% 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_1014__5377446_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:16.362

% 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 (78)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Complex__Ocomplex, type,
    fundam1201687030omplex : poly_complex > complex > poly_complex).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Nat__Onat, type,
    fundam170929432ly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    fundam1307691262omplex : poly_poly_complex > poly_complex > poly_poly_complex).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Nat__Onat, type,
    fundam1567013434ze_nat : poly_nat > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    fundam1956464160omplex : poly_poly_complex > nat).
thf(sy_c_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_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    one_on1331105667omplex : poly_poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    zero_z1200043727omplex : poly_p1267267526omplex).
thf(sy_c_If_001t__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    if_poly_complex : $o > poly_complex > poly_complex > poly_complex).
thf(sy_c_If_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    if_poly_nat : $o > poly_nat > poly_nat > poly_nat).
thf(sy_c_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    if_poly_poly_complex : $o > poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
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_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    degree_poly_nat : poly_poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    degree2006505739omplex : poly_p1267267526omplex > nat).
thf(sy_c_Polynomial_Ois__zero_001t__Complex__Ocomplex, type,
    is_zero_complex : poly_complex > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat, type,
    is_zero_nat : poly_nat > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    is_zero_poly_complex : poly_poly_complex > $o).
thf(sy_c_Polynomial_Omonom_001t__Complex__Ocomplex, type,
    monom_complex : complex > nat > poly_complex).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat, type,
    monom_nat : nat > nat > poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    monom_poly_complex : poly_complex > nat > poly_poly_complex).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    monom_poly_nat : poly_nat > nat > poly_poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    monom_1210178217omplex : poly_poly_complex > nat > poly_p1267267526omplex).
thf(sy_c_Polynomial_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    order_poly_complex : poly_complex > poly_poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    order_1735763309omplex : poly_poly_complex > poly_p1267267526omplex > nat).
thf(sy_c_Polynomial_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_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    pCons_1087637536omplex : poly_poly_complex > poly_p1267267526omplex > poly_p1267267526omplex).
thf(sy_c_Polynomial_Opcompose_001t__Complex__Ocomplex, type,
    pcompose_complex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pcompo1411605209omplex : poly_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_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_p282434315omplex : poly_p1267267526omplex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Complex__Ocomplex, type,
    coeff_complex : poly_complex > nat > complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    coeff_poly_complex : poly_poly_complex > nat > poly_complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    coeff_poly_nat : poly_poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    coeff_1429652124omplex : poly_p1267267526omplex > nat > poly_poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Complex__Ocomplex, type,
    poly_cutoff_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat, type,
    poly_cutoff_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_c622223248omplex : nat > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_cutoff_poly_nat : nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_c306577560omplex : nat > poly_p1267267526omplex > poly_p1267267526omplex).
thf(sy_c_Polynomial_Opoly__shift_001t__Complex__Ocomplex, type,
    poly_shift_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat, type,
    poly_shift_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_s558570093omplex : nat > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    reflec1997789704omplex : poly_p1267267526omplex > poly_p1267267526omplex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Complex__Ocomplex, type,
    synthe151143547omplex : poly_complex > complex > poly_complex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat, type,
    synthetic_div_nat : poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    synthe1985144195omplex : poly_poly_complex > poly_complex > poly_poly_complex).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).

% Relevant facts (246)
thf(fact_0_p, axiom,
    ((p = zero_z1746442943omplex))). % p
thf(fact_1_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_2_poly__0, axiom,
    ((![X : poly_poly_complex]: ((poly_p282434315omplex @ zero_z1200043727omplex @ X) = zero_z1040703943omplex)))). % poly_0
thf(fact_3_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_4_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_5_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_6_poly__all__0__iff__0, axiom,
    ((![P : poly_p1267267526omplex]: ((![X2 : poly_poly_complex]: ((poly_p282434315omplex @ P @ X2) = zero_z1040703943omplex)) = (P = zero_z1200043727omplex))))). % poly_all_0_iff_0
thf(fact_7_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P @ X2) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_8_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X2 : poly_complex]: ((poly_poly_complex2 @ P @ X2) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_9_fundamental__theorem__of__algebra, axiom,
