% 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_1058__5377844_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:36.755

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    poly_poly_poly_a : $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__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_a : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (77)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
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_Itf__a_J, type,
    fundam1343031620poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
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_Itf__a_J, type,
    fundam1032801442poly_a : poly_poly_a > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
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__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    one_one_poly_poly_a : poly_poly_a).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
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__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
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_Itf__a_J_J_J, type,
    zero_z2064990175poly_a : poly_poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
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_Itf__a_J_J, type,
    if_poly_poly_a : $o > poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_If_001t__Polynomial__Opoly_Itf__a_J, type,
    if_poly_a : $o > poly_a > poly_a > poly_a).
thf(sy_c_If_001tf__a, type,
    if_a : $o > a > a > a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > 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_Itf__a_J_J, type,
    degree_poly_poly_a : poly_poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_Itf__a_J, type,
    degree_poly_a : poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
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_Itf__a_J, type,
    is_zero_poly_a : poly_poly_a > $o).
thf(sy_c_Polynomial_Ois__zero_001tf__a, type,
    is_zero_a : poly_a > $o).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat, type,
    monom_nat : nat > nat > poly_nat).
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_Itf__a_J_J, type,
    monom_poly_poly_a : poly_poly_a > nat > poly_poly_poly_a).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_Itf__a_J, type,
    monom_poly_a : poly_a > nat > poly_poly_a).
thf(sy_c_Polynomial_Omonom_001tf__a, type,
    monom_a : a > nat > poly_a).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    order_poly_poly_a : poly_poly_a > poly_poly_poly_a > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_Itf__a_J, type,
    order_poly_a : poly_a > poly_poly_a > nat).
thf(sy_c_Polynomial_Oorder_001tf__a, type,
    order_a : a > poly_a > nat).
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__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_Itf__a_J_J, type,
    pCons_poly_poly_a : poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
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__Nat__Onat_J, type,
    pcompose_poly_nat : poly_poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    pcompose_poly_poly_a : poly_poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_Itf__a_J, type,
    pcompose_poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
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_Itf__a_J_J, type,
    poly_poly_poly_a2 : poly_poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
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__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_Itf__a_J_J, type,
    coeff_poly_poly_a : poly_poly_poly_a > nat > poly_poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_Itf__a_J, type,
    coeff_poly_a : poly_poly_a > nat > poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001tf__a, type,
    coeff_a : poly_a > nat > a).
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_Itf__a_J, type,
    poly_cutoff_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
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_Itf__a_J, type,
    poly_shift_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__shift_001tf__a, type,
    poly_shift_a : nat > poly_a > poly_a).
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__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    reflec581648976poly_a : poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    reflect_poly_poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001tf__a, type,
    reflect_poly_a : poly_a > poly_a).
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_Itf__a_J, type,
    synthetic_div_poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (244)
thf(fact_0_that, axiom,
    ((fundam236050252nt_a_a @ (poly_a2 @ p)))). % that
thf(fact_1_th, axiom,
    (((poly_a2 @ p) = (poly_a2 @ (pCons_a @ (poly_a2 @ p @ zero_zero_a) @ zero_zero_poly_a))))). % th
thf(fact_2__092_060open_062_092_060And_062x_O_Apoly_Ap_Ax_A_061_Apoly_Ap_A_I0_058_058_Ha_J_092_060close_062, axiom,
    ((![X : a]: ((poly_a2 @ p @ X) = (poly_a2 @ p @ zero_zero_a))))). % \<open>\<And>x. poly p x = poly p (0::'a)\<close>
thf(fact_3_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_4_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % poly_0
