% 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_972__5377038_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:34:50.454

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (42)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    minus_174331535omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    minus_minus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    one_on1331105667omplex : poly_poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    uminus1138659839omplex : poly_complex > poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pCons_poly_complex : poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_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_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__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_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).

% Relevant facts (165)
thf(fact_0_dsn, axiom,
    ((ord_less_nat @ (degree_complex @ s) @ na))). % dsn
thf(fact_1_assms_I2_J, axiom,
    (((degree_complex @ p) = n))). % assms(2)
thf(fact_2_dpn, axiom,
    (((degree_complex @ pa) = na))). % dpn
thf(fact_3__092_060open_062order_Aa_Ap_A_092_060le_062_An_092_060close_062, axiom,
    ((ord_less_eq_nat @ (order_complex @ a @ pa) @ na))). % \<open>order a p \<le> n\<close>
thf(fact_4_sne, axiom,
    ((~ ((s = zero_z1746442943omplex))))). % sne
thf(fact_5_pne, axiom,
    ((~ ((pa = zero_z1746442943omplex))))). % pne
thf(fact_6_u, axiom,
    (((power_184595776omplex @ r @ (degree_complex @ s)) = (times_1246143675omplex @ s @ u)))). % u
thf(fact_7_n0, axiom,
    ((~ ((na = zero_zero_nat))))). % n0
thf(fact_8_r, axiom,
    ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ r)))). % r
thf(fact_9_s, axiom,
    ((pa = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ s)))). % s
thf(fact_10__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_11_False, axiom,
    ((~ (((degree_complex @ s) = zero_zero_nat))))). % False
thf(fact_12_oop, axiom,
    ((![A : complex]: (ord_less_eq_nat @ (order_complex @ A @ pa) @ na)))). % oop
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_O_Aq_A_061_A_091_058_N_Aa_M_A1_058_093_A_K_Ar_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![R : poly_complex]: (~ ((qa = (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ R))))))))). % \<open>\<And>thesis. (\<And>r. q = [:- a, 1:] * r \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_Ap_A_061_A_091_058_N_Aa_M_A1_058_093_A_094_Aorder_Aa_Ap_A_K_As_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![S : poly_complex]: (~ ((pa = (times_1246143675omplex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ S))))))))). % \<open>\<And>thesis. (\<And>s. p = [:- a, 1:] ^ order a p * s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_15_that, axiom,
    ((~ (((order_complex @ a @ pa) = zero_zero_nat))))). % that
thf(fact_16_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N)) = one_one_poly_complex)))). % minus_one_mult_self
thf(fact_17_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N)) = one_one_complex)))). % minus_one_mult_self
thf(fact_18_left__minus__one__mult__self, axiom,
    ((![N : nat, A : poly_complex]: ((times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ (times_1246143675omplex @ (power_184595776omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_19_left__minus__one__mult__self, axiom,
    ((![N : nat, A : complex]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_20_diff__numeral__special_I12_J, axiom,
    (((minus_174331535omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ (uminus1138659839omplex @ one_one_poly_complex)) = zero_z1746442943omplex))). % diff_numeral_special(12)
thf(fact_21_diff__numeral__special_I12_J, axiom,
    (((minus_minus_complex @ (uminus1204672759omplex @ one_one_complex) @ (uminus1204672759omplex @ one_one_complex)) = zero_zero_complex))). % diff_numeral_special(12)
thf(fact_22_ap_I1_J, axiom,
    ((dvd_dvd_poly_complex @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (order_complex @ a @ pa)) @ pa))). % ap(1)
thf(fact_23_one__poly__eq__simps_I2_J, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % one_poly_eq_simps(2)
thf(fact_24_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_complex = (pCons_complex @ one_one_complex @ zero_z1746442943omplex)))). % one_poly_eq_simps(1)
thf(fact_25_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_26_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_27_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_28_pCons__eq__0__iff, axiom,
    ((![A : poly_complex, P : poly_poly_complex]: (((pCons_poly_complex @ A @ P) = zero_z1040703943omplex) = (((A = zero_z1746442943omplex)) & ((P = zero_z1040703943omplex))))))). % pCons_eq_0_iff
