% 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_959__5376944_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:43.232

% Could-be-implicit typings (10)
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__Real__Oreal_J_J, type,
    poly_poly_real : $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__Real__Oreal_J, type,
    poly_real : $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__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (85)
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_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Real__Oreal, type,
    fundam1947011094e_real : poly_real > nat).
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_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    zero_z1423781445y_real : poly_poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_less_poly_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_le1180086932y_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $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__Real__Oreal, type,
    degree_real : poly_real > 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_Ois__zero_001t__Real__Oreal, type,
    is_zero_real : poly_real > $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_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    order_poly_real : poly_real > poly_poly_real > nat).
thf(sy_c_Polynomial_Oorder_001t__Real__Oreal, type,
    order_real : real > poly_real > 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_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_poly_real2 : poly_poly_real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
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__Real__Oreal, type,
    poly_cutoff_real : nat > poly_real > poly_real).
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_Opoly__shift_001t__Real__Oreal, type,
    poly_shift_real : nat > poly_real > poly_real).
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_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    reflec1522834046y_real : poly_poly_real > poly_poly_real).
thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal, type,
    reflect_poly_real : poly_real > poly_real).
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_Polynomial_Osynthetic__div_001t__Real__Oreal, type,
    synthetic_div_real : poly_real > real > poly_real).
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_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    power_2108872382y_real : poly_real > nat > poly_real).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
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_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    dvd_dvd_poly_real : poly_real > poly_real > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal, type,
    dvd_dvd_real : real > real > $o).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal, type,
    arsinh_real : real > real).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal, type,
    artanh_real : real > real).
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 (242)
thf(fact_0_that, axiom,
    (((poly_complex2 @ s @ x) = zero_zero_complex))). % that
thf(fact_1_True, axiom,
    ((?[A : complex]: ((poly_complex2 @ pa @ A) = zero_zero_complex)))). % True
thf(fact_2_assms_I1_J, axiom,
    ((![X : complex]: (((poly_complex2 @ p @ X) = zero_zero_complex) => ((poly_complex2 @ q @ X) = zero_zero_complex))))). % assms(1)
thf(fact_3_pq0, axiom,
    ((![X : complex]: (((poly_complex2 @ pa @ X) = zero_zero_complex) => ((poly_complex2 @ qa @ X) = zero_zero_complex))))). % pq0
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_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_6_poly__0, axiom,
    ((![X2 : poly_real]: ((poly_poly_real2 @ zero_z1423781445y_real @ X2) = zero_zero_poly_real)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X2 : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X2) = zero_zero_poly_nat)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X2 : poly_poly_complex]: ((poly_p282434315omplex @ zero_z1200043727omplex @ X2) = zero_z1040703943omplex)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X2 : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X2) = zero_z1746442943omplex)))). % poly_0
thf(fact_10_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_11_poly__0, axiom,