thf(fact_5_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_8_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_9_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_10_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_11_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_12_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_13_pCons__eq__0__iff, axiom,
    ((![A : poly_nat, P : poly_poly_nat]: (((pCons_poly_nat @ A @ P) = zero_z1059985641ly_nat) = (((A = zero_zero_poly_nat)) & ((P = zero_z1059985641ly_nat))))))). % pCons_eq_0_iff
thf(fact_14_pCons__eq__0__iff, axiom,
    ((![A : poly_poly_a, P : poly_poly_poly_a]: (((pCons_poly_poly_a @ A @ P) = zero_z2064990175poly_a) = (((A = zero_z2096148049poly_a)) & ((P = zero_z2064990175poly_a))))))). % pCons_eq_0_iff
thf(fact_15_pCons__eq__0__iff, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((pCons_poly_a @ A @ P) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) & ((P = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_16_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_17_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_18_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_poly_a]: ((![X2 : poly_poly_a]: ((poly_poly_poly_a2 @ P @ X2) = zero_z2096148049poly_a)) = (P = zero_z2064990175poly_a))))). % poly_all_0_iff_0
thf(fact_19_poly__all__0__iff__0, axiom,
    ((![P : poly_a]: ((![X2 : a]: ((poly_a2 @ P @ X2) = zero_zero_a)) = (P = zero_zero_poly_a))))). % poly_all_0_iff_0
thf(fact_20_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_a]: ((![X2 : poly_a]: ((poly_poly_a2 @ P @ X2) = zero_zero_poly_a)) = (P = zero_z2096148049poly_a))))). % poly_all_0_iff_0
thf(fact_21_pCons__induct, axiom,
    ((![P2 : poly_poly_nat > $o, P : poly_poly_nat]: ((P2 @ zero_z1059985641ly_nat) => ((![A2 : poly_nat, P3 : poly_poly_nat]: (((~ ((A2 = zero_zero_poly_nat))) | (~ ((P3 = zero_z1059985641ly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_poly_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_22_pCons__induct, axiom,
    ((![P2 : poly_poly_poly_a > $o, P : poly_poly_poly_a]: ((P2 @ zero_z2064990175poly_a) => ((![A2 : poly_poly_a, P3 : poly_poly_poly_a]: (((~ ((A2 = zero_z2096148049poly_a))) | (~ ((P3 = zero_z2064990175poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_poly_poly_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_23_pCons__induct, axiom,
    ((![P2 : poly_poly_a > $o, P : poly_poly_a]: ((P2 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P3 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P3 = zero_z2096148049poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_poly_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_24_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_25_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_26_pCons__eq__iff, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: (((pCons_nat @ A @ P) = (pCons_nat @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_27_pCons__eq__iff, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: (((pCons_poly_a @ A @ P) = (pCons_poly_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_28_pCons__eq__iff, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: (((pCons_a @ A @ P) = (pCons_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_29_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_30_poly__induct2, axiom,
    ((![P2 : poly_a > poly_nat > $o, P : poly_a, Q : poly_nat]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A2 : a, P3 : poly_a, B2 : nat, Q2 : poly_nat]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_nat @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_31_poly__induct2, axiom,
    ((![P2 : poly_a > poly_poly_a > $o, P : poly_a, Q : poly_poly_a]: ((P2 @ zero_zero_poly_a @ zero_z2096148049poly_a) => ((![A2 : a, P3 : poly_a, B2 : poly_a, Q2 : poly_poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_poly_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_32_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_a > $o, P : poly_nat, Q : poly_a]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A2 : nat, P3 : poly_nat, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_33_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_nat > $o, P : poly_nat, Q : poly_nat]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat, B2 : nat, Q2 : poly_nat]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_nat @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_34_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_poly_a > $o, P : poly_nat, Q : poly_poly_a]: ((P2 @ zero_zero_poly_nat @ zero_z2096148049poly_a) => ((![A2 : nat, P3 : poly_nat, B2 : poly_a, Q2 : poly_poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_poly_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_35_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_a > $o, P : poly_poly_a, Q : poly_a]: ((P2 @ zero_z2096148049poly_a @ zero_zero_poly_a) => ((![A2 : poly_a, P3 : poly_poly_a, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_poly_a @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_36_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_nat > $o, P : poly_poly_a, Q : poly_nat]: ((P2 @ zero_z2096148049poly_a @ zero_zero_poly_nat) => ((![A2 : poly_a, P3 : poly_poly_a, B2 : nat, Q2 : poly_nat]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_poly_a @ A2 @ P3) @ (pCons_nat @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_37_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_poly_a > $o, P : poly_poly_a, Q : poly_poly_a]: ((P2 @ zero_z2096148049poly_a @ zero_z2096148049poly_a) => ((![A2 : poly_a, P3 : poly_poly_a, B2 : poly_a, Q2 : poly_poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_poly_a @ A2 @ P3) @ (pCons_poly_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_38_pderiv_Oinduct, axiom,
