% 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_969__5377028_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:49.039

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    poly_p1267267526omplex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    poly_poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (45)
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    times_1460995011omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    times_1465266917ly_nat : poly_poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    times_2061725899omplex : poly_p1267267526omplex > poly_p1267267526omplex > poly_p1267267526omplex).
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_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_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_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > 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_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_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex, type,
    dvd_dvd_complex : complex > complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    dvd_dvd_poly_complex : poly_complex > poly_complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    dvd_dvd_poly_nat : poly_nat > poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    dvd_dv598755940omplex : poly_poly_complex > poly_poly_complex > $o).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).
thf(sy_v_qa____, type,
    qa : poly_complex).
thf(sy_v_r____, type,
    r : poly_complex).
thf(sy_v_s____, type,
    s : poly_complex).
thf(sy_v_u____, type,
    u : poly_complex).
thf(sy_v_x, type,
    x : complex).

% Relevant facts (243)
thf(fact_0_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_1_u, axiom,
    (((power_184595776omplex @ r @ (degree_complex @ s)) = (times_1246143675omplex @ s @ u)))). % u
thf(fact_2_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_Ar_A_094_Adegree_As_A_061_As_A_K_Au_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U : poly_complex]: (~ (((power_184595776omplex @ r @ (degree_complex @ s)) = (times_1246143675omplex @ s @ U))))))))). % \<open>\<And>thesis. (\<And>u. r ^ degree s = s * u \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_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_5_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_6_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_7_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_8_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_9__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_10_poly__mult, axiom,
    ((![P : poly_poly_nat, Q : poly_poly_nat, X : poly_nat]: ((poly_poly_nat2 @ (times_1465266917ly_nat @ P @ Q) @ X) = (times_times_poly_nat @ (poly_poly_nat2 @ P @ X) @ (poly_poly_nat2 @ Q @ X)))))). % poly_mult
thf(fact_11_poly__mult, axiom,
    ((![P : poly_p1267267526omplex, Q : poly_p1267267526omplex, X : poly_poly_complex]: ((poly_p282434315omplex @ (times_2061725899omplex @ P @ Q) @ X) = (times_1460995011omplex @ (poly_p282434315omplex @ P @ X) @ (poly_p282434315omplex @ Q @ X)))))). % poly_mult
thf(fact_12_poly__mult, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex, X : poly_complex]: ((poly_poly_complex2 @ (times_1460995011omplex @ P @ Q) @ X) = (times_1246143675omplex @ (poly_poly_complex2 @ P @ X) @ (poly_poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_13_poly__mult, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q) @ X) = (times_times_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q @ X)))))). % poly_mult
thf(fact_14_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X) = (times_times_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_15__092_060open_062s_Advd_Ar_A_094_Adegree_As_092_060close_062, axiom,
    ((dvd_dvd_poly_complex @ s @ (power_184595776omplex @ r @ (degree_complex @ s))))). % \<open>s dvd r ^ degree s\<close>
thf(fact_16_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_17_power__commutes, axiom,
    ((![A : poly_nat, N : nat]: ((times_times_poly_nat @ (power_power_poly_nat @ A @ N) @ A) = (times_times_poly_nat @ A @ (power_power_poly_nat @ A @ N)))))). % power_commutes
thf(fact_18_power__commutes, axiom,
    ((![A : poly_poly_complex, N : nat]: ((times_1460995011omplex @ (power_432682568omplex @ A @ N) @ A) = (times_1460995011omplex @ A @ (power_432682568omplex @ A @ N)))))). % power_commutes
thf(fact_19_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_20_power__commutes, axiom,
    ((![A : poly_complex, N : nat]: ((times_1246143675omplex @ (power_184595776omplex @ A @ N) @ A) = (times_1246143675omplex @ A @ (power_184595776omplex @ A @ N)))))). % power_commutes
thf(fact_21_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_22_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_23_power__mult__distrib, axiom,