thf(fact_29_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_30_pCons__eq__0__iff, axiom,
    ((![A : complex, P : poly_complex]: (((pCons_complex @ A @ P) = zero_z1746442943omplex) = (((A = zero_zero_complex)) & ((P = zero_z1746442943omplex))))))). % pCons_eq_0_iff
thf(fact_31_mult__minus1, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ (uminus1138659839omplex @ one_one_poly_complex) @ Z) = (uminus1138659839omplex @ Z))))). % mult_minus1
thf(fact_32_mult__minus1, axiom,
    ((![Z : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z) = (uminus1204672759omplex @ Z))))). % mult_minus1
thf(fact_33_mult__minus1__right, axiom,
    ((![Z : poly_complex]: ((times_1246143675omplex @ Z @ (uminus1138659839omplex @ one_one_poly_complex)) = (uminus1138659839omplex @ Z))))). % mult_minus1_right
thf(fact_34_mult__minus1__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z))))). % mult_minus1_right
thf(fact_35_verit__minus__simplify_I3_J, axiom,
    ((![B : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ B) = (uminus1138659839omplex @ B))))). % verit_minus_simplify(3)
thf(fact_36_verit__minus__simplify_I3_J, axiom,
    ((![B : complex]: ((minus_minus_complex @ zero_zero_complex @ B) = (uminus1204672759omplex @ B))))). % verit_minus_simplify(3)
thf(fact_37_diff__0, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ A) = (uminus1138659839omplex @ A))))). % diff_0
thf(fact_38_diff__0, axiom,
    ((![A : complex]: ((minus_minus_complex @ zero_zero_complex @ A) = (uminus1204672759omplex @ A))))). % diff_0
thf(fact_39_assms_I3_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(3)
thf(fact_40_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_41_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_42_verit__minus__simplify_I4_J, axiom,
    ((![B : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_43_power__one__right, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_44_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_45_degree__minus, axiom,
    ((![P : poly_complex]: ((degree_complex @ (uminus1138659839omplex @ P)) = (degree_complex @ P))))). % degree_minus
thf(fact_46_pCons__eq__iff, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: (((pCons_complex @ A @ P) = (pCons_complex @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_47_order__uminus, axiom,
    ((![X : complex, P : poly_complex]: ((order_complex @ X @ (uminus1138659839omplex @ P)) = (order_complex @ X @ P))))). % order_uminus
thf(fact_48_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_49_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_50_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ A) = zero_z1746442943omplex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_51_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_52_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_53_diff__zero, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ zero_z1746442943omplex) = A)))). % diff_zero
thf(fact_54_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_55_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_56_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_57_diff__0__right, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ zero_z1746442943omplex) = A)))). % diff_0_right
thf(fact_58_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_59_diff__self, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ A) = zero_z1746442943omplex)))). % diff_self
thf(fact_60_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_61_neg__equal__0__iff__equal, axiom,
    ((![A : poly_complex]: (((uminus1138659839omplex @ A) = zero_z1746442943omplex) = (A = zero_z1746442943omplex))))). % neg_equal_0_iff_equal
thf(fact_62_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_63_neg__0__equal__iff__equal, axiom,
    ((![A : poly_complex]: ((zero_z1746442943omplex = (uminus1138659839omplex @ A)) = (zero_z1746442943omplex = A))))). % neg_0_equal_iff_equal
thf(fact_64_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_65_add_Oinverse__neutral, axiom,
    (((uminus1138659839omplex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % add.inverse_neutral
thf(fact_66_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_67_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_68_mult_Oright__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ A @ one_one_poly_complex) = A)))). % mult.right_neutral
thf(fact_69_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_70_mult_Oleft__neutral, axiom,
    ((![A : poly_complex]: ((times_1246143675omplex @ one_one_poly_complex @ A) = A)))). % mult.left_neutral