    ((![P2 : poly_a > $o, A0 : poly_a]: ((![A2 : a, P3 : poly_a]: (((~ ((P3 = zero_zero_poly_a))) => (P2 @ P3)) => (P2 @ (pCons_a @ A2 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_39_pderiv_Oinduct, axiom,
    ((![P2 : poly_nat > $o, A0 : poly_nat]: ((![A2 : nat, P3 : poly_nat]: (((~ ((P3 = zero_zero_poly_nat))) => (P2 @ P3)) => (P2 @ (pCons_nat @ A2 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_40_pderiv_Oinduct, axiom,
    ((![P2 : poly_poly_a > $o, A0 : poly_poly_a]: ((![A2 : poly_a, P3 : poly_poly_a]: (((~ ((P3 = zero_z2096148049poly_a))) => (P2 @ P3)) => (P2 @ (pCons_poly_a @ A2 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_41_offset__poly__single, axiom,
    ((![A : a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ H) = (pCons_a @ A @ zero_zero_poly_a))))). % offset_poly_single
thf(fact_42_offset__poly__single, axiom,
    ((![A : nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ H) = (pCons_nat @ A @ zero_zero_poly_nat))))). % offset_poly_single
thf(fact_43_offset__poly__single, axiom,
    ((![A : poly_a, H : poly_a]: ((fundam1343031620poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ H) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % offset_poly_single
thf(fact_44_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_a, Q : poly_a]: (((poly_a2 @ P) = (poly_a2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_45_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a]: (((poly_poly_a2 @ P) = (poly_poly_a2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_46_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_47_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A2 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_48_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A2 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_49_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X2 : a]: (![Y : a]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_50_offset__poly__eq__0__iff, axiom,
    ((![P : poly_a, H : a]: (((fundam1358810038poly_a @ P @ H) = zero_zero_poly_a) = (P = zero_zero_poly_a))))). % offset_poly_eq_0_iff
thf(fact_51_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_52_offset__poly__eq__0__iff, axiom,
    ((![P : poly_poly_a, H : poly_a]: (((fundam1343031620poly_a @ P @ H) = zero_z2096148049poly_a) = (P = zero_z2096148049poly_a))))). % offset_poly_eq_0_iff
thf(fact_53_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_54_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_55_offset__poly__0, axiom,
    ((![H : poly_a]: ((fundam1343031620poly_a @ zero_z2096148049poly_a @ H) = zero_z2096148049poly_a)))). % offset_poly_0
thf(fact_56_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_57_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_58_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_59_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_60_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_61_pderiv_Ocases, axiom,
    ((![X : poly_a]: (~ ((![A2 : a, P3 : poly_a]: (~ ((X = (pCons_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_62_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A2 : nat, P3 : poly_nat]: (~ ((X = (pCons_nat @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_63_pderiv_Ocases, axiom,
    ((![X : poly_poly_a]: (~ ((![A2 : poly_a, P3 : poly_poly_a]: (~ ((X = (pCons_poly_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_64_synthetic__div__pCons, axiom,
    ((![A : a, P : poly_a, C : a]: ((synthetic_div_a @ (pCons_a @ A @ P) @ C) = (pCons_a @ (poly_a2 @ P @ C) @ (synthetic_div_a @ P @ C)))))). % synthetic_div_pCons
thf(fact_65_synthetic__div__pCons, axiom,
    ((![A : nat, P : poly_nat, C : nat]: ((synthetic_div_nat @ (pCons_nat @ A @ P) @ C) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_pCons
thf(fact_66_synthetic__div__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, C : poly_a]: ((synthetic_div_poly_a @ (pCons_poly_a @ A @ P) @ C) = (pCons_poly_a @ (poly_poly_a2 @ P @ C) @ (synthetic_div_poly_a @ P @ C)))))). % synthetic_div_pCons