    ((![A : poly_complex, B : poly_complex, N : nat]: ((power_184595776omplex @ (times_1246143675omplex @ A @ B) @ N) = (times_1246143675omplex @ (power_184595776omplex @ A @ N) @ (power_184595776omplex @ B @ N)))))). % power_mult_distrib
thf(fact_24_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_25_power__mult__distrib, axiom,
    ((![A : poly_nat, B : poly_nat, N : nat]: ((power_power_poly_nat @ (times_times_poly_nat @ A @ B) @ N) = (times_times_poly_nat @ (power_power_poly_nat @ A @ N) @ (power_power_poly_nat @ B @ N)))))). % power_mult_distrib
thf(fact_26_power__mult__distrib, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, N : nat]: ((power_432682568omplex @ (times_1460995011omplex @ A @ B) @ N) = (times_1460995011omplex @ (power_432682568omplex @ A @ N) @ (power_432682568omplex @ B @ N)))))). % power_mult_distrib
thf(fact_27_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N))))))). % power_commuting_commutes
thf(fact_28_power__commuting__commutes, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: (((times_1246143675omplex @ X @ Y) = (times_1246143675omplex @ Y @ X)) => ((times_1246143675omplex @ (power_184595776omplex @ X @ N) @ Y) = (times_1246143675omplex @ Y @ (power_184595776omplex @ X @ N))))))). % power_commuting_commutes
thf(fact_29_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_30_power__commuting__commutes, axiom,
    ((![X : poly_nat, Y : poly_nat, N : nat]: (((times_times_poly_nat @ X @ Y) = (times_times_poly_nat @ Y @ X)) => ((times_times_poly_nat @ (power_power_poly_nat @ X @ N) @ Y) = (times_times_poly_nat @ Y @ (power_power_poly_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_31_power__commuting__commutes, axiom,
    ((![X : poly_poly_complex, Y : poly_poly_complex, N : nat]: (((times_1460995011omplex @ X @ Y) = (times_1460995011omplex @ Y @ X)) => ((times_1460995011omplex @ (power_432682568omplex @ X @ N) @ Y) = (times_1460995011omplex @ Y @ (power_432682568omplex @ X @ N))))))). % power_commuting_commutes
thf(fact_32_assms_I1_J, axiom,
    ((![X2 : complex]: (((poly_complex2 @ p @ X2) = zero_zero_complex) => ((poly_complex2 @ q @ X2) = zero_zero_complex))))). % assms(1)
thf(fact_33_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_34_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_35_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_36_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_37_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_38_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_39_True, axiom,
    ((?[A2 : complex]: ((poly_complex2 @ pa @ A2) = zero_zero_complex)))). % True
thf(fact_40_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_41_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_42__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_43_IH, axiom,
    ((![M : nat]: ((ord_less_nat @ M @ na) => (![P2 : poly_complex, Q2 : poly_complex]: ((![X3 : complex]: (((poly_complex2 @ P2 @ X3) = zero_zero_complex) => ((poly_complex2 @ Q2 @ X3) = zero_zero_complex))) => (((degree_complex @ P2) = M) => ((~ ((M = zero_zero_nat))) => (dvd_dvd_poly_complex @ P2 @ (power_184595776omplex @ Q2 @ M)))))))))). % IH
thf(fact_44_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_45_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_46__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_47_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_48_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_49_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_50_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_51_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_52_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_53_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_54_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_55_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_56_degree__power__eq, axiom,
    ((![P : poly_complex, N : nat]: ((~ ((P = zero_z1746442943omplex))) => ((degree_complex @ (power_184595776omplex @ P @ N)) = (times_times_nat @ N @ (degree_complex @ P))))))). % degree_power_eq
thf(fact_57_degree__power__eq, axiom,
    ((![P : poly_poly_complex, N : nat]: ((~ ((P = zero_z1040703943omplex))) => ((degree_poly_complex @ (power_432682568omplex @ P @ N)) = (times_times_nat @ N @ (degree_poly_complex @ P))))))). % degree_power_eq
thf(fact_58_degree__mult__eq__0, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((degree_complex @ (times_1246143675omplex @ P @ Q)) = zero_zero_nat) = (((P = zero_z1746442943omplex)) | ((((Q = zero_z1746442943omplex)) | ((((~ ((P = zero_z1746442943omplex)))) & ((((~ ((Q = zero_z1746442943omplex)))) & (((((degree_complex @ P) = zero_zero_nat)) & (((degree_complex @ Q) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_59_degree__mult__eq__0, axiom,
