% 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_961__5376958_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:44.857

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    poly_p1267267526omplex : $tType).
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__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    poly_poly_nat : $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 (75)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Nat__Onat, type,
    fundam1567013434ze_nat : poly_nat > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    fundam1956464160omplex : poly_poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_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_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_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    zero_z1200043727omplex : poly_p1267267526omplex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    degree_poly_complex : poly_poly_complex > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    degree_poly_nat : poly_poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    degree2006505739omplex : poly_p1267267526omplex > nat).
thf(sy_c_Polynomial_Ois__zero_001t__Complex__Ocomplex, type,
    is_zero_complex : poly_complex > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat, type,
    is_zero_nat : poly_nat > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    is_zero_poly_complex : poly_poly_complex > $o).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    order_poly_complex : poly_complex > poly_poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    order_1735763309omplex : poly_poly_complex > poly_p1267267526omplex > nat).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_p282434315omplex : poly_p1267267526omplex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Complex__Ocomplex, type,
    coeff_complex : poly_complex > nat > complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    coeff_poly_complex : poly_poly_complex > nat > poly_complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    coeff_poly_nat : poly_poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    coeff_1429652124omplex : poly_p1267267526omplex > nat > poly_poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Complex__Ocomplex, type,
    poly_cutoff_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat, type,
    poly_cutoff_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_c622223248omplex : nat > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_cutoff_poly_nat : nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_c306577560omplex : nat > poly_p1267267526omplex > poly_p1267267526omplex).
thf(sy_c_Polynomial_Opoly__shift_001t__Complex__Ocomplex, type,
    poly_shift_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat, type,
    poly_shift_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_s558570093omplex : nat > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    reflec1997789704omplex : poly_p1267267526omplex > poly_p1267267526omplex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Complex__Ocomplex, type,
    synthe151143547omplex : poly_complex > complex > poly_complex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat, type,
    synthetic_div_nat : poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    synthe1985144195omplex : poly_poly_complex > poly_complex > poly_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__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_s____, type,
    s : poly_complex).
thf(sy_v_x____, type,
    x : complex).

% Relevant facts (246)
thf(fact_0_True, axiom,
    ((?[A : complex]: ((poly_complex2 @ pa @ A) = zero_zero_complex)))). % True
thf(fact_1_pq0, axiom,
    ((![X : complex]: (((poly_complex2 @ pa @ X) = zero_zero_complex) => ((poly_complex2 @ qa @ X) = zero_zero_complex))))). % pq0
thf(fact_2__092_060open_062poly_Ap_Ax_A_061_A0_092_060close_062, axiom,
    (((poly_complex2 @ pa @ x) = zero_zero_complex))). % \<open>poly p x = 0\<close>
thf(fact_3_assms_I1_J, axiom,
    ((![X : complex]: (((poly_complex2 @ p @ X) = zero_zero_complex) => ((poly_complex2 @ q @ X) = zero_zero_complex))))). % assms(1)
thf(fact_4__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,
    ((~ ((![A : complex]: (~ (((poly_complex2 @ pa @ A) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>a. poly p a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_that, axiom,
    (((poly_complex2 @ s @ x) = zero_zero_complex))). % that
thf(fact_6_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_7_poly__0, axiom,
    ((![X2 : poly_poly_complex]: ((poly_p282434315omplex @ zero_z1200043727omplex @ X2) = zero_z1040703943omplex)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X2 : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X2) = zero_zero_poly_nat)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_10_poly__0, axiom,
    ((![X2 : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X2) = zero_z1746442943omplex)))). % poly_0
thf(fact_11_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_12_xa, axiom,
    ((~ ((x = a))))). % xa
thf(fact_13_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X3 : complex]: ((poly_complex2 @ P @ X3) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_14_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X3 : poly_complex]: ((poly_poly_complex2 @ P @ X3) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_15_poly__all__0__iff__0, axiom,