thf(fact_67_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_68_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_69_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_70_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_71_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_72_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_73_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_a]: (((poly_a2 @ (reflect_poly_a @ P) @ zero_zero_a) = zero_zero_a) = (P = zero_zero_poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_74_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_a]: (((poly_poly_a2 @ (reflect_poly_poly_a @ P) @ zero_zero_poly_a) = zero_zero_poly_a) = (P = zero_z2096148049poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_75_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_76_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_77_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_poly_a]: (((poly_poly_poly_a2 @ (reflec581648976poly_a @ P) @ zero_z2096148049poly_a) = zero_z2096148049poly_a) = (P = zero_z2064990175poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_78_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_79_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_80_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_shift_0
thf(fact_81_order__root, axiom,
    ((![P : poly_a, A : a]: (((poly_a2 @ P @ A) = zero_zero_a) = (((P = zero_zero_poly_a)) | ((~ (((order_a @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_82_order__root, axiom,
    ((![P : poly_poly_a, A : poly_a]: (((poly_poly_a2 @ P @ A) = zero_zero_poly_a) = (((P = zero_z2096148049poly_a)) | ((~ (((order_poly_a @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_83_order__root, axiom,
    ((![P : poly_poly_poly_a, A : poly_poly_a]: (((poly_poly_poly_a2 @ P @ A) = zero_z2096148049poly_a) = (((P = zero_z2064990175poly_a)) | ((~ (((order_poly_poly_a @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_84_reflect__poly__const, axiom,
    ((![A : a]: ((reflect_poly_a @ (pCons_a @ A @ zero_zero_poly_a)) = (pCons_a @ A @ zero_zero_poly_a))))). % reflect_poly_const
thf(fact_85_reflect__poly__const, axiom,
    ((![A : nat]: ((reflect_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = (pCons_nat @ A @ zero_zero_poly_nat))))). % reflect_poly_const
thf(fact_86_reflect__poly__const, axiom,
    ((![A : poly_a]: ((reflect_poly_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a)) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % reflect_poly_const
thf(fact_87_psize__eq__0__iff, axiom,
    ((![P : poly_a]: (((fundam247907092size_a @ P) = zero_zero_nat) = (P = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_88_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_89_psize__eq__0__iff, axiom,
    ((![P : poly_poly_a]: (((fundam1032801442poly_a @ P) = zero_zero_nat) = (P = zero_z2096148049poly_a))))). % psize_eq_0_iff
thf(fact_90_pcompose__const, axiom,
    ((![A : a, Q : poly_a]: ((pcompose_a @ (pCons_a @ A @ zero_zero_poly_a) @ Q) = (pCons_a @ A @ zero_zero_poly_a))))). % pcompose_const
thf(fact_91_pcompose__const, axiom,
    ((![A : nat, Q : poly_nat]: ((pcompose_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ Q) = (pCons_nat @ A @ zero_zero_poly_nat))))). % pcompose_const
thf(fact_92_pcompose__const, axiom,
    ((![A : poly_a, Q : poly_poly_a]: ((pcompose_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ Q) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % pcompose_const
thf(fact_93_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_94_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_95_pcompose__0, axiom,
    ((![Q : poly_poly_a]: ((pcompose_poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % pcompose_0
thf(fact_96_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_97_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_98_reflect__poly__0, axiom,
    (((reflect_poly_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % reflect_poly_0
thf(fact_99_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_100_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_101_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_102_poly__pcompose, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (pcompose_a @ P @ Q) @ X) = (poly_a2 @ P @ (poly_a2 @ Q @ X)))))). % poly_pcompose
thf(fact_103_poly__pcompose, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (pcompose_nat @ P @ Q) @ X) = (poly_nat2 @ P @ (poly_nat2 @ Q @ X)))))). % poly_pcompose
thf(fact_104_poly__pcompose, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (pcompose_poly_a @ P @ Q) @ X) = (poly_poly_a2 @ P @ (poly_poly_a2 @ Q @ X)))))). % poly_pcompose
thf(fact_105_order__0I, axiom,
    ((![P : poly_a, A : a]: ((~ (((poly_a2 @ P @ A) = zero_zero_a))) => ((order_a @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_106_order__0I, axiom,
    ((![P : poly_poly_a, A : poly_a]: ((~ (((poly_poly_a2 @ P @ A) = zero_zero_poly_a))) => ((order_poly_a @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_107_order__0I, axiom,
    ((![P : poly_poly_poly_a, A : poly_poly_a]: ((~ (((poly_poly_poly_a2 @ P @ A) = zero_z2096148049poly_a))) => ((order_poly_poly_a @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_108_pcompose__idR, axiom,
    ((![P : poly_a]: ((pcompose_a @ P @ (pCons_a @ zero_zero_a @ (pCons_a @ one_one_a @ zero_zero_poly_a))) = P)))). % pcompose_idR
thf(fact_109_pcompose__idR, axiom,
    ((![P : poly_poly_a]: ((pcompose_poly_a @ P @ (pCons_poly_a @ zero_zero_poly_a @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a))) = P)))). % pcompose_idR