    ((![X2 : real]: ((poly_real2 @ zero_zero_poly_real @ X2) = zero_zero_real)))). % poly_0
thf(fact_12_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_13_xa, axiom,
    ((~ ((x = a))))). % xa
thf(fact_14_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_15_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_16_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_17_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X3 : real]: ((poly_real2 @ P @ X3) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_18_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_real]: ((![X3 : poly_real]: ((poly_poly_real2 @ P @ X3) = zero_zero_poly_real)) = (P = zero_z1423781445y_real))))). % poly_all_0_iff_0
thf(fact_19_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_20_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_21_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_22_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_23_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_24_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_25_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_26_zero__reorient, axiom,
    ((![X2 : poly_complex]: ((zero_z1746442943omplex = X2) = (X2 = zero_z1746442943omplex))))). % zero_reorient
thf(fact_27_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_28_zero__reorient, axiom,
    ((![X2 : real]: ((zero_zero_real = X2) = (X2 = zero_zero_real))))). % zero_reorient
thf(fact_29_zero__reorient, axiom,
    ((![X2 : poly_real]: ((zero_zero_poly_real = X2) = (X2 = zero_zero_poly_real))))). % zero_reorient
thf(fact_30_zero__reorient, axiom,
    ((![X2 : poly_nat]: ((zero_zero_poly_nat = X2) = (X2 = zero_zero_poly_nat))))). % zero_reorient
thf(fact_31_zero__reorient, axiom,
    ((![X2 : poly_poly_complex]: ((zero_z1040703943omplex = X2) = (X2 = zero_z1040703943omplex))))). % zero_reorient
thf(fact_32_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_33_arsinh__0, axiom,
    (((arsinh_real @ zero_zero_real) = zero_zero_real))). % arsinh_0
thf(fact_34_artanh__0, axiom,
    (((artanh_real @ zero_zero_real) = zero_zero_real))). % artanh_0
thf(fact_35_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_36_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_37_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_38_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_39_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_40_degree__0, axiom,
    (((degree_real @ zero_zero_poly_real) = zero_zero_nat))). % degree_0
thf(fact_41_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_42_degree__0, axiom,
    (((degree_poly_complex @ zero_z1040703943omplex) = zero_zero_nat))). % degree_0
thf(fact_43_oa, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % oa
thf(fact_44_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X3 : complex]: (![Y : complex]: ((F @ X3) = (F @ Y)))))))). % constant_def
thf(fact_45_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_46_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_47_psize__eq__0__iff, axiom,
    ((![P : poly_real]: (((fundam1947011094e_real @ P) = zero_zero_nat) = (P = zero_zero_poly_real))))). % psize_eq_0_iff
thf(fact_48_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_49_psize__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((fundam1956464160omplex @ P) = zero_zero_nat) = (P = zero_z1040703943omplex))))). % psize_eq_0_iff
thf(fact_50_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_51_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_real, C : real]: (((synthetic_div_real @ P @ C) = zero_zero_poly_real) = ((degree_real @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_52_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_53_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_54_is__zero__null, axiom,
    ((is_zero_complex = (^[P2 : poly_complex]: (P2 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_55_is__zero__null, axiom,
    ((is_zero_real = (^[P2 : poly_real]: (P2 = zero_zero_poly_real))))). % is_zero_null
thf(fact_56_is__zero__null, axiom,
    ((is_zero_nat = (^[P2 : poly_nat]: (P2 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_57_is__zero__null, axiom,
    ((is_zero_poly_complex = (^[P2 : poly_poly_complex]: (P2 = zero_z1040703943omplex))))). % is_zero_null
