% 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_983__5377172_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:03.372

% 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 (56)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    minus_174331535omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    minus_minus_poly_nat : poly_nat > poly_nat > poly_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_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__Complex__Ocomplex_J_J, type,
    one_on1331105667omplex : poly_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__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    times_1460995011omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    uminus1138659839omplex : poly_complex > 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__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    if_poly_complex : $o > poly_complex > poly_complex > poly_complex).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    order_poly_complex : poly_complex > poly_poly_complex > nat).
thf(sy_c_Polynomial_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
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__Complex__Ocomplex_J, type,
    pCons_poly_complex : poly_complex > poly_poly_complex > poly_poly_complex).
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_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_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    dvd_dvd_poly_nat : poly_nat > poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    dvd_dv598755940omplex : poly_poly_complex > poly_poly_complex > $o).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).
thf(sy_v_qa____, type,
    qa : poly_complex).
thf(sy_v_r____, type,
    r : poly_complex).
thf(sy_v_s____, type,
    s : poly_complex).
thf(sy_v_u____, type,
    u : poly_complex).

% Relevant facts (236)
thf(fact_0_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_1_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_2_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_3_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_4_u, axiom,
    (((power_184595776omplex @ r @ (degree_complex @ s)) = (times_1246143675omplex @ s @ u)))). % u
thf(fact_5_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_Ar_A_094_Adegree_As_A_061_As_A_K_Au_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U : poly_complex]: (~ (((power_184595776omplex @ r @ (degree_complex @ s)) = (times_1246143675omplex @ s @ U))))))))). % \<open>\<And>thesis. (\<And>u. r ^ degree s = s * u \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_r, axiom,
    ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ r)))). % r
thf(fact_8__092_060open_062q_A_094_An_A_061_Ap_A_K_A_Iu_A_K_A_091_058_N_Aa_M_A1_058_093_A_094_A_In_A_N_Aorder_Aa_Ap_J_A_K_Ar_A_094_A_In_A_N_Adegree_As_J_J_092_060close_062, axiom,
    (((power_184595776omplex @ qa @ na) = (times_1246143675omplex @ pa @ (times_1246143675omplex @ (times_1246143675omplex @ u @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (minus_minus_nat @ na @ (order_complex @ a @ pa)))) @ (power_184595776omplex @ r @ (minus_minus_nat @ na @ (degree_complex @ s)))))))). % \<open>q ^ n = p * (u * [:- a, 1:] ^ (n - order a p) * r ^ (n - degree s))\<close>
thf(fact_9_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_10_s, axiom,
    ((pa = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ s)))). % s
thf(fact_11_oop, axiom,
    ((![A : complex]: (ord_less_eq_nat @ (order_complex @ A @ pa) @ na)))). % oop
thf(fact_12_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_13_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_14_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_15_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_16_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_17_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_18_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_19_power__commuting__commutes, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: (((times_1246143675omplex @ X @ Y) = (times_1246143675omplex @ Y @ X)) => ((times_1246143675omplex @ (power_184595776omplex @ X @ N) @ Y) = (times_1246143675omplex @ Y @ (power_184595776omplex @ X @ N))))))). % power_commuting_commutes
thf(fact_20_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N))))))). % power_commuting_commutes
thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_O_Aq_A_061_A_091_058_N_Aa_M_A1_058_093_A_K_Ar_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![R : poly_complex]: (~ ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ R))))))))). % \<open>\<And>thesis. (\<And>r. q = [:- a, 1:] * r \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_Ap_A_061_A_091_058_N_Aa_M_A1_058_093_A_094_Aorder_Aa_Ap_A_K_As_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![S : poly_complex]: (~ ((pa = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ S))))))))). % \<open>\<And>thesis. (\<And>s. p = [:- a, 1:] ^ order a p * s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_24_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_25_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_26_True, axiom,
