% 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_718__5374906_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:27.874

% Could-be-implicit typings (4)
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 (35)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex, type,
    invers502456322omplex : complex > complex).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
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_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_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
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_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_Osmult_001t__Complex__Ocomplex, type,
    smult_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
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_v_c____, type,
    c : complex).
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_thesis____, type,
    thesis : $o).

% Relevant facts (202)
thf(fact_0_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_1_a00, axiom,
    ((~ (((poly_complex2 @ q @ zero_zero_complex) = zero_zero_complex))))). % a00
thf(fact_2__092_060open_062_092_060exists_062k_Aa_Aqa_O_Aa_A_092_060noteq_062_A0_A_092_060and_062_Ak_A_092_060noteq_062_A0_A_092_060and_062_Apsize_Aqa_A_L_Ak_A_L_A1_A_061_Apsize_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A_092_060and_062_A_I_092_060forall_062z_O_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Az_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A0_A_L_Az_A_094_Ak_A_K_Apoly_A_IpCons_Aa_Aqa_J_Az_J_092_060close_062, axiom,
    ((?[K : nat, A : complex, Q : poly_complex]: ((~ ((A = zero_zero_complex))) & ((~ ((K = zero_zero_nat))) & (((plus_plus_nat @ (plus_plus_nat @ (fundam1709708056omplex @ Q) @ K) @ one_one_nat) = (fundam1709708056omplex @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))) & (![Z : complex]: ((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Z) = (plus_plus_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) @ (times_times_complex @ (power_power_complex @ Z @ K) @ (poly_complex2 @ (pCons_complex @ A @ Q) @ Z))))))))))). % \<open>\<exists>k a qa. a \<noteq> 0 \<and> k \<noteq> 0 \<and> psize qa + k + 1 = psize (smult (inverse (poly q 0)) q) \<and> (\<forall>z. poly (smult (inverse (poly q 0)) q) z = poly (smult (inverse (poly q 0)) q) 0 + z ^ k * poly (pCons a qa) z)\<close>
thf(fact_3__092_060open_062_I_092_060And_062x_Ay_O_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ax_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ay_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((![X : complex, Y : complex]: ((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ X) = (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Y))))))). % \<open>(\<And>x y. poly (smult (inverse (poly q 0)) q) x = poly (smult (inverse (poly q 0)) q) y) \<Longrightarrow> False\<close>
thf(fact_4__092_060open_062poly_Aq_A0_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A0_A_K_Apoly_Aq_A0_092_060close_062, axiom,
    (((poly_complex2 @ q @ zero_zero_complex) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) @ (poly_complex2 @ q @ zero_zero_complex))))). % \<open>poly q 0 = poly (smult (inverse (poly q 0)) q) 0 * poly q 0\<close>
thf(fact_5_qr, axiom,
    ((![Z : complex]: ((poly_complex2 @ q @ Z) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Z) @ (poly_complex2 @ q @ zero_zero_complex)))))). % qr
thf(fact_6_lgqr, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))). % lgqr
thf(fact_7_q_I1_J, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ pa)))). % q(1)
thf(fact_8__092_060open_062constant_A_Ipoly_Aq_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % \<open>constant (poly q) \<Longrightarrow> False\<close>
thf(fact_9_qnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % qnc
thf(fact_10_rnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))))). % rnc
thf(fact_11_poly__offset, axiom,
    ((![P : poly_complex, A2 : complex]: (?[Q : poly_complex]: (((fundam1709708056omplex @ Q) = (fundam1709708056omplex @ P)) & (![X2 : complex]: ((poly_complex2 @ Q @ X2) = (poly_complex2 @ P @ (plus_plus_complex @ A2 @ X2))))))))). % poly_offset
thf(fact_12_poly__pCons, axiom,
