% 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_941__5376832_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:29.859

% 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_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_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_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_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).

% Relevant facts (220)
thf(fact_0_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_1_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_2_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_3_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_4_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_5_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_6_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_7_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_8_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_9__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_10_r, axiom,
    ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ r)))). % r
thf(fact_11__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_12_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_13_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_14_power__eq__0__iff, axiom,
    ((![A : poly_complex, N : nat]: (((power_184595776omplex @ A @ N) = zero_z1746442943omplex) = (((A = zero_z1746442943omplex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_15_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_16_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_17_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_18_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_19_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_20_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_21_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_22_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_23_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_24_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_25_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_26_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_27_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_28_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_29_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_30_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_31_mult__minus1, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ Z) = (uminus1138659839omplex @ Z))))). % mult_minus1
thf(fact_32_mult__minus1, axiom,
    ((![Z : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z) = (uminus1204672759omplex @ Z))))). % mult_minus1
thf(fact_33_power__one__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_34_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_35_degree__minus, axiom,
    ((![P : poly_complex]: ((degree_complex @ (uminus1138659839omplex @ P)) = (degree_complex @ P))))). % degree_minus
thf(fact_36_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_37_order__uminus, axiom,
    ((![X : complex, P : poly_complex]: ((order_complex @ X @ (uminus1138659839omplex @ P)) = (order_complex @ X @ P))))). % order_uminus
thf(fact_38_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_39_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_40_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_41_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_42_minus__pCons, axiom,
    ((![A : complex, P : poly_complex]: ((uminus1138659839omplex @ (pCons_complex @ A @ P)) = (pCons_complex @ (uminus1204672759omplex @ A) @ (uminus1138659839omplex @ P)))))). % minus_pCons
thf(fact_43__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_44_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_45_mult__minus1__right, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ Z @ (uminus1138659839omplex @ one_one_poly_complex)) = (uminus1138659839omplex @ Z))))). % mult_minus1_right
thf(fact_46_mult__minus1__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z))))). % mult_minus1_right
thf(fact_47_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_48_order__1__eq__0, axiom,
    ((![X : complex]: ((order_complex @ X @ one_one_poly_complex) = zero_zero_nat)))). % order_1_eq_0
thf(fact_49_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_50_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_51_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_52_True, axiom,
    ((?[A2 : complex]: ((poly_complex2 @ pa @ A2) = zero_zero_complex)))). % True
thf(fact_53_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_54_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_55_IH, axiom,
    ((![M2 : nat]: ((ord_less_nat @ M2 @ na) => (![P2 : poly_complex, Q2 : poly_complex]: ((![X3 : complex]: (((poly_complex2 @ P2 @ X3) = zero_zero_complex) => ((poly_complex2 @ Q2 @ X3) = zero_zero_complex))) => (((degree_complex @ P2) = M2) => ((~ ((M2 = zero_zero_nat))) => (dvd_dvd_poly_complex @ P2 @ (power_184595776omplex @ Q2 @ M2)))))))))). % IH
thf(fact_56_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_57_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_58_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_59_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_73_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_74_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_75_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_76_dvd__power, axiom,
    ((![N : nat, X : complex]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_complex)) => (dvd_dvd_complex @ X @ (power_power_complex @ X @ N)))))). % dvd_power
thf(fact_77_dvd__power, axiom,
    ((![N : nat, X : poly_complex]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_poly_complex)) => (dvd_dvd_poly_complex @ X @ (power_184595776omplex @ X @ N)))))). % dvd_power
thf(fact_78_dvd__power, axiom,
    ((![N : nat, X : nat]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_nat)) => (dvd_dvd_nat @ X @ (power_power_nat @ X @ N)))))). % dvd_power
