% 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_966__5376990_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:47.659

% 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 (74)
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_Ooffset__poly_001t__Complex__Ocomplex, type,
    fundam1201687030omplex : poly_complex > complex > poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_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__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_Odegree_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    degree_poly_real : poly_poly_real > nat).
thf(sy_c_Polynomial_Odegree_001t__Real__Oreal, type,
    degree_real : poly_real > nat).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    order_poly_complex : poly_complex > poly_poly_complex > nat).
thf(sy_c_Polynomial_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_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_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    coeff_poly_real : poly_poly_real > nat > poly_real).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal, type,
    coeff_real : poly_real > nat > 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_Opseudo__mod_001t__Complex__Ocomplex, type,
    pseudo_mod_complex : poly_complex > poly_complex > 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__Real__Oreal, type,
    reflect_poly_real : poly_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__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    power_1336127338ly_nat : poly_poly_nat > nat > poly_poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    power_2001192272omplex : poly_p1267267526omplex > nat > poly_p1267267526omplex).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    power_393057350y_real : poly_poly_real > nat > poly_poly_real).
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_v_a____, type,
    a : complex).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).
thf(sy_v_qa____, type,
    qa : poly_complex).
thf(sy_v_r____, type,
    r : poly_complex).
thf(sy_v_s____, type,
    s : poly_complex).

% Relevant facts (243)
thf(fact_0_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_1_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_2__092_060open_062_092_060And_062x_O_Apoly_As_Ax_A_061_A0_A_092_060Longrightarrow_062_Apoly_Ar_Ax_A_061_A0_092_060close_062, axiom,
    ((![X : complex]: (((poly_complex2 @ s @ X) = zero_zero_complex) => ((poly_complex2 @ r @ X) = zero_zero_complex))))). % \<open>\<And>x. poly s x = 0 \<Longrightarrow> poly r x = 0\<close>
thf(fact_3_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_4_assms_I1_J, axiom,
    ((![X2 : complex]: (((poly_complex2 @ p @ X2) = zero_zero_complex) => ((poly_complex2 @ q @ X2) = zero_zero_complex))))). % assms(1)
thf(fact_5__092_060open_062_092_060lbrakk_062_092_060And_062x_O_Apoly_As_Ax_A_061_A0_A_092_060Longrightarrow_062_Apoly_Ar_Ax_A_061_A0_059_Adegree_As_A_061_Adegree_As_059_Adegree_As_A_092_060noteq_062_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_As_Advd_Ar_A_094_Adegree_As_092_060close_062, axiom,
    (((![X3 : complex]: (((poly_complex2 @ s @ X3) = zero_zero_complex) => ((poly_complex2 @ r @ X3) = zero_zero_complex))) => (((degree_complex @ s) = (degree_complex @ s)) => ((~ (((degree_complex @ s) = zero_zero_nat))) => (dvd_dvd_poly_complex @ s @ (power_184595776omplex @ r @ (degree_complex @ s)))))))). % \<open>\<lbrakk>\<And>x. poly s x = 0 \<Longrightarrow> poly r x = 0; degree s = degree s; degree s \<noteq> 0\<rbrakk> \<Longrightarrow> s dvd r ^ degree s\<close>
thf(fact_6_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_7_dvd__power__same, axiom,
    ((![X : complex, Y : complex, N : nat]: ((dvd_dvd_complex @ X @ Y) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)))))). % dvd_power_same
thf(fact_8_dvd__power__same, axiom,
    ((![X : poly_real, Y : poly_real, N : nat]: ((dvd_dvd_poly_real @ X @ Y) => (dvd_dvd_poly_real @ (power_2108872382y_real @ X @ N) @ (power_2108872382y_real @ Y @ N)))))). % dvd_power_same
thf(fact_9_dvd__power__same, axiom,
    ((![X : poly_nat, Y : poly_nat, N : nat]: ((dvd_dvd_poly_nat @ X @ Y) => (dvd_dvd_poly_nat @ (power_power_poly_nat @ X @ N) @ (power_power_poly_nat @ Y @ N)))))). % dvd_power_same
thf(fact_10_dvd__power__same, axiom,