thf(fact_58_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_59_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_real @ N @ zero_zero_poly_real) = zero_zero_poly_real)))). % poly_cutoff_0
thf(fact_60_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_61_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_c622223248omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_cutoff_0
thf(fact_62_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_63_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_64_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_65_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_real]: (((poly_real2 @ (reflect_poly_real @ P) @ zero_zero_real) = zero_zero_real) = (P = zero_zero_poly_real))))). % reflect_poly_at_0_eq_0_iff
thf(fact_66_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_real]: (((poly_poly_real2 @ (reflec1522834046y_real @ P) @ zero_zero_poly_real) = zero_zero_poly_real) = (P = zero_z1423781445y_real))))). % reflect_poly_at_0_eq_0_iff
thf(fact_67_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_68_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_69_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_70_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_71_order__root, axiom,
    ((![P : poly_real, A2 : real]: (((poly_real2 @ P @ A2) = zero_zero_real) = (((P = zero_zero_poly_real)) | ((~ (((order_real @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_72_order__root, axiom,
    ((![P : poly_poly_real, A2 : poly_real]: (((poly_poly_real2 @ P @ A2) = zero_zero_poly_real) = (((P = zero_z1423781445y_real)) | ((~ (((order_poly_real @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_73_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_74_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_75_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_shift_0
thf(fact_76_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_real @ N @ zero_zero_poly_real) = zero_zero_poly_real)))). % poly_shift_0
thf(fact_77_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_78_poly__shift__0, axiom,
    ((![N : nat]: ((poly_s558570093omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_shift_0
thf(fact_79_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_80_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_81_reflect__poly__0, axiom,
    (((reflect_poly_real @ zero_zero_poly_real) = zero_zero_poly_real))). % reflect_poly_0
thf(fact_82_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_83_reflect__poly__0, axiom,
    (((reflec309385472omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % reflect_poly_0
thf(fact_84_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_85_synthetic__div__0, axiom,
    ((![C : real]: ((synthetic_div_real @ zero_zero_poly_real @ C) = zero_zero_poly_real)))). % synthetic_div_0
thf(fact_86_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_87_synthetic__div__0, axiom,
    ((![C : poly_complex]: ((synthe1985144195omplex @ zero_z1040703943omplex @ C) = zero_z1040703943omplex)))). % synthetic_div_0
thf(fact_88_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_89_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_90_poly__power, axiom,
    ((![P : poly_real, N : nat, X2 : real]: ((poly_real2 @ (power_2108872382y_real @ P @ N) @ X2) = (power_power_real @ (poly_real2 @ P @ X2) @ N))))). % poly_power
thf(fact_91_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_92_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_93_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_94_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_95_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_96_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_97_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_98_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_99_order__0I, axiom,
    ((![P : poly_real, A2 : real]: ((~ (((poly_real2 @ P @ A2) = zero_zero_real))) => ((order_real @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_100_order__0I, axiom,
    ((![P : poly_poly_real, A2 : poly_real]: ((~ (((poly_poly_real2 @ P @ A2) = zero_zero_poly_real))) => ((order_poly_real @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_101_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_102_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_103_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_104_power__eq__0__iff, axiom,
