% 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_926__5376748_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:15.015

% 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 (45)
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__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_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__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_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > 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_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_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    divide1187762952omplex : poly_complex > poly_complex > 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__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__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_k____, type,
    k : 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).

% Relevant facts (177)
thf(fact_0_False, axiom,
    ((~ ((qa = zero_z1746442943omplex))))). % False
thf(fact_1_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_2_kpn, axiom,
    ((s = (pCons_complex @ k @ zero_z1746442943omplex)))). % kpn
thf(fact_3_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_4__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_5_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_6_r, axiom,
    ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ r)))). % r
thf(fact_7_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_8_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_9__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_10__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_O_As_A_061_A_091_058k_058_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![K : complex]: (~ ((s = (pCons_complex @ K @ zero_z1746442943omplex))))))))). % \<open>\<And>thesis. (\<And>k. s = [:k:] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_11_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_nonzero__divide__mult__cancel__left, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_19_nonzero__divide__mult__cancel__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ B @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ A)))))). % nonzero_divide_mult_cancel_right
thf(fact_20_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_21_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_22_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_23_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_24_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_25_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_26_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_27_True, axiom,
    (((degree_complex @ s) = zero_zero_nat))). % True
thf(fact_28_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_29_k, axiom,
    ((~ ((k = zero_zero_complex))))). % k
thf(fact_30_power__one__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_31_degree__minus, axiom,
    ((![P : poly_complex]: ((degree_complex @ (uminus1138659839omplex @ P)) = (degree_complex @ P))))). % degree_minus
thf(fact_32_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_33_order__uminus, axiom,
    ((![X : complex, P : poly_complex]: ((order_complex @ X @ (uminus1138659839omplex @ P)) = (order_complex @ X @ P))))). % order_uminus
thf(fact_34_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_35_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_36_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_37_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_38_times__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_39_divide__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_40_divide__divide__eq__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_41_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A) = (divide1210191872omplex @ (times_times_complex @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_42_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_43_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_44_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_45_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_46_minus__pCons, axiom,
    ((![A : complex, P : poly_complex]: ((uminus1138659839omplex @ (pCons_complex @ A @ P)) = (pCons_complex @ (uminus1204672759omplex @ A) @ (uminus1138659839omplex @ P)))))). % minus_pCons
thf(fact_47__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_48_diff__numeral__special_I9_J, axiom,
    (((minus_174331535omplex @ one_one_poly_complex @ one_one_poly_complex) = zero_z1746442943omplex))). % diff_numeral_special(9)