    ((![A2 : complex, P : poly_complex, X3 : complex]: ((poly_complex2 @ (pCons_complex @ A2 @ P) @ X3) = (plus_plus_complex @ A2 @ (times_times_complex @ X3 @ (poly_complex2 @ P @ X3))))))). % poly_pCons
thf(fact_13_poly__pCons, axiom,
    ((![A2 : nat, P : poly_nat, X3 : nat]: ((poly_nat2 @ (pCons_nat @ A2 @ P) @ X3) = (plus_plus_nat @ A2 @ (times_times_nat @ X3 @ (poly_nat2 @ P @ X3))))))). % poly_pCons
thf(fact_14_left__inverse, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A2) @ A2) = one_one_complex))))). % left_inverse
thf(fact_15_right__inverse, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((times_times_complex @ A2 @ (invers502456322omplex @ A2)) = one_one_complex))))). % right_inverse
thf(fact_16_poly__decompose, axiom,
    ((![P : poly_complex]: ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[K : nat, A : complex, Q : poly_complex]: ((~ ((A = zero_zero_complex))) & ((~ ((K = zero_zero_nat))) & (((plus_plus_nat @ (plus_plus_nat @ (fundam1709708056omplex @ Q) @ K) @ one_one_nat) = (fundam1709708056omplex @ P)) & (![Z : complex]: ((poly_complex2 @ P @ Z) = (plus_plus_complex @ (poly_complex2 @ P @ zero_zero_complex) @ (times_times_complex @ (power_power_complex @ Z @ K) @ (poly_complex2 @ (pCons_complex @ A @ Q) @ Z))))))))))))). % poly_decompose
thf(fact_17_poly__smult, axiom,
    ((![A2 : complex, P : poly_complex, X3 : complex]: ((poly_complex2 @ (smult_complex @ A2 @ P) @ X3) = (times_times_complex @ A2 @ (poly_complex2 @ P @ X3)))))). % poly_smult
thf(fact_18_poly__smult, axiom,
    ((![A2 : nat, P : poly_nat, X3 : nat]: ((poly_nat2 @ (smult_nat @ A2 @ P) @ X3) = (times_times_nat @ A2 @ (poly_nat2 @ P @ X3)))))). % poly_smult
thf(fact_19_smult__pCons, axiom,
    ((![A2 : complex, B : complex, P : poly_complex]: ((smult_complex @ A2 @ (pCons_complex @ B @ P)) = (pCons_complex @ (times_times_complex @ A2 @ B) @ (smult_complex @ A2 @ P)))))). % smult_pCons
thf(fact_20_smult__pCons, axiom,
    ((![A2 : nat, B : nat, P : poly_nat]: ((smult_nat @ A2 @ (pCons_nat @ B @ P)) = (pCons_nat @ (times_times_nat @ A2 @ B) @ (smult_nat @ A2 @ P)))))). % smult_pCons
thf(fact_21_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_22_mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex]: (((times_times_complex @ C @ A2) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_left2
thf(fact_23_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_24_mult__cancel__right2, axiom,
    ((![A2 : complex, C : complex]: (((times_times_complex @ A2 @ C) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_right2
thf(fact_25_r01, axiom,
    (((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) = one_one_complex))). % r01
thf(fact_26_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_27_inverse__inverse__eq, axiom,
    ((![A2 : complex]: ((invers502456322omplex @ (invers502456322omplex @ A2)) = A2)))). % inverse_inverse_eq
thf(fact_28_inverse__eq__iff__eq, axiom,
    ((![A2 : complex, B : complex]: (((invers502456322omplex @ A2) = (invers502456322omplex @ B)) = (A2 = B))))). % inverse_eq_iff_eq
thf(fact_29_pCons__eq__iff, axiom,
    ((![A2 : complex, P : poly_complex, B : complex, Q2 : poly_complex]: (((pCons_complex @ A2 @ P) = (pCons_complex @ B @ Q2)) = (((A2 = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_30_mult__smult__right, axiom,
    ((![P : poly_complex, A2 : complex, Q2 : poly_complex]: ((times_1246143675omplex @ P @ (smult_complex @ A2 @ Q2)) = (smult_complex @ A2 @ (times_1246143675omplex @ P @ Q2)))))). % mult_smult_right
thf(fact_31_mult__smult__left, axiom,
    ((![A2 : complex, P : poly_complex, Q2 : poly_complex]: ((times_1246143675omplex @ (smult_complex @ A2 @ P) @ Q2) = (smult_complex @ A2 @ (times_1246143675omplex @ P @ Q2)))))). % mult_smult_left
thf(fact_32_smult__0__right, axiom,
    ((![A2 : complex]: ((smult_complex @ A2 @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % smult_0_right
thf(fact_33_False, axiom,
    ((~ (((poly_complex2 @ pa @ c) = zero_zero_complex))))). % False
thf(fact_34_q_I2_J, axiom,
    ((![X2 : complex]: ((poly_complex2 @ q @ X2) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X2)))))). % q(2)