    ((![P : poly_complex]: ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra
thf(fact_10_fundamental__theorem__of__algebra__alt, axiom,
    ((![P : poly_complex]: ((~ ((?[A : complex, L : poly_complex]: ((~ ((A = zero_zero_complex))) & ((L = zero_z1746442943omplex) & (P = (pCons_complex @ A @ L))))))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra_alt
thf(fact_11_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_12_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_13_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_14_zero__reorient, axiom,
    ((![X : poly_poly_complex]: ((zero_z1040703943omplex = X) = (X = zero_z1040703943omplex))))). % zero_reorient
thf(fact_15_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_16_zero__reorient, axiom,
    ((![X : poly_complex]: ((zero_z1746442943omplex = X) = (X = zero_z1746442943omplex))))). % zero_reorient
thf(fact_17_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_18_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_nat]: (((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_19_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_p1267267526omplex]: (((poly_p282434315omplex @ (reflec1997789704omplex @ P) @ zero_z1040703943omplex) = zero_z1040703943omplex) = (P = zero_z1200043727omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_20_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_21_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_22_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_23_order__root, axiom,
    ((![P : poly_p1267267526omplex, A2 : poly_poly_complex]: (((poly_p282434315omplex @ P @ A2) = zero_z1040703943omplex) = (((P = zero_z1200043727omplex)) | ((~ (((order_1735763309omplex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_24_order__root, axiom,
    ((![P : poly_complex, A2 : complex]: (((poly_complex2 @ P @ A2) = zero_zero_complex) = (((P = zero_z1746442943omplex)) | ((~ (((order_complex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_25_order__root, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex]: (((poly_poly_complex2 @ P @ A2) = zero_z1746442943omplex) = (((P = zero_z1040703943omplex)) | ((~ (((order_poly_complex @ A2 @ P) = zero_zero_nat))))))))). % order_root
thf(fact_26_offset__poly__0, axiom,
    ((![H : complex]: ((fundam1201687030omplex @ zero_z1746442943omplex @ H) = zero_z1746442943omplex)))). % offset_poly_0
thf(fact_27_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_28_offset__poly__0, axiom,
    ((![H : poly_complex]: ((fundam1307691262omplex @ zero_z1040703943omplex @ H) = zero_z1040703943omplex)))). % offset_poly_0
thf(fact_29_offset__poly__eq__0__iff, axiom,
    ((![P : poly_complex, H : complex]: (((fundam1201687030omplex @ P @ H) = zero_z1746442943omplex) = (P = zero_z1746442943omplex))))). % offset_poly_eq_0_iff
thf(fact_30_offset__poly__eq__0__iff, axiom,
    ((![P : poly_nat, H : nat]: (((fundam170929432ly_nat @ P @ H) = zero_zero_poly_nat) = (P = zero_zero_poly_nat))))). % offset_poly_eq_0_iff
thf(fact_31_offset__poly__eq__0__iff, axiom,
    ((![P : poly_poly_complex, H : poly_complex]: (((fundam1307691262omplex @ P @ H) = zero_z1040703943omplex) = (P = zero_z1040703943omplex))))). % offset_poly_eq_0_iff
thf(fact_32_pCons__eq__iff, axiom,
    ((![A2 : complex, P : poly_complex, B : complex, Q : poly_complex]: (((pCons_complex @ A2 @ P) = (pCons_complex @ B @ Q)) = (((A2 = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_33_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_34_reflect__poly__0, axiom,
    (((reflec309385472omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % reflect_poly_0
thf(fact_35_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_36_pCons__eq__0__iff, axiom,
    ((![A2 : poly_nat, P : poly_poly_nat]: (((pCons_poly_nat @ A2 @ P) = zero_z1059985641ly_nat) = (((A2 = zero_zero_poly_nat)) & ((P = zero_z1059985641ly_nat))))))). % pCons_eq_0_iff
thf(fact_37_pCons__eq__0__iff, axiom,
    ((![A2 : poly_poly_complex, P : poly_p1267267526omplex]: (((pCons_1087637536omplex @ A2 @ P) = zero_z1200043727omplex) = (((A2 = zero_z1040703943omplex)) & ((P = zero_z1200043727omplex))))))). % pCons_eq_0_iff
thf(fact_38_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_39_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_40_pCons__eq__0__iff, axiom,
    ((![A2 : poly_complex, P : poly_poly_complex]: (((pCons_poly_complex @ A2 @ P) = zero_z1040703943omplex) = (((A2 = zero_z1746442943omplex)) & ((P = zero_z1040703943omplex))))))). % pCons_eq_0_iff
thf(fact_41_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_42_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_43_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_44_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_45_pCons__0__0, axiom,
    (((pCons_1087637536omplex @ zero_z1040703943omplex @ zero_z1200043727omplex) = zero_z1200043727omplex))). % pCons_0_0
thf(fact_46_reflect__poly__const, axiom,
    ((![A2 : nat]: ((reflect_poly_nat @ (pCons_nat @ A2 @ zero_zero_poly_nat)) = (pCons_nat @ A2 @ zero_zero_poly_nat))))). % reflect_poly_const
thf(fact_47_reflect__poly__const, axiom,
    ((![A2 : poly_complex]: ((reflec309385472omplex @ (pCons_poly_complex @ A2 @ zero_z1040703943omplex)) = (pCons_poly_complex @ A2 @ zero_z1040703943omplex))))). % reflect_poly_const
thf(fact_48_reflect__poly__const, axiom,
    ((![A2 : complex]: ((reflect_poly_complex @ (pCons_complex @ A2 @ zero_z1746442943omplex)) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % reflect_poly_const
thf(fact_49_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A : complex, Q2 : poly_complex]: (~ ((P = (pCons_complex @ A @ Q2)))))))))). % pCons_cases
thf(fact_50_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A : complex, P2 : poly_complex]: (~ ((X = (pCons_complex @ A @ P2)))))))))). % pderiv.cases
thf(fact_51_pderiv_Oinduct, axiom,