    ((![P : poly_nat, Q : poly_nat]: (((degree_nat @ (times_times_poly_nat @ P @ Q)) = zero_zero_nat) = (((P = zero_zero_poly_nat)) | ((((Q = zero_zero_poly_nat)) | ((((~ ((P = zero_zero_poly_nat)))) & ((((~ ((Q = zero_zero_poly_nat)))) & (((((degree_nat @ P) = zero_zero_nat)) & (((degree_nat @ Q) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_60_degree__mult__eq__0, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex]: (((degree_poly_complex @ (times_1460995011omplex @ P @ Q)) = zero_zero_nat) = (((P = zero_z1040703943omplex)) | ((((Q = zero_z1040703943omplex)) | ((((~ ((P = zero_z1040703943omplex)))) & ((((~ ((Q = zero_z1040703943omplex)))) & (((((degree_poly_complex @ P) = zero_zero_nat)) & (((degree_poly_complex @ Q) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_61_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_62_mult__poly__0__left, axiom,
    ((![Q : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % mult_poly_0_left
thf(fact_63_mult__poly__0__left, axiom,
    ((![Q : poly_poly_complex]: ((times_1460995011omplex @ zero_z1040703943omplex @ Q) = zero_z1040703943omplex)))). % mult_poly_0_left
thf(fact_64_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_65_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_66_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_67_mult__poly__0__right, axiom,
    ((![P : poly_nat]: ((times_times_poly_nat @ P @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_poly_0_right
thf(fact_68_mult__poly__0__right, axiom,
    ((![P : poly_poly_complex]: ((times_1460995011omplex @ P @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % mult_poly_0_right
thf(fact_69_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_70_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_71_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_184595776omplex @ zero_z1746442943omplex @ N) = zero_z1746442943omplex))))). % zero_power
thf(fact_72_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_73_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_74_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_432682568omplex @ zero_z1040703943omplex @ N) = zero_z1040703943omplex))))). % zero_power
thf(fact_75_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_76_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_77_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_78_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_79_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_80_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_81_power__mult, axiom,
    ((![A : complex, M2 : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M2 @ N)) = (power_power_complex @ (power_power_complex @ A @ M2) @ N))))). % power_mult
thf(fact_82_power__mult, axiom,
    ((![A : poly_complex, M2 : nat, N : nat]: ((power_184595776omplex @ A @ (times_times_nat @ M2 @ N)) = (power_184595776omplex @ (power_184595776omplex @ A @ M2) @ N))))). % power_mult
thf(fact_83_power__mult, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M2 @ N)) = (power_power_nat @ (power_power_nat @ A @ M2) @ N))))). % power_mult
thf(fact_84_power__mult, axiom,
    ((![A : poly_nat, M2 : nat, N : nat]: ((power_power_poly_nat @ A @ (times_times_nat @ M2 @ N)) = (power_power_poly_nat @ (power_power_poly_nat @ A @ M2) @ N))))). % power_mult
thf(fact_85_power__mult, axiom,
    ((![A : poly_poly_complex, M2 : nat, N : nat]: ((power_432682568omplex @ A @ (times_times_nat @ M2 @ N)) = (power_432682568omplex @ (power_432682568omplex @ A @ M2) @ N))))). % power_mult
thf(fact_86_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_87_power__not__zero, axiom,
    ((![A : poly_complex, N : nat]: ((~ ((A = zero_z1746442943omplex))) => (~ (((power_184595776omplex @ A @ N) = zero_z1746442943omplex))))))). % power_not_zero
thf(fact_88_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_89_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_90_power__not__zero, axiom,
    ((![A : poly_poly_complex, N : nat]: ((~ ((A = zero_z1040703943omplex))) => (~ (((power_432682568omplex @ A @ N) = zero_z1040703943omplex))))))). % power_not_zero
thf(fact_91_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_92_dvd__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((dvd_dvd_complex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((dvd_dvd_complex @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_93_dvd__mult__cancel__left, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ C @ A) @ (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((dvd_dvd_poly_complex @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_94_dvd__mult__cancel__left, axiom,