thf(fact_35_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_36_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_37_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_38_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_39_mult__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_40_mult__eq__0__iff, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_41_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_42_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_43_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_44_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_45_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_46_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_47_inverse__nonzero__iff__nonzero, axiom,
    ((![A2 : complex]: (((invers502456322omplex @ A2) = zero_zero_complex) = (A2 = zero_zero_complex))))). % inverse_nonzero_iff_nonzero
thf(fact_48_inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % inverse_zero
thf(fact_49_inverse__mult__distrib, axiom,
    ((![A2 : complex, B : complex]: ((invers502456322omplex @ (times_times_complex @ A2 @ B)) = (times_times_complex @ (invers502456322omplex @ A2) @ (invers502456322omplex @ B)))))). % inverse_mult_distrib
thf(fact_50_inverse__1, axiom,
    (((invers502456322omplex @ one_one_complex) = one_one_complex))). % inverse_1
thf(fact_51_inverse__eq__1__iff, axiom,
    ((![X3 : complex]: (((invers502456322omplex @ X3) = one_one_complex) = (X3 = one_one_complex))))). % inverse_eq_1_iff
thf(fact_52_pCons__eq__0__iff, axiom,
    ((![A2 : complex, P : poly_complex]: (((pCons_complex @ A2 @ P) = zero_z1746442943omplex) = (((A2 = zero_zero_complex)) & ((P = zero_z1746442943omplex))))))). % pCons_eq_0_iff
thf(fact_53_pCons__eq__0__iff, axiom,
    ((![A2 : nat, P : poly_nat]: (((pCons_nat @ A2 @ P) = zero_zero_poly_nat) = (((A2 = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_54_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_55_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_56_poly__0, axiom,
    ((![X3 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X3) = zero_zero_complex)))). % poly_0
thf(fact_57_poly__0, axiom,
    ((![X3 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X3) = zero_zero_nat)))). % poly_0
thf(fact_58_power__one__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_59_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_60_add__pCons, axiom,
    ((![A2 : nat, P : poly_nat, B : nat, Q2 : poly_nat]: ((plus_plus_poly_nat @ (pCons_nat @ A2 @ P) @ (pCons_nat @ B @ Q2)) = (pCons_nat @ (plus_plus_nat @ A2 @ B) @ (plus_plus_poly_nat @ P @ Q2)))))). % add_pCons
thf(fact_61_add__pCons, axiom,
    ((![A2 : complex, P : poly_complex, B : complex, Q2 : poly_complex]: ((plus_p1547158847omplex @ (pCons_complex @ A2 @ P) @ (pCons_complex @ B @ Q2)) = (pCons_complex @ (plus_plus_complex @ A2 @ B) @ (plus_p1547158847omplex @ P @ Q2)))))). % add_pCons
thf(fact_62_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_63_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_64_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_65_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_66_smult__eq__0__iff, axiom,
    ((![A2 : complex, P : poly_complex]: (((smult_complex @ A2 @ P) = zero_z1746442943omplex) = (((A2 = zero_zero_complex)) | ((P = zero_z1746442943omplex))))))). % smult_eq_0_iff
thf(fact_67_smult__eq__0__iff, axiom,
    ((![A2 : nat, P : poly_nat]: (((smult_nat @ A2 @ P) = zero_zero_poly_nat) = (((A2 = zero_zero_nat)) | ((P = zero_zero_poly_nat))))))). % smult_eq_0_iff
thf(fact_68_smult__0__left, axiom,
    ((![P : poly_complex]: ((smult_complex @ zero_zero_complex @ P) = zero_z1746442943omplex)))). % smult_0_left
thf(fact_69_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_70_poly__mult, axiom,
    ((![P : poly_complex, Q2 : poly_complex, X3 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q2) @ X3) = (times_times_complex @ (poly_complex2 @ P @ X3) @ (poly_complex2 @ Q2 @ X3)))))). % poly_mult
thf(fact_71_poly__mult, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X3 : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q2) @ X3) = (times_times_nat @ (poly_nat2 @ P @ X3) @ (poly_nat2 @ Q2 @ X3)))))). % poly_mult
thf(fact_72_poly__add, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X3 : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q2) @ X3) = (plus_plus_nat @ (poly_nat2 @ P @ X3) @ (poly_nat2 @ Q2 @ X3)))))). % poly_add
