% 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_1060__5377856_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:38.201

% 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 (65)
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_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_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__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_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_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_v_p, type,
    p : poly_a).

% Relevant facts (240)
thf(fact_0_that, axiom,
    ((fundam236050252nt_a_a @ (poly_a2 @ p)))). % that
thf(fact_1__092_060open_062p_A_061_A_091_058poly_Ap_A_I0_058_058_Ha_J_058_093_092_060close_062, axiom,
    ((p = (pCons_a @ (poly_a2 @ p @ zero_zero_a) @ zero_zero_poly_a)))). % \<open>p = [:poly p (0::'a):]\<close>
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_th, axiom,
    (((poly_a2 @ p) = (poly_a2 @ (pCons_a @ (poly_a2 @ p @ zero_zero_a) @ zero_zero_poly_a))))). % th
thf(fact_4_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_5_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_9_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_10_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_11_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_12_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_13_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_48_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A2 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_49_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A2 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_50_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_51_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_52_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_53_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X2 : a]: (![Y : a]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_54_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_55_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_56_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_57_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_58_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_59_offset__poly__0, axiom,
    ((![H : poly_a]: ((fundam1343031620poly_a @ zero_z2096148049poly_a @ H) = zero_z2096148049poly_a)))). % offset_poly_0
thf(fact_60_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_61_degree__eq__zeroE, axiom,
    ((![P : poly_a]: (((degree_a @ P) = zero_zero_nat) => (~ ((![A2 : a]: (~ ((P = (pCons_a @ A2 @ zero_zero_poly_a))))))))))). % degree_eq_zeroE
thf(fact_62_degree__eq__zeroE, axiom,
    ((![P : poly_nat]: (((degree_nat @ P) = zero_zero_nat) => (~ ((![A2 : nat]: (~ ((P = (pCons_nat @ A2 @ zero_zero_poly_nat))))))))))). % degree_eq_zeroE
thf(fact_63_degree__eq__zeroE, axiom,
    ((![P : poly_poly_a]: (((degree_poly_a @ P) = zero_zero_nat) => (~ ((![A2 : poly_a]: (~ ((P = (pCons_poly_a @ A2 @ zero_z2096148049poly_a))))))))))). % degree_eq_zeroE
thf(fact_64_degree__pCons__0, axiom,
    ((![A : a]: ((degree_a @ (pCons_a @ A @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_65_degree__pCons__0, axiom,
    ((![A : nat]: ((degree_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_66_degree__pCons__0, axiom,
    ((![A : poly_a]: ((degree_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_67_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_68_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_69_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_70_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_71_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_72_pderiv_Ocases, axiom,
    ((![X : poly_a]: (~ ((![A2 : a, P3 : poly_a]: (~ ((X = (pCons_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_73_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A2 : nat, P3 : poly_nat]: (~ ((X = (pCons_nat @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_74_pderiv_Ocases, axiom,
    ((![X : poly_poly_a]: (~ ((![A2 : poly_a, P3 : poly_poly_a]: (~ ((X = (pCons_poly_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_75_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_76_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_77_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_78_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_79_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_80_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_81_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_82_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_83_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_degree__pCons__eq__if, axiom,
    ((![P : poly_a, A : a]: (((P = zero_zero_poly_a) => ((degree_a @ (pCons_a @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P)) = (suc @ (degree_a @ P)))))))). % degree_pCons_eq_if
thf(fact_93_degree__pCons__eq__if, axiom,
    ((![P : poly_nat, A : nat]: (((P = zero_zero_poly_nat) => ((degree_nat @ (pCons_nat @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A @ P)) = (suc @ (degree_nat @ P)))))))). % degree_pCons_eq_if
thf(fact_94_degree__pCons__eq__if, axiom,
    ((![P : poly_poly_a, A : poly_a]: (((P = zero_z2096148049poly_a) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = (suc @ (degree_poly_a @ P)))))))). % degree_pCons_eq_if
thf(fact_95_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_96_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_97_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_shift_0