thf(fact_71_minus__diff__eq, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (minus_minus_complex @ A @ B)) = (minus_minus_complex @ B @ A))))). % minus_diff_eq
thf(fact_72_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_73_power__one, axiom,
    ((![N : nat]: ((power_184595776omplex @ one_one_poly_complex @ N) = one_one_poly_complex)))). % power_one
thf(fact_74_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_75_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_76_diff__pCons, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: ((minus_174331535omplex @ (pCons_complex @ A @ P) @ (pCons_complex @ B @ Q)) = (pCons_complex @ (minus_minus_complex @ A @ B) @ (minus_174331535omplex @ P @ Q)))))). % diff_pCons
thf(fact_77_diff__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: ((minus_minus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q)) = (pCons_nat @ (minus_minus_nat @ A @ B) @ (minus_minus_poly_nat @ P @ Q)))))). % diff_pCons
thf(fact_78__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_79_minus__pCons, axiom,
    ((![A : complex, P : poly_complex]: ((uminus1138659839omplex @ (pCons_complex @ A @ P)) = (pCons_complex @ (uminus1204672759omplex @ A) @ (uminus1138659839omplex @ P)))))). % minus_pCons
thf(fact_80__092_060open_062_091_058_N_Aa_M_A1_058_093_Advd_Aq_092_060close_062, axiom,
    ((dvd_dvd_poly_complex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ qa))). % \<open>[:- a, 1:] dvd q\<close>
thf(fact_81_diff__numeral__special_I9_J, axiom,
    (((minus_174331535omplex @ one_one_poly_complex @ one_one_poly_complex) = zero_z1746442943omplex))). % diff_numeral_special(9)
thf(fact_82_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_83_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_84_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_85_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_86_order__1__eq__0, axiom,
    ((![X : complex]: ((order_complex @ X @ one_one_poly_complex) = zero_zero_nat)))). % order_1_eq_0
thf(fact_87_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_88_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_89_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_90_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_91_const__poly__dvd__const__poly__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: ((dvd_dv598755940omplex @ (pCons_poly_complex @ A @ zero_z1040703943omplex) @ (pCons_poly_complex @ B @ zero_z1040703943omplex)) = (dvd_dvd_poly_complex @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_92_const__poly__dvd__const__poly__iff, axiom,
    ((![A : complex, B : complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ A @ zero_z1746442943omplex) @ (pCons_complex @ B @ zero_z1746442943omplex)) = (dvd_dvd_complex @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_93_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_94_power__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_eq_nat @ X @ Y)))))). % power_increasing_iff
thf(fact_95_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_96_power__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_97_True, axiom,
    ((?[A2 : complex]: ((poly_complex2 @ pa @ A2) = zero_zero_complex)))). % True
thf(fact_98_a, axiom,
    (((poly_complex2 @ pa @ a) = zero_zero_complex))). % a
thf(fact_99_pq0, axiom,
    ((![X2 : complex]: (((poly_complex2 @ pa @ X2) = zero_zero_complex) => ((poly_complex2 @ qa @ X2) = zero_zero_complex))))). % pq0
thf(fact_100_IH, axiom,
    ((![M2 : nat]: ((ord_less_nat @ M2 @ na) => (![P2 : poly_complex, Q2 : poly_complex]: ((![X3 : complex]: (((poly_complex2 @ P2 @ X3) = zero_zero_complex) => ((poly_complex2 @ Q2 @ X3) = zero_zero_complex))) => (((degree_complex @ P2) = M2) => ((~ ((M2 = zero_zero_nat))) => (dvd_dvd_poly_complex @ P2 @ (power_184595776omplex @ Q2 @ M2)))))))))). % IH
thf(fact_101_verit__comp__simplify1_I3_J, axiom,
    ((![B2 : nat, A3 : nat]: ((~ ((ord_less_eq_nat @ B2 @ A3))) = (ord_less_nat @ A3 @ B2))))). % verit_comp_simplify1(3)
thf(fact_102_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_103_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_104_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_105_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : poly_complex]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M) @ (power_184595776omplex @ A @ N)))))). % le_imp_power_dvd
thf(fact_106_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : nat]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % le_imp_power_dvd
thf(fact_107_power__le__dvd, axiom,
    ((![A : poly_complex, N : nat, B : poly_complex, M : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M) @ B)))))). % power_le_dvd
