & ((dvd_dvd_nat @ B3 @ B) & (dvd_dvd_nat @ C2 @ C)))))))). % division_decomp
thf(fact_186_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_187_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_188_exists__least__lemma, axiom,
    ((![P2 : nat > $o]: ((~ ((P2 @ zero_zero_nat))) => ((?[X_1 : nat]: (P2 @ X_1)) => (?[N2 : nat]: ((~ ((P2 @ N2))) & (P2 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_189_dvd__field__iff, axiom,
    ((dvd_dvd_complex = (^[A2 : complex]: (^[B2 : complex]: (((A2 = zero_zero_complex)) => ((B2 = zero_zero_complex)))))))). % dvd_field_iff
thf(fact_190_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_complex, C : complex]: (((synthe151143547omplex @ P @ C) = zero_z1746442943omplex) = ((degree_complex @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_191_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_192_is__zero__null, axiom,
    ((is_zero_complex = (^[P3 : poly_complex]: (P3 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_193_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_194_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_195_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_196_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_197_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = one_one_poly_complex)))))). % poly_cutoff_1
thf(fact_198_mult_Oleft__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ one_one_poly_complex @ A) = A)))). % mult.left_neutral
thf(fact_199_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_200_mult_Oright__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ one_one_poly_complex) = A)))). % mult.right_neutral
thf(fact_201_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_202_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_203_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_204_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_205_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_206_mult__cancel__right2, axiom,
    ((![A : poly_complex, C : poly_complex]: (((times_1246143675omplex @ A @ C) = C) = (((C = zero_z1746442943omplex)) | ((A = one_one_poly_complex))))))). % mult_cancel_right2
thf(fact_207_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_208_mult__cancel__right1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_right1
thf(fact_209_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_210_mult__cancel__left2, axiom,
    ((![C : poly_complex, A : poly_complex]: (((times_1246143675omplex @ C @ A) = C) = (((C = zero_z1746442943omplex)) | ((A = one_one_poly_complex))))))). % mult_cancel_left2
thf(fact_211_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_212_mult__cancel__left1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_left1
thf(fact_213_unit__prod, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ one_one_poly_complex) => ((dvd_dvd_poly_complex @ B @ one_one_poly_complex) => (dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ one_one_poly_complex)))))). % unit_prod
thf(fact_214_unit__prod, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ (times_times_nat @ A @ B) @ one_one_nat)))))). % unit_prod
thf(fact_215_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_216_poly__1, axiom,
    ((![X2 : complex]: ((poly_complex2 @ one_one_poly_complex @ X2) = one_one_complex)))). % poly_1
thf(fact_217_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ one_one_poly_complex))))). % not_is_unit_0
thf(fact_218_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_219_left__right__inverse__power, axiom,
    ((![X2 : poly_complex, Y2 : poly_complex, N : nat]: (((times_1246143675omplex @ X2 @ Y2) = one_one_poly_complex) => ((times_1246143675omplex @ (power_184595776omplex @ X2 @ N) @ (power_184595776omplex @ Y2 @ N)) = one_one_poly_complex))))). % left_right_inverse_power
thf(fact_220_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

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $true @ X2 @ Y2) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ($false)).