    ((![X : poly_poly_complex, Y : poly_poly_complex, N : nat]: ((dvd_dv598755940omplex @ X @ Y) => (dvd_dv598755940omplex @ (power_432682568omplex @ X @ N) @ (power_432682568omplex @ Y @ N)))))). % dvd_power_same
thf(fact_11_dvd__power__same, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ X @ Y) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ N)))))). % dvd_power_same
thf(fact_12_dvd__power__same, axiom,
    ((![X : nat, Y : nat, N : nat]: ((dvd_dvd_nat @ X @ Y) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)))))). % dvd_power_same
thf(fact_13_dvd__power__same, axiom,
    ((![X : real, Y : real, N : nat]: ((dvd_dvd_real @ X @ Y) => (dvd_dvd_real @ (power_power_real @ X @ N) @ (power_power_real @ Y @ N)))))). % dvd_power_same
thf(fact_14_poly__power, axiom,
    ((![P : poly_poly_real, N : nat, X : poly_real]: ((poly_poly_real2 @ (power_393057350y_real @ P @ N) @ X) = (power_2108872382y_real @ (poly_poly_real2 @ P @ X) @ N))))). % poly_power
thf(fact_15_poly__power, axiom,
    ((![P : poly_poly_nat, N : nat, X : poly_nat]: ((poly_poly_nat2 @ (power_1336127338ly_nat @ P @ N) @ X) = (power_power_poly_nat @ (poly_poly_nat2 @ P @ X) @ N))))). % poly_power
thf(fact_16_poly__power, axiom,
    ((![P : poly_p1267267526omplex, N : nat, X : poly_poly_complex]: ((poly_p282434315omplex @ (power_2001192272omplex @ P @ N) @ X) = (power_432682568omplex @ (poly_p282434315omplex @ P @ X) @ N))))). % poly_power
thf(fact_17_poly__power, axiom,
    ((![P : poly_poly_complex, N : nat, X : poly_complex]: ((poly_poly_complex2 @ (power_432682568omplex @ P @ N) @ X) = (power_184595776omplex @ (poly_poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_18_poly__power, axiom,
    ((![P : poly_nat, N : nat, X : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X) = (power_power_nat @ (poly_nat2 @ P @ X) @ N))))). % poly_power
thf(fact_19_poly__power, axiom,
    ((![P : poly_real, N : nat, X : real]: ((poly_real2 @ (power_2108872382y_real @ P @ N) @ X) = (power_power_real @ (poly_real2 @ P @ X) @ N))))). % poly_power
thf(fact_20_poly__power, axiom,
    ((![P : poly_complex, N : nat, X : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X) = (power_power_complex @ (poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_21_IH, axiom,
    ((![M : nat]: ((ord_less_nat @ M @ na) => (![P2 : poly_complex, Q : poly_complex]: ((![X3 : complex]: (((poly_complex2 @ P2 @ X3) = zero_zero_complex) => ((poly_complex2 @ Q @ X3) = zero_zero_complex))) => (((degree_complex @ P2) = M) => ((~ ((M = zero_zero_nat))) => (dvd_dvd_poly_complex @ P2 @ (power_184595776omplex @ Q @ M)))))))))). % IH
thf(fact_22_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_23_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_24_dvd__refl, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ A)))). % dvd_refl
thf(fact_25_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_26_dvd__refl, axiom,
    ((![A : real]: (dvd_dvd_real @ A @ A)))). % dvd_refl
thf(fact_27_dvd__trans, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((dvd_dvd_poly_complex @ B @ C) => (dvd_dvd_poly_complex @ A @ C)))))). % dvd_trans
thf(fact_28_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_29_dvd__trans, axiom,
    ((![A : real, B : real, C : real]: ((dvd_dvd_real @ A @ B) => ((dvd_dvd_real @ B @ C) => (dvd_dvd_real @ A @ C)))))). % dvd_trans
thf(fact_30_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_31_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_32_dvd__0__right, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_33_dvd__0__right, axiom,
    ((![A : real]: (dvd_dvd_real @ A @ zero_zero_real)))). % dvd_0_right
thf(fact_34_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_35_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_36_dvd__0__left__iff, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) = (A = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_37_dvd__0__left__iff, axiom,
    ((![A : real]: ((dvd_dvd_real @ zero_zero_real @ A) = (A = zero_zero_real))))). % dvd_0_left_iff
thf(fact_38_degree__offset__poly, axiom,
    ((![P : poly_complex, H : complex]: ((degree_complex @ (fundam1201687030omplex @ P @ H)) = (degree_complex @ P))))). % degree_offset_poly
thf(fact_39_lead__coeff__power, axiom,
    ((![P : poly_poly_real, N : nat]: ((coeff_poly_real @ (power_393057350y_real @ P @ N) @ (degree_poly_real @ (power_393057350y_real @ P @ N))) = (power_2108872382y_real @ (coeff_poly_real @ P @ (degree_poly_real @ P)) @ N))))). % lead_coeff_power