    ((![C : poly_poly_complex, A : poly_poly_complex, B : poly_poly_complex]: ((dvd_dv598755940omplex @ (times_1460995011omplex @ C @ A) @ (times_1460995011omplex @ C @ B)) = (((C = zero_z1040703943omplex)) | ((dvd_dv598755940omplex @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_95_dvd__mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((dvd_dvd_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((dvd_dvd_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_96_dvd__mult__cancel__right, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ C) @ (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((dvd_dvd_poly_complex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_97_dvd__mult__cancel__right, axiom,
    ((![A : poly_poly_complex, C : poly_poly_complex, B : poly_poly_complex]: ((dvd_dv598755940omplex @ (times_1460995011omplex @ A @ C) @ (times_1460995011omplex @ B @ C)) = (((C = zero_z1040703943omplex)) | ((dvd_dv598755940omplex @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_98_dvd__times__left__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ A @ B) @ (times_1246143675omplex @ A @ C)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_99_dvd__times__left__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_100_dvd__times__left__cancel__iff, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, C : poly_poly_complex]: ((~ ((A = zero_z1040703943omplex))) => ((dvd_dv598755940omplex @ (times_1460995011omplex @ A @ B) @ (times_1460995011omplex @ A @ C)) = (dvd_dv598755940omplex @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_101_dvd__times__right__cancel__iff, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (times_1246143675omplex @ B @ A) @ (times_1246143675omplex @ C @ A)) = (dvd_dvd_poly_complex @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_102_dvd__times__right__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_103_dvd__times__right__cancel__iff, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, C : poly_poly_complex]: ((~ ((A = zero_z1040703943omplex))) => ((dvd_dv598755940omplex @ (times_1460995011omplex @ B @ A) @ (times_1460995011omplex @ C @ A)) = (dvd_dv598755940omplex @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_104_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_105_mult__less__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M2 @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M2 @ N))))))). % mult_less_cancel2
thf(fact_106_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_107_nat__0__less__mult__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M2 @ N)) = (((ord_less_nat @ zero_zero_nat @ M2)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_108_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_109_mult__zero__left, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_110_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_111_mult__zero__left, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ A) = zero_zero_poly_nat)))). % mult_zero_left
thf(fact_112_mult__zero__left, axiom,
    ((![A : poly_poly_complex]: ((times_1460995011omplex @ zero_z1040703943omplex @ A) = zero_z1040703943omplex)))). % mult_zero_left
thf(fact_113_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_114_mult__zero__right, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right
thf(fact_115_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_116_mult__zero__right, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_zero_right
thf(fact_117_mult__zero__right, axiom,
    ((![A : poly_poly_complex]: ((times_1460995011omplex @ A @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % mult_zero_right
thf(fact_118_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_119_mult__eq__0__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex) = (((A = zero_z1746442943omplex)) | ((B = zero_z1746442943omplex))))))). % mult_eq_0_iff
thf(fact_120_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_121_mult__eq__0__iff, axiom,
    ((![A : poly_nat, B : poly_nat]: (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat) = (((A = zero_zero_poly_nat)) | ((B = zero_zero_poly_nat))))))). % mult_eq_0_iff
thf(fact_122_mult__eq__0__iff, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex]: (((times_1460995011omplex @ A @ B) = zero_z1040703943omplex) = (((A = zero_z1040703943omplex)) | ((B = zero_z1040703943omplex))))))). % mult_eq_0_iff
thf(fact_123_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_124_mult__cancel__left, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ C @ A) = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((A = B))))))). % mult_cancel_left
thf(fact_125_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_126_mult__cancel__left, axiom,
    ((![C : poly_poly_complex, A : poly_poly_complex, B : poly_poly_complex]: (((times_1460995011omplex @ C @ A) = (times_1460995011omplex @ C @ B)) = (((C = zero_z1040703943omplex)) | ((A = B))))))). % mult_cancel_left
thf(fact_127_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_128_mult__cancel__right, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ C) = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((A = B))))))). % mult_cancel_right