thf(fact_108_power__le__dvd, axiom,
    ((![A : nat, N : nat, B : nat, M : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ B)))))). % power_le_dvd
thf(fact_109_dvd__power__le, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat, M : nat]: ((dvd_dvd_poly_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ M))))))). % dvd_power_le
thf(fact_110_dvd__power__le, axiom,
    ((![X : nat, Y : nat, N : nat, M : nat]: ((dvd_dvd_nat @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ M))))))). % dvd_power_le
thf(fact_111_is__unit__iff__degree, axiom,
    ((![P : poly_complex]: ((~ ((P = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) = ((degree_complex @ P) = zero_zero_nat)))))). % is_unit_iff_degree
thf(fact_112_degree__diff__less, axiom,
    ((![P : poly_complex, N : nat, Q : poly_complex]: ((ord_less_nat @ (degree_complex @ P) @ N) => ((ord_less_nat @ (degree_complex @ Q) @ N) => (ord_less_nat @ (degree_complex @ (minus_174331535omplex @ P @ Q)) @ N)))))). % degree_diff_less
thf(fact_113_degree__diff__le, axiom,
    ((![P : poly_complex, N : nat, Q : poly_complex]: ((ord_less_eq_nat @ (degree_complex @ P) @ N) => ((ord_less_eq_nat @ (degree_complex @ Q) @ N) => (ord_less_eq_nat @ (degree_complex @ (minus_174331535omplex @ P @ Q)) @ N)))))). % degree_diff_le
thf(fact_114_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_115_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_116_power__mult, axiom,
    ((![A : poly_complex, M : nat, N : nat]: ((power_184595776omplex @ A @ (times_times_nat @ M @ N)) = (power_184595776omplex @ (power_184595776omplex @ A @ M) @ N))))). % power_mult
thf(fact_117_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_118_dvd__power, axiom,
    ((![N : nat, X : complex]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_complex)) => (dvd_dvd_complex @ X @ (power_power_complex @ X @ N)))))). % dvd_power
thf(fact_119_dvd__power, axiom,
    ((![N : nat, X : poly_complex]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_poly_complex)) => (dvd_dvd_poly_complex @ X @ (power_184595776omplex @ X @ N)))))). % dvd_power
thf(fact_120_dvd__power, axiom,
    ((![N : nat, X : nat]: (((ord_less_nat @ zero_zero_nat @ N) | (X = one_one_nat)) => (dvd_dvd_nat @ X @ (power_power_nat @ X @ N)))))). % dvd_power
thf(fact_121_is__unit__power__iff, axiom,
    ((![A : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A @ N) @ one_one_poly_complex) = (((dvd_dvd_poly_complex @ A @ one_one_poly_complex)) | ((N = zero_zero_nat))))))). % is_unit_power_iff
thf(fact_122_is__unit__power__iff, axiom,
    ((![A : nat, N : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ one_one_nat) = (((dvd_dvd_nat @ A @ one_one_nat)) | ((N = zero_zero_nat))))))). % is_unit_power_iff
thf(fact_123_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_124_self__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ A @ (power_power_nat @ A @ N))))))). % self_le_power
thf(fact_125_power__minus__mult, axiom,
    ((![N : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_times_nat @ (power_power_nat @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_power_nat @ A @ N)))))). % power_minus_mult
thf(fact_126_power__minus__mult, axiom,
    ((![N : nat, A : poly_complex]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_1246143675omplex @ (power_184595776omplex @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_184595776omplex @ A @ N)))))). % power_minus_mult
thf(fact_127_is__unit__const__poly__iff, axiom,
    ((![C : poly_complex]: ((dvd_dv598755940omplex @ (pCons_poly_complex @ C @ zero_z1040703943omplex) @ one_on1331105667omplex) = (dvd_dvd_poly_complex @ C @ one_one_poly_complex))))). % is_unit_const_poly_iff
thf(fact_128_is__unit__const__poly__iff, axiom,
    ((![C : complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ C @ zero_z1746442943omplex) @ one_one_poly_complex) = (dvd_dvd_complex @ C @ one_one_complex))))). % is_unit_const_poly_iff
thf(fact_129_is__unit__poly__iff, axiom,
    ((![P : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ one_on1331105667omplex) = (?[C2 : poly_complex]: (((P = (pCons_poly_complex @ C2 @ zero_z1040703943omplex))) & ((dvd_dvd_poly_complex @ C2 @ one_one_poly_complex)))))))). % is_unit_poly_iff
thf(fact_130_is__unit__poly__iff, axiom,
    ((![P : poly_complex]: ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) = (?[C2 : complex]: (((P = (pCons_complex @ C2 @ zero_z1746442943omplex))) & ((dvd_dvd_complex @ C2 @ one_one_complex)))))))). % is_unit_poly_iff
thf(fact_131_is__unit__polyE, axiom,