    ((![P3 : poly_complex > $o, A0 : poly_complex]: ((![A : complex, P2 : poly_complex]: (((~ ((P2 = zero_z1746442943omplex))) => (P3 @ P2)) => (P3 @ (pCons_complex @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_52_pderiv_Oinduct, axiom,
    ((![P3 : poly_nat > $o, A0 : poly_nat]: ((![A : nat, P2 : poly_nat]: (((~ ((P2 = zero_zero_poly_nat))) => (P3 @ P2)) => (P3 @ (pCons_nat @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_53_pderiv_Oinduct, axiom,
    ((![P3 : poly_poly_complex > $o, A0 : poly_poly_complex]: ((![A : poly_complex, P2 : poly_poly_complex]: (((~ ((P2 = zero_z1040703943omplex))) => (P3 @ P2)) => (P3 @ (pCons_poly_complex @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_54_poly__induct2, axiom,
    ((![P3 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P3 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A : complex, P2 : poly_complex, B2 : complex, Q2 : poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_complex @ A @ P2) @ (pCons_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_55_poly__induct2, axiom,
    ((![P3 : poly_complex > poly_nat > $o, P : poly_complex, Q : poly_nat]: ((P3 @ zero_z1746442943omplex @ zero_zero_poly_nat) => ((![A : complex, P2 : poly_complex, B2 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_complex @ A @ P2) @ (pCons_nat @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_56_poly__induct2, axiom,
    ((![P3 : poly_complex > poly_poly_complex > $o, P : poly_complex, Q : poly_poly_complex]: ((P3 @ zero_z1746442943omplex @ zero_z1040703943omplex) => ((![A : complex, P2 : poly_complex, B2 : poly_complex, Q2 : poly_poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_complex @ A @ P2) @ (pCons_poly_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_57_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_complex > $o, P : poly_nat, Q : poly_complex]: ((P3 @ zero_zero_poly_nat @ zero_z1746442943omplex) => ((![A : nat, P2 : poly_nat, B2 : complex, Q2 : poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_58_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_nat > $o, P : poly_nat, Q : poly_nat]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A : nat, P2 : poly_nat, B2 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_nat @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_59_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_poly_complex > $o, P : poly_nat, Q : poly_poly_complex]: ((P3 @ zero_zero_poly_nat @ zero_z1040703943omplex) => ((![A : nat, P2 : poly_nat, B2 : poly_complex, Q2 : poly_poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_poly_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_60_poly__induct2, axiom,
    ((![P3 : poly_poly_complex > poly_complex > $o, P : poly_poly_complex, Q : poly_complex]: ((P3 @ zero_z1040703943omplex @ zero_z1746442943omplex) => ((![A : poly_complex, P2 : poly_poly_complex, B2 : complex, Q2 : poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_complex @ A @ P2) @ (pCons_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_61_poly__induct2, axiom,
    ((![P3 : poly_poly_complex > poly_nat > $o, P : poly_poly_complex, Q : poly_nat]: ((P3 @ zero_z1040703943omplex @ zero_zero_poly_nat) => ((![A : poly_complex, P2 : poly_poly_complex, B2 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_complex @ A @ P2) @ (pCons_nat @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_62_poly__induct2, axiom,
    ((![P3 : poly_poly_complex > poly_poly_complex > $o, P : poly_poly_complex, Q : poly_poly_complex]: ((P3 @ zero_z1040703943omplex @ zero_z1040703943omplex) => ((![A : poly_complex, P2 : poly_poly_complex, B2 : poly_complex, Q2 : poly_poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_complex @ A @ P2) @ (pCons_poly_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_63_offset__poly__single, axiom,
    ((![A2 : complex, H : complex]: ((fundam1201687030omplex @ (pCons_complex @ A2 @ zero_z1746442943omplex) @ H) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % offset_poly_single
thf(fact_64_offset__poly__single, axiom,
    ((![A2 : nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A2 @ zero_zero_poly_nat) @ H) = (pCons_nat @ A2 @ zero_zero_poly_nat))))). % offset_poly_single
thf(fact_65_offset__poly__single, axiom,
    ((![A2 : poly_complex, H : poly_complex]: ((fundam1307691262omplex @ (pCons_poly_complex @ A2 @ zero_z1040703943omplex) @ H) = (pCons_poly_complex @ A2 @ zero_z1040703943omplex))))). % offset_poly_single
thf(fact_66_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y : complex]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_67_pCons__induct, axiom,
    ((![P3 : poly_poly_nat > $o, P : poly_poly_nat]: ((P3 @ zero_z1059985641ly_nat) => ((![A : poly_nat, P2 : poly_poly_nat]: (((~ ((A = zero_zero_poly_nat))) | (~ ((P2 = zero_z1059985641ly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_poly_nat @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_68_pCons__induct, axiom,
    ((![P3 : poly_p1267267526omplex > $o, P : poly_p1267267526omplex]: ((P3 @ zero_z1200043727omplex) => ((![A : poly_poly_complex, P2 : poly_p1267267526omplex]: (((~ ((A = zero_z1040703943omplex))) | (~ ((P2 = zero_z1200043727omplex)))) => ((P3 @ P2) => (P3 @ (pCons_1087637536omplex @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_69_pCons__induct, axiom,
    ((![P3 : poly_complex > $o, P : poly_complex]: ((P3 @ zero_z1746442943omplex) => ((![A : complex, P2 : poly_complex]: (((~ ((A = zero_zero_complex))) | (~ ((P2 = zero_z1746442943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_complex @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_70_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A : nat, P2 : poly_nat]: (((~ ((A = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_71_pCons__induct, axiom,
    ((![P3 : poly_poly_complex > $o, P : poly_poly_complex]: ((P3 @ zero_z1040703943omplex) => ((![A : poly_complex, P2 : poly_poly_complex]: (((~ ((A = zero_z1746442943omplex))) | (~ ((P2 = zero_z1040703943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_poly_complex @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_72_order__0I, axiom,