    ((?[A2 : complex]: ((poly_complex2 @ pa @ A2) = zero_zero_complex)))). % True
thf(fact_27_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_28_assms_I1_J, axiom,
    ((![X2 : complex]: (((poly_complex2 @ p @ X2) = zero_zero_complex) => ((poly_complex2 @ q @ X2) = zero_zero_complex))))). % assms(1)
thf(fact_29__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_O_Apoly_Ap_Aa_A_061_A0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![A2 : complex]: (~ (((poly_complex2 @ pa @ A2) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>a. poly p a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_30__092_060open_062_092_060And_062x_O_Apoly_As_Ax_A_061_A0_A_092_060Longrightarrow_062_Apoly_Ar_Ax_A_061_A0_092_060close_062, axiom,
    ((![X : complex]: (((poly_complex2 @ s @ X) = zero_zero_complex) => ((poly_complex2 @ r @ X) = zero_zero_complex))))). % \<open>\<And>x. poly s x = 0 \<Longrightarrow> poly r x = 0\<close>
thf(fact_31_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_32_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_33_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_34__092_060open_062order_Aa_Ap_A_092_060le_062_An_092_060close_062, axiom,
    ((ord_less_eq_nat @ (order_complex @ a @ pa) @ na))). % \<open>order a p \<le> n\<close>
thf(fact_35_u_H, axiom,
    ((![X : complex]: ((times_times_complex @ (poly_complex2 @ s @ X) @ (poly_complex2 @ u @ X)) = (power_power_complex @ (poly_complex2 @ r @ X) @ (degree_complex @ s)))))). % u'
thf(fact_36_left__minus__one__mult__self, axiom,
    ((![N : nat, A : poly_complex]: ((times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ (times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_37_left__minus__one__mult__self, axiom,
    ((![N : nat, A : complex]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_38_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N)) = one_one_poly_complex)))). % minus_one_mult_self
thf(fact_39_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N)) = one_one_complex)))). % minus_one_mult_self
thf(fact_40_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_decreasing
thf(fact_41_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_42_fundamental__theorem__of__algebra__alt, axiom,
    ((![P : poly_complex]: ((~ ((?[A2 : complex, L : poly_complex]: ((~ ((A2 = zero_zero_complex))) & ((L = zero_z1746442943omplex) & (P = (pCons_complex @ A2 @ L))))))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra_alt
thf(fact_43_power__le__one, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ one_one_nat)))))). % power_le_one
thf(fact_44_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_184595776omplex @ zero_z1746442943omplex @ N) = one_one_poly_complex)) & ((~ ((N = zero_zero_nat))) => ((power_184595776omplex @ zero_z1746442943omplex @ N) = zero_z1746442943omplex)))))). % power_0_left
thf(fact_45_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_46_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_47_one__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A @ N)))))). % one_le_power
thf(fact_48_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_49_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_50_power__0, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ zero_zero_nat) = one_one_poly_complex)))). % power_0
thf(fact_51_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_52_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_53_power__eq__if, axiom,
    ((power_power_nat = (^[P2 : nat]: (^[M : nat]: (if_nat @ (M = zero_zero_nat) @ one_one_nat @ (times_times_nat @ P2 @ (power_power_nat @ P2 @ (minus_minus_nat @ M @ one_one_nat))))))))). % power_eq_if
thf(fact_54_power__eq__if, axiom,
    ((power_184595776omplex = (^[P2 : poly_complex]: (^[M : nat]: (if_poly_complex @ (M = zero_zero_nat) @ one_one_poly_complex @ (times_1246143675omplex @ P2 @ (power_184595776omplex @ P2 @ (minus_minus_nat @ M @ one_one_nat))))))))). % power_eq_if
thf(fact_55_power__eq__if, axiom,
    ((power_power_complex = (^[P2 : complex]: (^[M : nat]: (if_complex @ (M = zero_zero_nat) @ one_one_complex @ (times_times_complex @ P2 @ (power_power_complex @ P2 @ (minus_minus_nat @ M @ one_one_nat))))))))). % power_eq_if
thf(fact_56_power__mult, axiom,
    ((![A : poly_complex, M2 : nat, N : nat]: ((power_184595776omplex @ A @ (times_times_nat @ M2 @ N)) = (power_184595776omplex @ (power_184595776omplex @ A @ M2) @ N))))). % power_mult
thf(fact_57_power__mult, axiom,
    ((![A : complex, M2 : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M2 @ N)) = (power_power_complex @ (power_power_complex @ A @ M2) @ N))))). % power_mult
thf(fact_58_power__mult, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M2 @ N)) = (power_power_nat @ (power_power_nat @ A @ M2) @ N))))). % power_mult