    ((![P : poly_p1267267526omplex]: ((![X3 : poly_poly_complex]: ((poly_p282434315omplex @ P @ X3) = zero_z1040703943omplex)) = (P = zero_z1200043727omplex))))). % poly_all_0_iff_0
thf(fact_16_fundamental__theorem__of__algebra, axiom,
    ((![P : poly_complex]: ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra
thf(fact_17_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_18_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex]: (((poly_poly_complex2 @ P) = (poly_poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_19_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_20_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_21_zero__reorient, axiom,
    ((![X2 : poly_complex]: ((zero_z1746442943omplex = X2) = (X2 = zero_z1746442943omplex))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X2 : poly_poly_complex]: ((zero_z1040703943omplex = X2) = (X2 = zero_z1040703943omplex))))). % zero_reorient
thf(fact_23_zero__reorient, axiom,
    ((![X2 : poly_nat]: ((zero_zero_poly_nat = X2) = (X2 = zero_zero_poly_nat))))). % zero_reorient
thf(fact_24_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_25_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_26_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_27_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_28_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_p1267267526omplex]: (((poly_p282434315omplex @ (reflec1997789704omplex @ P) @ zero_z1040703943omplex) = zero_z1040703943omplex) = (P = zero_z1200043727omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_29_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_nat]: (((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_30_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_31_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_32_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_33_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_34_reflect__poly__0, axiom,
    (((reflec309385472omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % reflect_poly_0
thf(fact_35_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_36_oa, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % oa
thf(fact_37_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_38_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_39_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X3 : complex]: (![Y : complex]: ((F @ X3) = (F @ Y)))))))). % constant_def
thf(fact_40_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_41_psize__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((fundam1956464160omplex @ P) = zero_zero_nat) = (P = zero_z1040703943omplex))))). % psize_eq_0_iff
thf(fact_42_psize__eq__0__iff, axiom,
    ((![P : poly_nat]: (((fundam1567013434ze_nat @ P) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % psize_eq_0_iff
thf(fact_43_is__zero__null, axiom,
    ((is_zero_complex = (^[P2 : poly_complex]: (P2 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_44_is__zero__null, axiom,
    ((is_zero_poly_complex = (^[P2 : poly_poly_complex]: (P2 = zero_z1040703943omplex))))). % is_zero_null
thf(fact_45_is__zero__null, axiom,
    ((is_zero_nat = (^[P2 : poly_nat]: (P2 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_46_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_47_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_c622223248omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_cutoff_0
thf(fact_48_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_49_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_complex]: (((coeff_complex @ (reflect_poly_complex @ P) @ zero_zero_nat) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_50_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_51_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_complex]: (((coeff_poly_complex @ (reflec309385472omplex @ P) @ zero_zero_nat) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_52_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_p1267267526omplex]: (((coeff_1429652124omplex @ (reflec1997789704omplex @ P) @ zero_zero_nat) = zero_z1040703943omplex) = (P = zero_z1200043727omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_53_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_nat]: (((coeff_poly_nat @ (reflec781175074ly_nat @ P) @ zero_zero_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_54_order__root, axiom,
    ((![P : poly_complex, A2 : complex]: (((poly_complex2 @ P @ A2) = zero_zero_complex) = (((P = zero_z1746442943omplex)) | ((~ (((order_complex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_55_order__root, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex]: (((poly_poly_complex2 @ P @ A2) = zero_z1746442943omplex) = (((P = zero_z1040703943omplex)) | ((~ (((order_poly_complex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_56_order__root, axiom,
    ((![P : poly_p1267267526omplex, A2 : poly_poly_complex]: (((poly_p282434315omplex @ P @ A2) = zero_z1040703943omplex) = (((P = zero_z1200043727omplex)) | ((~ (((order_1735763309omplex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_57_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_58_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_shift_0
thf(fact_59_poly__shift__0, axiom,
    ((![N : nat]: ((poly_s558570093omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_shift_0