    ((![P : poly_complex, A2 : complex]: ((~ (((poly_complex2 @ P @ A2) = zero_zero_complex))) => ((order_complex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_73_order__0I, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex]: ((~ (((poly_poly_complex2 @ P @ A2) = zero_z1746442943omplex))) => ((order_poly_complex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_74_order__0I, axiom,
    ((![P : poly_p1267267526omplex, A2 : poly_poly_complex]: ((~ (((poly_p282434315omplex @ P @ A2) = zero_z1040703943omplex))) => ((order_1735763309omplex @ A2 @ P) = zero_zero_nat))))). % order_0I
thf(fact_75_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_76_psize__eq__0__iff, axiom,
    ((![P : poly_nat]: (((fundam1567013434ze_nat @ P) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % psize_eq_0_iff
thf(fact_77_psize__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((fundam1956464160omplex @ P) = zero_zero_nat) = (P = zero_z1040703943omplex))))). % psize_eq_0_iff
thf(fact_78_synthetic__div__pCons, axiom,
    ((![A2 : nat, P : poly_nat, C : nat]: ((synthetic_div_nat @ (pCons_nat @ A2 @ P) @ C) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_pCons
thf(fact_79_synthetic__div__pCons, axiom,
    ((![A2 : poly_complex, P : poly_poly_complex, C : poly_complex]: ((synthe1985144195omplex @ (pCons_poly_complex @ A2 @ P) @ C) = (pCons_poly_complex @ (poly_poly_complex2 @ P @ C) @ (synthe1985144195omplex @ P @ C)))))). % synthetic_div_pCons
thf(fact_80_synthetic__div__pCons, axiom,
    ((![A2 : complex, P : poly_complex, C : complex]: ((synthe151143547omplex @ (pCons_complex @ A2 @ P) @ C) = (pCons_complex @ (poly_complex2 @ P @ C) @ (synthe151143547omplex @ P @ C)))))). % synthetic_div_pCons
thf(fact_81_is__zero__null, axiom,
    ((is_zero_complex = (^[P4 : poly_complex]: (P4 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_82_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_83_is__zero__null, axiom,
    ((is_zero_poly_complex = (^[P4 : poly_poly_complex]: (P4 = zero_z1040703943omplex))))). % is_zero_null
thf(fact_84_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_85_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_86_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_c622223248omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_cutoff_0
thf(fact_87_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_nat]: (((coeff_poly_nat @ (reflec781175074ly_nat @ P) @ zero_zero_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_88_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_p1267267526omplex]: (((coeff_1429652124omplex @ (reflec1997789704omplex @ P) @ zero_zero_nat) = zero_z1040703943omplex) = (P = zero_z1200043727omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_89_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_90_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_complex]: (((coeff_poly_complex @ (reflec309385472omplex @ P) @ zero_zero_nat) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_91_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_complex]: (((coeff_complex @ (reflect_poly_complex @ P) @ zero_zero_nat) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_92_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_shift_0
thf(fact_93_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_94_poly__shift__0, axiom,
    ((![N : nat]: ((poly_s558570093omplex @ N @ zero_z1040703943omplex) = zero_z1040703943omplex)))). % poly_shift_0
thf(fact_95_monom__eq__const__iff, axiom,
    ((![C : poly_nat, N : nat, D : poly_nat]: (((monom_poly_nat @ C @ N) = (pCons_poly_nat @ D @ zero_z1059985641ly_nat)) = (((C = D)) & ((((C = zero_zero_poly_nat)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_96_monom__eq__const__iff, axiom,
    ((![C : poly_poly_complex, N : nat, D : poly_poly_complex]: (((monom_1210178217omplex @ C @ N) = (pCons_1087637536omplex @ D @ zero_z1200043727omplex)) = (((C = D)) & ((((C = zero_z1040703943omplex)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_97_monom__eq__const__iff, axiom,
    ((![C : complex, N : nat, D : complex]: (((monom_complex @ C @ N) = (pCons_complex @ D @ zero_z1746442943omplex)) = (((C = D)) & ((((C = zero_zero_complex)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_98_monom__eq__const__iff, axiom,
    ((![C : nat, N : nat, D : nat]: (((monom_nat @ C @ N) = (pCons_nat @ D @ zero_zero_poly_nat)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_99_monom__eq__const__iff, axiom,
    ((![C : poly_complex, N : nat, D : poly_complex]: (((monom_poly_complex @ C @ N) = (pCons_poly_complex @ D @ zero_z1040703943omplex)) = (((C = D)) & ((((C = zero_z1746442943omplex)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_100_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_nat]: ((~ (((coeff_poly_nat @ P @ zero_zero_nat) = zero_zero_poly_nat))) => ((reflec781175074ly_nat @ (reflec781175074ly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_101_reflect__poly__reflect__poly, axiom,
    ((![P : poly_p1267267526omplex]: ((~ (((coeff_1429652124omplex @ P @ zero_zero_nat) = zero_z1040703943omplex))) => ((reflec1997789704omplex @ (reflec1997789704omplex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_102_reflect__poly__reflect__poly, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((reflect_poly_nat @ (reflect_poly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_103_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((reflec309385472omplex @ (reflec309385472omplex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_104_reflect__poly__reflect__poly, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((reflect_poly_complex @ (reflect_poly_complex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_105_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_106_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_107_synthetic__div__0, axiom,