    ((![A2 : poly_real, N : nat]: (((power_2108872382y_real @ A2 @ N) = zero_zero_poly_real) = (((A2 = zero_zero_poly_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_105_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_106_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_107_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_108_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_109_power__eq__0__iff, axiom,
    ((![A2 : real, N : nat]: (((power_power_real @ A2 @ N) = zero_zero_real) = (((A2 = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_110_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_111_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_112_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_113_oop, axiom,
    ((![A2 : complex]: (ord_less_eq_nat @ (order_complex @ A2 @ pa) @ na)))). % oop
thf(fact_114_dvd__0__right, axiom,
    ((![A2 : complex]: (dvd_dvd_complex @ A2 @ zero_zero_complex)))). % dvd_0_right
thf(fact_115_dvd__0__right, axiom,
    ((![A2 : poly_complex]: (dvd_dvd_poly_complex @ A2 @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_116_dvd__0__right, axiom,
    ((![A2 : nat]: (dvd_dvd_nat @ A2 @ zero_zero_nat)))). % dvd_0_right
thf(fact_117_dvd__0__right, axiom,
    ((![A2 : real]: (dvd_dvd_real @ A2 @ zero_zero_real)))). % dvd_0_right
thf(fact_118_dvd__0__right, axiom,
    ((![A2 : poly_real]: (dvd_dvd_poly_real @ A2 @ zero_zero_poly_real)))). % dvd_0_right
thf(fact_119_dvd__0__right, axiom,
    ((![A2 : poly_nat]: (dvd_dvd_poly_nat @ A2 @ zero_zero_poly_nat)))). % dvd_0_right
thf(fact_120_dvd__0__right, axiom,
    ((![A2 : poly_poly_complex]: (dvd_dv598755940omplex @ A2 @ zero_z1040703943omplex)))). % dvd_0_right
thf(fact_121_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_122_dvd__0__left__iff, axiom,
    ((![A2 : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A2) = (A2 = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_123_dvd__0__left__iff, axiom,
    ((![A2 : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A2) = (A2 = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_124_dvd__0__left__iff, axiom,
    ((![A2 : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A2) = (A2 = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_125_dvd__0__left__iff, axiom,
    ((![A2 : real]: ((dvd_dvd_real @ zero_zero_real @ A2) = (A2 = zero_zero_real))))). % dvd_0_left_iff
thf(fact_126_dvd__0__left__iff, axiom,
    ((![A2 : poly_real]: ((dvd_dvd_poly_real @ zero_zero_poly_real @ A2) = (A2 = zero_zero_poly_real))))). % dvd_0_left_iff
thf(fact_127_dvd__0__left__iff, axiom,
    ((![A2 : poly_nat]: ((dvd_dvd_poly_nat @ zero_zero_poly_nat @ A2) = (A2 = zero_zero_poly_nat))))). % dvd_0_left_iff
thf(fact_128_dvd__0__left__iff, axiom,
    ((![A2 : poly_poly_complex]: ((dvd_dv598755940omplex @ zero_z1040703943omplex @ A2) = (A2 = zero_z1040703943omplex))))). % dvd_0_left_iff
thf(fact_129_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_130_bot__nat__0_Oextremum, axiom,
    ((![A2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ A2)))). % bot_nat_0.extremum
thf(fact_131_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_132_power__mono__iff, axiom,
    ((![A2 : poly_real, B : poly_real, N : nat]: ((ord_le1180086932y_real @ zero_zero_poly_real @ A2) => ((ord_le1180086932y_real @ zero_zero_poly_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_le1180086932y_real @ (power_2108872382y_real @ A2 @ N) @ (power_2108872382y_real @ B @ N)) = (ord_le1180086932y_real @ A2 @ B)))))))). % power_mono_iff
thf(fact_133_power__mono__iff, axiom,
    ((![A2 : real, B : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A2) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ (power_power_real @ A2 @ N) @ (power_power_real @ B @ N)) = (ord_less_eq_real @ A2 @ B)))))))). % power_mono_iff
thf(fact_134_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_135_dvd__imp__le, axiom,
    ((![K : nat, N : nat]: ((dvd_dvd_nat @ K @ N) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ K @ N)))))). % dvd_imp_le
thf(fact_136_gcd__nat_Oextremum, axiom,
    ((![A2 : nat]: (dvd_dvd_nat @ A2 @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_137_gcd__nat_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A2) & (~ ((zero_zero_nat = A2))))))))). % gcd_nat.extremum_strict