thf(fact_73_poly__add, axiom,
    ((![P : poly_complex, Q2 : poly_complex, X3 : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P @ Q2) @ X3) = (plus_plus_complex @ (poly_complex2 @ P @ X3) @ (poly_complex2 @ Q2 @ X3)))))). % poly_add
thf(fact_74_poly__1, axiom,
    ((![X3 : nat]: ((poly_nat2 @ one_one_poly_nat @ X3) = one_one_nat)))). % poly_1
thf(fact_75_poly__1, axiom,
    ((![X3 : complex]: ((poly_complex2 @ one_one_poly_complex @ X3) = one_one_complex)))). % poly_1
thf(fact_76_smult__smult, axiom,
    ((![A2 : complex, B : complex, P : poly_complex]: ((smult_complex @ A2 @ (smult_complex @ B @ P)) = (smult_complex @ (times_times_complex @ A2 @ B) @ P))))). % smult_smult
thf(fact_77_smult__smult, axiom,
    ((![A2 : nat, B : nat, P : poly_nat]: ((smult_nat @ A2 @ (smult_nat @ B @ P)) = (smult_nat @ (times_times_nat @ A2 @ B) @ P))))). % smult_smult
thf(fact_78_smult__1__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ one_one_nat @ P) = P)))). % smult_1_left
thf(fact_79_smult__1__left, axiom,
    ((![P : poly_complex]: ((smult_complex @ one_one_complex @ P) = P)))). % smult_1_left
thf(fact_80_poly__power, axiom,
    ((![P : poly_complex, N : nat, X3 : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X3) = (power_power_complex @ (poly_complex2 @ P @ X3) @ N))))). % poly_power
thf(fact_81_poly__power, axiom,
    ((![P : poly_nat, N : nat, X3 : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X3) = (power_power_nat @ (poly_nat2 @ P @ X3) @ N))))). % poly_power
thf(fact_82_smult__one, axiom,
    ((![C : complex]: ((smult_complex @ C @ one_one_poly_complex) = (pCons_complex @ C @ zero_z1746442943omplex))))). % smult_one
thf(fact_83_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_84_mult__pCons__right, axiom,
    ((![P : poly_complex, A2 : complex, Q2 : poly_complex]: ((times_1246143675omplex @ P @ (pCons_complex @ A2 @ Q2)) = (plus_p1547158847omplex @ (smult_complex @ A2 @ P) @ (pCons_complex @ zero_zero_complex @ (times_1246143675omplex @ P @ Q2))))))). % mult_pCons_right
thf(fact_85_mult__pCons__right, axiom,
    ((![P : poly_nat, A2 : nat, Q2 : poly_nat]: ((times_times_poly_nat @ P @ (pCons_nat @ A2 @ Q2)) = (plus_plus_poly_nat @ (smult_nat @ A2 @ P) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q2))))))). % mult_pCons_right
thf(fact_86_mult__pCons__left, axiom,
    ((![A2 : complex, P : poly_complex, Q2 : poly_complex]: ((times_1246143675omplex @ (pCons_complex @ A2 @ P) @ Q2) = (plus_p1547158847omplex @ (smult_complex @ A2 @ Q2) @ (pCons_complex @ zero_zero_complex @ (times_1246143675omplex @ P @ Q2))))))). % mult_pCons_left
thf(fact_87_mult__pCons__left, axiom,
    ((![A2 : nat, P : poly_nat, Q2 : poly_nat]: ((times_times_poly_nat @ (pCons_nat @ A2 @ P) @ Q2) = (plus_plus_poly_nat @ (smult_nat @ A2 @ Q2) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q2))))))). % mult_pCons_left
thf(fact_88_power__mult, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A2 @ M) @ N))))). % power_mult
thf(fact_89_power__mult, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A2 @ M) @ N))))). % power_mult
thf(fact_90_smult__add__right, axiom,
    ((![A2 : complex, P : poly_complex, Q2 : poly_complex]: ((smult_complex @ A2 @ (plus_p1547158847omplex @ P @ Q2)) = (plus_p1547158847omplex @ (smult_complex @ A2 @ P) @ (smult_complex @ A2 @ Q2)))))). % smult_add_right
thf(fact_91_pderiv_Oinduct, axiom,
    ((![P2 : poly_complex > $o, A0 : poly_complex]: ((![A : complex, P3 : poly_complex]: (((~ ((P3 = zero_z1746442943omplex))) => (P2 @ P3)) => (P2 @ (pCons_complex @ A @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_92_poly__induct2, axiom,
    ((![P2 : poly_complex > poly_complex > $o, P : poly_complex, Q2 : poly_complex]: ((P2 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A : complex, P3 : poly_complex, B2 : complex, Q : poly_complex]: ((P2 @ P3 @ Q) => (P2 @ (pCons_complex @ A @ P3) @ (pCons_complex @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_93_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X4 : complex]: (![Y2 : complex]: ((F @ X4) = (F @ Y2)))))))). % constant_def
thf(fact_94_offset__poly__eq__0__lemma, axiom,