thf(fact_59_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_60_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_61_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_62_power__minus, axiom,
    ((![A : poly_complex, N : nat]: ((power_184595776omplex @ (uminus1138659839omplex @ A) @ N) = (times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ (power_184595776omplex @ A @ N)))))). % power_minus
thf(fact_63_power__minus, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ N) = (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ A @ N)))))). % power_minus
thf(fact_64_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_65_left__right__inverse__power, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: (((times_1246143675omplex @ X @ Y) = one_one_poly_complex) => ((times_1246143675omplex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ N)) = one_one_poly_complex))))). % left_right_inverse_power
thf(fact_66_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_67_diff__numeral__special_I12_J, axiom,
    (((minus_174331535omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ (uminus1138659839omplex @ one_one_poly_complex)) = zero_z1746442943omplex))). % diff_numeral_special(12)
thf(fact_68_diff__numeral__special_I12_J, axiom,
    (((minus_minus_complex @ (uminus1204672759omplex @ one_one_complex) @ (uminus1204672759omplex @ one_one_complex)) = zero_zero_complex))). % diff_numeral_special(12)
thf(fact_69_ap_I1_J, axiom,
    ((dvd_dvd_poly_complex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ pa))). % ap(1)
thf(fact_70_poly__power, axiom,
    ((![P : poly_poly_complex, N : nat, X : poly_complex]: ((poly_poly_complex2 @ (power_432682568omplex @ P @ N) @ X) = (power_184595776omplex @ (poly_poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_71_poly__power, axiom,
    ((![P : poly_nat, N : nat, X : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X) = (power_power_nat @ (poly_nat2 @ P @ X) @ N))))). % poly_power
thf(fact_72_poly__power, axiom,
    ((![P : poly_complex, N : nat, X : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X) = (power_power_complex @ (poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_73_poly__mult, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex, X : poly_complex]: ((poly_poly_complex2 @ (times_1460995011omplex @ P @ Q) @ X) = (times_1246143675omplex @ (poly_poly_complex2 @ P @ X) @ (poly_poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_74_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X) = (times_times_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_75_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_76_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_77_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_78_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_79_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_80_one__poly__eq__simps_I2_J, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % one_poly_eq_simps(2)
thf(fact_81_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_82_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_complex = (pCons_complex @ one_one_complex @ zero_z1746442943omplex)))). % one_poly_eq_simps(1)
thf(fact_83_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_84_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_85_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_86_degree__minus, axiom,
    ((![P : poly_complex]: ((degree_complex @ (uminus1138659839omplex @ P)) = (degree_complex @ P))))). % degree_minus
thf(fact_87_pCons__eq__iff, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: (((pCons_complex @ A @ P) = (pCons_complex @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_88_order__uminus, axiom,
    ((![X : complex, P : poly_complex]: ((order_complex @ X @ (uminus1138659839omplex @ P)) = (order_complex @ X @ P))))). % order_uminus
thf(fact_89__092_060open_062s_Advd_Ar_A_094_Adegree_As_092_060close_062, axiom,
    ((dvd_dvd_poly_complex @ s @ (power_184595776omplex @ r @ (degree_complex @ s))))). % \<open>s dvd r ^ degree s\<close>
thf(fact_90_power__one__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_91_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_92_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_93_diff__pCons, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: ((minus_174331535omplex @ (pCons_complex @ A @ P) @ (pCons_complex @ B @ Q)) = (pCons_complex @ (minus_minus_complex @ A @ B) @ (minus_174331535omplex @ P @ Q)))))). % diff_pCons
thf(fact_94_diff__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: ((minus_minus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q)) = (pCons_nat @ (minus_minus_nat @ A @ B) @ (minus_minus_poly_nat @ P @ Q)))))). % diff_pCons
thf(fact_95_minus__pCons, axiom,