thf(fact_138_gcd__nat_Oextremum__unique, axiom,
    ((![A2 : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A2) = (A2 = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_139_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_140_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A2 : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A2) => (A2 = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_141_less__mono__imp__le__mono, axiom,
    ((![F2 : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F2 @ I2) @ (F2 @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F2 @ I) @ (F2 @ J))))))). % less_mono_imp_le_mono
thf(fact_142_le__neq__implies__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((~ ((M2 = N))) => (ord_less_nat @ M2 @ N)))))). % le_neq_implies_less
thf(fact_143_less__or__eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: (((ord_less_nat @ M2 @ N) | (M2 = N)) => (ord_less_eq_nat @ M2 @ N))))). % less_or_eq_imp_le
thf(fact_144_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N2 : nat]: (((ord_less_nat @ M3 @ N2)) | ((M3 = N2)))))))). % le_eq_less_or_eq
thf(fact_145_less__imp__le__nat, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_eq_nat @ M2 @ N))))). % less_imp_le_nat
thf(fact_146_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M3 @ N2)) & ((~ ((M3 = N2)))))))))). % nat_less_le
thf(fact_147_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A2 : nat]: ((ord_less_eq_nat @ A2 @ zero_zero_nat) => (A2 = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_148_bot__nat__0_Oextremum__unique, axiom,
    ((![A2 : nat]: ((ord_less_eq_nat @ A2 @ zero_zero_nat) = (A2 = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_149_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_150_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_151_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_152_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_153_eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: ((M2 = N) => (ord_less_eq_nat @ M2 @ N))))). % eq_imp_le
thf(fact_154_le__antisym, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((ord_less_eq_nat @ N @ M2) => (M2 = N)))))). % le_antisym
thf(fact_155_nat__le__linear, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) | (ord_less_eq_nat @ N @ M2))))). % nat_le_linear
thf(fact_156_Nat_Oex__has__greatest__nat, axiom,
    ((![P4 : nat > $o, K : nat, B : nat]: ((P4 @ K) => ((![Y2 : nat]: ((P4 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (?[X4 : nat]: ((P4 @ X4) & (![Y3 : nat]: ((P4 @ Y3) => (ord_less_eq_nat @ Y3 @ X4)))))))))). % Nat.ex_has_greatest_nat
thf(fact_157_zero__le, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X2)))). % zero_le
thf(fact_158_zero__le__power, axiom,
    ((![A2 : poly_real, N : nat]: ((ord_le1180086932y_real @ zero_zero_poly_real @ A2) => (ord_le1180086932y_real @ zero_zero_poly_real @ (power_2108872382y_real @ A2 @ N)))))). % zero_le_power
thf(fact_159_zero__le__power, axiom,
    ((![A2 : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A2) => (ord_less_eq_real @ zero_zero_real @ (power_power_real @ A2 @ N)))))). % zero_le_power
thf(fact_160_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_161_power__mono, axiom,
    ((![A2 : poly_real, B : poly_real, N : nat]: ((ord_le1180086932y_real @ A2 @ B) => ((ord_le1180086932y_real @ zero_zero_poly_real @ A2) => (ord_le1180086932y_real @ (power_2108872382y_real @ A2 @ N) @ (power_2108872382y_real @ B @ N))))))). % power_mono
thf(fact_162_power__mono, axiom,
    ((![A2 : real, B : real, N : nat]: ((ord_less_eq_real @ A2 @ B) => ((ord_less_eq_real @ zero_zero_real @ A2) => (ord_less_eq_real @ (power_power_real @ A2 @ N) @ (power_power_real @ B @ N))))))). % power_mono
thf(fact_163_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_164_nat__power__less__imp__less, axiom,
    ((![I : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M2) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M2 @ N)))))). % nat_power_less_imp_less
thf(fact_165_nat__dvd__not__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ M2) => ((ord_less_nat @ M2 @ N) => (~ ((dvd_dvd_nat @ N @ M2)))))))). % nat_dvd_not_less
thf(fact_166_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_167_dvd__power__le, axiom,