    ((![C : complex, P : poly_complex, A2 : complex]: (((plus_p1547158847omplex @ (smult_complex @ C @ P) @ (pCons_complex @ A2 @ P)) = zero_z1746442943omplex) => (P = zero_z1746442943omplex))))). % offset_poly_eq_0_lemma
thf(fact_95_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_96_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_97_synthetic__div__unique__lemma, axiom,
    ((![C : complex, P : poly_complex, A2 : complex]: (((smult_complex @ C @ P) = (pCons_complex @ A2 @ P)) => (P = zero_z1746442943omplex))))). % synthetic_div_unique_lemma
thf(fact_98_inverse__eq__imp__eq, axiom,
    ((![A2 : complex, B : complex]: (((invers502456322omplex @ A2) = (invers502456322omplex @ B)) => (A2 = B))))). % inverse_eq_imp_eq
thf(fact_99_pderiv_Ocases, axiom,
    ((![X3 : poly_complex]: (~ ((![A : complex, P3 : poly_complex]: (~ ((X3 = (pCons_complex @ A @ P3)))))))))). % pderiv.cases
thf(fact_100_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A : complex, Q : poly_complex]: (~ ((P = (pCons_complex @ A @ Q)))))))))). % pCons_cases
thf(fact_101_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q2 : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q2)) = (P = Q2))))). % poly_eq_poly_eq_iff
thf(fact_102_mult__right__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_103_mult__right__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_104_mult__left__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_105_mult__left__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_106_no__zero__divisors, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A2 @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_107_no__zero__divisors, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A2 @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_108_divisors__zero, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) => ((A2 = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_109_divisors__zero, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) => ((A2 = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_110_mult__not__zero, axiom,
    ((![A2 : complex, B : complex]: ((~ (((times_times_complex @ A2 @ B) = zero_zero_complex))) => ((~ ((A2 = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_111_mult__not__zero, axiom,
    ((![A2 : nat, B : nat]: ((~ (((times_times_nat @ A2 @ B) = zero_zero_nat))) => ((~ ((A2 = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_112_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_113_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_114_combine__common__factor, axiom,
    ((![A2 : complex, E : complex, B : complex, C : complex]: ((plus_plus_complex @ (times_times_complex @ A2 @ E) @ (plus_plus_complex @ (times_times_complex @ B @ E) @ C)) = (plus_plus_complex @ (times_times_complex @ (plus_plus_complex @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_115_combine__common__factor, axiom,
    ((![A2 : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A2 @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_116_distrib__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % distrib_right
thf(fact_117_distrib__right, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_118_distrib__left, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A2 @ B) @ (times_times_complex @ A2 @ C)))))). % distrib_left
thf(fact_119_distrib__left, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ A2 @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A2 @ B) @ (times_times_nat @ A2 @ C)))))). % distrib_left
thf(fact_120_comm__semiring__class_Odistrib, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_121_comm__semiring__class_Odistrib, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_122_ring__class_Oring__distribs_I1_J, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A2 @ B) @ (times_times_complex @ A2 @ C)))))). % ring_class.ring_distribs(1)
thf(fact_123_ring__class_Oring__distribs_I2_J, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_124_power__not__zero, axiom,
    ((![A2 : complex, N : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_125_power__not__zero, axiom,
    ((![A2 : nat, N : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_126_power__commuting__commutes, axiom,