    ((![C : poly_complex]: ((synthe1985144195omplex @ zero_z1040703943omplex @ C) = zero_z1040703943omplex)))). % synthetic_div_0
thf(fact_108_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_109_coeff__0, axiom,
    ((![N : nat]: ((coeff_1429652124omplex @ zero_z1200043727omplex @ N) = zero_z1040703943omplex)))). % coeff_0
thf(fact_110_coeff__0, axiom,
    ((![N : nat]: ((coeff_complex @ zero_z1746442943omplex @ N) = zero_zero_complex)))). % coeff_0
thf(fact_111_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_112_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_complex @ zero_z1040703943omplex @ N) = zero_z1746442943omplex)))). % coeff_0
thf(fact_113_coeff__pCons__0, axiom,
    ((![A2 : complex, P : poly_complex]: ((coeff_complex @ (pCons_complex @ A2 @ P) @ zero_zero_nat) = A2)))). % coeff_pCons_0
thf(fact_114_monom__eq__0, axiom,
    ((![N : nat]: ((monom_complex @ zero_zero_complex @ N) = zero_z1746442943omplex)))). % monom_eq_0
thf(fact_115_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_complex @ zero_z1746442943omplex @ N) = zero_z1040703943omplex)))). % monom_eq_0
thf(fact_116_monom__eq__0, axiom,
    ((![N : nat]: ((monom_nat @ zero_zero_nat @ N) = zero_zero_poly_nat)))). % monom_eq_0
thf(fact_117_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_nat @ zero_zero_poly_nat @ N) = zero_z1059985641ly_nat)))). % monom_eq_0
thf(fact_118_monom__eq__0, axiom,
    ((![N : nat]: ((monom_1210178217omplex @ zero_z1040703943omplex @ N) = zero_z1200043727omplex)))). % monom_eq_0
thf(fact_119_monom__eq__0__iff, axiom,
    ((![A2 : poly_nat, N : nat]: (((monom_poly_nat @ A2 @ N) = zero_z1059985641ly_nat) = (A2 = zero_zero_poly_nat))))). % monom_eq_0_iff
thf(fact_120_monom__eq__0__iff, axiom,
    ((![A2 : poly_poly_complex, N : nat]: (((monom_1210178217omplex @ A2 @ N) = zero_z1200043727omplex) = (A2 = zero_z1040703943omplex))))). % monom_eq_0_iff
thf(fact_121_monom__eq__0__iff, axiom,
    ((![A2 : complex, N : nat]: (((monom_complex @ A2 @ N) = zero_z1746442943omplex) = (A2 = zero_zero_complex))))). % monom_eq_0_iff
thf(fact_122_monom__eq__0__iff, axiom,
    ((![A2 : nat, N : nat]: (((monom_nat @ A2 @ N) = zero_zero_poly_nat) = (A2 = zero_zero_nat))))). % monom_eq_0_iff
thf(fact_123_monom__eq__0__iff, axiom,
    ((![A2 : poly_complex, N : nat]: (((monom_poly_complex @ A2 @ N) = zero_z1040703943omplex) = (A2 = zero_z1746442943omplex))))). % monom_eq_0_iff
thf(fact_124_coeff__monom, axiom,
    ((![M : nat, N : nat, A2 : complex]: (((M = N) => ((coeff_complex @ (monom_complex @ A2 @ M) @ N) = A2)) & ((~ ((M = N))) => ((coeff_complex @ (monom_complex @ A2 @ M) @ N) = zero_zero_complex)))))). % coeff_monom
thf(fact_125_coeff__monom, axiom,
    ((![M : nat, N : nat, A2 : poly_complex]: (((M = N) => ((coeff_poly_complex @ (monom_poly_complex @ A2 @ M) @ N) = A2)) & ((~ ((M = N))) => ((coeff_poly_complex @ (monom_poly_complex @ A2 @ M) @ N) = zero_z1746442943omplex)))))). % coeff_monom
thf(fact_126_coeff__monom, axiom,
    ((![M : nat, N : nat, A2 : nat]: (((M = N) => ((coeff_nat @ (monom_nat @ A2 @ M) @ N) = A2)) & ((~ ((M = N))) => ((coeff_nat @ (monom_nat @ A2 @ M) @ N) = zero_zero_nat)))))). % coeff_monom
thf(fact_127_coeff__monom, axiom,
    ((![M : nat, N : nat, A2 : poly_nat]: (((M = N) => ((coeff_poly_nat @ (monom_poly_nat @ A2 @ M) @ N) = A2)) & ((~ ((M = N))) => ((coeff_poly_nat @ (monom_poly_nat @ A2 @ M) @ N) = zero_zero_poly_nat)))))). % coeff_monom
thf(fact_128_coeff__monom, axiom,
    ((![M : nat, N : nat, A2 : poly_poly_complex]: (((M = N) => ((coeff_1429652124omplex @ (monom_1210178217omplex @ A2 @ M) @ N) = A2)) & ((~ ((M = N))) => ((coeff_1429652124omplex @ (monom_1210178217omplex @ A2 @ M) @ N) = zero_z1040703943omplex)))))). % coeff_monom
thf(fact_129_order__0__monom, axiom,
    ((![C : complex, N : nat]: ((~ ((C = zero_zero_complex))) => ((order_complex @ zero_zero_complex @ (monom_complex @ C @ N)) = N))))). % order_0_monom
thf(fact_130_order__0__monom, axiom,
    ((![C : poly_complex, N : nat]: ((~ ((C = zero_z1746442943omplex))) => ((order_poly_complex @ zero_z1746442943omplex @ (monom_poly_complex @ C @ N)) = N))))). % order_0_monom
thf(fact_131_order__0__monom, axiom,
    ((![C : poly_poly_complex, N : nat]: ((~ ((C = zero_z1040703943omplex))) => ((order_1735763309omplex @ zero_z1040703943omplex @ (monom_1210178217omplex @ C @ N)) = N))))). % order_0_monom
thf(fact_132_monom_Orep__eq, axiom,
    ((![X : complex, Xa : nat]: ((coeff_complex @ (monom_complex @ X @ Xa)) = (^[N2 : nat]: (if_complex @ (Xa = N2) @ X @ zero_zero_complex)))))). % monom.rep_eq
thf(fact_133_monom_Orep__eq, axiom,
    ((![X : poly_complex, Xa : nat]: ((coeff_poly_complex @ (monom_poly_complex @ X @ Xa)) = (^[N2 : nat]: (if_poly_complex @ (Xa = N2) @ X @ zero_z1746442943omplex)))))). % monom.rep_eq
thf(fact_134_monom_Orep__eq, axiom,
    ((![X : nat, Xa : nat]: ((coeff_nat @ (monom_nat @ X @ Xa)) = (^[N2 : nat]: (if_nat @ (Xa = N2) @ X @ zero_zero_nat)))))). % monom.rep_eq
thf(fact_135_monom_Orep__eq, axiom,
    ((![X : poly_nat, Xa : nat]: ((coeff_poly_nat @ (monom_poly_nat @ X @ Xa)) = (^[N2 : nat]: (if_poly_nat @ (Xa = N2) @ X @ zero_zero_poly_nat)))))). % monom.rep_eq
thf(fact_136_monom_Orep__eq, axiom,
    ((![X : poly_poly_complex, Xa : nat]: ((coeff_1429652124omplex @ (monom_1210178217omplex @ X @ Xa)) = (^[N2 : nat]: (if_poly_poly_complex @ (Xa = N2) @ X @ zero_z1040703943omplex)))))). % monom.rep_eq
thf(fact_137_monom__eq__iff_H, axiom,
    ((![C : complex, N : nat, D : complex, M : nat]: (((monom_complex @ C @ N) = (monom_complex @ D @ M)) = (((C = D)) & ((((C = zero_zero_complex)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_138_monom__eq__iff_H, axiom,
    ((![C : poly_complex, N : nat, D : poly_complex, M : nat]: (((monom_poly_complex @ C @ N) = (monom_poly_complex @ D @ M)) = (((C = D)) & ((((C = zero_z1746442943omplex)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_139_monom__eq__iff_H, axiom,