thf(fact_60_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_61_reflect__poly__reflect__poly, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((reflect_poly_complex @ (reflect_poly_complex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_62_reflect__poly__reflect__poly, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((reflect_poly_nat @ (reflect_poly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_63_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((reflec309385472omplex @ (reflec309385472omplex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_64_reflect__poly__reflect__poly, axiom,
    ((![P : poly_p1267267526omplex]: ((~ (((coeff_1429652124omplex @ P @ zero_zero_nat) = zero_z1040703943omplex))) => ((reflec1997789704omplex @ (reflec1997789704omplex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_65_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_nat]: ((~ (((coeff_poly_nat @ P @ zero_zero_nat) = zero_zero_poly_nat))) => ((reflec781175074ly_nat @ (reflec781175074ly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_66_coeff__0, axiom,
    ((![N : nat]: ((coeff_1429652124omplex @ zero_z1200043727omplex @ N) = zero_z1040703943omplex)))). % coeff_0
thf(fact_67_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_68_coeff__0, axiom,
    ((![N : nat]: ((coeff_complex @ zero_z1746442943omplex @ N) = zero_zero_complex)))). % coeff_0
thf(fact_69_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_complex @ zero_z1040703943omplex @ N) = zero_z1746442943omplex)))). % coeff_0
thf(fact_70_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_71_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_72_degree__0, axiom,
    (((degree_poly_complex @ zero_z1040703943omplex) = zero_zero_nat))). % degree_0
thf(fact_73_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_74_leading__coeff__0__iff, axiom,
    ((![P : poly_complex]: (((coeff_complex @ P @ (degree_complex @ P)) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % leading_coeff_0_iff
thf(fact_75_leading__coeff__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % leading_coeff_0_iff
thf(fact_76_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_complex]: (((coeff_poly_complex @ P @ (degree_poly_complex @ P)) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % leading_coeff_0_iff
thf(fact_77_leading__coeff__0__iff, axiom,
    ((![P : poly_p1267267526omplex]: (((coeff_1429652124omplex @ P @ (degree2006505739omplex @ P)) = zero_z1040703943omplex) = (P = zero_z1200043727omplex))))). % leading_coeff_0_iff
thf(fact_78_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_nat]: (((coeff_poly_nat @ P @ (degree_poly_nat @ P)) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % leading_coeff_0_iff
thf(fact_79_coeff__0__reflect__poly, axiom,
    ((![P : poly_complex]: ((coeff_complex @ (reflect_poly_complex @ P) @ zero_zero_nat) = (coeff_complex @ P @ (degree_complex @ P)))))). % coeff_0_reflect_poly
thf(fact_80_degree__reflect__poly__eq, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((degree_complex @ (reflect_poly_complex @ P)) = (degree_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_81_degree__reflect__poly__eq, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((degree_nat @ (reflect_poly_nat @ P)) = (degree_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_82_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((degree_poly_complex @ (reflec309385472omplex @ P)) = (degree_poly_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_83_degree__reflect__poly__eq, axiom,
    ((![P : poly_p1267267526omplex]: ((~ (((coeff_1429652124omplex @ P @ zero_zero_nat) = zero_z1040703943omplex))) => ((degree2006505739omplex @ (reflec1997789704omplex @ P)) = (degree2006505739omplex @ P)))))). % degree_reflect_poly_eq
thf(fact_84_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_nat]: ((~ (((coeff_poly_nat @ P @ zero_zero_nat) = zero_zero_poly_nat))) => ((degree_poly_nat @ (reflec781175074ly_nat @ P)) = (degree_poly_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_85_poly__reflect__poly__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = (coeff_complex @ P @ (degree_complex @ P)))))). % poly_reflect_poly_0
thf(fact_86_poly__reflect__poly__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % poly_reflect_poly_0
thf(fact_87_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ (degree_poly_complex @ P)))))). % poly_reflect_poly_0
thf(fact_88_poly__reflect__poly__0, axiom,
    ((![P : poly_p1267267526omplex]: ((poly_p282434315omplex @ (reflec1997789704omplex @ P) @ zero_z1040703943omplex) = (coeff_1429652124omplex @ P @ (degree2006505739omplex @ P)))))). % poly_reflect_poly_0
thf(fact_89_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_nat]: ((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = (coeff_poly_nat @ P @ (degree_poly_nat @ P)))))). % poly_reflect_poly_0
thf(fact_90_leading__coeff__neq__0, axiom,
    ((![P : poly_p1267267526omplex]: ((~ ((P = zero_z1200043727omplex))) => (~ (((coeff_1429652124omplex @ P @ (degree2006505739omplex @ P)) = zero_z1040703943omplex))))))). % leading_coeff_neq_0
thf(fact_91_leading__coeff__neq__0, axiom,
    ((![P : poly_poly_nat]: ((~ ((P = zero_z1059985641ly_nat))) => (~ (((coeff_poly_nat @ P @ (degree_poly_nat @ P)) = zero_zero_poly_nat))))))). % leading_coeff_neq_0
thf(fact_92_leading__coeff__neq__0, axiom,
    ((![P : poly_complex]: ((~ ((P = zero_z1746442943omplex))) => (~ (((coeff_complex @ P @ (degree_complex @ P)) = zero_zero_complex))))))). % leading_coeff_neq_0
thf(fact_93_leading__coeff__neq__0, axiom,
    ((![P : poly_poly_complex]: ((~ ((P = zero_z1040703943omplex))) => (~ (((coeff_poly_complex @ P @ (degree_poly_complex @ P)) = zero_z1746442943omplex))))))). % leading_coeff_neq_0
thf(fact_94_leading__coeff__neq__0, axiom,