thf(fact_40_lead__coeff__power, axiom,
    ((![P : poly_poly_nat, N : nat]: ((coeff_poly_nat @ (power_1336127338ly_nat @ P @ N) @ (degree_poly_nat @ (power_1336127338ly_nat @ P @ N))) = (power_power_poly_nat @ (coeff_poly_nat @ P @ (degree_poly_nat @ P)) @ N))))). % lead_coeff_power
thf(fact_41_lead__coeff__power, axiom,
    ((![P : poly_p1267267526omplex, N : nat]: ((coeff_1429652124omplex @ (power_2001192272omplex @ P @ N) @ (degree2006505739omplex @ (power_2001192272omplex @ P @ N))) = (power_432682568omplex @ (coeff_1429652124omplex @ P @ (degree2006505739omplex @ P)) @ N))))). % lead_coeff_power
thf(fact_42_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_43_lead__coeff__power, axiom,
    ((![P : poly_real, N : nat]: ((coeff_real @ (power_2108872382y_real @ P @ N) @ (degree_real @ (power_2108872382y_real @ P @ N))) = (power_power_real @ (coeff_real @ P @ (degree_real @ P)) @ N))))). % lead_coeff_power
thf(fact_44_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_45_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_46_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_47_True, axiom,
    ((?[A2 : complex]: ((poly_complex2 @ pa @ A2) = zero_zero_complex)))). % True
thf(fact_48_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_49_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_50__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_O_Apoly_Ap_Aa_A_061_A0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![A2 : complex]: (~ (((poly_complex2 @ pa @ A2) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>a. poly p a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_51_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_52_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_53_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_54_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_complex @ zero_z1040703943omplex @ N) = zero_z1746442943omplex)))). % coeff_0
thf(fact_55_coeff__0, axiom,
    ((![N : nat]: ((coeff_real @ zero_zero_poly_real @ N) = zero_zero_real)))). % coeff_0
thf(fact_56_coeff__0, axiom,
    ((![N : nat]: ((coeff_complex @ zero_z1746442943omplex @ N) = zero_zero_complex)))). % coeff_0
thf(fact_57_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_58_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_59_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_60_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_61_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_62_power__eq__0__iff, axiom,
    ((![A : poly_complex, N : nat]: (((power_184595776omplex @ A @ N) = zero_z1746442943omplex) = (((A = zero_z1746442943omplex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_63_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_64_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_65_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_66_power__eq__0__iff, axiom,
    ((![A : poly_real, N : nat]: (((power_2108872382y_real @ A @ N) = zero_zero_poly_real) = (((A = zero_zero_poly_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_67_power__eq__0__iff, axiom,
    ((![A : poly_nat, N : nat]: (((power_power_poly_nat @ A @ N) = zero_zero_poly_nat) = (((A = zero_zero_poly_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_68_power__eq__0__iff, axiom,
    ((![A : poly_poly_complex, N : nat]: (((power_432682568omplex @ A @ N) = zero_z1040703943omplex) = (((A = zero_z1040703943omplex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_69_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_70_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_71_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_72_leading__coeff__0__iff, axiom,
    ((![P : poly_real]: (((coeff_real @ P @ (degree_real @ P)) = zero_zero_real) = (P = zero_zero_poly_real))))). % leading_coeff_0_iff
thf(fact_73_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_74_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_75_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_76_coeff__eq__0, axiom,
    ((![P : poly_real, N : nat]: ((ord_less_nat @ (degree_real @ P) @ N) => ((coeff_real @ P @ N) = zero_zero_real))))). % coeff_eq_0
thf(fact_77_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_78_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_79_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_80_poly__0__coeff__0, axiom,