thf(fact_49_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_50_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_51_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_52_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_53_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_54_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_55_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_56_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_57_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_58_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_59_mult__minus1__right, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ Z @ (uminus1138659839omplex @ one_one_poly_complex)) = (uminus1138659839omplex @ Z))))). % mult_minus1_right
thf(fact_60_mult__minus1__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z))))). % mult_minus1_right
thf(fact_61_mult__minus1, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ Z) = (uminus1138659839omplex @ Z))))). % mult_minus1
thf(fact_62_mult__minus1, axiom,
    ((![Z : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z) = (uminus1204672759omplex @ Z))))). % mult_minus1
thf(fact_63_divide__minus1, axiom,
    ((![X : complex]: ((divide1210191872omplex @ X @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ X))))). % divide_minus1
thf(fact_64_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_65_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_66_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_67_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_68_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_69_order__1__eq__0, axiom,
    ((![X : complex]: ((order_complex @ X @ one_one_poly_complex) = zero_zero_nat)))). % order_1_eq_0
thf(fact_70_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_71_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_72_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_73_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_74_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_75_dvd__field__iff, axiom,
    ((dvd_dvd_complex = (^[A2 : complex]: (^[B2 : complex]: (((A2 = zero_zero_complex)) => ((B2 = zero_zero_complex)))))))). % dvd_field_iff
thf(fact_76_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_77_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_78_divide__poly__0, axiom,
    ((![F : poly_complex]: ((divide1187762952omplex @ F @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % divide_poly_0
thf(fact_79_poly__div__mult__right, axiom,
    ((![X : poly_complex, Y : poly_complex, Z : poly_complex]: ((divide1187762952omplex @ X @ (times_1246143675omplex @ Y @ Z)) = (divide1187762952omplex @ (divide1187762952omplex @ X @ Y) @ Z))))). % poly_div_mult_right
thf(fact_80_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_81_degree__eq__zeroE, axiom,
    ((![P : poly_complex]: (((degree_complex @ P) = zero_zero_nat) => (~ ((![A3 : complex]: (~ ((P = (pCons_complex @ A3 @ zero_z1746442943omplex))))))))))). % degree_eq_zeroE
thf(fact_82_degree__pCons__0, axiom,
    ((![A : complex]: ((degree_complex @ (pCons_complex @ A @ zero_z1746442943omplex)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_div__power, axiom,
    ((![B : poly_complex, A : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ B @ A) => ((power_184595776omplex @ (divide1187762952omplex @ A @ B) @ N) = (divide1187762952omplex @ (power_184595776omplex @ A @ N) @ (power_184595776omplex @ B @ N))))))). % div_power
thf(fact_94_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_95_power__0, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ zero_zero_nat) = one_one_poly_complex)))). % power_0
thf(fact_96_divide__poly, axiom,
    ((![G : poly_complex, F : poly_complex]: ((~ ((G = zero_z1746442943omplex))) => ((divide1187762952omplex @ (times_1246143675omplex @ F @ G) @ G) = F))))). % divide_poly
thf(fact_97_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_98_power__eq__if, axiom,
    ((power_power_complex = (^[P2 : complex]: (^[M2 : nat]: (if_complex @ (M2 = zero_zero_nat) @ one_one_complex @ (times_times_complex @ P2 @ (power_power_complex @ P2 @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_99_power__eq__if, axiom,
    ((power_184595776omplex = (^[P2 : poly_complex]: (^[M2 : nat]: (if_poly_complex @ (M2 = zero_zero_nat) @ one_one_poly_complex @ (times_1246143675omplex @ P2 @ (power_184595776omplex @ P2 @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_100_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A3 : complex, P3 : poly_complex]: (~ ((X = (pCons_complex @ A3 @ P3)))))))))). % pderiv.cases
thf(fact_101_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A3 : complex, Q2 : poly_complex]: (~ ((P = (pCons_complex @ A3 @ Q2)))))))))). % pCons_cases
thf(fact_102_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_103_order__decomp, axiom,
