% 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_1051__5377808_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:32.331

% Could-be-implicit typings (6)
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 (48)
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_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_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_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_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_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_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_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_001tf__a, type,
    is_zero_a : poly_a > $o).
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_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_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_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_It__Nat__Onat_J, type,
    poly_cutoff_poly_nat : nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_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_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_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_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (195)
thf(fact_0_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X : a]: (![Y : a]: ((F @ X) = (F @ Y)))))))). % constant_def
thf(fact_1_poly__0, axiom,
    ((![X2 : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X2) = zero_zero_poly_nat)))). % poly_0
thf(fact_2_poly__0, axiom,
    ((![X2 : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X2) = zero_zero_poly_a)))). % poly_0
thf(fact_3_poly__0, axiom,
    ((![X2 : a]: ((poly_a2 @ zero_zero_poly_a @ X2) = zero_zero_a)))). % poly_0
thf(fact_4_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_5_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_a]: ((![X : poly_a]: ((poly_poly_a2 @ P @ X) = zero_zero_poly_a)) = (P = zero_z2096148049poly_a))))). % poly_all_0_iff_0
thf(fact_6_poly__all__0__iff__0, axiom,
    ((![P : poly_a]: ((![X : a]: ((poly_a2 @ P @ X) = zero_zero_a)) = (P = zero_zero_poly_a))))). % poly_all_0_iff_0
thf(fact_7_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_8_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_9_zero__reorient, axiom,
    ((![X2 : poly_nat]: ((zero_zero_poly_nat = X2) = (X2 = zero_zero_poly_nat))))). % zero_reorient
thf(fact_10_zero__reorient, axiom,
    ((![X2 : poly_a]: ((zero_zero_poly_a = X2) = (X2 = zero_zero_poly_a))))). % zero_reorient
thf(fact_11_zero__reorient, axiom,
    ((![X2 : a]: ((zero_zero_a = X2) = (X2 = zero_zero_a))))). % zero_reorient
thf(fact_12_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_13_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_14_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_15_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_16_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_17_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_18_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_19_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_20_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_21_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_22_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_23_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_24_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_25_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_26_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_27_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_28_is__zero__null, axiom,
    ((is_zero_nat = (^[P2 : poly_nat]: (P2 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_29_is__zero__null, axiom,
    ((is_zero_a = (^[P2 : poly_a]: (P2 = zero_zero_poly_a))))). % is_zero_null
thf(fact_30_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_31_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_32_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_33_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_34_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_35_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_36_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_37_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_38_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_39_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_40_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_41_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_42_order__degree, axiom,