thf(fact_79_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A2 : complex, P3 : poly_complex]: (~ ((X = (pCons_complex @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_80_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A2 : complex, Q3 : poly_complex]: (~ ((P = (pCons_complex @ A2 @ Q3)))))))))). % pCons_cases
thf(fact_81_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_82_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_83_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_84_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_85_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_86_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_one__neq__neg__one, axiom,
    ((~ ((one_one_complex = (uminus1204672759omplex @ one_one_complex)))))). % one_neq_neg_one
thf(fact_95_pderiv_Oinduct, axiom,
    ((![P4 : poly_complex > $o, A0 : poly_complex]: ((![A2 : complex, P3 : poly_complex]: (((~ ((P3 = zero_z1746442943omplex))) => (P4 @ P3)) => (P4 @ (pCons_complex @ A2 @ P3)))) => (P4 @ A0))))). % pderiv.induct
thf(fact_96_poly__induct2, axiom,
    ((![P4 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P4 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A2 : complex, P3 : poly_complex, B2 : complex, Q3 : poly_complex]: ((P4 @ P3 @ Q3) => (P4 @ (pCons_complex @ A2 @ P3) @ (pCons_complex @ B2 @ Q3)))) => (P4 @ P @ Q)))))). % poly_induct2
thf(fact_97_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_98_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_99_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_100_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_101_zero__neq__neg__one, axiom,
    ((~ ((zero_z1746442943omplex = (uminus1138659839omplex @ one_one_poly_complex)))))). % zero_neq_neg_one
thf(fact_102_zero__neq__neg__one, axiom,
    ((~ ((zero_zero_complex = (uminus1204672759omplex @ one_one_complex)))))). % zero_neq_neg_one
thf(fact_103_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_104_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_105_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_106_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_107_power__0, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ zero_zero_nat) = one_one_poly_complex)))). % power_0
thf(fact_108_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_109_pCons__induct, axiom,
    ((![P4 : poly_poly_complex > $o, P : poly_poly_complex]: ((P4 @ zero_z1040703943omplex) => ((![A2 : poly_complex, P3 : poly_poly_complex]: (((~ ((A2 = zero_z1746442943omplex))) | (~ ((P3 = zero_z1040703943omplex)))) => ((P4 @ P3) => (P4 @ (pCons_poly_complex @ A2 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_110_pCons__induct, axiom,
    ((![P4 : poly_nat > $o, P : poly_nat]: ((P4 @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P4 @ P3) => (P4 @ (pCons_nat @ A2 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_111_pCons__induct, axiom,
    ((![P4 : poly_complex > $o, P : poly_complex]: ((P4 @ zero_z1746442943omplex) => ((![A2 : complex, P3 : poly_complex]: (((~ ((A2 = zero_zero_complex))) | (~ ((P3 = zero_z1746442943omplex)))) => ((P4 @ P3) => (P4 @ (pCons_complex @ A2 @ P3))))) => (P4 @ P)))))). % pCons_induct
thf(fact_112_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_113_power__less__power__Suc, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_less_power_Suc
thf(fact_114_power__gt1__lemma, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ one_one_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_gt1_lemma
thf(fact_115_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_116_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_117_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_118_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_119_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_120_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_121_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_122_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_123_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_184595776omplex @ zero_z1746442943omplex @ N) = zero_z1746442943omplex))))). % zero_power
thf(fact_124_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_125_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_126_degree__pCons__0, axiom,
    ((![A : complex]: ((degree_complex @ (pCons_complex @ A @ zero_z1746442943omplex)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_127_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_128_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_129_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_130_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_131_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_132_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_133_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_134_pow__divides__pow__iff, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (dvd_dvd_nat @ A @ B)))))). % pow_divides_pow_iff
thf(fact_135_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_136_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_137_dvd__times__right__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_138_dvd__times__right__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ B @ A) @ (times_1246143675omplex @ C @ A)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_139_dvd__times__left__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_140_dvd__times__left__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ (times_1246143675omplex @ A @ C)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_141_dvd__mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((dvd_dvd_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((dvd_dvd_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_142_dvd__mult__cancel__right, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ C) @ (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((dvd_dvd_poly_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_143_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_144_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_145_assms_I1_J, axiom,