thf(fact_98_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_99_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_100_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_101_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_102_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_103_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_104_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_105_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_106_reflect__poly__0, axiom,
    (((reflect_poly_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % reflect_poly_0
thf(fact_107_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_108_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_109_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_110_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_111_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_poly_a @ zero_z2064990175poly_a @ N) = zero_z2096148049poly_a)))). % coeff_0
thf(fact_112_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_113_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_114_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_115_coeff__pCons__0, axiom,
    ((![A : a, P : poly_a]: ((coeff_a @ (pCons_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_116_coeff__pCons__0, axiom,
    ((![A : nat, P : poly_nat]: ((coeff_nat @ (pCons_nat @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_117_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_118_coeff__pCons__Suc, axiom,
    ((![A : a, P : poly_a, N : nat]: ((coeff_a @ (pCons_a @ A @ P) @ (suc @ N)) = (coeff_a @ P @ N))))). % coeff_pCons_Suc
thf(fact_119_coeff__pCons__Suc, axiom,
    ((![A : nat, P : poly_nat, N : nat]: ((coeff_nat @ (pCons_nat @ A @ P) @ (suc @ N)) = (coeff_nat @ P @ N))))). % coeff_pCons_Suc
thf(fact_120_coeff__pCons__Suc, axiom,
    ((![A : poly_a, P : poly_poly_a, N : nat]: ((coeff_poly_a @ (pCons_poly_a @ A @ P) @ (suc @ N)) = (coeff_poly_a @ P @ N))))). % coeff_pCons_Suc
thf(fact_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_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_coeff__0__reflect__poly, axiom,
    ((![P : poly_a]: ((coeff_a @ (reflect_poly_a @ P) @ zero_zero_nat) = (coeff_a @ P @ (degree_a @ P)))))). % coeff_0_reflect_poly
thf(fact_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_154_zero__poly_Orep__eq, axiom,
    (((coeff_poly_poly_a @ zero_z2064990175poly_a) = (^[Uu : nat]: zero_z2096148049poly_a)))). % zero_poly.rep_eq
thf(fact_155_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_156_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_157_zero__poly_Orep__eq, axiom,
    (((coeff_poly_a @ zero_z2096148049poly_a) = (^[Uu : nat]: zero_zero_poly_a)))). % zero_poly.rep_eq
thf(fact_158_poly__0__coeff__0, axiom,
    ((![P : poly_a]: ((poly_a2 @ P @ zero_zero_a) = (coeff_a @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_159_poly__0__coeff__0, axiom,
    ((![P : poly_poly_a]: ((poly_poly_a2 @ P @ zero_zero_poly_a) = (coeff_poly_a @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_160_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_161_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_162_poly__0__coeff__0, axiom,
    ((![P : poly_poly_poly_a]: ((poly_poly_poly_a2 @ P @ zero_z2096148049poly_a) = (coeff_poly_poly_a @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_degree__pCons__eq, axiom,
    ((![P : poly_a, A : a]: ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P)) = (suc @ (degree_a @ P))))))). % degree_pCons_eq
thf(fact_172_degree__pCons__eq, axiom,
    ((![P : poly_nat, A : nat]: ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A @ P)) = (suc @ (degree_nat @ P))))))). % degree_pCons_eq
thf(fact_173_degree__pCons__eq, axiom,
    ((![P : poly_poly_a, A : poly_a]: ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = (suc @ (degree_poly_a @ P))))))). % degree_pCons_eq
thf(fact_174_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_175_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_176_psize__def, axiom,
    ((fundam247907092size_a = (^[P4 : poly_a]: (if_nat @ (P4 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ P4))))))). % psize_def
thf(fact_177_psize__def, axiom,
    ((fundam1567013434ze_nat = (^[P4 : poly_nat]: (if_nat @ (P4 = zero_zero_poly_nat) @ zero_zero_nat @ (suc @ (degree_nat @ P4))))))). % psize_def
thf(fact_178_psize__def, axiom,
    ((fundam1032801442poly_a = (^[P4 : poly_poly_a]: (if_nat @ (P4 = zero_z2096148049poly_a) @ zero_zero_nat @ (suc @ (degree_poly_a @ P4))))))). % psize_def
thf(fact_179_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_180_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_181_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_182_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_183_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_184_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_185_Suc__inject, axiom,
    ((![X : nat, Y3 : nat]: (((suc @ X) = (suc @ Y3)) => (X = Y3))))). % Suc_inject
thf(fact_186_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_187_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_188_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_189_old_Onat_Oexhaust, axiom,