    ((![P : poly_real]: ((poly_real2 @ P @ zero_zero_real) = (coeff_real @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_81_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_82_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_83_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_84_less__degree__imp, axiom,
    ((![N : nat, P : poly_real]: ((ord_less_nat @ N @ (degree_real @ P)) => (?[I : nat]: ((ord_less_nat @ N @ I) & (~ (((coeff_real @ P @ I) = zero_zero_real))))))))). % less_degree_imp
thf(fact_85_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X4 : complex]: ((poly_complex2 @ P @ X4) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_86_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X4 : poly_complex]: ((poly_poly_complex2 @ P @ X4) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_87_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X4 : real]: ((poly_real2 @ P @ X4) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_88_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_89_zero__poly_Orep__eq, axiom,
    (((coeff_poly_complex @ zero_z1040703943omplex) = (^[Uu : nat]: zero_z1746442943omplex)))). % zero_poly.rep_eq
thf(fact_90_zero__poly_Orep__eq, axiom,
    (((coeff_real @ zero_zero_poly_real) = (^[Uu : nat]: zero_zero_real)))). % zero_poly.rep_eq
thf(fact_91_zero__poly_Orep__eq, axiom,
    (((coeff_complex @ zero_z1746442943omplex) = (^[Uu : nat]: zero_zero_complex)))). % zero_poly.rep_eq
thf(fact_92_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_93_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_94_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_95_leading__coeff__neq__0, axiom,
    ((![P : poly_real]: ((~ ((P = zero_zero_poly_real))) => (~ (((coeff_real @ P @ (degree_real @ P)) = zero_zero_real))))))). % leading_coeff_neq_0
thf(fact_96_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_97_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q2 : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q2)) = (P = Q2))))). % poly_eq_poly_eq_iff
thf(fact_98_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q2 : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q2)) = (P = Q2))))). % poly_eq_poly_eq_iff
thf(fact_99_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_poly_complex, Q2 : poly_poly_complex]: (((poly_poly_complex2 @ P) = (poly_poly_complex2 @ Q2)) = (P = Q2))))). % poly_eq_poly_eq_iff
thf(fact_100_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_101_offset__poly__0, axiom,
    ((![H : complex]: ((fundam1201687030omplex @ zero_z1746442943omplex @ H) = zero_z1746442943omplex)))). % offset_poly_0
thf(fact_102_offset__poly__eq__0__iff, axiom,
    ((![P : poly_complex, H : complex]: (((fundam1201687030omplex @ P @ H) = zero_z1746442943omplex) = (P = zero_z1746442943omplex))))). % offset_poly_eq_0_iff
thf(fact_103_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_184595776omplex @ zero_z1746442943omplex @ N) = zero_z1746442943omplex))))). % zero_power
thf(fact_104_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_105_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_106_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_107_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_2108872382y_real @ zero_zero_poly_real @ N) = zero_zero_poly_real))))). % zero_power
thf(fact_108_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_poly_nat @ zero_zero_poly_nat @ N) = zero_zero_poly_nat))))). % zero_power
thf(fact_109_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_432682568omplex @ zero_z1040703943omplex @ N) = zero_z1040703943omplex))))). % zero_power
thf(fact_110_zero__less__power, axiom,
    ((![A : poly_real, N : nat]: ((ord_less_poly_real @ zero_zero_poly_real @ A) => (ord_less_poly_real @ zero_zero_poly_real @ (power_2108872382y_real @ A @ N)))))). % zero_less_power
thf(fact_111_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_112_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_113_coeff__0__power, axiom,
    ((![P : poly_poly_real, N : nat]: ((coeff_poly_real @ (power_393057350y_real @ P @ N) @ zero_zero_nat) = (power_2108872382y_real @ (coeff_poly_real @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_114_coeff__0__power, axiom,
    ((![P : poly_poly_nat, N : nat]: ((coeff_poly_nat @ (power_1336127338ly_nat @ P @ N) @ zero_zero_nat) = (power_power_poly_nat @ (coeff_poly_nat @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_115_coeff__0__power, axiom,