    ((![P : poly_a, A : a]: ((~ ((P = zero_zero_poly_a))) => (ord_less_eq_nat @ (order_a @ A @ P) @ (degree_a @ P)))))). % order_degree
thf(fact_43_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_44_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_45_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_46_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_47_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_48_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_49_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_50_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_51_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_52_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_53_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_72_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_73_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_74_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_75_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_76_Nat_Oex__has__greatest__nat, axiom,
    ((![P3 : nat > $o, K : nat, B : nat]: ((P3 @ K) => ((![Y2 : nat]: ((P3 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (?[X3 : nat]: ((P3 @ X3) & (![Y3 : nat]: ((P3 @ Y3) => (ord_less_eq_nat @ Y3 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_77_degree__reflect__poly__le, axiom,
    ((![P : poly_a]: (ord_less_eq_nat @ (degree_a @ (reflect_poly_a @ P)) @ (degree_a @ P))))). % degree_reflect_poly_le
thf(fact_78_le__degree, axiom,
    ((![P : poly_nat, N : nat]: ((~ (((coeff_nat @ P @ N) = zero_zero_nat))) => (ord_less_eq_nat @ N @ (degree_nat @ P)))))). % le_degree
thf(fact_79_le__degree, axiom,
    ((![P : poly_poly_nat, N : nat]: ((~ (((coeff_poly_nat @ P @ N) = zero_zero_poly_nat))) => (ord_less_eq_nat @ N @ (degree_poly_nat @ P)))))). % le_degree
thf(fact_80_le__degree, axiom,
    ((![P : poly_poly_a, N : nat]: ((~ (((coeff_poly_a @ P @ N) = zero_zero_poly_a))) => (ord_less_eq_nat @ N @ (degree_poly_a @ P)))))). % le_degree
thf(fact_81_le__degree, axiom,
    ((![P : poly_a, N : nat]: ((~ (((coeff_a @ P @ N) = zero_zero_a))) => (ord_less_eq_nat @ N @ (degree_a @ P)))))). % le_degree
thf(fact_82_zero__le, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X2)))). % zero_le
thf(fact_83_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_84_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_85_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_86_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_87_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_88_zero__poly_Orep__eq, axiom,
    (((coeff_poly_a @ zero_z2096148049poly_a) = (^[Uu : nat]: zero_zero_poly_a)))). % zero_poly.rep_eq
thf(fact_89_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_90_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_order__refl, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ X2 @ X2)))). % order_refl
thf(fact_100_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_101_eq__zero__or__degree__less, axiom,
    ((![P : poly_nat, N : nat]: ((ord_less_eq_nat @ (degree_nat @ P) @ N) => (((coeff_nat @ P @ N) = zero_zero_nat) => ((P = zero_zero_poly_nat) | (ord_less_nat @ (degree_nat @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_102_eq__zero__or__degree__less, axiom,
    ((![P : poly_poly_nat, N : nat]: ((ord_less_eq_nat @ (degree_poly_nat @ P) @ N) => (((coeff_poly_nat @ P @ N) = zero_zero_poly_nat) => ((P = zero_z1059985641ly_nat) | (ord_less_nat @ (degree_poly_nat @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_103_eq__zero__or__degree__less, axiom,
    ((![P : poly_poly_a, N : nat]: ((ord_less_eq_nat @ (degree_poly_a @ P) @ N) => (((coeff_poly_a @ P @ N) = zero_zero_poly_a) => ((P = zero_z2096148049poly_a) | (ord_less_nat @ (degree_poly_a @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_104_eq__zero__or__degree__less, axiom,
    ((![P : poly_a, N : nat]: ((ord_less_eq_nat @ (degree_a @ P) @ N) => (((coeff_a @ P @ N) = zero_zero_a) => ((P = zero_zero_poly_a) | (ord_less_nat @ (degree_a @ P) @ N))))))). % eq_zero_or_degree_less
thf(fact_105_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_106_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_107_psize__def, axiom,
    ((fundam1567013434ze_nat = (^[P2 : poly_nat]: (if_nat @ (P2 = zero_zero_poly_nat) @ zero_zero_nat @ (suc @ (degree_nat @ P2))))))). % psize_def
thf(fact_108_psize__def, axiom,
    ((fundam247907092size_a = (^[P2 : poly_a]: (if_nat @ (P2 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ P2))))))). % psize_def
thf(fact_109_degree__le, axiom,