    ((![X3 : complex, Y3 : complex, N : nat]: (((times_times_complex @ X3 @ Y3) = (times_times_complex @ Y3 @ X3)) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ Y3) = (times_times_complex @ Y3 @ (power_power_complex @ X3 @ N))))))). % power_commuting_commutes
thf(fact_127_power__commuting__commutes, axiom,
    ((![X3 : nat, Y3 : nat, N : nat]: (((times_times_nat @ X3 @ Y3) = (times_times_nat @ Y3 @ X3)) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ Y3) = (times_times_nat @ Y3 @ (power_power_nat @ X3 @ N))))))). % power_commuting_commutes
thf(fact_128_power__mult__distrib, axiom,
    ((![A2 : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B) @ N) = (times_times_complex @ (power_power_complex @ A2 @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_129_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N) = (times_times_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_130_power__commutes, axiom,
    ((![A2 : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A2 @ N) @ A2) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N)))))). % power_commutes
thf(fact_131_power__commutes, axiom,
    ((![A2 : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N)))))). % power_commutes
thf(fact_132_nonzero__imp__inverse__nonzero, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => (~ (((invers502456322omplex @ A2) = zero_zero_complex))))))). % nonzero_imp_inverse_nonzero
thf(fact_133_nonzero__inverse__inverse__eq, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((invers502456322omplex @ (invers502456322omplex @ A2)) = A2))))). % nonzero_inverse_inverse_eq
thf(fact_134_nonzero__inverse__eq__imp__eq, axiom,
    ((![A2 : complex, B : complex]: (((invers502456322omplex @ A2) = (invers502456322omplex @ B)) => ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (A2 = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_135_inverse__zero__imp__zero, axiom,
    ((![A2 : complex]: (((invers502456322omplex @ A2) = zero_zero_complex) => (A2 = zero_zero_complex))))). % inverse_zero_imp_zero
thf(fact_136_field__class_Ofield__inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % field_class.field_inverse_zero
thf(fact_137_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y3 : complex, X3 : complex]: (((times_times_complex @ Y3 @ X3) = (times_times_complex @ X3 @ Y3)) => ((times_times_complex @ (invers502456322omplex @ Y3) @ X3) = (times_times_complex @ X3 @ (invers502456322omplex @ Y3))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_138_pCons__induct, axiom,
    ((![P2 : poly_complex > $o, P : poly_complex]: ((P2 @ zero_z1746442943omplex) => ((![A : complex, P3 : poly_complex]: (((~ ((A = zero_zero_complex))) | (~ ((P3 = zero_z1746442943omplex)))) => ((P2 @ P3) => (P2 @ (pCons_complex @ A @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_139_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A : nat, P3 : poly_nat]: (((~ ((A = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_140_power__inverse, axiom,
    ((![A2 : complex, N : nat]: ((power_power_complex @ (invers502456322omplex @ A2) @ N) = (invers502456322omplex @ (power_power_complex @ A2 @ N)))))). % power_inverse
thf(fact_141_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_142_smult__add__left, axiom,
    ((![A2 : nat, B : nat, P : poly_nat]: ((smult_nat @ (plus_plus_nat @ A2 @ B) @ P) = (plus_plus_poly_nat @ (smult_nat @ A2 @ P) @ (smult_nat @ B @ P)))))). % smult_add_left
thf(fact_143_smult__add__left, axiom,
    ((![A2 : complex, B : complex, P : poly_complex]: ((smult_complex @ (plus_plus_complex @ A2 @ B) @ P) = (plus_p1547158847omplex @ (smult_complex @ A2 @ P) @ (smult_complex @ B @ P)))))). % smult_add_left
thf(fact_144_left__right__inverse__power, axiom,
    ((![X3 : complex, Y3 : complex, N : nat]: (((times_times_complex @ X3 @ Y3) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ (power_power_complex @ Y3 @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_145_left__right__inverse__power, axiom,