    ((![X2 : poly_complex, Y4 : poly_complex, N : nat, M2 : nat]: ((dvd_dvd_poly_complex @ X2 @ Y4) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X2 @ N) @ (power_184595776omplex @ Y4 @ M2))))))). % dvd_power_le
thf(fact_168_dvd__power__le, axiom,
    ((![X2 : nat, Y4 : nat, N : nat, M2 : nat]: ((dvd_dvd_nat @ X2 @ Y4) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_nat @ (power_power_nat @ X2 @ N) @ (power_power_nat @ Y4 @ M2))))))). % dvd_power_le
thf(fact_169_dvd__power__le, axiom,
    ((![X2 : real, Y4 : real, N : nat, M2 : nat]: ((dvd_dvd_real @ X2 @ Y4) => ((ord_less_eq_nat @ N @ M2) => (dvd_dvd_real @ (power_power_real @ X2 @ N) @ (power_power_real @ Y4 @ M2))))))). % dvd_power_le
thf(fact_170_power__le__dvd, axiom,
    ((![A2 : poly_complex, N : nat, B : poly_complex, M2 : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A2 @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A2 @ M2) @ B)))))). % power_le_dvd
thf(fact_171_power__le__dvd, axiom,
    ((![A2 : nat, N : nat, B : nat, M2 : nat]: ((dvd_dvd_nat @ (power_power_nat @ A2 @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_nat @ (power_power_nat @ A2 @ M2) @ B)))))). % power_le_dvd
thf(fact_172_power__le__dvd, axiom,
    ((![A2 : real, N : nat, B : real, M2 : nat]: ((dvd_dvd_real @ (power_power_real @ A2 @ N) @ B) => ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_real @ (power_power_real @ A2 @ M2) @ B)))))). % power_le_dvd
thf(fact_173_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A2 : poly_complex]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A2 @ M2) @ (power_184595776omplex @ A2 @ N)))))). % le_imp_power_dvd
thf(fact_174_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_175_le__imp__power__dvd, axiom,
    ((![M2 : nat, N : nat, A2 : real]: ((ord_less_eq_nat @ M2 @ N) => (dvd_dvd_real @ (power_power_real @ A2 @ M2) @ (power_power_real @ A2 @ N)))))). % le_imp_power_dvd
thf(fact_176_ex__least__nat__le, axiom,
    ((![P4 : nat > $o, N : nat]: ((P4 @ N) => ((~ ((P4 @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N) & ((![I3 : nat]: ((ord_less_nat @ I3 @ K2) => (~ ((P4 @ I3))))) & (P4 @ K2))))))))). % ex_least_nat_le
thf(fact_177_power__less__imp__less__base, axiom,
    ((![A2 : poly_real, N : nat, B : poly_real]: ((ord_less_poly_real @ (power_2108872382y_real @ A2 @ N) @ (power_2108872382y_real @ B @ N)) => ((ord_le1180086932y_real @ zero_zero_poly_real @ B) => (ord_less_poly_real @ A2 @ B)))))). % power_less_imp_less_base
thf(fact_178_power__less__imp__less__base, axiom,
    ((![A2 : real, N : nat, B : real]: ((ord_less_real @ (power_power_real @ A2 @ N) @ (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_real @ A2 @ B)))))). % power_less_imp_less_base
thf(fact_179_power__less__imp__less__base, axiom,
    ((![A2 : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A2 @ B)))))). % power_less_imp_less_base
thf(fact_180_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_181_power__eq__imp__eq__base, axiom,
    ((![A2 : poly_real, N : nat, B : poly_real]: (((power_2108872382y_real @ A2 @ N) = (power_2108872382y_real @ B @ N)) => ((ord_le1180086932y_real @ zero_zero_poly_real @ A2) => ((ord_le1180086932y_real @ zero_zero_poly_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A2 = B)))))))). % power_eq_imp_eq_base
thf(fact_182_power__eq__imp__eq__base, axiom,
    ((![A2 : real, N : nat, B : real]: (((power_power_real @ A2 @ N) = (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ A2) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A2 = B)))))))). % power_eq_imp_eq_base
thf(fact_183_power__eq__imp__eq__base, axiom,
    ((![A2 : nat, N : nat, B : nat]: (((power_power_nat @ A2 @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A2) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A2 = B)))))))). % power_eq_imp_eq_base
thf(fact_184_power__eq__iff__eq__base, axiom,
    ((![N : nat, A2 : poly_real, B : poly_real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_le1180086932y_real @ zero_zero_poly_real @ A2) => ((ord_le1180086932y_real @ zero_zero_poly_real @ B) => (((power_2108872382y_real @ A2 @ N) = (power_2108872382y_real @ B @ N)) = (A2 = B)))))))). % power_eq_iff_eq_base
thf(fact_185_power__eq__iff__eq__base, axiom,