    ((![A : complex, P : poly_complex]: ((uminus1138659839omplex @ (pCons_complex @ A @ P)) = (pCons_complex @ (uminus1204672759omplex @ A) @ (uminus1138659839omplex @ P)))))). % minus_pCons
thf(fact_96_poly__diff, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (minus_174331535omplex @ P @ Q) @ X) = (minus_minus_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_diff
thf(fact_97_poly__minus, axiom,
    ((![P : poly_complex, X : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P) @ X) = (uminus1204672759omplex @ (poly_complex2 @ P @ X)))))). % poly_minus
thf(fact_98__092_060open_062_091_058_N_Aa_M_A1_058_093_Advd_Aq_092_060close_062, axiom,
    ((dvd_dvd_poly_complex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ qa))). % \<open>[:- a, 1:] dvd q\<close>
thf(fact_99_diff__numeral__special_I9_J, axiom,
    (((minus_174331535omplex @ one_one_poly_complex @ one_one_poly_complex) = zero_z1746442943omplex))). % diff_numeral_special(9)
thf(fact_100_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_101_mult__minus1, axiom,
    ((![Z2 : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ Z2) = (uminus1138659839omplex @ Z2))))). % mult_minus1
thf(fact_102_mult__minus1, axiom,
    ((![Z2 : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z2) = (uminus1204672759omplex @ Z2))))). % mult_minus1
thf(fact_103_mult__minus1__right, axiom,
    ((![Z2 : poly_complex]: ((times_1246143675omplex @ Z2 @ (uminus1138659839omplex @ one_one_poly_complex)) = (uminus1138659839omplex @ Z2))))). % mult_minus1_right
thf(fact_104_mult__minus1__right, axiom,
    ((![Z2 : complex]: ((times_times_complex @ Z2 @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z2))))). % mult_minus1_right
thf(fact_105__092_060open_062_092_060lbrakk_062_092_060And_062x_O_Apoly_As_Ax_A_061_A0_A_092_060Longrightarrow_062_Apoly_Ar_Ax_A_061_A0_059_Adegree_As_A_061_Adegree_As_059_Adegree_As_A_092_060noteq_062_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_As_Advd_Ar_A_094_Adegree_As_092_060close_062, axiom,
    (((![X3 : complex]: (((poly_complex2 @ s @ X3) = zero_zero_complex) => ((poly_complex2 @ r @ X3) = zero_zero_complex))) => (((degree_complex @ s) = (degree_complex @ s)) => ((~ (((degree_complex @ s) = zero_zero_nat))) => (dvd_dvd_poly_complex @ s @ (power_184595776omplex @ r @ (degree_complex @ s)))))))). % \<open>\<lbrakk>\<And>x. poly s x = 0 \<Longrightarrow> poly r x = 0; degree s = degree s; degree s \<noteq> 0\<rbrakk> \<Longrightarrow> s dvd r ^ degree s\<close>
thf(fact_106_pCons__eq__0__iff, axiom,
    ((![A : poly_complex, P : poly_poly_complex]: (((pCons_poly_complex @ A @ P) = zero_z1040703943omplex) = (((A = zero_z1746442943omplex)) & ((P = zero_z1040703943omplex))))))). % pCons_eq_0_iff
thf(fact_107_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_108_pCons__eq__0__iff, axiom,
    ((![A : complex, P : poly_complex]: (((pCons_complex @ A @ P) = zero_z1746442943omplex) = (((A = zero_zero_complex)) & ((P = zero_z1746442943omplex))))))). % pCons_eq_0_iff
thf(fact_109_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_110_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_111_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_112_order__1__eq__0, axiom,
    ((![X : complex]: ((order_complex @ X @ one_one_poly_complex) = zero_zero_nat)))). % order_1_eq_0
thf(fact_113_const__poly__dvd__const__poly__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dv598755940omplex @ (pCons_poly_complex @ A @ zero_z1040703943omplex) @ (pCons_poly_complex @ B @ zero_z1040703943omplex)) = (dvd_dvd_poly_complex @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_114_const__poly__dvd__const__poly__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ (pCons_nat @ B @ zero_zero_poly_nat)) = (dvd_dvd_nat @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_115_const__poly__dvd__const__poly__iff, axiom,
    ((![A : complex, B : complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ A @ zero_z1746442943omplex) @ (pCons_complex @ B @ zero_z1746442943omplex)) = (dvd_dvd_complex @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_116_IH, axiom,
    ((![M3 : nat]: ((ord_less_nat @ M3 @ na) => (![P3 : poly_complex, Q2 : poly_complex]: ((![X3 : complex]: (((poly_complex2 @ P3 @ X3) = zero_zero_complex) => ((poly_complex2 @ Q2 @ X3) = zero_zero_complex))) => (((degree_complex @ P3) = M3) => ((~ ((M3 = zero_zero_nat))) => (dvd_dvd_poly_complex @ P3 @ (power_184595776omplex @ Q2 @ M3)))))))))). % IH
thf(fact_117_is__unit__const__poly__iff, axiom,
    ((![C : poly_complex]: ((dvd_dv598755940omplex @ (pCons_poly_complex @ C @ zero_z1040703943omplex) @ one_on1331105667omplex) = (dvd_dvd_poly_complex @ C @ one_one_poly_complex))))). % is_unit_const_poly_iff
thf(fact_118_is__unit__const__poly__iff, axiom,
    ((![C : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ one_one_poly_nat) = (dvd_dvd_nat @ C @ one_one_nat))))). % is_unit_const_poly_iff
thf(fact_119_is__unit__const__poly__iff, axiom,