    ((![X3 : nat, Y3 : nat, N : nat]: (((times_times_nat @ X3 @ Y3) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ (power_power_nat @ Y3 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_146_power__0, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_147_power__0, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_148_nonzero__inverse__mult__distrib, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((invers502456322omplex @ (times_times_complex @ A2 @ B)) = (times_times_complex @ (invers502456322omplex @ B) @ (invers502456322omplex @ A2)))))))). % nonzero_inverse_mult_distrib
thf(fact_149_power__add, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A2 @ M) @ (power_power_complex @ A2 @ N)))))). % power_add
thf(fact_150_power__add, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A2 @ M) @ (power_power_nat @ A2 @ N)))))). % power_add
thf(fact_151_inverse__unique, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = one_one_complex) => ((invers502456322omplex @ A2) = B))))). % inverse_unique
thf(fact_152_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_153_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_154_division__ring__inverse__add, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((plus_plus_complex @ (invers502456322omplex @ A2) @ (invers502456322omplex @ B)) = (times_times_complex @ (times_times_complex @ (invers502456322omplex @ A2) @ (plus_plus_complex @ A2 @ B)) @ (invers502456322omplex @ B)))))))). % division_ring_inverse_add
thf(fact_155_inverse__add, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((plus_plus_complex @ (invers502456322omplex @ A2) @ (invers502456322omplex @ B)) = (times_times_complex @ (times_times_complex @ (plus_plus_complex @ A2 @ B) @ (invers502456322omplex @ A2)) @ (invers502456322omplex @ B)))))))). % inverse_add
thf(fact_156_field__class_Ofield__inverse, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A2) @ A2) = one_one_complex))))). % field_class.field_inverse
thf(fact_157_pqc0, axiom,
    (((poly_complex2 @ pa @ c) = (poly_complex2 @ q @ zero_zero_complex)))). % pqc0
thf(fact_158__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_062psize_Aq_A_061_Apsize_Ap_059_A_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Q : poly_complex]: (((fundam1709708056omplex @ Q) = (fundam1709708056omplex @ pa)) => (~ ((![X2 : complex]: ((poly_complex2 @ Q @ X2) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X2)))))))))))). % \<open>\<And>thesis. (\<And>q. \<lbrakk>psize q = psize p; \<forall>x. poly q x = poly p (c + x)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_159__092_060open_062_092_060exists_062q_O_Apsize_Aq_A_061_Apsize_Ap_A_092_060and_062_A_I_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_J_092_060close_062, axiom,
    ((?[Q : poly_complex]: (((fundam1709708056omplex @ Q) = (fundam1709708056omplex @ pa)) & (![X2 : complex]: ((poly_complex2 @ Q @ X2) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X2)))))))). % \<open>\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (c + x))\<close>
thf(fact_160_less_Ohyps, axiom,
    ((![P : poly_complex]: ((ord_less_nat @ (fundam1709708056omplex @ P) @ (fundam1709708056omplex @ pa)) => ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z2 : complex]: ((poly_complex2 @ P @ Z2) = zero_zero_complex))))))). % less.hyps
thf(fact_161_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_162_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_163_mult_Oright__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ one_one_complex) = A2)))). % mult.right_neutral
thf(fact_164_mult_Oright__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ one_one_nat) = A2)))). % mult.right_neutral
thf(fact_165_mult_Oleft__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ one_one_complex @ A2) = A2)))). % mult.left_neutral
thf(fact_166_mult_Oleft__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ one_one_nat @ A2) = A2)))). % mult.left_neutral
thf(fact_167_add__left__cancel, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_168_add__left__cancel, axiom,
    ((![A2 : complex, B : complex, C : complex]: (((plus_plus_complex @ A2 @ B) = (plus_plus_complex @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_169_add__right__cancel, axiom,
    ((![B : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B @ A2) = (plus_plus_nat @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_170_add__right__cancel, axiom,