    ((![N : nat, A2 : real, B : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ zero_zero_real @ A2) => ((ord_less_eq_real @ zero_zero_real @ B) => (((power_power_real @ A2 @ N) = (power_power_real @ B @ N)) = (A2 = B)))))))). % power_eq_iff_eq_base
thf(fact_186_power__eq__iff__eq__base, axiom,
    ((![N : nat, A2 : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A2) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A2 @ N) = (power_power_nat @ B @ N)) = (A2 = B)))))))). % power_eq_iff_eq_base
thf(fact_187_linorder__neqE__linordered__idom, axiom,
    ((![X2 : real, Y4 : real]: ((~ ((X2 = Y4))) => ((~ ((ord_less_real @ X2 @ Y4))) => (ord_less_real @ Y4 @ X2)))))). % linorder_neqE_linordered_idom
thf(fact_188_dvd__refl, axiom,
    ((![A2 : poly_complex]: (dvd_dvd_poly_complex @ A2 @ A2)))). % dvd_refl
thf(fact_189_dvd__refl, axiom,
    ((![A2 : nat]: (dvd_dvd_nat @ A2 @ A2)))). % dvd_refl
thf(fact_190_dvd__trans, axiom,
    ((![A2 : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A2 @ B) => ((dvd_dvd_poly_complex @ B @ C) => (dvd_dvd_poly_complex @ A2 @ C)))))). % dvd_trans
thf(fact_191_dvd__trans, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A2 @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A2 @ C)))))). % dvd_trans
thf(fact_192_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_193_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_194_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_195_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_196_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_197_nat__less__induct, axiom,
    ((![P4 : nat > $o, N : nat]: ((![N3 : nat]: ((![M : nat]: ((ord_less_nat @ M @ N3) => (P4 @ M))) => (P4 @ N3))) => (P4 @ N))))). % nat_less_induct
thf(fact_198_infinite__descent, axiom,
    ((![P4 : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P4 @ N3))) => (?[M : nat]: ((ord_less_nat @ M @ N3) & (~ ((P4 @ M))))))) => (P4 @ N))))). % infinite_descent
thf(fact_199_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((X2 = Y4))) => ((~ ((ord_less_nat @ X2 @ Y4))) => (ord_less_nat @ Y4 @ X2)))))). % linorder_neqE_nat
thf(fact_200_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_201_divides__degree, axiom,
    ((![P : poly_real, Q : poly_real]: ((dvd_dvd_poly_real @ P @ Q) => ((ord_less_eq_nat @ (degree_real @ P) @ (degree_real @ Q)) | (Q = zero_zero_poly_real)))))). % divides_degree
thf(fact_202_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_203_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_204_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_205_dvd__imp__degree__le, axiom,
    ((![P : poly_real, Q : poly_real]: ((dvd_dvd_poly_real @ P @ Q) => ((~ ((Q = zero_zero_poly_real))) => (ord_less_eq_nat @ (degree_real @ P) @ (degree_real @ Q))))))). % dvd_imp_degree_le
thf(fact_206_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_207_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_208_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_209_order__degree, axiom,
    ((![P : poly_real, A2 : real]: ((~ ((P = zero_zero_poly_real))) => (ord_less_eq_nat @ (order_real @ A2 @ P) @ (degree_real @ P)))))). % order_degree
thf(fact_210_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_211_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_212_dvd__imp__order__le, axiom,
    ((![Q : poly_real, P : poly_real, A2 : real]: ((~ ((Q = zero_zero_poly_real))) => ((dvd_dvd_poly_real @ P @ Q) => (ord_less_eq_nat @ (order_real @ A2 @ P) @ (order_real @ A2 @ Q))))))). % dvd_imp_order_le
thf(fact_213_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_214_power__strict__mono, axiom,
    ((![A2 : real, B : real, N : nat]: ((ord_less_real @ A2 @ B) => ((ord_less_eq_real @ zero_zero_real @ A2) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ (power_power_real @ A2 @ N) @ (power_power_real @ B @ N)))))))). % power_strict_mono
thf(fact_215_power__strict__mono, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((ord_less_nat @ A2 @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A2) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_216_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_217_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_218_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_219_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_220_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_221_infinite__descent0, axiom,