    ((![P : poly_complex, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (?[Q2 : poly_complex]: ((P = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ A @ P)) @ Q2)) & (~ ((dvd_dvd_poly_complex @ (pCons_complex @ (uminus1204672759omplex @ A) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ Q2))))))))). % order_decomp
thf(fact_104_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_105_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_106_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_107_divide__divide__eq__left_H, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ C @ B)))))). % divide_divide_eq_left'
thf(fact_108_divide__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ W) @ (times_times_complex @ Y @ Z)))))). % divide_divide_times_eq
thf(fact_109_times__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((times_times_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ Z) @ (times_times_complex @ Y @ W)))))). % times_divide_times_eq
thf(fact_110_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_111_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_112_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_113_one__neq__neg__one, axiom,
    ((~ ((one_one_complex = (uminus1204672759omplex @ one_one_complex)))))). % one_neq_neg_one
thf(fact_114_diff__divide__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (divide1210191872omplex @ A @ C) @ (divide1210191872omplex @ B @ C)))))). % diff_divide_distrib
thf(fact_115_minus__divide__left, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (divide1210191872omplex @ A @ B)) = (divide1210191872omplex @ (uminus1204672759omplex @ A) @ B))))). % minus_divide_left
thf(fact_116_minus__divide__divide, axiom,
    ((![A : complex, B : complex]: ((divide1210191872omplex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (divide1210191872omplex @ A @ B))))). % minus_divide_divide
thf(fact_117_minus__divide__right, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (divide1210191872omplex @ A @ B)) = (divide1210191872omplex @ A @ (uminus1204672759omplex @ B)))))). % minus_divide_right
thf(fact_118_power__divide, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_119_pderiv_Oinduct, axiom,
    ((![P4 : poly_complex > $o, A0 : poly_complex]: ((![A3 : complex, P3 : poly_complex]: (((~ ((P3 = zero_z1746442943omplex))) => (P4 @ P3)) => (P4 @ (pCons_complex @ A3 @ P3)))) => (P4 @ A0))))). % pderiv.induct
thf(fact_120_poly__induct2, axiom,
    ((![P4 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P4 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A3 : complex, P3 : poly_complex, B3 : complex, Q2 : poly_complex]: ((P4 @ P3 @ Q2) => (P4 @ (pCons_complex @ A3 @ P3) @ (pCons_complex @ B3 @ Q2)))) => (P4 @ P @ Q)))))). % poly_induct2
thf(fact_121_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_122_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_123_nonzero__eq__divide__eq, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((A = (divide1210191872omplex @ B @ C)) = ((times_times_complex @ A @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_124_nonzero__divide__eq__eq, axiom,
    ((![C : complex, B : complex, A : complex]: ((~ ((C = zero_zero_complex))) => (((divide1210191872omplex @ B @ C) = A) = (B = (times_times_complex @ A @ C))))))). % nonzero_divide_eq_eq
thf(fact_125_eq__divide__imp, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = B) => (A = (divide1210191872omplex @ B @ C))))))). % eq_divide_imp
thf(fact_126_divide__eq__imp, axiom,
    ((![C : complex, B : complex, A : complex]: ((~ ((C = zero_zero_complex))) => ((B = (times_times_complex @ A @ C)) => ((divide1210191872omplex @ B @ C) = A)))))). % divide_eq_imp
thf(fact_127_eq__divide__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((A = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ A @ C) = B)))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_divide_eq
thf(fact_128_divide__eq__eq, axiom,
    ((![B : complex, C : complex, A : complex]: (((divide1210191872omplex @ B @ C) = A) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ A @ C))))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % divide_eq_eq
thf(fact_129_frac__eq__eq, axiom,
    ((![Y : complex, Z : complex, X : complex, W : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z = zero_zero_complex))) => (((divide1210191872omplex @ X @ Y) = (divide1210191872omplex @ W @ Z)) = ((times_times_complex @ X @ Z) = (times_times_complex @ W @ Y)))))))). % frac_eq_eq
thf(fact_130_right__inverse__eq, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => (((divide1210191872omplex @ A @ B) = one_one_complex) = (A = B)))))). % right_inverse_eq
thf(fact_131_zero__neq__neg__one, axiom,
    ((~ ((zero_z1746442943omplex = (uminus1138659839omplex @ one_one_poly_complex)))))). % zero_neq_neg_one
thf(fact_132_zero__neq__neg__one, axiom,