thf(fact_129_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_130_mult__cancel__right, axiom,
    ((![A : poly_poly_complex, C : poly_poly_complex, B : poly_poly_complex]: (((times_1460995011omplex @ A @ C) = (times_1460995011omplex @ B @ C)) = (((C = zero_z1040703943omplex)) | ((A = B))))))). % mult_cancel_right
thf(fact_131_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_132_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_133_dvd__0__left__iff, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) = (A = zero_z1746442943omplex))))). % dvd_0_left_iff
thf(fact_134_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_135_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_136_dvd__0__right, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ zero_z1746442943omplex)))). % dvd_0_right
thf(fact_137_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_138_mult__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: (((times_times_nat @ M2 @ K) = (times_times_nat @ N @ K)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_139_mult__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_140_mult__0__right, axiom,
    ((![M2 : nat]: ((times_times_nat @ M2 @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_141_mult__is__0, axiom,
    ((![M2 : nat, N : nat]: (((times_times_nat @ M2 @ N) = zero_zero_nat) = (((M2 = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_142_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_143_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_144_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_145_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_146_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_147_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_148_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_149_dvd__mult__cancel, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((dvd_dvd_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) => ((ord_less_nat @ zero_zero_nat @ K) => (dvd_dvd_nat @ M2 @ N)))))). % dvd_mult_cancel
thf(fact_150_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_151_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_152_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_153_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_154_dvd__refl, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ A)))). % dvd_refl
thf(fact_155_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_156_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_157_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_158_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_159_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_160_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_161_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_162_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_163_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_164_mult__not__zero, axiom,
    ((![A : complex, B : complex]: ((~ (((times_times_complex @ A @ B) = zero_zero_complex))) => ((~ ((A = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_165_mult__not__zero, axiom,
    ((![A : poly_complex, B : poly_complex]: ((~ (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex))) => ((~ ((A = zero_z1746442943omplex))) & (~ ((B = zero_z1746442943omplex)))))))). % mult_not_zero
thf(fact_166_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_167_mult__not__zero, axiom,
    ((![A : poly_nat, B : poly_nat]: ((~ (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat))) => ((~ ((A = zero_zero_poly_nat))) & (~ ((B = zero_zero_poly_nat)))))))). % mult_not_zero
thf(fact_168_mult__not__zero, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex]: ((~ (((times_1460995011omplex @ A @ B) = zero_z1040703943omplex))) => ((~ ((A = zero_z1040703943omplex))) & (~ ((B = zero_z1040703943omplex)))))))). % mult_not_zero
thf(fact_169_divisors__zero, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) => ((A = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_170_divisors__zero, axiom,
    ((![A : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex) => ((A = zero_z1746442943omplex) | (B = zero_z1746442943omplex)))))). % divisors_zero
thf(fact_171_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_172_divisors__zero, axiom,
    ((![A : poly_nat, B : poly_nat]: (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat) => ((A = zero_zero_poly_nat) | (B = zero_zero_poly_nat)))))). % divisors_zero
thf(fact_173_divisors__zero, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex]: (((times_1460995011omplex @ A @ B) = zero_z1040703943omplex) => ((A = zero_z1040703943omplex) | (B = zero_z1040703943omplex)))))). % divisors_zero
thf(fact_174_no__zero__divisors, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_175_no__zero__divisors, axiom,
    ((![A : poly_complex, B : poly_complex]: ((~ ((A = zero_z1746442943omplex))) => ((~ ((B = zero_z1746442943omplex))) => (~ (((times_1246143675omplex @ A @ B) = zero_z1746442943omplex)))))))). % no_zero_divisors