    ((![P : poly_p1267267526omplex, N : nat]: ((coeff_1429652124omplex @ (power_2001192272omplex @ P @ N) @ zero_zero_nat) = (power_432682568omplex @ (coeff_1429652124omplex @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_116_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_117_coeff__0__power, axiom,
    ((![P : poly_real, N : nat]: ((coeff_real @ (power_2108872382y_real @ P @ N) @ zero_zero_nat) = (power_power_real @ (coeff_real @ P @ zero_zero_nat) @ N))))). % coeff_0_power
thf(fact_118_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_119_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_120_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_121_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_122_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_123_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_124_power__not__zero, axiom,
    ((![A : poly_real, N : nat]: ((~ ((A = zero_zero_poly_real))) => (~ (((power_2108872382y_real @ A @ N) = zero_zero_poly_real))))))). % power_not_zero
thf(fact_125_power__not__zero, axiom,
    ((![A : poly_nat, N : nat]: ((~ ((A = zero_zero_poly_nat))) => (~ (((power_power_poly_nat @ A @ N) = zero_zero_poly_nat))))))). % power_not_zero
thf(fact_126_power__not__zero, axiom,
    ((![A : poly_poly_complex, N : nat]: ((~ ((A = zero_z1040703943omplex))) => (~ (((power_432682568omplex @ A @ N) = zero_z1040703943omplex))))))). % power_not_zero
thf(fact_127_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_128_dvd__0__left, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) => (A = zero_zero_complex))))). % dvd_0_left
thf(fact_129_dvd__0__left, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) => (A = zero_z1746442943omplex))))). % dvd_0_left
thf(fact_130_dvd__0__left, axiom,
    ((![A : real]: ((dvd_dvd_real @ zero_zero_real @ A) => (A = zero_zero_real))))). % dvd_0_left
thf(fact_131_pow__divides__pow__iff, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (dvd_dvd_nat @ A @ B)))))). % pow_divides_pow_iff
thf(fact_132_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_133_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_134_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_135_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_136_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_137_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_138_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_139_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_real]: (((ord_less_nat @ K @ N) => ((coeff_real @ (poly_cutoff_real @ N @ P) @ K) = (coeff_real @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_real @ (poly_cutoff_real @ N @ P) @ K) = zero_zero_real)))))). % coeff_poly_cutoff
thf(fact_140_pseudo__mod_I2_J, axiom,
    ((![G : poly_complex, F : poly_complex]: ((~ ((G = zero_z1746442943omplex))) => (((pseudo_mod_complex @ F @ G) = zero_z1746442943omplex) | (ord_less_nat @ (degree_complex @ (pseudo_mod_complex @ F @ G)) @ (degree_complex @ G))))))). % pseudo_mod(2)
thf(fact_141_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_142_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_143_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_144_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_145_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_146_zero__reorient, axiom,
    ((![X : poly_complex]: ((zero_z1746442943omplex = X) = (X = zero_z1746442943omplex))))). % zero_reorient
thf(fact_147_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_148_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_149_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_150_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_151_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_152_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_153_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_154_infinite__descent, axiom,
    ((![P3 : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P3 @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P3 @ M))))))) => (P3 @ N))))). % infinite_descent
thf(fact_155_nat__less__induct, axiom,
    ((![P3 : nat > $o, N : nat]: ((![N2 : nat]: ((![M : nat]: ((ord_less_nat @ M @ N2) => (P3 @ M))) => (P3 @ N2))) => (P3 @ N))))). % nat_less_induct