    ((![C : nat, N : nat, D : nat, M : nat]: (((monom_nat @ C @ N) = (monom_nat @ D @ M)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_140_monom__eq__iff_H, axiom,
    ((![C : poly_nat, N : nat, D : poly_nat, M : nat]: (((monom_poly_nat @ C @ N) = (monom_poly_nat @ D @ M)) = (((C = D)) & ((((C = zero_zero_poly_nat)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_141_monom__eq__iff_H, axiom,
    ((![C : poly_poly_complex, N : nat, D : poly_poly_complex, M : nat]: (((monom_1210178217omplex @ C @ N) = (monom_1210178217omplex @ D @ M)) = (((C = D)) & ((((C = zero_z1040703943omplex)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_142_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_143_zero__poly_Orep__eq, axiom,
    (((coeff_1429652124omplex @ zero_z1200043727omplex) = (^[Uu : nat]: zero_z1040703943omplex)))). % zero_poly.rep_eq
thf(fact_144_zero__poly_Orep__eq, axiom,
    (((coeff_complex @ zero_z1746442943omplex) = (^[Uu : nat]: zero_zero_complex)))). % zero_poly.rep_eq
thf(fact_145_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_146_zero__poly_Orep__eq, axiom,
    (((coeff_poly_complex @ zero_z1040703943omplex) = (^[Uu : nat]: zero_z1746442943omplex)))). % zero_poly.rep_eq
thf(fact_147_poly__0__coeff__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ P @ zero_zero_complex) = (coeff_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_148_poly__0__coeff__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ P @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_149_poly__0__coeff__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ P @ zero_zero_nat) = (coeff_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_150_poly__0__coeff__0, axiom,
    ((![P : poly_poly_nat]: ((poly_poly_nat2 @ P @ zero_zero_poly_nat) = (coeff_poly_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_151_poly__0__coeff__0, axiom,
    ((![P : poly_p1267267526omplex]: ((poly_p282434315omplex @ P @ zero_z1040703943omplex) = (coeff_1429652124omplex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_152_monom__0, axiom,
    ((![A2 : complex]: ((monom_complex @ A2 @ zero_zero_nat) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % monom_0
thf(fact_153_monom__0, axiom,
    ((![A2 : nat]: ((monom_nat @ A2 @ zero_zero_nat) = (pCons_nat @ A2 @ zero_zero_poly_nat))))). % monom_0
thf(fact_154_monom__0, axiom,
    ((![A2 : poly_complex]: ((monom_poly_complex @ A2 @ zero_zero_nat) = (pCons_poly_complex @ A2 @ zero_z1040703943omplex))))). % monom_0
thf(fact_155_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_nat]: ((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = (coeff_poly_nat @ P @ (degree_poly_nat @ P)))))). % poly_reflect_poly_0
thf(fact_156_poly__reflect__poly__0, axiom,
    ((![P : poly_p1267267526omplex]: ((poly_p282434315omplex @ (reflec1997789704omplex @ P) @ zero_z1040703943omplex) = (coeff_1429652124omplex @ P @ (degree2006505739omplex @ P)))))). % poly_reflect_poly_0
thf(fact_157_poly__reflect__poly__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % poly_reflect_poly_0
thf(fact_158_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ (degree_poly_complex @ P)))))). % poly_reflect_poly_0
thf(fact_159_poly__reflect__poly__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = (coeff_complex @ P @ (degree_complex @ P)))))). % poly_reflect_poly_0
thf(fact_160_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = one_one_poly_complex)))))). % poly_cutoff_1
thf(fact_161_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = zero_zero_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = one_one_poly_nat)))))). % poly_cutoff_1
thf(fact_162_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_c622223248omplex @ N @ one_on1331105667omplex) = zero_z1040703943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_c622223248omplex @ N @ one_on1331105667omplex) = one_on1331105667omplex)))))). % poly_cutoff_1
thf(fact_163_pcompose__0_H, axiom,
    ((![P : poly_complex]: ((pcompose_complex @ P @ zero_z1746442943omplex) = (pCons_complex @ (coeff_complex @ P @ zero_zero_nat) @ zero_z1746442943omplex))))). % pcompose_0'
thf(fact_164_pcompose__0_H, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ zero_zero_poly_nat) = (pCons_nat @ (coeff_nat @ P @ zero_zero_nat) @ zero_zero_poly_nat))))). % pcompose_0'
thf(fact_165_pcompose__0_H, axiom,
    ((![P : poly_poly_complex]: ((pcompo1411605209omplex @ P @ zero_z1040703943omplex) = (pCons_poly_complex @ (coeff_poly_complex @ P @ zero_zero_nat) @ zero_z1040703943omplex))))). % pcompose_0'
thf(fact_166_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_nat]: ((~ (((coeff_poly_nat @ P @ zero_zero_nat) = zero_zero_poly_nat))) => ((degree_poly_nat @ (reflec781175074ly_nat @ P)) = (degree_poly_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_167_degree__reflect__poly__eq, axiom,
    ((![P : poly_p1267267526omplex]: ((~ (((coeff_1429652124omplex @ P @ zero_zero_nat) = zero_z1040703943omplex))) => ((degree2006505739omplex @ (reflec1997789704omplex @ P)) = (degree2006505739omplex @ P)))))). % degree_reflect_poly_eq
thf(fact_168_degree__reflect__poly__eq, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((degree_nat @ (reflect_poly_nat @ P)) = (degree_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_169_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((degree_poly_complex @ (reflec309385472omplex @ P)) = (degree_poly_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_170_degree__reflect__poly__eq, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((degree_complex @ (reflect_poly_complex @ P)) = (degree_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_171_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_complex @ N @ one_one_poly_complex) = one_one_poly_complex)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)))))). % poly_shift_1
thf(fact_172_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_nat @ N @ one_one_poly_nat) = one_one_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_nat @ N @ one_one_poly_nat) = zero_zero_poly_nat)))))). % poly_shift_1