    ((![P4 : nat > $o, N : nat]: ((P4 @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P4 @ N3))) => (?[M : nat]: ((ord_less_nat @ M @ N3) & (~ ((P4 @ M)))))))) => (P4 @ N)))))). % infinite_descent0
thf(fact_222_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_223_nat__descend__induct, axiom,
    ((![N : nat, P4 : nat > $o, M2 : nat]: ((![K2 : nat]: ((ord_less_nat @ N @ K2) => (P4 @ K2))) => ((![K2 : nat]: ((ord_less_eq_nat @ K2 @ N) => ((![I3 : nat]: ((ord_less_nat @ K2 @ I3) => (P4 @ I3))) => (P4 @ K2)))) => (P4 @ M2)))))). % nat_descend_induct
thf(fact_224_realpow__pos__nth, axiom,
    ((![N : nat, A2 : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A2) => (?[R : real]: ((ord_less_real @ zero_zero_real @ R) & ((power_power_real @ R @ N) = A2)))))))). % realpow_pos_nth
thf(fact_225_poly__IVT__neg, axiom,
    ((![A2 : real, B : real, P : poly_real]: ((ord_less_real @ A2 @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A2)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X4 : real]: ((ord_less_real @ A2 @ X4) & ((ord_less_real @ X4 @ B) & ((poly_real2 @ P @ X4) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_226_poly__IVT__pos, axiom,
    ((![A2 : real, B : real, P : poly_real]: ((ord_less_real @ A2 @ B) => ((ord_less_real @ (poly_real2 @ P @ A2) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X4 : real]: ((ord_less_real @ A2 @ X4) & ((ord_less_real @ X4 @ B) & ((poly_real2 @ P @ X4) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_227_real__sup__exists, axiom,
    ((![P4 : real > $o]: ((?[X_1 : real]: (P4 @ X_1)) => ((?[Z2 : real]: (![X4 : real]: ((P4 @ X4) => (ord_less_real @ X4 @ Z2)))) => (?[S2 : real]: (![Y3 : real]: ((?[X3 : real]: (((P4 @ X3)) & ((ord_less_real @ Y3 @ X3)))) = (ord_less_real @ Y3 @ S2))))))))). % real_sup_exists
thf(fact_228_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = B))) => ((dvd_dvd_nat @ A2 @ B) => ((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B))))))))). % gcd_nat.not_eq_order_implies_strict
thf(fact_229_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A2 : nat, B : nat]: (((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B)))) => (~ ((A2 = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_230_gcd__nat_Ostrict__implies__order, axiom,
    ((![A2 : nat, B : nat]: (((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B)))) => (dvd_dvd_nat @ A2 @ B))))). % gcd_nat.strict_implies_order
thf(fact_231_gcd__nat_Ostrict__iff__order, axiom,
    ((![A2 : nat, B : nat]: ((((dvd_dvd_nat @ A2 @ B)) & ((~ ((A2 = B))))) = (((dvd_dvd_nat @ A2 @ B)) & ((~ ((A2 = B))))))))). % gcd_nat.strict_iff_order
thf(fact_232_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A3 : nat]: (^[B2 : nat]: (((((dvd_dvd_nat @ A3 @ B2)) & ((~ ((A3 = B2)))))) | ((A3 = B2)))))))). % gcd_nat.order_iff_strict
thf(fact_233_gcd__nat_Ostrict__trans2, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A2 @ C) & (~ ((A2 = C))))))))). % gcd_nat.strict_trans2
thf(fact_234_gcd__nat_Ostrict__trans1, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A2 @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A2 @ C) & (~ ((A2 = C))))))))). % gcd_nat.strict_trans1
thf(fact_235_gcd__nat_Ostrict__trans, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A2 @ C) & (~ ((A2 = C))))))))). % gcd_nat.strict_trans
thf(fact_236_gcd__nat_Oantisym, axiom,
    ((![A2 : nat, B : nat]: ((dvd_dvd_nat @ A2 @ B) => ((dvd_dvd_nat @ B @ A2) => (A2 = B)))))). % gcd_nat.antisym
thf(fact_237_gcd__nat_Oirrefl, axiom,
    ((![A2 : nat]: (~ (((dvd_dvd_nat @ A2 @ A2) & (~ ((A2 = A2))))))))). % gcd_nat.irrefl
thf(fact_238_gcd__nat_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z3 : nat]: (Y5 = Z3))) = (^[A3 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A3 @ B2)) & ((dvd_dvd_nat @ B2 @ A3)))))))). % gcd_nat.eq_iff
thf(fact_239_gcd__nat_Otrans, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A2 @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A2 @ C)))))). % gcd_nat.trans
thf(fact_240_gcd__nat_Orefl, axiom,
    ((![A2 : nat]: (dvd_dvd_nat @ A2 @ A2)))). % gcd_nat.refl
thf(fact_241_gcd__nat_Oasym, axiom,
    ((![A2 : nat, B : nat]: (((dvd_dvd_nat @ A2 @ B) & (~ ((A2 = B)))) => (~ (((dvd_dvd_nat @ B @ A2) & (~ ((B = A2)))))))))). % gcd_nat.asym

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