    ((![P : poly_nat]: ((~ ((P = zero_zero_poly_nat))) => (~ (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat))))))). % leading_coeff_neq_0
thf(fact_95_zero__poly_Orep__eq, axiom,
    (((coeff_1429652124omplex @ zero_z1200043727omplex) = (^[Uu : nat]: zero_z1040703943omplex)))). % zero_poly.rep_eq
thf(fact_96_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_97_zero__poly_Orep__eq, axiom,
    (((coeff_complex @ zero_z1746442943omplex) = (^[Uu : nat]: zero_zero_complex)))). % zero_poly.rep_eq
thf(fact_98_zero__poly_Orep__eq, axiom,
    (((coeff_poly_complex @ zero_z1040703943omplex) = (^[Uu : nat]: zero_z1746442943omplex)))). % zero_poly.rep_eq
thf(fact_99_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_100_poly__0__coeff__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ P @ zero_zero_complex) = (coeff_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_101_poly__0__coeff__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ P @ zero_zero_nat) = (coeff_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_102_poly__0__coeff__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ P @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_103_poly__0__coeff__0, axiom,
    ((![P : poly_p1267267526omplex]: ((poly_p282434315omplex @ P @ zero_z1040703943omplex) = (coeff_1429652124omplex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_104_poly__0__coeff__0, axiom,
    ((![P : poly_poly_nat]: ((poly_poly_nat2 @ P @ zero_zero_poly_nat) = (coeff_poly_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_105_order__0I, axiom,
    ((![P : poly_complex, A2 : complex]: ((~ (((poly_complex2 @ P @ A2) = zero_zero_complex))) => ((order_complex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_106_order__0I, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex]: ((~ (((poly_poly_complex2 @ P @ A2) = zero_z1746442943omplex))) => ((order_poly_complex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_107_order__0I, axiom,
    ((![P : poly_p1267267526omplex, A2 : poly_poly_complex]: ((~ (((poly_p282434315omplex @ P @ A2) = zero_z1040703943omplex))) => ((order_1735763309omplex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_108_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_109_oop, axiom,
    ((![A2 : complex]: (ord_less_eq_nat @ (order_complex @ A2 @ pa) @ na)))). % oop
thf(fact_110_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_complex, C : complex]: (((synthe151143547omplex @ P @ C) = zero_z1746442943omplex) = ((degree_complex @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_111_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_poly_complex, C : poly_complex]: (((synthe1985144195omplex @ P @ C) = zero_z1040703943omplex) = ((degree_poly_complex @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_112_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_nat, C : nat]: (((synthetic_div_nat @ P @ C) = zero_zero_poly_nat) = ((degree_nat @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_113_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = one_one_poly_complex)))))). % poly_cutoff_1
thf(fact_114_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_c622223248omplex @ N @ one_on1331105667omplex) = zero_z1040703943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_c622223248omplex @ N @ one_on1331105667omplex) = one_on1331105667omplex)))))). % poly_cutoff_1
thf(fact_115_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = zero_zero_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = one_one_poly_nat)))))). % poly_cutoff_1
thf(fact_116_psize__def, axiom,
    ((fundam1709708056omplex = (^[P2 : poly_complex]: (if_nat @ (P2 = zero_z1746442943omplex) @ zero_zero_nat @ (suc @ (degree_complex @ P2))))))). % psize_def
thf(fact_117_psize__def, axiom,
    ((fundam1956464160omplex = (^[P2 : poly_poly_complex]: (if_nat @ (P2 = zero_z1040703943omplex) @ zero_zero_nat @ (suc @ (degree_poly_complex @ P2))))))). % psize_def
thf(fact_118_psize__def, axiom,
    ((fundam1567013434ze_nat = (^[P2 : poly_nat]: (if_nat @ (P2 = zero_zero_poly_nat) @ zero_zero_nat @ (suc @ (degree_nat @ P2))))))). % psize_def
thf(fact_119_IH, axiom,
    ((![M : nat]: ((ord_less_nat @ M @ na) => (![P3 : poly_complex, Q2 : poly_complex]: ((![X4 : complex]: (((poly_complex2 @ P3 @ X4) = zero_zero_complex) => ((poly_complex2 @ Q2 @ X4) = zero_zero_complex))) => (((degree_complex @ P3) = M) => ((~ ((M = zero_zero_nat))) => (dvd_dvd_poly_complex @ P3 @ (power_184595776omplex @ Q2 @ M)))))))))). % IH
thf(fact_120_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_complex @ N @ one_one_poly_complex) = one_one_poly_complex)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)))))). % poly_shift_1
thf(fact_121_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_s558570093omplex @ N @ one_on1331105667omplex) = one_on1331105667omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_s558570093omplex @ N @ one_on1331105667omplex) = zero_z1040703943omplex)))))). % poly_shift_1
thf(fact_122_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_nat @ N @ one_one_poly_nat) = one_one_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_nat @ N @ one_one_poly_nat) = zero_zero_poly_nat)))))). % poly_shift_1
thf(fact_123_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_124_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_125_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_126_synthetic__div__0, axiom,
    ((![C : poly_complex]: ((synthe1985144195omplex @ zero_z1040703943omplex @ C) = zero_z1040703943omplex)))). % synthetic_div_0