thf(fact_173_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_s558570093omplex @ N @ one_on1331105667omplex) = one_on1331105667omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_s558570093omplex @ N @ one_on1331105667omplex) = zero_z1040703943omplex)))))). % poly_shift_1
thf(fact_174_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_complex]: (((ord_less_nat @ K @ N) => ((coeff_complex @ (poly_cutoff_complex @ N @ P) @ K) = (coeff_complex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_complex @ (poly_cutoff_complex @ N @ P) @ K) = zero_zero_complex)))))). % coeff_poly_cutoff
thf(fact_175_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_complex]: (((ord_less_nat @ K @ N) => ((coeff_poly_complex @ (poly_c622223248omplex @ N @ P) @ K) = (coeff_poly_complex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_complex @ (poly_c622223248omplex @ N @ P) @ K) = zero_z1746442943omplex)))))). % coeff_poly_cutoff
thf(fact_176_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = (coeff_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = zero_zero_nat)))))). % coeff_poly_cutoff
thf(fact_177_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = (coeff_poly_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = zero_zero_poly_nat)))))). % coeff_poly_cutoff
thf(fact_178_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_p1267267526omplex]: (((ord_less_nat @ K @ N) => ((coeff_1429652124omplex @ (poly_c306577560omplex @ N @ P) @ K) = (coeff_1429652124omplex @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_1429652124omplex @ (poly_c306577560omplex @ N @ P) @ K) = zero_z1040703943omplex)))))). % coeff_poly_cutoff
thf(fact_179_coeff__0__reflect__poly, axiom,
    ((![P : poly_nat]: ((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % coeff_0_reflect_poly
thf(fact_180_coeff__0__reflect__poly, axiom,
    ((![P : poly_poly_complex]: ((coeff_poly_complex @ (reflec309385472omplex @ P) @ zero_zero_nat) = (coeff_poly_complex @ P @ (degree_poly_complex @ P)))))). % coeff_0_reflect_poly
thf(fact_181_coeff__0__reflect__poly, axiom,
    ((![P : poly_complex]: ((coeff_complex @ (reflect_poly_complex @ P) @ zero_zero_nat) = (coeff_complex @ P @ (degree_complex @ P)))))). % coeff_0_reflect_poly
thf(fact_182_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_183_pcompose__0, axiom,
    ((![Q : poly_complex]: ((pcompose_complex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % pcompose_0
thf(fact_184_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_185_pcompose__0, axiom,
    ((![Q : poly_poly_complex]: ((pcompo1411605209omplex @ zero_z1040703943omplex @ Q) = zero_z1040703943omplex)))). % pcompose_0
thf(fact_186_reflect__poly__1, axiom,
    (((reflect_poly_nat @ one_one_poly_nat) = one_one_poly_nat))). % reflect_poly_1
thf(fact_187_reflect__poly__1, axiom,
    (((reflec309385472omplex @ one_on1331105667omplex) = one_on1331105667omplex))). % reflect_poly_1
thf(fact_188_reflect__poly__1, axiom,
    (((reflect_poly_complex @ one_one_poly_complex) = one_one_poly_complex))). % reflect_poly_1
thf(fact_189_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_190_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_191_degree__0, axiom,
    (((degree_poly_complex @ zero_z1040703943omplex) = zero_zero_nat))). % degree_0
thf(fact_192_degree__1, axiom,
    (((degree_complex @ one_one_poly_complex) = zero_zero_nat))). % degree_1
thf(fact_193_lead__coeff__monom, axiom,
    ((![C : complex, N : nat]: ((coeff_complex @ (monom_complex @ C @ N) @ (degree_complex @ (monom_complex @ C @ N))) = C)))). % lead_coeff_monom
thf(fact_194_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_195_poly__1, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ one_on1331105667omplex @ X) = one_one_poly_complex)))). % poly_1
thf(fact_196_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_197_pcompose__const, axiom,
    ((![A2 : nat, Q : poly_nat]: ((pcompose_nat @ (pCons_nat @ A2 @ zero_zero_poly_nat) @ Q) = (pCons_nat @ A2 @ zero_zero_poly_nat))))). % pcompose_const
thf(fact_198_pcompose__const, axiom,
    ((![A2 : poly_complex, Q : poly_poly_complex]: ((pcompo1411605209omplex @ (pCons_poly_complex @ A2 @ zero_z1040703943omplex) @ Q) = (pCons_poly_complex @ A2 @ zero_z1040703943omplex))))). % pcompose_const
thf(fact_199_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_200_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_201_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_202__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062p_A_061_A0_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062p_A_092_060noteq_062_A0_059_Adegree_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_059_A_092_060And_062n_O_A_092_060lbrakk_062p_A_092_060noteq_062_A0_059_Adegree_Ap_A_061_ASuc_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    (((~ ((p = zero_z1746442943omplex))) => (((~ ((p = zero_z1746442943omplex))) => (~ (((degree_complex @ p) = zero_zero_nat)))) => (~ (((~ ((p = zero_z1746442943omplex))) => (![N3 : nat]: (~ (((degree_complex @ p) = (suc @ N3)))))))))))). % \<open>\<And>thesis. \<lbrakk>p = 0 \<Longrightarrow> thesis; \<lbrakk>p \<noteq> 0; degree p = 0\<rbrakk> \<Longrightarrow> thesis; \<And>n. \<lbrakk>p \<noteq> 0; degree p = Suc n\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_203_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_204_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_205_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_206_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_207_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_208_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_209_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_210_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_211_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J : nat]: ((ord_less_nat @ I @ J) => (~ ((K = (suc @ J))))))))))))). % Nat.lessE