    ((![P : poly_poly_complex]: ((dvd_dv598755940omplex @ P @ one_on1331105667omplex) => (~ ((![C3 : poly_complex]: ((P = (pCons_poly_complex @ C3 @ zero_z1040703943omplex)) => (~ ((dvd_dvd_poly_complex @ C3 @ one_one_poly_complex))))))))))). % is_unit_polyE
thf(fact_132_is__unit__polyE, axiom,
    ((![P : poly_complex]: ((dvd_dvd_poly_complex @ P @ one_one_poly_complex) => (~ ((![C3 : complex]: ((P = (pCons_complex @ C3 @ zero_z1746442943omplex)) => (~ ((dvd_dvd_complex @ C3 @ one_one_complex))))))))))). % is_unit_polyE
thf(fact_133_is__unit__pCons__iff, axiom,
    ((![A : complex, P : poly_complex]: ((dvd_dvd_poly_complex @ (pCons_complex @ A @ P) @ one_one_poly_complex) = (((P = zero_z1746442943omplex)) & ((~ ((A = zero_zero_complex))))))))). % is_unit_pCons_iff
thf(fact_134_is__unit__triv, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (dvd_dvd_poly_complex @ (pCons_complex @ A @ zero_z1746442943omplex) @ one_one_poly_complex))))). % is_unit_triv
thf(fact_135_dvd__imp__degree__le, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((~ ((Q = zero_z1746442943omplex))) => (ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q))))))). % dvd_imp_degree_le
thf(fact_136_divides__degree, axiom,
    ((![P : poly_complex, Q : poly_complex]: ((dvd_dvd_poly_complex @ P @ Q) => ((ord_less_eq_nat @ (degree_complex @ P) @ (degree_complex @ Q)) | (Q = zero_z1746442943omplex)))))). % divides_degree
thf(fact_137_dvd__imp__order__le, axiom,
    ((![Q : poly_complex, P : poly_complex, A : complex]: ((~ ((Q = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ P @ Q) => (ord_less_eq_nat @ (order_complex @ A @ P) @ (order_complex @ A @ Q))))))). % dvd_imp_order_le
thf(fact_138_degree__power__le, axiom,
    ((![P : poly_complex, N : nat]: (ord_less_eq_nat @ (degree_complex @ (power_184595776omplex @ P @ N)) @ (times_times_nat @ (degree_complex @ P) @ N))))). % degree_power_le
thf(fact_139_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_140_power__0, axiom,
    ((![A : poly_complex]: ((power_184595776omplex @ A @ zero_zero_nat) = one_one_poly_complex)))). % power_0
thf(fact_141_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_142_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_143_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_144_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_145_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_146_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_147_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_148_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_149_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_150_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_151_dvd__power__iff, axiom,
    ((![X : poly_complex, M : nat, N : nat]: ((~ ((X = zero_z1746442943omplex))) => ((dvd_dvd_poly_complex @ (power_184595776omplex @ X @ M) @ (power_184595776omplex @ X @ N)) = (((dvd_dvd_poly_complex @ X @ one_one_poly_complex)) | ((ord_less_eq_nat @ M @ N)))))))). % dvd_power_iff
thf(fact_152_dvd__power__iff, axiom,
    ((![X : nat, M : nat, N : nat]: ((~ ((X = zero_zero_nat))) => ((dvd_dvd_nat @ (power_power_nat @ X @ M) @ (power_power_nat @ X @ N)) = (((dvd_dvd_nat @ X @ one_one_nat)) | ((ord_less_eq_nat @ M @ N)))))))). % dvd_power_iff
thf(fact_153_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_154_power__le__imp__le__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_eq_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_155_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_156_power__eq__imp__eq__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_157_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_158_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_159_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_160_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_161_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_162_degree__eq__zeroE, axiom,
    ((![P : poly_complex]: (((degree_complex @ P) = zero_zero_nat) => (~ ((![A2 : complex]: (~ ((P = (pCons_complex @ A2 @ zero_z1746442943omplex))))))))))). % degree_eq_zeroE
thf(fact_163_degree__pCons__0, axiom,
    ((![A : complex]: ((degree_complex @ (pCons_complex @ A @ zero_z1746442943omplex)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_164_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

% Conjectures (1)
thf(conj_0, conjecture,
    (((power_184595776omplex @ qa @ na) = (times_1246143675omplex @ pa @ (times_1246143675omplex @ (times_1246143675omplex @ u @ (power_184595776omplex @ (pCons_complex @ (uminus1204672759omplex @ a) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (minus_minus_nat @ na @ (order_complex @ a @ pa)))) @ (power_184595776omplex @ r @ (minus_minus_nat @ na @ (degree_complex @ s)))))))).