thf(fact_176_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_177_no__zero__divisors, axiom,
    ((![A : poly_nat, B : poly_nat]: ((~ ((A = zero_zero_poly_nat))) => ((~ ((B = zero_zero_poly_nat))) => (~ (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat)))))))). % no_zero_divisors
thf(fact_178_no__zero__divisors, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex]: ((~ ((A = zero_z1040703943omplex))) => ((~ ((B = zero_z1040703943omplex))) => (~ (((times_1460995011omplex @ A @ B) = zero_z1040703943omplex)))))))). % no_zero_divisors
thf(fact_179_mult__left__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_180_mult__left__cancel, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ C @ A) = (times_1246143675omplex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_181_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_182_mult__left__cancel, axiom,
    ((![C : poly_poly_complex, A : poly_poly_complex, B : poly_poly_complex]: ((~ ((C = zero_z1040703943omplex))) => (((times_1460995011omplex @ C @ A) = (times_1460995011omplex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_183_mult__right__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_184_mult__right__cancel, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ A @ C) = (times_1246143675omplex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_185_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_186_mult__right__cancel, axiom,
    ((![C : poly_poly_complex, A : poly_poly_complex, B : poly_poly_complex]: ((~ ((C = zero_z1040703943omplex))) => (((times_1460995011omplex @ A @ C) = (times_1460995011omplex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_187_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_188_dvd__0__left, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) => (A = zero_zero_complex))))). % dvd_0_left
thf(fact_189_dvd__0__left, axiom,
    ((![A : poly_complex]: ((dvd_dvd_poly_complex @ zero_z1746442943omplex @ A) => (A = zero_z1746442943omplex))))). % dvd_0_left
thf(fact_190_dvdE, axiom,
    ((![B : complex, A : complex]: ((dvd_dvd_complex @ B @ A) => (~ ((![K2 : complex]: (~ ((A = (times_times_complex @ B @ K2))))))))))). % dvdE
thf(fact_191_dvdE, axiom,
    ((![B : poly_complex, A : poly_complex]: ((dvd_dvd_poly_complex @ B @ A) => (~ ((![K2 : poly_complex]: (~ ((A = (times_1246143675omplex @ B @ K2))))))))))). % dvdE
thf(fact_192_dvdE, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ A) => (~ ((![K2 : nat]: (~ ((A = (times_times_nat @ B @ K2))))))))))). % dvdE
thf(fact_193_dvdE, axiom,
    ((![B : poly_nat, A : poly_nat]: ((dvd_dvd_poly_nat @ B @ A) => (~ ((![K2 : poly_nat]: (~ ((A = (times_times_poly_nat @ B @ K2))))))))))). % dvdE
thf(fact_194_dvdE, axiom,
    ((![B : poly_poly_complex, A : poly_poly_complex]: ((dvd_dv598755940omplex @ B @ A) => (~ ((![K2 : poly_poly_complex]: (~ ((A = (times_1460995011omplex @ B @ K2))))))))))). % dvdE
thf(fact_195_dvdI, axiom,
    ((![A : complex, B : complex, K : complex]: ((A = (times_times_complex @ B @ K)) => (dvd_dvd_complex @ B @ A))))). % dvdI
thf(fact_196_dvdI, axiom,
    ((![A : poly_complex, B : poly_complex, K : poly_complex]: ((A = (times_1246143675omplex @ B @ K)) => (dvd_dvd_poly_complex @ B @ A))))). % dvdI
thf(fact_197_dvdI, axiom,
    ((![A : nat, B : nat, K : nat]: ((A = (times_times_nat @ B @ K)) => (dvd_dvd_nat @ B @ A))))). % dvdI
thf(fact_198_dvdI, axiom,
    ((![A : poly_nat, B : poly_nat, K : poly_nat]: ((A = (times_times_poly_nat @ B @ K)) => (dvd_dvd_poly_nat @ B @ A))))). % dvdI
thf(fact_199_dvdI, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, K : poly_poly_complex]: ((A = (times_1460995011omplex @ B @ K)) => (dvd_dv598755940omplex @ B @ A))))). % dvdI
thf(fact_200_dvd__def, axiom,
    ((dvd_dvd_complex = (^[B2 : complex]: (^[A3 : complex]: (?[K3 : complex]: (A3 = (times_times_complex @ B2 @ K3)))))))). % dvd_def
thf(fact_201_dvd__def, axiom,
    ((dvd_dvd_poly_complex = (^[B2 : poly_complex]: (^[A3 : poly_complex]: (?[K3 : poly_complex]: (A3 = (times_1246143675omplex @ B2 @ K3)))))))). % dvd_def
thf(fact_202_dvd__def, axiom,
    ((dvd_dvd_nat = (^[B2 : nat]: (^[A3 : nat]: (?[K3 : nat]: (A3 = (times_times_nat @ B2 @ K3)))))))). % dvd_def
thf(fact_203_dvd__def, axiom,
    ((dvd_dvd_poly_nat = (^[B2 : poly_nat]: (^[A3 : poly_nat]: (?[K3 : poly_nat]: (A3 = (times_times_poly_nat @ B2 @ K3)))))))). % dvd_def
thf(fact_204_dvd__def, axiom,
    ((dvd_dv598755940omplex = (^[B2 : poly_poly_complex]: (^[A3 : poly_poly_complex]: (?[K3 : poly_poly_complex]: (A3 = (times_1460995011omplex @ B2 @ K3)))))))). % dvd_def
thf(fact_205_dvd__productE, axiom,
    ((![P : nat, A : nat, B : nat]: ((dvd_dvd_nat @ P @ (times_times_nat @ A @ B)) => (~ ((![X3 : nat, Y2 : nat]: ((P = (times_times_nat @ X3 @ Y2)) => ((dvd_dvd_nat @ X3 @ A) => (~ ((dvd_dvd_nat @ Y2 @ B)))))))))))). % dvd_productE