thf(fact_212_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_213_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J : nat]: ((ord_less_nat @ I @ J) => (~ ((K = (suc @ J)))))))))))). % Suc_lessE
thf(fact_214_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_215_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_216_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_217_Ex__less__Suc, axiom,
    ((![N : nat, P3 : nat > $o]: ((?[I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) & ((P3 @ I2)))) = (((P3 @ N)) | ((?[I2 : nat]: (((ord_less_nat @ I2 @ N)) & ((P3 @ I2)))))))))). % Ex_less_Suc
thf(fact_218_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_219_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_220_All__less__Suc, axiom,
    ((![N : nat, P3 : nat > $o]: ((![I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) => ((P3 @ I2)))) = (((P3 @ N)) & ((![I2 : nat]: (((ord_less_nat @ I2 @ N)) => ((P3 @ I2)))))))))). % All_less_Suc
thf(fact_221_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M2 : nat]: (((M = (suc @ M2))) & ((ord_less_nat @ N @ M2)))))))). % Suc_less_eq2
thf(fact_222_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_223_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_224_less__trans__Suc, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ J2) => ((ord_less_nat @ J2 @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_225_less__Suc__induct, axiom,
    ((![I : nat, J2 : nat, P3 : nat > nat > $o]: ((ord_less_nat @ I @ J2) => ((![I3 : nat]: (P3 @ I3 @ (suc @ I3))) => ((![I3 : nat, J : nat, K2 : nat]: ((ord_less_nat @ I3 @ J) => ((ord_less_nat @ J @ K2) => ((P3 @ I3 @ J) => ((P3 @ J @ K2) => (P3 @ I3 @ K2)))))) => (P3 @ I @ J2))))))). % less_Suc_induct
thf(fact_226_strict__inc__induct, axiom,
    ((![I : nat, J2 : nat, P3 : nat > $o]: ((ord_less_nat @ I @ J2) => ((![I3 : nat]: ((J2 = (suc @ I3)) => (P3 @ I3))) => ((![I3 : nat]: ((ord_less_nat @ I3 @ J2) => ((P3 @ (suc @ I3)) => (P3 @ I3)))) => (P3 @ I))))))). % strict_inc_induct
thf(fact_227_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_228_nat__induct__non__zero, axiom,
    ((![N : nat, P3 : nat > $o]: ((ord_less_nat @ zero_zero_nat @ N) => ((P3 @ one_one_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((P3 @ N3) => (P3 @ (suc @ N3))))) => (P3 @ N))))))). % nat_induct_non_zero
thf(fact_229_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_230_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_231_Suc__inject, axiom,
    ((![X : nat, Y3 : nat]: (((suc @ X) = (suc @ Y3)) => (X = Y3))))). % Suc_inject
thf(fact_232_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_233_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_234_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_235_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_236_nat__induct, axiom,
    ((![P3 : nat > $o, N : nat]: ((P3 @ zero_zero_nat) => ((![N3 : nat]: ((P3 @ N3) => (P3 @ (suc @ N3)))) => (P3 @ N)))))). % nat_induct
thf(fact_237_diff__induct, axiom,
    ((![P3 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P3 @ X3 @ zero_zero_nat)) => ((![Y4 : nat]: (P3 @ zero_zero_nat @ (suc @ Y4))) => ((![X3 : nat, Y4 : nat]: ((P3 @ X3 @ Y4) => (P3 @ (suc @ X3) @ (suc @ Y4)))) => (P3 @ M @ N))))))). % diff_induct
thf(fact_238_zero__induct, axiom,
    ((![P3 : nat > $o, K : nat]: ((P3 @ K) => ((![N3 : nat]: ((P3 @ (suc @ N3)) => (P3 @ N3))) => (P3 @ zero_zero_nat)))))). % zero_induct
thf(fact_239_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_240_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_241_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_242_old_Onat_Oexhaust, axiom,
    ((![Y3 : nat]: ((~ ((Y3 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y3 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_243_old_Onat_Oinducts, axiom,
    ((![P3 : nat > $o, Nat : nat]: ((P3 @ zero_zero_nat) => ((![Nat3 : nat]: ((P3 @ Nat3) => (P3 @ (suc @ Nat3)))) => (P3 @ Nat)))))). % old.nat.inducts
thf(fact_244_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M3 : nat]: (N = (suc @ M3))))))). % not0_implies_Suc
thf(fact_245_Ex__less__Suc2, axiom,
    ((![N : nat, P3 : nat > $o]: ((?[I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) & ((P3 @ I2)))) = (((P3 @ zero_zero_nat)) | ((?[I2 : nat]: (((ord_less_nat @ I2 @ N)) & ((P3 @ (suc @ I2))))))))))). % Ex_less_Suc2

% Helper facts (11)
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y3 : nat]: ((if_nat @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y3 : nat]: ((if_nat @ $true @ X @ Y3) = X)))).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y3 : complex]: ((if_complex @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y3 : complex]: ((if_complex @ $true @ X @ Y3) = X)))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T, axiom,
    ((![X : poly_nat, Y3 : poly_nat]: ((if_poly_nat @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T, axiom,
    ((![X : poly_nat, Y3 : poly_nat]: ((if_poly_nat @ $true @ X @ Y3) = X)))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_T, axiom,
    ((![X : poly_complex, Y3 : poly_complex]: ((if_poly_complex @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_T, axiom,
    ((![X : poly_complex, Y3 : poly_complex]: ((if_poly_complex @ $true @ X @ Y3) = X)))).
thf(help_If_3_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_T, axiom,
    ((![P3 : $o]: ((P3 = $true) | (P3 = $false))))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_T, axiom,
    ((![X : poly_poly_complex, Y3 : poly_poly_complex]: ((if_poly_poly_complex @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_T, axiom,
    ((![X : poly_poly_complex, Y3 : poly_poly_complex]: ((if_poly_poly_complex @ $true @ X @ Y3) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((![X2 : complex]: ((((poly_complex2 @ p @ X2) = zero_zero_complex)) => (((poly_complex2 @ q @ X2) = zero_zero_complex)))) = (q = zero_z1746442943omplex)))).