thf(fact_110_pcompose__idR, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P)))). % pcompose_idR
thf(fact_111_pcompose__idR, axiom,
    ((![P : poly_poly_nat]: ((pcompose_poly_nat @ P @ (pCons_poly_nat @ zero_zero_poly_nat @ (pCons_poly_nat @ one_one_poly_nat @ zero_z1059985641ly_nat))) = P)))). % pcompose_idR
thf(fact_112_pcompose__idR, axiom,
    ((![P : poly_poly_poly_a]: ((pcompose_poly_poly_a @ P @ (pCons_poly_poly_a @ zero_z2096148049poly_a @ (pCons_poly_poly_a @ one_one_poly_poly_a @ zero_z2064990175poly_a))) = P)))). % pcompose_idR
thf(fact_113_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_a @ N @ one_one_poly_a) = zero_zero_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_a @ N @ one_one_poly_a) = one_one_poly_a)))))). % poly_cutoff_1
thf(fact_114_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_115_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_poly_a @ N @ one_one_poly_poly_a) = zero_z2096148049poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_poly_a @ N @ one_one_poly_poly_a) = one_one_poly_poly_a)))))). % poly_cutoff_1
thf(fact_116_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_a]: (((coeff_a @ (reflect_poly_a @ P) @ zero_zero_nat) = zero_zero_a) = (P = zero_zero_poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_117_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_a]: (((coeff_poly_a @ (reflect_poly_poly_a @ P) @ zero_zero_nat) = zero_zero_poly_a) = (P = zero_z2096148049poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_118_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_119_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_120_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_poly_a]: (((coeff_poly_poly_a @ (reflec581648976poly_a @ P) @ zero_zero_nat) = zero_z2096148049poly_a) = (P = zero_z2064990175poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_121_pcompose__0_H, axiom,
    ((![P : poly_a]: ((pcompose_a @ P @ zero_zero_poly_a) = (pCons_a @ (coeff_a @ P @ zero_zero_nat) @ zero_zero_poly_a))))). % pcompose_0'
thf(fact_122_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_123_pcompose__0_H, axiom,
    ((![P : poly_poly_a]: ((pcompose_poly_a @ P @ zero_z2096148049poly_a) = (pCons_poly_a @ (coeff_poly_a @ P @ zero_zero_nat) @ zero_z2096148049poly_a))))). % pcompose_0'
thf(fact_124_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_125_monom__eq__const__iff, axiom,
    ((![C : poly_poly_a, N : nat, D : poly_poly_a]: (((monom_poly_poly_a @ C @ N) = (pCons_poly_poly_a @ D @ zero_z2064990175poly_a)) = (((C = D)) & ((((C = zero_z2096148049poly_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_126_monom__eq__const__iff, axiom,
    ((![C : a, N : nat, D : a]: (((monom_a @ C @ N) = (pCons_a @ D @ zero_zero_poly_a)) = (((C = D)) & ((((C = zero_zero_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_127_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_128_monom__eq__const__iff, axiom,
    ((![C : poly_a, N : nat, D : poly_a]: (((monom_poly_a @ C @ N) = (pCons_poly_a @ D @ zero_z2096148049poly_a)) = (((C = D)) & ((((C = zero_zero_poly_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_129_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_a @ N @ one_one_poly_a) = one_one_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_a @ N @ one_one_poly_a) = zero_zero_poly_a)))))). % poly_shift_1
thf(fact_130_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_131_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_poly_a @ N @ one_one_poly_poly_a) = one_one_poly_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_poly_a @ N @ one_one_poly_poly_a) = zero_z2096148049poly_a)))))). % poly_shift_1
thf(fact_132_reflect__poly__reflect__poly, axiom,
    ((![P : poly_a]: ((~ (((coeff_a @ P @ zero_zero_nat) = zero_zero_a))) => ((reflect_poly_a @ (reflect_poly_a @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_133_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_a]: ((~ (((coeff_poly_a @ P @ zero_zero_nat) = zero_zero_poly_a))) => ((reflect_poly_poly_a @ (reflect_poly_poly_a @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_134_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_135_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_136_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_poly_a]: ((~ (((coeff_poly_poly_a @ P @ zero_zero_nat) = zero_z2096148049poly_a))) => ((reflec581648976poly_a @ (reflec581648976poly_a @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_137_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_a, C : a]: (((synthetic_div_a @ P @ C) = zero_zero_poly_a) = ((degree_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_138_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_nat, C : nat]: (((synthetic_div_nat @ P @ C) = zero_zero_poly_nat) = ((degree_nat @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_139_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_poly_a, C : poly_a]: (((synthetic_div_poly_a @ P @ C) = zero_z2096148049poly_a) = ((degree_poly_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_140_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_141_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_poly_a @ zero_z2064990175poly_a @ N) = zero_z2096148049poly_a)))). % coeff_0