    ((![C : complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ C @ zero_z1746442943omplex) @ one_one_poly_complex) = (dvd_dvd_complex @ C @ one_one_complex))))). % is_unit_const_poly_iff
thf(fact_120_is__unit__poly__iff, axiom,
    ((![P : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ one_on1331105667omplex) = (?[C2 : poly_complex]: (((P = (pCons_poly_complex @ C2 @ zero_z1040703943omplex))) & ((dvd_dvd_poly_complex @ C2 @ one_one_poly_complex)))))))). % is_unit_poly_iff
thf(fact_121_is__unit__poly__iff, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) = (?[C2 : nat]: (((P = (pCons_nat @ C2 @ zero_zero_poly_nat))) & ((dvd_dvd_nat @ C2 @ one_one_nat)))))))). % is_unit_poly_iff
thf(fact_122_is__unit__poly__iff, axiom,
    ((![P : poly_complex]: ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) = (?[C2 : complex]: (((P = (pCons_complex @ C2 @ zero_z1746442943omplex))) & ((dvd_dvd_complex @ C2 @ one_one_complex)))))))). % is_unit_poly_iff
thf(fact_123_is__unit__polyE, axiom,
    ((![P : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ one_on1331105667omplex) => (~ ((![C3 : poly_complex]: ((P = (pCons_poly_complex @ C3 @ zero_z1040703943omplex)) => (~ ((dvd_dvd_poly_complex @ C3 @ one_one_poly_complex))))))))))). % is_unit_polyE
thf(fact_124_is__unit__polyE, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) => (~ ((![C3 : nat]: ((P = (pCons_nat @ C3 @ zero_zero_poly_nat)) => (~ ((dvd_dvd_nat @ C3 @ one_one_nat))))))))))). % is_unit_polyE
thf(fact_125_is__unit__polyE, axiom,
    ((![P : poly_complex]: ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) => (~ ((![C3 : complex]: ((P = (pCons_complex @ C3 @ zero_z1746442943omplex)) => (~ ((dvd_dvd_complex @ C3 @ one_one_complex))))))))))). % is_unit_polyE
thf(fact_126_dvd__power__same, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ X @ Y) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ N)))))). % dvd_power_same
thf(fact_127_dvd__power__same, axiom,
    ((![X : complex, Y : complex, N : nat]: ((dvd_dvd_complex @ X @ Y) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)))))). % dvd_power_same
thf(fact_128_dvd__power__same, axiom,
    ((![X : nat, Y : nat, N : nat]: ((dvd_dvd_nat @ X @ Y) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)))))). % dvd_power_same
thf(fact_129_is__unit__pCons__iff, axiom,
    ((![A : complex, P : poly_complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ A @ P) @ one_one_poly_complex) = (((P = zero_z1746442943omplex)) & ((~ ((A = zero_zero_complex))))))))). % is_unit_pCons_iff
thf(fact_130_is__unit__triv, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (dvd_dvd_poly_complex @ (pCons_complex @ A @ zero_z1746442943omplex) @ one_one_poly_complex))))). % is_unit_triv
thf(fact_131_is__unit__iff__degree, axiom,
    ((![P : poly_complex]: ((~ ((P = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) = ((degree_complex @ P) = zero_zero_nat)))))). % is_unit_iff_degree
thf(fact_132_degree__diff__le, axiom,
    ((![P : poly_complex, N : nat, Q : poly_complex]: ((ord_less_eq_nat @ (degree_complex @ P) @ N) => ((ord_less_eq_nat @ (degree_complex @ Q) @ N) => (ord_less_eq_nat @ (degree_complex @ (minus_174331535omplex @ P @ Q)) @ N)))))). % degree_diff_le
thf(fact_133_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A : poly_complex]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M2) @ (power_184595776omplex @ A @ N)))))). % le_imp_power_dvd
thf(fact_134_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A : complex]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M2) @ (power_power_complex @ A @ N)))))). % le_imp_power_dvd
thf(fact_135_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A : nat]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M2) @ (power_power_nat @ A @ N)))))). % le_imp_power_dvd
thf(fact_136_power__le__dvd, axiom,
    ((![A : poly_complex, N : nat, B : poly_complex, M2 : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M2) @ B)))))). % power_le_dvd
thf(fact_137_power__le__dvd, axiom,
    ((![A : complex, N : nat, B : complex, M2 : nat]: ((dvd_dvd_complex @ (power_power_complex @ A @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M2) @ B)))))). % power_le_dvd
thf(fact_138_power__le__dvd, axiom,
    ((![A : nat, N : nat, B : nat, M2 : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M2) @ B)))))). % power_le_dvd
thf(fact_139_dvd__power__le, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat, M2 : nat]: ((dvd_dvd_poly_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ M2))))))). % dvd_power_le
thf(fact_140_dvd__power__le, axiom,
    ((![X : complex, Y : complex, N : nat, M2 : nat]: ((dvd_dvd_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ M2))))))). % dvd_power_le
thf(fact_141_dvd__power__le, axiom,
    ((![X : nat, Y : nat, N : nat, M2 : nat]: ((dvd_dvd_nat @ X @ Y) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ M2))))))). % dvd_power_le