thf(fact_127_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_128_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_129_poly__1, axiom,
    ((![X2 : complex]: ((poly_complex2 @ one_one_poly_complex @ X2) = one_one_complex)))). % poly_1
thf(fact_130_poly__1, axiom,
    ((![X2 : poly_complex]: ((poly_poly_complex2 @ one_on1331105667omplex @ X2) = one_one_poly_complex)))). % poly_1
thf(fact_131_poly__1, axiom,
    ((![X2 : nat]: ((poly_nat2 @ one_one_poly_nat @ X2) = one_one_nat)))). % poly_1
thf(fact_132_poly__power, axiom,
    ((![P : poly_poly_complex, N : nat, X2 : poly_complex]: ((poly_poly_complex2 @ (power_432682568omplex @ P @ N) @ X2) = (power_184595776omplex @ (poly_poly_complex2 @ P @ X2) @ N))))). % poly_power
thf(fact_133_poly__power, axiom,
    ((![P : poly_nat, N : nat, X2 : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X2) = (power_power_nat @ (poly_nat2 @ P @ X2) @ N))))). % poly_power
thf(fact_134_poly__power, axiom,
    ((![P : poly_complex, N : nat, X2 : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X2) = (power_power_complex @ (poly_complex2 @ P @ X2) @ N))))). % poly_power
thf(fact_135_order__1__eq__0, axiom,
    ((![X2 : complex]: ((order_complex @ X2 @ one_one_poly_complex) = zero_zero_nat)))). % order_1_eq_0
thf(fact_136_lead__coeff__1, axiom,
    (((coeff_complex @ one_one_poly_complex @ (degree_complex @ one_one_poly_complex)) = one_one_complex))). % lead_coeff_1
thf(fact_137_lead__coeff__1, axiom,
    (((coeff_nat @ one_one_poly_nat @ (degree_nat @ one_one_poly_nat)) = one_one_nat))). % lead_coeff_1
thf(fact_138_one__reorient, axiom,
    ((![X2 : nat]: ((one_one_nat = X2) = (X2 = one_one_nat))))). % one_reorient
thf(fact_139_coeff__0__power, axiom,
    ((![P : poly_poly_complex, N : nat]: ((coeff_poly_complex @ (power_432682568omplex @ P @ N) @ zero_zero_nat) = (power_184595776omplex @ (coeff_poly_complex @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_140_coeff__0__power, axiom,
    ((![P : poly_nat, N : nat]: ((coeff_nat @ (power_power_poly_nat @ P @ N) @ zero_zero_nat) = (power_power_nat @ (coeff_nat @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_141_coeff__0__power, axiom,
    ((![P : poly_complex, N : nat]: ((coeff_complex @ (power_184595776omplex @ P @ N) @ zero_zero_nat) = (power_power_complex @ (coeff_complex @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_142_lead__coeff__power, axiom,
    ((![P : poly_poly_complex, N : nat]: ((coeff_poly_complex @ (power_432682568omplex @ P @ N) @ (degree_poly_complex @ (power_432682568omplex @ P @ N))) = (power_184595776omplex @ (coeff_poly_complex @ P @ (degree_poly_complex @ P)) @ N))))). % lead_coeff_power
thf(fact_143_lead__coeff__power, axiom,
    ((![P : poly_nat, N : nat]: ((coeff_nat @ (power_power_poly_nat @ P @ N) @ (degree_nat @ (power_power_poly_nat @ P @ N))) = (power_power_nat @ (coeff_nat @ P @ (degree_nat @ P)) @ N))))). % lead_coeff_power
thf(fact_144_lead__coeff__power, axiom,
    ((![P : poly_complex, N : nat]: ((coeff_complex @ (power_184595776omplex @ P @ N) @ (degree_complex @ (power_184595776omplex @ P @ N))) = (power_power_complex @ (coeff_complex @ P @ (degree_complex @ P)) @ N))))). % lead_coeff_power
thf(fact_145_dvd__imp__degree__le, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((~ ((Q = zero_z1746442943omplex))) => (ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q))))))). % dvd_imp_degree_le
thf(fact_146_dvd__imp__degree__le, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ Q) => ((~ ((Q = zero_z1040703943omplex))) => (ord_less_eq_nat @ (degree_poly_complex @ P) @ (degree_poly_complex @ Q))))))). % dvd_imp_degree_le
thf(fact_147_dvd__imp__degree__le, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((dvd_dvd_poly_nat @ P @ Q) => ((~ ((Q = zero_zero_poly_nat))) => (ord_less_eq_nat @ (degree_nat @ P) @ (degree_nat @ Q))))))). % dvd_imp_degree_le
thf(fact_148_divides__degree, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q)) | (Q = zero_z1746442943omplex)))))). % divides_degree
thf(fact_149_divides__degree, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ Q) => ((ord_less_eq_nat @ (degree_poly_complex @ P) @ (degree_poly_complex @ Q)) | (Q = zero_z1040703943omplex)))))). % divides_degree
thf(fact_150_divides__degree, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((dvd_dvd_poly_nat @ P @ Q) => ((ord_less_eq_nat @ (degree_nat @ P) @ (degree_nat @ Q)) | (Q = zero_zero_poly_nat)))))). % divides_degree
thf(fact_151_dvd__imp__order__le, axiom,
    ((![Q : poly_complex, P : poly_complex, A2 : complex]: ((~ ((Q = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ P @ Q) => (ord_less_eq_nat @ (order_complex @ A2 @ P) @ (order_complex @ A2 @ Q))))))). % dvd_imp_order_le
thf(fact_152_dvd__imp__order__le, axiom,
    ((![Q : poly_poly_complex, P : poly_poly_complex, A2 : poly_complex]: ((~ ((Q = zero_z1040703943omplex))) => ((dvd_dv598755940omplex @ P @ Q) => (ord_less_eq_nat @ (order_poly_complex @ A2 @ P) @ (order_poly_complex @ A2 @ Q))))))). % dvd_imp_order_le
thf(fact_153_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_154_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_155_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_156_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_157_zero__le, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X2)))). % zero_le
thf(fact_158_reflect__poly__power, axiom,
    ((![P : poly_complex, N : nat]: ((reflect_poly_complex @ (power_184595776omplex @ P @ N)) = (power_184595776omplex @ (reflect_poly_complex @ P) @ N))))). % reflect_poly_power