    ((![Y3 : nat]: ((~ ((Y3 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y3 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_190_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_191_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_192_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_193_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_194_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M2 : 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 @ M2 @ N))))))). % diff_induct
thf(fact_195_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((P2 @ N2) => (P2 @ (suc @ N2)))) => (P2 @ N)))))). % nat_induct
thf(fact_196_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_197_exists__least__lemma, axiom,
    ((![P2 : nat > $o]: ((~ ((P2 @ zero_zero_nat))) => ((?[X_1 : nat]: (P2 @ X_1)) => (?[N2 : nat]: ((~ ((P2 @ N2))) & (P2 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_198_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_199_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_200_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_201_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_202_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_203_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_204_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_205_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_206_pcompose__0, axiom,
    ((![Q : poly_poly_a]: ((pcompose_poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % pcompose_0
thf(fact_207_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_208_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_209_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_210_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_211_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_212_Suc__mono, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ (suc @ M2) @ (suc @ N)))))). % Suc_mono
thf(fact_213_Suc__less__eq, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ (suc @ N)) = (ord_less_nat @ M2 @ N))))). % Suc_less_eq
thf(fact_214_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_215_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_216_nat__induct__non__zero, axiom,
    ((![N : nat, P2 : nat > $o]: ((ord_less_nat @ zero_zero_nat @ N) => ((P2 @ one_one_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((P2 @ N2) => (P2 @ (suc @ N2))))) => (P2 @ N))))))). % nat_induct_non_zero
thf(fact_217_linorder__neqE__nat, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_neqE_nat
thf(fact_218_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P2 @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P2 @ M3))))))) => (P2 @ N))))). % infinite_descent
thf(fact_219_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N2) => (P2 @ M3))) => (P2 @ N2))) => (P2 @ N))))). % nat_less_induct
thf(fact_220_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_221_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_222_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_223_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_224_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_225_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_226_Suc__lessD, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ N) => (ord_less_nat @ M2 @ N))))). % Suc_lessD
thf(fact_227_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_228_Suc__lessI, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => ((~ (((suc @ M2) = N))) => (ord_less_nat @ (suc @ M2) @ N)))))). % Suc_lessI
thf(fact_229_less__SucE, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ (suc @ N)) => ((~ ((ord_less_nat @ M2 @ N))) => (M2 = N)))))). % less_SucE
thf(fact_230_less__SucI, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ M2 @ (suc @ N)))))). % less_SucI
thf(fact_231_Ex__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((?[I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) & ((P2 @ I2)))) = (((P2 @ N)) | ((?[I2 : nat]: (((ord_less_nat @ I2 @ N)) & ((P2 @ I2)))))))))). % Ex_less_Suc
thf(fact_232_less__Suc__eq, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ (suc @ N)) = (((ord_less_nat @ M2 @ N)) | ((M2 = N))))))). % less_Suc_eq
thf(fact_233_not__less__eq, axiom,
    ((![M2 : nat, N : nat]: ((~ ((ord_less_nat @ M2 @ N))) = (ord_less_nat @ N @ (suc @ M2)))))). % not_less_eq
thf(fact_234_All__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((![I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) => ((P2 @ I2)))) = (((P2 @ N)) & ((![I2 : nat]: (((ord_less_nat @ I2 @ N)) => ((P2 @ I2)))))))))). % All_less_Suc
thf(fact_235_Suc__less__eq2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ (suc @ N) @ M2) = (?[M4 : nat]: (((M2 = (suc @ M4))) & ((ord_less_nat @ N @ M4)))))))). % Suc_less_eq2
thf(fact_236_less__antisym, axiom,
    ((![N : nat, M2 : nat]: ((~ ((ord_less_nat @ N @ M2))) => ((ord_less_nat @ N @ (suc @ M2)) => (M2 = N)))))). % less_antisym
thf(fact_237_Suc__less__SucD, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ (suc @ N)) => (ord_less_nat @ M2 @ N))))). % Suc_less_SucD
thf(fact_238_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_239_less__Suc__induct, axiom,
    ((![I : nat, J2 : nat, P2 : nat > nat > $o]: ((ord_less_nat @ I @ J2) => ((![I3 : nat]: (P2 @ I3 @ (suc @ I3))) => ((![I3 : nat, J : nat, K2 : nat]: ((ord_less_nat @ I3 @ J) => ((ord_less_nat @ J @ K2) => ((P2 @ I3 @ J) => ((P2 @ J @ K2) => (P2 @ I3 @ K2)))))) => (P2 @ I @ J2))))))). % less_Suc_induct

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
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)))).

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