thf(fact_142_dvd__imp__degree__le, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((~ ((Q = zero_z1746442943omplex))) => (ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q))))))). % dvd_imp_degree_le
thf(fact_143_divides__degree, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q)) | (Q = zero_z1746442943omplex)))))). % divides_degree
thf(fact_144_dvd__imp__order__le, axiom,
    ((![Q : poly_complex, P : poly_complex, A : complex]: ((~ ((Q = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ P @ Q) => (ord_less_eq_nat @ (order_complex @ A @ P) @ (order_complex @ A @ Q))))))). % dvd_imp_order_le
thf(fact_145_is__unit__power__iff, axiom,
    ((![A : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A @ N) @ one_one_poly_complex) = (((dvd_dvd_poly_complex @ A @ one_one_poly_complex)) | ((N = zero_zero_nat))))))). % is_unit_power_iff
thf(fact_146_is__unit__power__iff, axiom,
    ((![A : nat, N : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ one_one_nat) = (((dvd_dvd_nat @ A @ one_one_nat)) | ((N = zero_zero_nat))))))). % is_unit_power_iff
thf(fact_147_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A2 : complex, P4 : poly_complex]: (~ ((X = (pCons_complex @ A2 @ P4)))))))))). % pderiv.cases
thf(fact_148_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A2 : complex, Q3 : poly_complex]: (~ ((P = (pCons_complex @ A2 @ Q3)))))))))). % pCons_cases
thf(fact_149_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_150_dvd__power__iff, axiom,
    ((![X : poly_complex, M2 : nat, N : nat]: ((~ ((X = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (power_184595776omplex @ X @ M2) @ (power_184595776omplex @ X @ N)) = (((dvd_dvd_poly_complex @ X @ one_one_poly_complex)) | ((ord_less_eq_nat @ M2 @ N)))))))). % dvd_power_iff
thf(fact_151_dvd__power__iff, axiom,
    ((![X : nat, M2 : nat, N : nat]: ((~ ((X = zero_zero_nat))) => ((dvd_dvd_nat @ (power_power_nat @ X @ M2) @ (power_power_nat @ X @ N)) = (((dvd_dvd_nat @ X @ one_one_nat)) | ((ord_less_eq_nat @ M2 @ N)))))))). % dvd_power_iff
thf(fact_152_dvd__iff__poly__eq__0, axiom,
    ((![C : poly_complex, P : poly_poly_complex]: ((dvd_dv598755940omplex @ (pCons_poly_complex @ C @ (pCons_poly_complex @ one_one_poly_complex @ zero_z1040703943omplex)) @ P) = ((poly_poly_complex2 @ P @ (uminus1138659839omplex @ C)) = zero_z1746442943omplex))))). % dvd_iff_poly_eq_0
thf(fact_153_dvd__iff__poly__eq__0, axiom,
    ((![C : complex, P : poly_complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ C @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ P) = ((poly_complex2 @ P @ (uminus1204672759omplex @ C)) = zero_zero_complex))))). % dvd_iff_poly_eq_0
thf(fact_154_poly__eq__0__iff__dvd, axiom,
    ((![P : poly_poly_complex, C : poly_complex]: (((poly_poly_complex2 @ P @ C) = zero_z1746442943omplex) = (dvd_dv598755940omplex @ (pCons_poly_complex @ (uminus1138659839omplex @ C) @ (pCons_poly_complex @ one_one_poly_complex @ zero_z1040703943omplex)) @ P))))). % poly_eq_0_iff_dvd
thf(fact_155_poly__eq__0__iff__dvd, axiom,
    ((![P : poly_complex, C : complex]: (((poly_complex2 @ P @ C) = zero_zero_complex) = (dvd_dvd_poly_complex @ (pCons_complex @ (uminus1204672759omplex @ C) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ P))))). % poly_eq_0_iff_dvd
thf(fact_156_order__1, axiom,
    ((![A : complex, P : poly_complex]: (dvd_dvd_poly_complex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ A @ P)) @ P)))). % order_1
thf(fact_157_order__divides, axiom,
    ((![A : complex, N : nat, P : poly_complex]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ N) @ P) = (((P = zero_z1746442943omplex)) | ((ord_less_eq_nat @ N @ (order_complex @ A @ P)))))))). % order_divides
thf(fact_158_order__decomp, axiom,
    ((![P : poly_complex, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (?[Q3 : poly_complex]: ((P = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ A @ P)) @ Q3)) & (~ ((dvd_dvd_poly_complex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ Q3))))))))). % order_decomp
thf(fact_159_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_160_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_161_one__neq__neg__one, axiom,
    ((~ ((one_one_complex = (uminus1204672759omplex @ one_one_complex)))))). % one_neq_neg_one
thf(fact_162_pderiv_Oinduct, axiom,
    ((![P5 : poly_complex > $o, A0 : poly_complex]: ((![A2 : complex, P4 : poly_complex]: (((~ ((P4 = zero_z1746442943omplex))) => (P5 @ P4)) => (P5 @ (pCons_complex @ A2 @ P4)))) => (P5 @ A0))))). % pderiv.induct
thf(fact_163_poly__induct2, axiom,
    ((![P5 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P5 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A2 : complex, P4 : poly_complex, B2 : complex, Q3 : poly_complex]: ((P5 @ P4 @ Q3) => (P5 @ (pCons_complex @ A2 @ P4) @ (pCons_complex @ B2 @ Q3)))) => (P5 @ P @ Q)))))). % poly_induct2
thf(fact_164_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_165_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_166_zero__neq__neg__one, axiom,