    ((~ ((zero_zero_complex = (uminus1204672759omplex @ one_one_complex)))))). % zero_neq_neg_one
thf(fact_133_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_134_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_135_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_136_nonzero__minus__divide__divide, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_minus_divide_divide
thf(fact_137_nonzero__minus__divide__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((uminus1204672759omplex @ (divide1210191872omplex @ A @ B)) = (divide1210191872omplex @ A @ (uminus1204672759omplex @ B))))))). % nonzero_minus_divide_right
thf(fact_138_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_139_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_140_power__one__over, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ one_one_complex @ A) @ N) = (divide1210191872omplex @ one_one_complex @ (power_power_complex @ A @ N)))))). % power_one_over
thf(fact_141_pCons__induct, axiom,
    ((![P4 : poly_poly_complex > $o, P : poly_poly_complex]: ((P4 @ zero_z1040703943omplex) => ((![A3 : poly_complex, P3 : poly_poly_complex]: (((~ ((A3 = zero_z1746442943omplex))) | (~ ((P3 = zero_z1040703943omplex)))) => ((P4 @ P3) => (P4 @ (pCons_poly_complex @ A3 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_142_pCons__induct, axiom,
    ((![P4 : poly_nat > $o, P : poly_nat]: ((P4 @ zero_zero_poly_nat) => ((![A3 : nat, P3 : poly_nat]: (((~ ((A3 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P4 @ P3) => (P4 @ (pCons_nat @ A3 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_143_pCons__induct, axiom,
    ((![P4 : poly_complex > $o, P : poly_complex]: ((P4 @ zero_z1746442943omplex) => ((![A3 : complex, P3 : poly_complex]: (((~ ((A3 = zero_zero_complex))) | (~ ((P3 = zero_z1746442943omplex)))) => ((P4 @ P3) => (P4 @ (pCons_complex @ A3 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_144_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_145_divide__diff__eq__iff, axiom,
    ((![Z : complex, X : complex, Y : complex]: ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ (divide1210191872omplex @ X @ Z) @ Y) = (divide1210191872omplex @ (minus_minus_complex @ X @ (times_times_complex @ Y @ Z)) @ Z)))))). % divide_diff_eq_iff
thf(fact_146_diff__divide__eq__iff, axiom,
    ((![Z : complex, X : complex, Y : complex]: ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ X @ (divide1210191872omplex @ Y @ Z)) = (divide1210191872omplex @ (minus_minus_complex @ (times_times_complex @ X @ Z) @ Y) @ Z)))))). % diff_divide_eq_iff
thf(fact_147_diff__frac__eq, axiom,
    ((![Y : complex, Z : complex, X : complex, W : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ W @ Z)) = (divide1210191872omplex @ (minus_minus_complex @ (times_times_complex @ X @ Z) @ (times_times_complex @ W @ Y)) @ (times_times_complex @ Y @ Z)))))))). % diff_frac_eq
thf(fact_148_add__divide__eq__if__simps_I4_J, axiom,
    ((![Z : complex, A : complex, B : complex]: (((Z = zero_zero_complex) => ((minus_minus_complex @ A @ (divide1210191872omplex @ B @ Z)) = A)) & ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ A @ (divide1210191872omplex @ B @ Z)) = (divide1210191872omplex @ (minus_minus_complex @ (times_times_complex @ A @ Z) @ B) @ Z))))))). % add_divide_eq_if_simps(4)
thf(fact_149_nonzero__neg__divide__eq__eq2, axiom,
    ((![B : complex, C : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((C = (uminus1204672759omplex @ (divide1210191872omplex @ A @ B))) = ((times_times_complex @ C @ B) = (uminus1204672759omplex @ A))))))). % nonzero_neg_divide_eq_eq2
thf(fact_150_nonzero__neg__divide__eq__eq, axiom,
    ((![B : complex, A : complex, C : complex]: ((~ ((B = zero_zero_complex))) => (((uminus1204672759omplex @ (divide1210191872omplex @ A @ B)) = C) = ((uminus1204672759omplex @ A) = (times_times_complex @ C @ B))))))). % nonzero_neg_divide_eq_eq
thf(fact_151_minus__divide__eq__eq, axiom,
    ((![B : complex, C : complex, A : complex]: (((uminus1204672759omplex @ (divide1210191872omplex @ B @ C)) = A) = (((((~ ((C = zero_zero_complex)))) => (((uminus1204672759omplex @ B) = (times_times_complex @ A @ C))))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % minus_divide_eq_eq
thf(fact_152_eq__minus__divide__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((A = (uminus1204672759omplex @ (divide1210191872omplex @ B @ C))) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ A @ C) = (uminus1204672759omplex @ B))))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_minus_divide_eq
thf(fact_153_divide__eq__minus__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = (uminus1204672759omplex @ one_one_complex)) = (((~ ((B = zero_zero_complex)))) & ((A = (uminus1204672759omplex @ B)))))))). % divide_eq_minus_1_iff