thf(fact_159_degree__le, axiom,
    ((![N : nat, P : poly_complex]: ((![I : nat]: ((ord_less_nat @ N @ I) => ((coeff_complex @ P @ I) = zero_zero_complex))) => (ord_less_eq_nat @ (degree_complex @ P) @ N))))). % degree_le
thf(fact_160_degree__le, axiom,
    ((![N : nat, P : poly_nat]: ((![I : nat]: ((ord_less_nat @ N @ I) => ((coeff_nat @ P @ I) = zero_zero_nat))) => (ord_less_eq_nat @ (degree_nat @ P) @ N))))). % degree_le
thf(fact_161_degree__le, axiom,
    ((![N : nat, P : poly_poly_complex]: ((![I : nat]: ((ord_less_nat @ N @ I) => ((coeff_poly_complex @ P @ I) = zero_z1746442943omplex))) => (ord_less_eq_nat @ (degree_poly_complex @ P) @ N))))). % degree_le
thf(fact_162_degree__le, axiom,
    ((![N : nat, P : poly_p1267267526omplex]: ((![I : nat]: ((ord_less_nat @ N @ I) => ((coeff_1429652124omplex @ P @ I) = zero_z1040703943omplex))) => (ord_less_eq_nat @ (degree2006505739omplex @ P) @ N))))). % degree_le
thf(fact_163_degree__le, axiom,
    ((![N : nat, P : poly_poly_nat]: ((![I : nat]: ((ord_less_nat @ N @ I) => ((coeff_poly_nat @ P @ I) = zero_zero_poly_nat))) => (ord_less_eq_nat @ (degree_poly_nat @ P) @ N))))). % degree_le
thf(fact_164_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_165_eq__zero__or__degree__less, axiom,
    ((![P : poly_complex, N : nat]: ((ord_less_eq_nat @ (degree_complex @ P) @ N) => (((coeff_complex @ P @ N) = zero_zero_complex) => ((P = zero_z1746442943omplex) | (ord_less_nat @ (degree_complex @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_166_eq__zero__or__degree__less, axiom,
    ((![P : poly_nat, N : nat]: ((ord_less_eq_nat @ (degree_nat @ P) @ N) => (((coeff_nat @ P @ N) = zero_zero_nat) => ((P = zero_zero_poly_nat) | (ord_less_nat @ (degree_nat @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_167_eq__zero__or__degree__less, axiom,
    ((![P : poly_poly_complex, N : nat]: ((ord_less_eq_nat @ (degree_poly_complex @ P) @ N) => (((coeff_poly_complex @ P @ N) = zero_z1746442943omplex) => ((P = zero_z1040703943omplex) | (ord_less_nat @ (degree_poly_complex @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_168_eq__zero__or__degree__less, axiom,
    ((![P : poly_p1267267526omplex, N : nat]: ((ord_less_eq_nat @ (degree2006505739omplex @ P) @ N) => (((coeff_1429652124omplex @ P @ N) = zero_z1040703943omplex) => ((P = zero_z1200043727omplex) | (ord_less_nat @ (degree2006505739omplex @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_169_eq__zero__or__degree__less, axiom,
    ((![P : poly_poly_nat, N : nat]: ((ord_less_eq_nat @ (degree_poly_nat @ P) @ N) => (((coeff_poly_nat @ P @ N) = zero_zero_poly_nat) => ((P = zero_z1059985641ly_nat) | (ord_less_nat @ (degree_poly_nat @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_170_degree__reflect__poly__le, axiom,
    ((![P : poly_complex]: (ord_less_eq_nat @ (degree_complex @ (reflect_poly_complex @ P)) @ (degree_complex @ P))))). % degree_reflect_poly_le
thf(fact_171_less__degree__imp, axiom,
    ((![N : nat, P : poly_complex]: ((ord_less_nat @ N @ (degree_complex @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_complex @ P @ I) = zero_zero_complex))))))))). % less_degree_imp
thf(fact_172_less__degree__imp, axiom,
    ((![N : nat, P : poly_nat]: ((ord_less_nat @ N @ (degree_nat @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_nat @ P @ I) = zero_zero_nat))))))))). % less_degree_imp
thf(fact_173_less__degree__imp, axiom,
    ((![N : nat, P : poly_poly_complex]: ((ord_less_nat @ N @ (degree_poly_complex @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_poly_complex @ P @ I) = zero_z1746442943omplex))))))))). % less_degree_imp
thf(fact_174_less__degree__imp, axiom,
    ((![N : nat, P : poly_p1267267526omplex]: ((ord_less_nat @ N @ (degree2006505739omplex @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_1429652124omplex @ P @ I) = zero_z1040703943omplex))))))))). % less_degree_imp
thf(fact_175_less__degree__imp, axiom,
    ((![N : nat, P : poly_poly_nat]: ((ord_less_nat @ N @ (degree_poly_nat @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_poly_nat @ P @ I) = zero_zero_poly_nat))))))))). % less_degree_imp
thf(fact_176_coeff__eq__0, axiom,
    ((![P : poly_complex, N : nat]: ((ord_less_nat @ (degree_complex @ P) @ N) => ((coeff_complex @ P @ N) = zero_zero_complex))))). % coeff_eq_0
thf(fact_177_coeff__eq__0, axiom,
    ((![P : poly_nat, N : nat]: ((ord_less_nat @ (degree_nat @ P) @ N) => ((coeff_nat @ P @ N) = zero_zero_nat))))). % coeff_eq_0
thf(fact_178_coeff__eq__0, axiom,
    ((![P : poly_poly_complex, N : nat]: ((ord_less_nat @ (degree_poly_complex @ P) @ N) => ((coeff_poly_complex @ P @ N) = zero_z1746442943omplex))))). % coeff_eq_0
thf(fact_179_coeff__eq__0, axiom,
    ((![P : poly_p1267267526omplex, N : nat]: ((ord_less_nat @ (degree2006505739omplex @ P) @ N) => ((coeff_1429652124omplex @ P @ N) = zero_z1040703943omplex))))). % coeff_eq_0
thf(fact_180_coeff__eq__0, axiom,
    ((![P : poly_poly_nat, N : nat]: ((ord_less_nat @ (degree_poly_nat @ P) @ N) => ((coeff_poly_nat @ P @ N) = zero_zero_poly_nat))))). % coeff_eq_0
thf(fact_181_le__degree, axiom,
    ((![P : poly_complex, N : nat]: ((~ (((coeff_complex @ P @ N) = zero_zero_complex))) => (ord_less_eq_nat @ N @ (degree_complex @ P)))))). % le_degree
thf(fact_182_le__degree, axiom,
    ((![P : poly_nat, N : nat]: ((~ (((coeff_nat @ P @ N) = zero_zero_nat))) => (ord_less_eq_nat @ N @ (degree_nat @ P)))))). % le_degree
thf(fact_183_le__degree, axiom,