thf(fact_142_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_143_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_144_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_145_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_146_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_147_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_148_coeff__pCons__0, axiom,
    ((![A : a, P : poly_a]: ((coeff_a @ (pCons_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_149_coeff__pCons__0, axiom,
    ((![A : nat, P : poly_nat]: ((coeff_nat @ (pCons_nat @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_150_coeff__pCons__0, axiom,
    ((![A : poly_a, P : poly_poly_a]: ((coeff_poly_a @ (pCons_poly_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_151_monom__eq__0__iff, axiom,
    ((![A : poly_nat, N : nat]: (((monom_poly_nat @ A @ N) = zero_z1059985641ly_nat) = (A = zero_zero_poly_nat))))). % monom_eq_0_iff
thf(fact_152_monom__eq__0__iff, axiom,
    ((![A : poly_poly_a, N : nat]: (((monom_poly_poly_a @ A @ N) = zero_z2064990175poly_a) = (A = zero_z2096148049poly_a))))). % monom_eq_0_iff
thf(fact_153_monom__eq__0__iff, axiom,
    ((![A : a, N : nat]: (((monom_a @ A @ N) = zero_zero_poly_a) = (A = zero_zero_a))))). % monom_eq_0_iff
thf(fact_154_monom__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((monom_nat @ A @ N) = zero_zero_poly_nat) = (A = zero_zero_nat))))). % monom_eq_0_iff
thf(fact_155_monom__eq__0__iff, axiom,
    ((![A : poly_a, N : nat]: (((monom_poly_a @ A @ N) = zero_z2096148049poly_a) = (A = zero_zero_poly_a))))). % monom_eq_0_iff
thf(fact_156_monom__eq__0, axiom,
    ((![N : nat]: ((monom_a @ zero_zero_a @ N) = zero_zero_poly_a)))). % monom_eq_0
thf(fact_157_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_a @ zero_zero_poly_a @ N) = zero_z2096148049poly_a)))). % monom_eq_0
thf(fact_158_monom__eq__0, axiom,
    ((![N : nat]: ((monom_nat @ zero_zero_nat @ N) = zero_zero_poly_nat)))). % monom_eq_0
thf(fact_159_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_nat @ zero_zero_poly_nat @ N) = zero_z1059985641ly_nat)))). % monom_eq_0
thf(fact_160_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_poly_a @ zero_z2096148049poly_a @ N) = zero_z2064990175poly_a)))). % monom_eq_0
thf(fact_161_coeff__monom, axiom,
    ((![M : nat, N : nat, A : a]: (((M = N) => ((coeff_a @ (monom_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_a @ (monom_a @ A @ M) @ N) = zero_zero_a)))))). % coeff_monom
thf(fact_162_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_a]: (((M = N) => ((coeff_poly_a @ (monom_poly_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_a @ (monom_poly_a @ A @ M) @ N) = zero_zero_poly_a)))))). % coeff_monom
thf(fact_163_coeff__monom, axiom,
    ((![M : nat, N : nat, A : nat]: (((M = N) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = zero_zero_nat)))))). % coeff_monom
thf(fact_164_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_nat]: (((M = N) => ((coeff_poly_nat @ (monom_poly_nat @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_nat @ (monom_poly_nat @ A @ M) @ N) = zero_zero_poly_nat)))))). % coeff_monom
thf(fact_165_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_poly_a]: (((M = N) => ((coeff_poly_poly_a @ (monom_poly_poly_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_poly_a @ (monom_poly_poly_a @ A @ M) @ N) = zero_z2096148049poly_a)))))). % coeff_monom
thf(fact_166_poly__1, axiom,
    ((![X : a]: ((poly_a2 @ one_one_poly_a @ X) = one_one_a)))). % poly_1
thf(fact_167_poly__1, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ one_one_poly_poly_a @ X) = one_one_poly_a)))). % poly_1
thf(fact_168_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_169_order__0__monom, axiom,
    ((![C : a, N : nat]: ((~ ((C = zero_zero_a))) => ((order_a @ zero_zero_a @ (monom_a @ C @ N)) = N))))). % order_0_monom
thf(fact_170_order__0__monom, axiom,
    ((![C : poly_a, N : nat]: ((~ ((C = zero_zero_poly_a))) => ((order_poly_a @ zero_zero_poly_a @ (monom_poly_a @ C @ N)) = N))))). % order_0_monom
thf(fact_171_order__0__monom, axiom,
    ((![C : poly_poly_a, N : nat]: ((~ ((C = zero_z2096148049poly_a))) => ((order_poly_poly_a @ zero_z2096148049poly_a @ (monom_poly_poly_a @ C @ N)) = N))))). % order_0_monom
thf(fact_172_leading__coeff__0__iff, axiom,
    ((![P : poly_a]: (((coeff_a @ P @ (degree_a @ P)) = zero_zero_a) = (P = zero_zero_poly_a))))). % leading_coeff_0_iff
thf(fact_173_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_a]: (((coeff_poly_a @ P @ (degree_poly_a @ P)) = zero_zero_poly_a) = (P = zero_z2096148049poly_a))))). % leading_coeff_0_iff
thf(fact_174_leading__coeff__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % leading_coeff_0_iff
thf(fact_175_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_nat]: (((coeff_poly_nat @ P @ (degree_poly_nat @ P)) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % leading_coeff_0_iff
thf(fact_176_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_poly_a]: (((coeff_poly_poly_a @ P @ (degree_poly_poly_a @ P)) = zero_z2096148049poly_a) = (P = zero_z2064990175poly_a))))). % leading_coeff_0_iff