thf(fact_154_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_155_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_156_minus__divide__diff__eq__iff, axiom,
    ((![Z : complex, X : complex, Y : complex]: ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ (uminus1204672759omplex @ (divide1210191872omplex @ X @ Z)) @ Y) = (divide1210191872omplex @ (minus_minus_complex @ (uminus1204672759omplex @ X) @ (times_times_complex @ Y @ Z)) @ Z)))))). % minus_divide_diff_eq_iff
thf(fact_157_add__divide__eq__if__simps_I5_J, axiom,
    ((![Z : complex, A : complex, B : complex]: (((Z = zero_zero_complex) => ((minus_minus_complex @ (divide1210191872omplex @ A @ Z) @ B) = (uminus1204672759omplex @ B))) & ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ (divide1210191872omplex @ A @ Z) @ B) = (divide1210191872omplex @ (minus_minus_complex @ A @ (times_times_complex @ B @ Z)) @ Z))))))). % add_divide_eq_if_simps(5)
thf(fact_158_add__divide__eq__if__simps_I6_J, axiom,
    ((![Z : complex, A : complex, B : complex]: (((Z = zero_zero_complex) => ((minus_minus_complex @ (uminus1204672759omplex @ (divide1210191872omplex @ A @ Z)) @ B) = (uminus1204672759omplex @ B))) & ((~ ((Z = zero_zero_complex))) => ((minus_minus_complex @ (uminus1204672759omplex @ (divide1210191872omplex @ A @ Z)) @ B) = (divide1210191872omplex @ (minus_minus_complex @ (uminus1204672759omplex @ A) @ (times_times_complex @ B @ Z)) @ Z))))))). % add_divide_eq_if_simps(6)
thf(fact_159_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_160_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_161_unit__div__mult__self, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ one_one_poly_complex) => ((times_1246143675omplex @ (divide1187762952omplex @ B @ A) @ A) = B))))). % unit_div_mult_self
thf(fact_162_unit__mult__div__div, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ one_one_poly_complex) => ((times_1246143675omplex @ B @ (divide1187762952omplex @ one_one_poly_complex @ A)) = (divide1187762952omplex @ B @ A)))))). % unit_mult_div_div
thf(fact_163_div__diff, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ C @ A) => ((dvd_dvd_poly_complex @ C @ B) => ((divide1187762952omplex @ (minus_174331535omplex @ A @ B) @ C) = (minus_174331535omplex @ (divide1187762952omplex @ A @ C) @ (divide1187762952omplex @ B @ C)))))))). % div_diff
thf(fact_164_unit__div__1__div__1, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ A @ one_one_poly_complex) => ((divide1187762952omplex @ one_one_poly_complex @ (divide1187762952omplex @ one_one_poly_complex @ A)) = A))))). % unit_div_1_div_1
thf(fact_165_unit__div__1__unit, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ A @ one_one_poly_complex) => (dvd_dvd_poly_complex @ (divide1187762952omplex @ one_one_poly_complex @ A) @ one_one_poly_complex))))). % unit_div_1_unit
thf(fact_166_unit__div, 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 @ (divide1187762952omplex @ A @ B) @ one_one_poly_complex)))))). % unit_div
thf(fact_167_dvd__mult__div__cancel, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((times_1246143675omplex @ A @ (divide1187762952omplex @ B @ A)) = B))))). % dvd_mult_div_cancel
thf(fact_168_dvd__div__mult__self, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((times_1246143675omplex @ (divide1187762952omplex @ B @ A) @ A) = B))))). % dvd_div_mult_self
thf(fact_169_assms_I1_J, axiom,
    ((![X2 : complex]: (((poly_complex2 @ p @ X2) = zero_zero_complex) => ((poly_complex2 @ q @ X2) = zero_zero_complex))))). % assms(1)
thf(fact_170__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,
    ((~ ((![A3 : complex]: (~ (((poly_complex2 @ pa @ A3) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>a. poly p a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_171_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_172_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_173_mult__zero__left, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_174_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_175_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_176_mult__zero__right, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right

% Helper facts (5)
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,
    ((![P4 : $o]: ((P4 = $true) | (P4 = $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,
    (((power_184595776omplex @ (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ r) @ na) = (times_1246143675omplex @ pa @ (times_1246143675omplex @ (times_1246143675omplex @ (pCons_complex @ (divide1210191872omplex @ one_one_complex @ k) @ zero_z1746442943omplex) @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (minus_minus_nat @ na @ (order_complex @ a @ pa)))) @ (power_184595776omplex @ r @ na)))))).