    ((![P : poly_poly_complex, N : nat]: ((~ (((coeff_poly_complex @ P @ N) = zero_z1746442943omplex))) => (ord_less_eq_nat @ N @ (degree_poly_complex @ P)))))). % le_degree
thf(fact_184_le__degree, axiom,
    ((![P : poly_p1267267526omplex, N : nat]: ((~ (((coeff_1429652124omplex @ P @ N) = zero_z1040703943omplex))) => (ord_less_eq_nat @ N @ (degree2006505739omplex @ P)))))). % le_degree
thf(fact_185_le__degree, axiom,
    ((![P : poly_poly_nat, N : nat]: ((~ (((coeff_poly_nat @ P @ N) = zero_zero_poly_nat))) => (ord_less_eq_nat @ N @ (degree_poly_nat @ P)))))). % le_degree
thf(fact_186_order__degree, axiom,
    ((![P : poly_complex, A2 : complex]: ((~ ((P = zero_z1746442943omplex))) => (ord_less_eq_nat @ (order_complex @ A2 @ P) @ (degree_complex @ P)))))). % order_degree
thf(fact_187_order__degree, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex]: ((~ ((P = zero_z1040703943omplex))) => (ord_less_eq_nat @ (order_poly_complex @ A2 @ P) @ (degree_poly_complex @ P)))))). % order_degree
thf(fact_188_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_complex]: (((ord_less_nat @ K @ N) => ((coeff_complex @ (poly_cutoff_complex @ N @ P) @ K) = (coeff_complex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_complex @ (poly_cutoff_complex @ N @ P) @ K) = zero_zero_complex)))))). % coeff_poly_cutoff
thf(fact_189_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = (coeff_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = zero_zero_nat)))))). % coeff_poly_cutoff
thf(fact_190_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_complex]: (((ord_less_nat @ K @ N) => ((coeff_poly_complex @ (poly_c622223248omplex @ N @ P) @ K) = (coeff_poly_complex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_complex @ (poly_c622223248omplex @ N @ P) @ K) = zero_z1746442943omplex)))))). % coeff_poly_cutoff
thf(fact_191_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_p1267267526omplex]: (((ord_less_nat @ K @ N) => ((coeff_1429652124omplex @ (poly_c306577560omplex @ N @ P) @ K) = (coeff_1429652124omplex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_1429652124omplex @ (poly_c306577560omplex @ N @ P) @ K) = zero_z1040703943omplex)))))). % coeff_poly_cutoff
thf(fact_192_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = (coeff_poly_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = zero_zero_poly_nat)))))). % coeff_poly_cutoff
thf(fact_193_power__decreasing__iff, axiom,
    ((![B : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M2) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M2))))))). % power_decreasing_iff
thf(fact_194_power__mono__iff, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A2) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A2 @ B)))))))). % power_mono_iff
thf(fact_195_power__increasing__iff, axiom,
    ((![B : nat, X2 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X2) @ (power_power_nat @ B @ Y2)) = (ord_less_eq_nat @ X2 @ Y2)))))). % power_increasing_iff
thf(fact_196_power__strict__decreasing__iff, axiom,
    ((![B : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M2) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M2))))))). % power_strict_decreasing_iff
thf(fact_197_pow__divides__pow__iff, axiom,
    ((![N : nat, A2 : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((dvd_dvd_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)) = (dvd_dvd_nat @ A2 @ B)))))). % pow_divides_pow_iff
thf(fact_198_power__one__right, axiom,
    ((![A2 : poly_complex]: ((power_184595776omplex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_199_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_200_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_201_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_202_nat__power__eq__Suc__0__iff, axiom,
    ((![X2 : nat, M2 : nat]: (((power_power_nat @ X2 @ M2) = (suc @ zero_zero_nat)) = (((M2 = zero_zero_nat)) | ((X2 = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_203_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_204_nat__zero__less__power__iff, axiom,
    ((![X2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X2 @ N)) = (((ord_less_nat @ zero_zero_nat @ X2)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_205_power__inject__exp, axiom,
    ((![A2 : nat, M2 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A2) => (((power_power_nat @ A2 @ M2) = (power_power_nat @ A2 @ N)) = (M2 = N)))))). % power_inject_exp
thf(fact_206_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N)) = zero_zero_complex)))). % power_0_Suc
thf(fact_207_power__0__Suc, axiom,
    ((![N : nat]: ((power_432682568omplex @ zero_z1040703943omplex @ (suc @ N)) = zero_z1040703943omplex)))). % power_0_Suc
thf(fact_208_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_poly_nat @ zero_zero_poly_nat @ (suc @ N)) = zero_zero_poly_nat)))). % power_0_Suc
thf(fact_209_power__0__Suc, axiom,
    ((![N : nat]: ((power_184595776omplex @ zero_z1746442943omplex @ (suc @ N)) = zero_z1746442943omplex)))). % power_0_Suc
thf(fact_210_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_211_power__Suc0__right, axiom,
    ((![A2 : poly_complex]: ((power_184595776omplex @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_212_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_213_power__strict__increasing__iff, axiom,
    ((![B : nat, X2 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X2) @ (power_power_nat @ B @ Y2)) = (ord_less_nat @ X2 @ Y2)))))). % power_strict_increasing_iff
thf(fact_214_power__eq__0__iff, axiom,