    ((![B : complex, A2 : complex, C : complex]: (((plus_plus_complex @ B @ A2) = (plus_plus_complex @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_171_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_172_add_Oleft__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % add.left_neutral
thf(fact_173_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_174_add_Oright__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % add.right_neutral
thf(fact_175_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_176_add__cancel__left__left, axiom,
    ((![B : complex, A2 : complex]: (((plus_plus_complex @ B @ A2) = A2) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_177_add__cancel__left__left, axiom,
    ((![B : nat, A2 : nat]: (((plus_plus_nat @ B @ A2) = A2) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_178_add__cancel__left__right, axiom,
    ((![A2 : complex, B : complex]: (((plus_plus_complex @ A2 @ B) = A2) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_179_add__cancel__left__right, axiom,
    ((![A2 : nat, B : nat]: (((plus_plus_nat @ A2 @ B) = A2) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_180_add__cancel__right__left, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ B @ A2)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_181_add__cancel__right__left, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ B @ A2)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_182_add__cancel__right__right, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ A2 @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_183_add__cancel__right__right, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ A2 @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_184_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y3 : nat]: (((plus_plus_nat @ X3 @ Y3) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_185_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y3 : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y3)) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_186_add__less__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A2 @ B))))). % add_less_cancel_left
thf(fact_187_add__less__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A2 @ B))))). % add_less_cancel_right
thf(fact_188_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_189_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_190_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_191_nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X3 @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_192_nat__add__left__cancel__less, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K2 @ M) @ (plus_plus_nat @ K2 @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_193_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_194_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_195_mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel1
thf(fact_196_mult__cancel2, axiom,
    ((![M : nat, K2 : nat, N : nat]: (((times_times_nat @ M @ K2) = (times_times_nat @ N @ K2)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel2
thf(fact_197_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_198_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_199_add__less__same__cancel1, axiom,
    ((![B : nat, A2 : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A2) @ B) = (ord_less_nat @ A2 @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_200_add__less__same__cancel2, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ B) @ B) = (ord_less_nat @ A2 @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_201_less__add__same__cancel1, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ (plus_plus_nat @ A2 @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![A3 : complex, K3 : nat, S : poly_complex]: ((~ ((A3 = zero_zero_complex))) => ((~ ((K3 = zero_zero_nat))) => (((plus_plus_nat @ (plus_plus_nat @ (fundam1709708056omplex @ S) @ K3) @ one_one_nat) = (fundam1709708056omplex @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))) => ((![Z2 : complex]: ((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Z2) = (plus_plus_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) @ (times_times_complex @ (power_power_complex @ Z2 @ K3) @ (poly_complex2 @ (pCons_complex @ A3 @ S) @ Z2))))) => thesis))))))).
thf(conj_1, conjecture,
    (thesis)).