thf(fact_177_lead__coeff__pCons_I1_J, axiom,
    ((![P : poly_a, A : a]: ((~ ((P = zero_zero_poly_a))) => ((coeff_a @ (pCons_a @ A @ P) @ (degree_a @ (pCons_a @ A @ P))) = (coeff_a @ P @ (degree_a @ P))))))). % lead_coeff_pCons(1)
thf(fact_178_lead__coeff__pCons_I1_J, axiom,
    ((![P : poly_nat, A : nat]: ((~ ((P = zero_zero_poly_nat))) => ((coeff_nat @ (pCons_nat @ A @ P) @ (degree_nat @ (pCons_nat @ A @ P))) = (coeff_nat @ P @ (degree_nat @ P))))))). % lead_coeff_pCons(1)
thf(fact_179_lead__coeff__pCons_I1_J, axiom,
    ((![P : poly_poly_a, A : poly_a]: ((~ ((P = zero_z2096148049poly_a))) => ((coeff_poly_a @ (pCons_poly_a @ A @ P) @ (degree_poly_a @ (pCons_poly_a @ A @ P))) = (coeff_poly_a @ P @ (degree_poly_a @ P))))))). % lead_coeff_pCons(1)
thf(fact_180_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_a, A : a]: ((P = zero_zero_poly_a) => ((coeff_a @ (pCons_a @ A @ P) @ (degree_a @ (pCons_a @ A @ P))) = A))))). % lead_coeff_pCons(2)
thf(fact_181_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_nat, A : nat]: ((P = zero_zero_poly_nat) => ((coeff_nat @ (pCons_nat @ A @ P) @ (degree_nat @ (pCons_nat @ A @ P))) = A))))). % lead_coeff_pCons(2)
thf(fact_182_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_poly_a, A : poly_a]: ((P = zero_z2096148049poly_a) => ((coeff_poly_a @ (pCons_poly_a @ A @ P) @ (degree_poly_a @ (pCons_poly_a @ A @ P))) = A))))). % lead_coeff_pCons(2)
thf(fact_183_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_a = (pCons_a @ one_one_a @ zero_zero_poly_a)))). % one_poly_eq_simps(1)
thf(fact_184_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_185_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_poly_a = (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)))). % one_poly_eq_simps(1)
thf(fact_186_one__poly__eq__simps_I2_J, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % one_poly_eq_simps(2)
thf(fact_187_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_188_one__poly__eq__simps_I2_J, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % one_poly_eq_simps(2)
thf(fact_189_monom__eq__1, axiom,
    (((monom_nat @ one_one_nat @ zero_zero_nat) = one_one_poly_nat))). % monom_eq_1
thf(fact_190_lead__coeff__1, axiom,
    (((coeff_nat @ one_one_poly_nat @ (degree_nat @ one_one_poly_nat)) = one_one_nat))). % lead_coeff_1
thf(fact_191_degree__reflect__poly__eq, axiom,
    ((![P : poly_a]: ((~ (((coeff_a @ P @ zero_zero_nat) = zero_zero_a))) => ((degree_a @ (reflect_poly_a @ P)) = (degree_a @ P)))))). % degree_reflect_poly_eq
thf(fact_192_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_a]: ((~ (((coeff_poly_a @ P @ zero_zero_nat) = zero_zero_poly_a))) => ((degree_poly_a @ (reflect_poly_poly_a @ P)) = (degree_poly_a @ P)))))). % degree_reflect_poly_eq
thf(fact_193_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_194_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_195_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_poly_a]: ((~ (((coeff_poly_poly_a @ P @ zero_zero_nat) = zero_z2096148049poly_a))) => ((degree_poly_poly_a @ (reflec581648976poly_a @ P)) = (degree_poly_poly_a @ P)))))). % degree_reflect_poly_eq
thf(fact_196_poly__reflect__poly__0, axiom,
    ((![P : poly_a]: ((poly_a2 @ (reflect_poly_a @ P) @ zero_zero_a) = (coeff_a @ P @ (degree_a @ P)))))). % poly_reflect_poly_0
thf(fact_197_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_a]: ((poly_poly_a2 @ (reflect_poly_poly_a @ P) @ zero_zero_poly_a) = (coeff_poly_a @ P @ (degree_poly_a @ P)))))). % poly_reflect_poly_0
thf(fact_198_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_199_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_200_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_poly_a]: ((poly_poly_poly_a2 @ (reflec581648976poly_a @ P) @ zero_z2096148049poly_a) = (coeff_poly_poly_a @ P @ (degree_poly_poly_a @ P)))))). % poly_reflect_poly_0
thf(fact_201_monom_Orep__eq, axiom,
    ((![X : a, Xa : nat]: ((coeff_a @ (monom_a @ X @ Xa)) = (^[N2 : nat]: (if_a @ (Xa = N2) @ X @ zero_zero_a)))))). % monom.rep_eq
thf(fact_202_monom_Orep__eq, axiom,
    ((![X : poly_a, Xa : nat]: ((coeff_poly_a @ (monom_poly_a @ X @ Xa)) = (^[N2 : nat]: (if_poly_a @ (Xa = N2) @ X @ zero_zero_poly_a)))))). % monom.rep_eq
thf(fact_203_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_204_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_205_monom_Orep__eq, axiom,
    ((![X : poly_poly_a, Xa : nat]: ((coeff_poly_poly_a @ (monom_poly_poly_a @ X @ Xa)) = (^[N2 : nat]: (if_poly_poly_a @ (Xa = N2) @ X @ zero_z2096148049poly_a)))))). % monom.rep_eq
thf(fact_206_monom__eq__1__iff, axiom,
    ((![C : nat, N : nat]: (((monom_nat @ C @ N) = one_one_poly_nat) = (((C = one_one_nat)) & ((N = zero_zero_nat))))))). % monom_eq_1_iff
thf(fact_207_degree__monom__eq, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => ((degree_a @ (monom_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_208_degree__monom__eq, axiom,
    ((![A : poly_a, N : nat]: ((~ ((A = zero_zero_poly_a))) => ((degree_poly_a @ (monom_poly_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_209_degree__monom__eq, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => ((degree_nat @ (monom_nat @ A @ N)) = N))))). % degree_monom_eq