thf(fact_156_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_157_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_158_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_159_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_160_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_161_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_162_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_163_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_164_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_165_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_166_infinite__descent0, axiom,
    ((![P3 : nat > $o, N : nat]: ((P3 @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P3 @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P3 @ M)))))))) => (P3 @ N)))))). % infinite_descent0
thf(fact_167_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_168_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_169_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_170_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_171_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_172_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_173_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X3 : real]: (((ord_less_real @ zero_zero_real @ X3) & ((power_power_real @ X3 @ N) = A)) & (![Y2 : real]: (((ord_less_real @ zero_zero_real @ Y2) & ((power_power_real @ Y2 @ N) = A)) => (Y2 = X3)))))))))). % realpow_pos_nth_unique
thf(fact_174_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R : real]: ((ord_less_real @ zero_zero_real @ R) & ((power_power_real @ R @ N) = A)))))))). % realpow_pos_nth
thf(fact_175_dvd__field__iff, axiom,
    ((dvd_dvd_complex = (^[A3 : complex]: (^[B2 : complex]: (((A3 = zero_zero_complex)) => ((B2 = zero_zero_complex)))))))). % dvd_field_iff
thf(fact_176_dvd__field__iff, axiom,
    ((dvd_dvd_real = (^[A3 : real]: (^[B2 : real]: (((A3 = zero_zero_real)) => ((B2 = zero_zero_real)))))))). % dvd_field_iff
thf(fact_177_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_178_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_179_dvd__antisym, axiom,
    ((![M2 : nat, N : nat]: ((dvd_dvd_nat @ M2 @ N) => ((dvd_dvd_nat @ N @ M2) => (M2 = N)))))). % dvd_antisym
thf(fact_180_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_181_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_182_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_183_gcd__nat_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z2 : nat]: (Y3 = Z2))) = (^[A3 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A3 @ B2)) & ((dvd_dvd_nat @ B2 @ A3)))))))). % gcd_nat.eq_iff
thf(fact_184_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_185_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_186_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_187_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_188_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_189_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_190_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_191_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_192_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_193_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_194_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_195_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ B) & (~ ((A = B))))))))). % gcd_nat.not_eq_order_implies_strict
thf(fact_196_real__sup__exists, axiom,
    ((![P3 : real > $o]: ((?[X_1 : real]: (P3 @ X_1)) => ((?[Z3 : real]: (![X3 : real]: ((P3 @ X3) => (ord_less_real @ X3 @ Z3)))) => (?[S2 : real]: (![Y2 : real]: ((?[X4 : real]: (((P3 @ X4)) & ((ord_less_real @ Y2 @ X4)))) = (ord_less_real @ Y2 @ S2))))))))). % real_sup_exists
thf(fact_197_constant__def, axiom,
    ((fundam1158420650omplex = (^[F2 : complex > complex]: (![X4 : complex]: (![Y4 : complex]: ((F2 @ X4) = (F2 @ Y4)))))))). % constant_def
thf(fact_198_linordered__field__no__lb, axiom,
    ((![X2 : real]: (?[Y5 : real]: (ord_less_real @ Y5 @ X2))))). % linordered_field_no_lb
thf(fact_199_linordered__field__no__ub, axiom,
    ((![X2 : real]: (?[X_12 : real]: (ord_less_real @ X2 @ X_12))))). % linordered_field_no_ub
thf(fact_200_order__0I, axiom,
    ((![P : poly_complex, A : complex]: ((~ (((poly_complex2 @ P @ A) = zero_zero_complex))) => ((order_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_201_order__0I, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: ((~ (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex))) => ((order_poly_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_202_order__0I, axiom,