    ((![A2 : complex, N : nat]: (((power_power_complex @ A2 @ N) = zero_zero_complex) = (((A2 = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_215_power__eq__0__iff, axiom,
    ((![A2 : poly_poly_complex, N : nat]: (((power_432682568omplex @ A2 @ N) = zero_z1040703943omplex) = (((A2 = zero_z1040703943omplex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_216_power__eq__0__iff, axiom,
    ((![A2 : poly_nat, N : nat]: (((power_power_poly_nat @ A2 @ N) = zero_zero_poly_nat) = (((A2 = zero_zero_poly_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_217_power__eq__0__iff, axiom,
    ((![A2 : poly_complex, N : nat]: (((power_184595776omplex @ A2 @ N) = zero_z1746442943omplex) = (((A2 = zero_z1746442943omplex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_218_power__eq__0__iff, axiom,
    ((![A2 : nat, N : nat]: (((power_power_nat @ A2 @ N) = zero_zero_nat) = (((A2 = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_219_gcd__nat_Oextremum, axiom,
    ((![A2 : nat]: (dvd_dvd_nat @ A2 @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_220_gcd__nat_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A2) & (~ ((zero_zero_nat = A2))))))))). % gcd_nat.extremum_strict
thf(fact_221_gcd__nat_Oextremum__unique, axiom,
    ((![A2 : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A2) = (A2 = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_222_gcd__nat_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (((dvd_dvd_nat @ A2 @ zero_zero_nat)) & ((~ ((A2 = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_223_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A2 : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A2) => (A2 = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_224_power__dvd__imp__le, axiom,
    ((![I2 : nat, M2 : nat, N : nat]: ((dvd_dvd_nat @ (power_power_nat @ I2 @ M2) @ (power_power_nat @ I2 @ N)) => ((ord_less_nat @ one_one_nat @ I2) => (ord_less_eq_nat @ M2 @ N)))))). % power_dvd_imp_le
thf(fact_225_nat__power__less__imp__less, axiom,
    ((![I2 : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I2) => ((ord_less_nat @ (power_power_nat @ I2 @ M2) @ (power_power_nat @ I2 @ N)) => (ord_less_nat @ M2 @ N)))))). % nat_power_less_imp_less
thf(fact_226_dvd__pos__nat, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((dvd_dvd_nat @ M2 @ N) => (ord_less_nat @ zero_zero_nat @ M2)))))). % dvd_pos_nat
thf(fact_227_power__gt__expt, axiom,
    ((![N : nat, K : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => (ord_less_nat @ K @ (power_power_nat @ N @ K)))))). % power_gt_expt
thf(fact_228_nat__one__le__power, axiom,
    ((![I2 : nat, N : nat]: ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ I2) => (ord_less_eq_nat @ (suc @ zero_zero_nat) @ (power_power_nat @ I2 @ N)))))). % nat_one_le_power
thf(fact_229_power__not__zero, axiom,
    ((![A2 : complex, N : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_230_power__not__zero, axiom,
    ((![A2 : poly_poly_complex, N : nat]: ((~ ((A2 = zero_z1040703943omplex))) => (~ (((power_432682568omplex @ A2 @ N) = zero_z1040703943omplex))))))). % power_not_zero
thf(fact_231_power__not__zero, axiom,
    ((![A2 : poly_nat, N : nat]: ((~ ((A2 = zero_zero_poly_nat))) => (~ (((power_power_poly_nat @ A2 @ N) = zero_zero_poly_nat))))))). % power_not_zero
thf(fact_232_power__not__zero, axiom,
    ((![A2 : poly_complex, N : nat]: ((~ ((A2 = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A2 @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_233_power__not__zero, axiom,
    ((![A2 : nat, N : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_234_dvd__power__same, axiom,
    ((![X2 : poly_complex, Y2 : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ X2 @ Y2) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X2 @ N) @ (power_184595776omplex @ Y2 @ N)))))). % dvd_power_same
thf(fact_235_dvd__power__same, axiom,
    ((![X2 : nat, Y2 : nat, N : nat]: ((dvd_dvd_nat @ X2 @ Y2) => (dvd_dvd_nat @ (power_power_nat @ X2 @ N) @ (power_power_nat @ Y2 @ N)))))). % dvd_power_same
thf(fact_236_zero__le__power, axiom,
    ((![A2 : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A2) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A2 @ N)))))). % zero_le_power
thf(fact_237_power__mono, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A2 @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A2) => (ord_less_eq_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_238_zero__less__power, axiom,
    ((![A2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A2) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A2 @ N)))))). % zero_less_power
thf(fact_239_one__le__power, axiom,
    ((![A2 : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A2) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A2 @ N)))))). % one_le_power
thf(fact_240_power__0, axiom,
    ((![A2 : poly_complex]: ((power_184595776omplex @ A2 @ zero_zero_nat) = one_one_poly_complex)))). % power_0
thf(fact_241_power__0, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_242_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A2 : nat]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_nat @ (power_power_nat @ A2 @ M2) @ (power_power_nat @ A2 @ N)))))). % le_imp_power_dvd
thf(fact_243_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_244_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_245_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P4 : $o]: ((P4 = $true) | (P4 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $true @ X2 @ Y2) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((poly_complex2 @ qa @ x) = zero_zero_complex))).