    ((~ ((zero_z1746442943omplex = (uminus1138659839omplex @ one_one_poly_complex)))))). % zero_neq_neg_one
thf(fact_167_zero__neq__neg__one, axiom,
    ((~ ((zero_zero_complex = (uminus1204672759omplex @ one_one_complex)))))). % zero_neq_neg_one
thf(fact_168_pCons__induct, axiom,
    ((![P5 : poly_poly_complex > $o, P : poly_poly_complex]: ((P5 @ zero_z1040703943omplex) => ((![A2 : poly_complex, P4 : poly_poly_complex]: (((~ ((A2 = zero_z1746442943omplex))) | (~ ((P4 = zero_z1040703943omplex)))) => ((P5 @ P4) => (P5 @ (pCons_poly_complex @ A2 @ P4))))) => (P5 @ P)))))). % pCons_induct
thf(fact_169_pCons__induct, axiom,
    ((![P5 : poly_nat > $o, P : poly_nat]: ((P5 @ zero_zero_poly_nat) => ((![A2 : nat, P4 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P4 = zero_zero_poly_nat)))) => ((P5 @ P4) => (P5 @ (pCons_nat @ A2 @ P4))))) => (P5 @ P)))))). % pCons_induct
thf(fact_170_pCons__induct, axiom,
    ((![P5 : poly_complex > $o, P : poly_complex]: ((P5 @ zero_z1746442943omplex) => ((![A2 : complex, P4 : poly_complex]: (((~ ((A2 = zero_zero_complex))) | (~ ((P4 = zero_z1746442943omplex)))) => ((P5 @ P4) => (P5 @ (pCons_complex @ A2 @ P4))))) => (P5 @ P)))))). % pCons_induct
thf(fact_171_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_172_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_173_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X4 : poly_complex]: ((poly_poly_complex2 @ P @ X4) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_174_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X4 : complex]: ((poly_complex2 @ P @ X4) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_175_degree__power__le, axiom,
    ((![P : poly_complex, N : nat]: (ord_less_eq_nat @ (degree_complex @ (power_184595776omplex @ P @ N)) @ (times_times_nat @ (degree_complex @ P) @ N))))). % degree_power_le
thf(fact_176_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_177_degree__eq__zeroE, axiom,
    ((![P : poly_complex]: (((degree_complex @ P) = zero_zero_nat) => (~ ((![A2 : complex]: (~ ((P = (pCons_complex @ A2 @ zero_z1746442943omplex))))))))))). % degree_eq_zeroE
thf(fact_178_degree__pCons__0, axiom,
    ((![A : complex]: ((degree_complex @ (pCons_complex @ A @ zero_z1746442943omplex)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_179_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_180_degree__mult__right__le, axiom,
    ((![Q : poly_complex, P : poly_complex]: ((~ ((Q = zero_z1746442943omplex))) => (ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ (times_1246143675omplex @ P @ Q))))))). % degree_mult_right_le
thf(fact_181_order__0I, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: ((~ (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex))) => ((order_poly_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_182_order__0I, axiom,
    ((![P : poly_complex, A : complex]: ((~ (((poly_complex2 @ P @ A) = zero_zero_complex))) => ((order_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_183_order__degree, axiom,
    ((![P : poly_complex, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (ord_less_eq_nat @ (order_complex @ A @ P) @ (degree_complex @ P)))))). % order_degree
thf(fact_184_order__root, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex) = (((P = zero_z1040703943omplex)) | ((~ (((order_poly_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_185_order__root, axiom,
    ((![P : poly_complex, A : complex]: (((poly_complex2 @ P @ A) = zero_zero_complex) = (((P = zero_z1746442943omplex)) | ((~ (((order_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_186_degree__linear__power, axiom,
    ((![A : nat, N : nat]: ((degree_nat @ (power_power_poly_nat @ (pCons_nat @ A @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat)) @ N)) = N)))). % degree_linear_power
thf(fact_187_degree__linear__power, axiom,
    ((![A : complex, N : nat]: ((degree_complex @ (power_184595776omplex @ (pCons_complex @ A @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ N)) = N)))). % degree_linear_power
thf(fact_188_order__power__n__n, axiom,
    ((![A : complex, N : nat]: ((order_complex @ A @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ N)) = N)))). % order_power_n_n
thf(fact_189_poly__root__induct, axiom,
    ((![Q4 : poly_poly_complex > $o, P5 : poly_complex > $o, P : poly_poly_complex]: ((Q4 @ zero_z1040703943omplex) => ((![P4 : poly_poly_complex]: ((![A3 : poly_complex]: ((P5 @ A3) => (~ (((poly_poly_complex2 @ P4 @ A3) = zero_z1746442943omplex))))) => (Q4 @ P4))) => ((![A2 : poly_complex, P4 : poly_poly_complex]: ((P5 @ A2) => ((Q4 @ P4) => (Q4 @ (times_1460995011omplex @ (pCons_poly_complex @ A2 @ (pCons_poly_complex @ (uminus1138659839omplex @ one_one_poly_complex) @ zero_z1040703943omplex)) @ P4))))) => (Q4 @ P))))))). % poly_root_induct
thf(fact_190_poly__root__induct, axiom,