    ((![P : poly_real, A : real]: ((~ (((poly_real2 @ P @ A) = zero_zero_real))) => ((order_real @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_203_order__root, axiom,
    ((![P : poly_complex, A : complex]: (((poly_complex2 @ P @ A) = zero_zero_complex) = (((P = zero_z1746442943omplex)) | ((~ (((order_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_204_order__root, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex) = (((P = zero_z1040703943omplex)) | ((~ (((order_poly_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_205_order__root, axiom,
    ((![P : poly_real, A : real]: (((poly_real2 @ P @ A) = zero_zero_real) = (((P = zero_zero_poly_real)) | ((~ (((order_real @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_206_oop, axiom,
    ((![A : complex]: (ord_less_eq_nat @ (order_complex @ A @ pa) @ na)))). % oop
thf(fact_207_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_208_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_209_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_210_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_211_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_212_poly__reflect__poly__0, axiom,
    ((![P : poly_real]: ((poly_real2 @ (reflect_poly_real @ P) @ zero_zero_real) = (coeff_real @ P @ (degree_real @ P)))))). % poly_reflect_poly_0
thf(fact_213_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_214_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_215_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_216_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_217_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_218_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_219_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_220_reflect__poly__reflect__poly, axiom,
    ((![P : poly_real]: ((~ (((coeff_real @ P @ zero_zero_nat) = zero_zero_real))) => ((reflect_poly_real @ (reflect_poly_real @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_221_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_222_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_223_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_224_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_225_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_226_power__mono__iff, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) = (ord_less_eq_real @ A @ B)))))))). % power_mono_iff
thf(fact_227_power__mono__iff, axiom,
    ((![A : poly_real, B : poly_real, N : nat]: ((ord_le1180086932y_real @ zero_zero_poly_real @ A) => ((ord_le1180086932y_real @ zero_zero_poly_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_le1180086932y_real @ (power_2108872382y_real @ A @ N) @ (power_2108872382y_real @ B @ N)) = (ord_le1180086932y_real @ A @ B)))))))). % power_mono_iff
thf(fact_228_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_229_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_real]: (((coeff_real @ (reflect_poly_real @ P) @ zero_zero_nat) = zero_zero_real) = (P = zero_zero_poly_real))))). % coeff_0_reflect_poly_0_iff
thf(fact_230_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_231_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_232_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_233_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_234_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_eq_nat @ M3 @ N3)) & ((~ ((M3 = N3)))))))))). % nat_less_le
thf(fact_235_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_236_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_nat @ M3 @ N3)) | ((M3 = N3)))))))). % le_eq_less_or_eq
thf(fact_237_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_238_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_239_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I2 : nat, J : nat]: ((![I : nat, J2 : nat]: ((ord_less_nat @ I @ J2) => (ord_less_nat @ (F @ I) @ (F @ J2)))) => ((ord_less_eq_nat @ I2 @ J) => (ord_less_eq_nat @ (F @ I2) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_240_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_241_le__trans, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I2 @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I2 @ K)))))). % le_trans
thf(fact_242_eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: ((M2 = N) => (ord_less_eq_nat @ M2 @ N))))). % eq_imp_le

% Conjectures (1)
thf(conj_0, conjecture,
    ((dvd_dvd_poly_complex @ s @ (power_184595776omplex @ r @ (degree_complex @ s))))).