    ((![X2 : complex]: (((poly_complex2 @ p @ X2) = zero_zero_complex) => ((poly_complex2 @ q @ X2) = zero_zero_complex))))). % assms(1)
thf(fact_146__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_147_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_148_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_149_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_150_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_151_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_152_mult__zero__left, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_153_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_154_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_155_mult__zero__right, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_add_Oinverse__neutral, axiom,
    (((uminus1138659839omplex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % add.inverse_neutral
thf(fact_166_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_167_neg__0__equal__iff__equal, axiom,
    ((![A : poly_complex]: ((zero_z1746442943omplex = (uminus1138659839omplex @ A)) = (zero_z1746442943omplex = A))))). % neg_0_equal_iff_equal
thf(fact_168_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_169_neg__equal__0__iff__equal, axiom,
    ((![A : poly_complex]: (((uminus1138659839omplex @ A) = zero_z1746442943omplex) = (A = zero_z1746442943omplex))))). % neg_equal_0_iff_equal
thf(fact_170_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_171_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_172_mult_Oleft__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ one_one_poly_complex @ A) = A)))). % mult.left_neutral
thf(fact_173_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_174_mult_Oright__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ one_one_poly_complex) = A)))). % mult.right_neutral
thf(fact_175_dvd__0__right, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_176_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_177_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_178_dvd__0__left__iff, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) = (A = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_179_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_180_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_dvd__minus__iff, axiom,
    ((![X : complex, Y : complex]: ((dvd_dvd_complex @ X @ (uminus1204672759omplex @ Y)) = (dvd_dvd_complex @ X @ Y))))). % dvd_minus_iff
thf(fact_189_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_190_minus__dvd__iff, axiom,
    ((![X : complex, Y : complex]: ((dvd_dvd_complex @ (uminus1204672759omplex @ X) @ Y) = (dvd_dvd_complex @ X @ Y))))). % minus_dvd_iff
thf(fact_191_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_192_mult__cancel__left1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_left1
thf(fact_193_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_194_mult__cancel__left2, axiom,
    ((![C : poly_complex, A : poly_complex]: (((times_1246143675omplex @ C @ A) = C) = (((C = zero_z1746442943omplex)) | ((A = one_one_poly_complex))))))). % mult_cancel_left2
thf(fact_195_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_196_mult__cancel__right1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_right1
thf(fact_197_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_198_mult__cancel__right2, axiom,
    ((![A : poly_complex, C : poly_complex]: (((times_1246143675omplex @ A @ C) = C) = (((C = zero_z1746442943omplex)) | ((A = one_one_poly_complex))))))). % mult_cancel_right2
thf(fact_199_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N)) = zero_zero_complex)))). % power_0_Suc
thf(fact_200_power__0__Suc, axiom,
    ((![N : nat]: ((power_184595776omplex @ zero_z1746442943omplex @ (suc @ N)) = zero_z1746442943omplex)))). % power_0_Suc
thf(fact_201_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_202_power__Suc0__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_203_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_204_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_205_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_206_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_207_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_208_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_209_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_210_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_211_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_212_poly__minus, axiom,
    ((![P : poly_complex, X : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P) @ X) = (uminus1204672759omplex @ (poly_complex2 @ P @ X)))))). % poly_minus
thf(fact_213_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_214_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_215_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_216_degree__pCons__eq__if, axiom,
    ((![P : poly_complex, A : complex]: (((P = zero_z1746442943omplex) => ((degree_complex @ (pCons_complex @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_z1746442943omplex))) => ((degree_complex @ (pCons_complex @ A @ P)) = (suc @ (degree_complex @ P)))))))). % degree_pCons_eq_if
thf(fact_217_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_218_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_219_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum

% Conjectures (5)
thf(conj_0, hypothesis,
    ($true)).
thf(conj_1, hypothesis,
    ((ord_less_nat @ zero_zero_nat @ (order_complex @ a @ pa)))).
thf(conj_2, hypothesis,
    ((ord_less_nat @ zero_zero_nat @ (degree_complex @ s)))).
thf(conj_3, hypothesis,
    ((na = (degree_complex @ pa)))).
thf(conj_4, conjecture,
    ((ord_less_nat @ (degree_complex @ s) @ (degree_complex @ (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ s))))).