    ((![Q4 : poly_complex > $o, P5 : complex > $o, P : poly_complex]: ((Q4 @ zero_z1746442943omplex) => ((![P4 : poly_complex]: ((![A3 : complex]: ((P5 @ A3) => (~ (((poly_complex2 @ P4 @ A3) = zero_zero_complex))))) => (Q4 @ P4))) => ((![A2 : complex, P4 : poly_complex]: ((P5 @ A2) => ((Q4 @ P4) => (Q4 @ (times_1246143675omplex @ (pCons_complex @ A2 @ (pCons_complex @ (uminus1204672759omplex @ one_one_complex) @ zero_z1746442943omplex)) @ P4))))) => (Q4 @ P))))))). % poly_root_induct
thf(fact_191_ap_I2_J, axiom,
    ((~ ((dvd_dvd_poly_complex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (suc @ (order_complex @ a @ pa))) @ pa))))). % ap(2)
thf(fact_192_diff__is__0__eq_H, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((minus_minus_nat @ M2 @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_193_diff__is__0__eq, axiom,
    ((![M2 : nat, N : nat]: (((minus_minus_nat @ M2 @ N) = zero_zero_nat) = (ord_less_eq_nat @ M2 @ N))))). % diff_is_0_eq
thf(fact_194_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_195_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_196_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_197_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_198_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_199_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_200_nat__dvd__1__iff__1, axiom,
    ((![M2 : nat]: ((dvd_dvd_nat @ M2 @ one_one_nat) = (M2 = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_201_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_212_mult__zero__right, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right
thf(fact_213_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_214_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_215_mult__zero__left, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_216_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_217_dvd__0__right, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_218_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_219_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_220_dvd__0__left__iff, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) = (A = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_221_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_222_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_223_mult__minus__right, axiom,
    ((![A : poly_complex, B : poly_complex]: ((times_1246143675omplex @ A @ (uminus1138659839omplex @ B)) = (uminus1138659839omplex @ (times_1246143675omplex @ A @ B)))))). % mult_minus_right
thf(fact_224_mult__minus__right, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ A @ (uminus1204672759omplex @ B)) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_right
thf(fact_225_minus__mult__minus, axiom,
    ((![A : poly_complex, B : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ A) @ (uminus1138659839omplex @ B)) = (times_1246143675omplex @ A @ B))))). % minus_mult_minus
thf(fact_226_minus__mult__minus, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (times_times_complex @ A @ B))))). % minus_mult_minus
thf(fact_227_mult__minus__left, axiom,
    ((![A : poly_complex, B : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ A) @ B) = (uminus1138659839omplex @ (times_1246143675omplex @ A @ B)))))). % mult_minus_left
thf(fact_228_mult__minus__left, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_left
thf(fact_229_dvd__minus__iff, axiom,
    ((![X : poly_complex, Y : poly_complex]: ((dvd_dvd_poly_complex @ X @ (uminus1138659839omplex @ Y)) = (dvd_dvd_poly_complex @ X @ Y))))). % dvd_minus_iff
thf(fact_230_dvd__minus__iff, axiom,
    ((![X : complex, Y : complex]: ((dvd_dvd_complex @ X @ (uminus1204672759omplex @ Y)) = (dvd_dvd_complex @ X @ Y))))). % dvd_minus_iff
thf(fact_231_minus__dvd__iff, axiom,
    ((![X : poly_complex, Y : poly_complex]: ((dvd_dvd_poly_complex @ (uminus1138659839omplex @ X) @ Y) = (dvd_dvd_poly_complex @ X @ Y))))). % minus_dvd_iff
thf(fact_232_minus__dvd__iff, axiom,
    ((![X : complex, Y : complex]: ((dvd_dvd_complex @ (uminus1204672759omplex @ X) @ Y) = (dvd_dvd_complex @ X @ Y))))). % minus_dvd_iff
thf(fact_233_dvd__1__left, axiom,
    ((![K : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K)))). % dvd_1_left
thf(fact_234_dvd__1__iff__1, axiom,
    ((![M2 : nat]: ((dvd_dvd_nat @ M2 @ (suc @ zero_zero_nat)) = (M2 = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_235_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0

% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $true @ X @ Y) = X)))).
thf(help_If_3_1_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_T, axiom,
    ((![P5 : $o]: ((P5 = $true) | (P5 = $false))))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_T, axiom,
    ((![X : poly_complex, Y : poly_complex]: ((if_poly_complex @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_T, axiom,
    ((![X : poly_complex, Y : poly_complex]: ((if_poly_complex @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[K2 : poly_complex]: ((power_184595776omplex @ qa @ na) = (times_1246143675omplex @ pa @ K2))))).