thf(fact_206_dvd__mult, axiom,
    ((![A : complex, C : complex, B : complex]: ((dvd_dvd_complex @ A @ C) => (dvd_dvd_complex @ A @ (times_times_complex @ B @ C)))))). % dvd_mult
thf(fact_207_dvd__mult, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ A @ C) => (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ B @ C)))))). % dvd_mult
thf(fact_208_dvd__mult, axiom,
    ((![A : nat, C : nat, B : nat]: ((dvd_dvd_nat @ A @ C) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult
thf(fact_209_dvd__mult, axiom,
    ((![A : poly_nat, C : poly_nat, B : poly_nat]: ((dvd_dvd_poly_nat @ A @ C) => (dvd_dvd_poly_nat @ A @ (times_times_poly_nat @ B @ C)))))). % dvd_mult
thf(fact_210_dvd__mult, axiom,
    ((![A : poly_poly_complex, C : poly_poly_complex, B : poly_poly_complex]: ((dvd_dv598755940omplex @ A @ C) => (dvd_dv598755940omplex @ A @ (times_1460995011omplex @ B @ C)))))). % dvd_mult
thf(fact_211_division__decomp, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ (times_times_nat @ B @ C)) => (?[B3 : nat, C2 : nat]: ((A = (times_times_nat @ B3 @ C2)) & ((dvd_dvd_nat @ B3 @ B) & (dvd_dvd_nat @ C2 @ C)))))))). % division_decomp
thf(fact_212_dvd__mult2, axiom,
    ((![A : complex, B : complex, C : complex]: ((dvd_dvd_complex @ A @ B) => (dvd_dvd_complex @ A @ (times_times_complex @ B @ C)))))). % dvd_mult2
thf(fact_213_dvd__mult2, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => (dvd_dvd_poly_complex @ A @ (times_1246143675omplex @ B @ C)))))). % dvd_mult2
thf(fact_214_dvd__mult2, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult2
thf(fact_215_dvd__mult2, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((dvd_dvd_poly_nat @ A @ B) => (dvd_dvd_poly_nat @ A @ (times_times_poly_nat @ B @ C)))))). % dvd_mult2
thf(fact_216_dvd__mult2, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, C : poly_poly_complex]: ((dvd_dv598755940omplex @ A @ B) => (dvd_dv598755940omplex @ A @ (times_1460995011omplex @ B @ C)))))). % dvd_mult2
thf(fact_217_dvd__mult__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ A @ C))))). % dvd_mult_left
thf(fact_218_dvd__mult__left, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((dvd_dvd_poly_nat @ (times_times_poly_nat @ A @ B) @ C) => (dvd_dvd_poly_nat @ A @ C))))). % dvd_mult_left
thf(fact_219_dvd__mult__left, axiom,
    ((![A : poly_poly_complex, B : poly_poly_complex, C : poly_poly_complex]: ((dvd_dv598755940omplex @ (times_1460995011omplex @ A @ B) @ C) => (dvd_dv598755940omplex @ A @ C))))). % dvd_mult_left
thf(fact_220_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_221_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_222_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_223_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_224_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_225_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_226_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_227_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_228_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_229_nat__mult__less__cancel__disj, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M2 @ N))))))). % nat_mult_less_cancel_disj
thf(fact_230_nat__mult__dvd__cancel__disj, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((dvd_dvd_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((dvd_dvd_nat @ M2 @ N))))))). % nat_mult_dvd_cancel_disj
thf(fact_231_nat__mult__dvd__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((dvd_dvd_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (dvd_dvd_nat @ M2 @ N)))))). % nat_mult_dvd_cancel1
thf(fact_232_nat__mult__less__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (ord_less_nat @ M2 @ N)))))). % nat_mult_less_cancel1
thf(fact_233_nat__mult__eq__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (M2 = N)))))). % nat_mult_eq_cancel1
thf(fact_234_dvd__antisym, axiom,
    ((![M2 : nat, N : nat]: ((dvd_dvd_nat @ M2 @ N) => ((dvd_dvd_nat @ N @ M2) => (M2 = N)))))). % dvd_antisym
thf(fact_235_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_236_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_237_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_238_gcd__nat_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z : nat]: (Y3 = Z))) = (^[A3 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A3 @ B2)) & ((dvd_dvd_nat @ B2 @ A3)))))))). % gcd_nat.eq_iff
thf(fact_239_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_240_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_241_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_242_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

% Conjectures (1)
thf(conj_0, conjecture,
    (((times_times_complex @ (poly_complex2 @ s @ x) @ (poly_complex2 @ u @ x)) = (power_power_complex @ (poly_complex2 @ r @ x) @ (degree_complex @ s))))).