thf(fact_210_degree__monom__eq, axiom,
    ((![A : poly_nat, N : nat]: ((~ ((A = zero_zero_poly_nat))) => ((degree_poly_nat @ (monom_poly_nat @ A @ N)) = N))))). % degree_monom_eq
thf(fact_211_degree__monom__eq, axiom,
    ((![A : poly_poly_a, N : nat]: ((~ ((A = zero_z2096148049poly_a))) => ((degree_poly_poly_a @ (monom_poly_poly_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_212_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_213_leading__coeff__neq__0, axiom,
    ((![P : poly_poly_nat]: ((~ ((P = zero_z1059985641ly_nat))) => (~ (((coeff_poly_nat @ P @ (degree_poly_nat @ P)) = zero_zero_poly_nat))))))). % leading_coeff_neq_0
thf(fact_214_leading__coeff__neq__0, axiom,
    ((![P : poly_poly_poly_a]: ((~ ((P = zero_z2064990175poly_a))) => (~ (((coeff_poly_poly_a @ P @ (degree_poly_poly_a @ P)) = zero_z2096148049poly_a))))))). % leading_coeff_neq_0
thf(fact_215_leading__coeff__neq__0, axiom,
    ((![P : poly_a]: ((~ ((P = zero_zero_poly_a))) => (~ (((coeff_a @ P @ (degree_a @ P)) = zero_zero_a))))))). % leading_coeff_neq_0
thf(fact_216_leading__coeff__neq__0, axiom,
    ((![P : poly_nat]: ((~ ((P = zero_zero_poly_nat))) => (~ (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat))))))). % leading_coeff_neq_0
thf(fact_217_leading__coeff__neq__0, axiom,
    ((![P : poly_poly_a]: ((~ ((P = zero_z2096148049poly_a))) => (~ (((coeff_poly_a @ P @ (degree_poly_a @ P)) = zero_zero_poly_a))))))). % leading_coeff_neq_0
thf(fact_218_monom__eq__iff_H, axiom,
    ((![C : a, N : nat, D : a, M : nat]: (((monom_a @ C @ N) = (monom_a @ D @ M)) = (((C = D)) & ((((C = zero_zero_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_219_monom__eq__iff_H, axiom,
    ((![C : poly_a, N : nat, D : poly_a, M : nat]: (((monom_poly_a @ C @ N) = (monom_poly_a @ D @ M)) = (((C = D)) & ((((C = zero_zero_poly_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_220_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_221_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_222_monom__eq__iff_H, axiom,
    ((![C : poly_poly_a, N : nat, D : poly_poly_a, M : nat]: (((monom_poly_poly_a @ C @ N) = (monom_poly_poly_a @ D @ M)) = (((C = D)) & ((((C = zero_z2096148049poly_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_223_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_224_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_225_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_226_nat__dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ one_one_nat) = (M = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_227_dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ (suc @ zero_zero_nat)) = (M = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_228_dvd__1__left, axiom,
    ((![K : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K)))). % dvd_1_left
thf(fact_229_Suc__inject, axiom,
    ((![X : nat, Y3 : nat]: (((suc @ X) = (suc @ Y3)) => (X = Y3))))). % Suc_inject
thf(fact_230_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_231_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_232_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_233_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_234_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_235_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N3 : nat]: ((P2 @ N3) => (P2 @ (suc @ N3)))) => (P2 @ N)))))). % nat_induct
thf(fact_236_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P2 @ X3 @ zero_zero_nat)) => ((![Y4 : nat]: (P2 @ zero_zero_nat @ (suc @ Y4))) => ((![X3 : nat, Y4 : nat]: ((P2 @ X3 @ Y4) => (P2 @ (suc @ X3) @ (suc @ Y4)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_237_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N3 : nat]: ((P2 @ (suc @ N3)) => (P2 @ N3))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_238_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_239_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_240_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_241_old_Onat_Oexhaust, axiom,
    ((![Y3 : nat]: ((~ ((Y3 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y3 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_242_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_243_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc

% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T, axiom,
    ((![X : a, Y3 : a]: ((if_a @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001tf__a_T, axiom,
    ((![X : a, Y3 : a]: ((if_a @ $true @ X @ Y3) = X)))).
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__Polynomial__Opoly_Itf__a_J_T, axiom,
    ((![X : poly_a, Y3 : poly_a]: ((if_poly_a @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_Itf__a_J_T, axiom,
    ((![X : poly_a, Y3 : poly_a]: ((if_poly_a @ $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_3_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![X : poly_poly_a, Y3 : poly_poly_a]: ((if_poly_poly_a @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![X : poly_poly_a, Y3 : poly_poly_a]: ((if_poly_poly_a @ $true @ X @ Y3) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((p = (pCons_a @ (poly_a2 @ p @ zero_zero_a) @ zero_zero_poly_a)))).