    ((![N : nat, P : poly_nat]: ((![I2 : nat]: ((ord_less_nat @ N @ I2) => ((coeff_nat @ P @ I2) = zero_zero_nat))) => (ord_less_eq_nat @ (degree_nat @ P) @ N))))). % degree_le
thf(fact_110_degree__le, axiom,
    ((![N : nat, P : poly_poly_nat]: ((![I2 : nat]: ((ord_less_nat @ N @ I2) => ((coeff_poly_nat @ P @ I2) = zero_zero_poly_nat))) => (ord_less_eq_nat @ (degree_poly_nat @ P) @ N))))). % degree_le
thf(fact_111_degree__le, axiom,
    ((![N : nat, P : poly_poly_a]: ((![I2 : nat]: ((ord_less_nat @ N @ I2) => ((coeff_poly_a @ P @ I2) = zero_zero_poly_a))) => (ord_less_eq_nat @ (degree_poly_a @ P) @ N))))). % degree_le
thf(fact_112_degree__le, axiom,
    ((![N : nat, P : poly_a]: ((![I2 : nat]: ((ord_less_nat @ N @ I2) => ((coeff_a @ P @ I2) = zero_zero_a))) => (ord_less_eq_nat @ (degree_a @ P) @ N))))). % degree_le
thf(fact_113_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_114_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_115_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = (coeff_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_nat @ (poly_cutoff_nat @ N @ P) @ K) = zero_zero_nat)))))). % coeff_poly_cutoff
thf(fact_116_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_nat]: (((ord_less_nat @ K @ N) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = (coeff_poly_nat @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_nat @ (poly_cutoff_poly_nat @ N @ P) @ K) = zero_zero_poly_nat)))))). % coeff_poly_cutoff
thf(fact_117_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_poly_a]: (((ord_less_nat @ K @ N) => ((coeff_poly_a @ (poly_cutoff_poly_a @ N @ P) @ K) = (coeff_poly_a @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_poly_a @ (poly_cutoff_poly_a @ N @ P) @ K) = zero_zero_poly_a)))))). % coeff_poly_cutoff
thf(fact_118_coeff__poly__cutoff, axiom,
    ((![K : nat, N : nat, P : poly_a]: (((ord_less_nat @ K @ N) => ((coeff_a @ (poly_cutoff_a @ N @ P) @ K) = (coeff_a @ P @ K))) & ((~ ((ord_less_nat @ K @ N))) => ((coeff_a @ (poly_cutoff_a @ N @ P) @ K) = zero_zero_a)))))). % coeff_poly_cutoff
thf(fact_119_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_120_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_121_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_122_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_123_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_124_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_125_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_126_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_127_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_128_Suc__le__mono, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N) @ (suc @ M)) = (ord_less_eq_nat @ N @ M))))). % Suc_le_mono
thf(fact_129_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_130_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_131_degree__1, axiom,
    (((degree_a @ one_one_poly_a) = zero_zero_nat))). % degree_1
thf(fact_132_poly__1, axiom,
    ((![X2 : a]: ((poly_a2 @ one_one_poly_a @ X2) = one_one_a)))). % poly_1
thf(fact_133_poly__1, axiom,
    ((![X2 : nat]: ((poly_nat2 @ one_one_poly_nat @ X2) = one_one_nat)))). % poly_1
thf(fact_134_order__1__eq__0, axiom,
    ((![X2 : a]: ((order_a @ X2 @ one_one_poly_a) = zero_zero_nat)))). % order_1_eq_0
thf(fact_135_lead__coeff__1, axiom,
    (((coeff_a @ one_one_poly_a @ (degree_a @ one_one_poly_a)) = one_one_a))). % lead_coeff_1
thf(fact_136_Suc__leI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ (suc @ M) @ N))))). % Suc_leI
thf(fact_137_Suc__le__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) = (ord_less_nat @ M @ N))))). % Suc_le_eq
thf(fact_138_dec__induct, axiom,
    ((![I : nat, J : nat, P3 : nat > $o]: ((ord_less_eq_nat @ I @ J) => ((P3 @ I) => ((![N2 : nat]: ((ord_less_eq_nat @ I @ N2) => ((ord_less_nat @ N2 @ J) => ((P3 @ N2) => (P3 @ (suc @ N2)))))) => (P3 @ J))))))). % dec_induct
thf(fact_139_inc__induct, axiom,
    ((![I : nat, J : nat, P3 : nat > $o]: ((ord_less_eq_nat @ I @ J) => ((P3 @ J) => ((![N2 : nat]: ((ord_less_eq_nat @ I @ N2) => ((ord_less_nat @ N2 @ J) => ((P3 @ (suc @ N2)) => (P3 @ N2))))) => (P3 @ I))))))). % inc_induct
thf(fact_140_Suc__le__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_le_lessD
thf(fact_141_le__less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % le_less_Suc_eq
thf(fact_142_less__Suc__eq__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (ord_less_eq_nat @ M @ N))))). % less_Suc_eq_le
thf(fact_143_less__eq__Suc__le, axiom,
    ((ord_less_nat = (^[N3 : nat]: (ord_less_eq_nat @ (suc @ N3)))))). % less_eq_Suc_le
thf(fact_144_le__imp__less__Suc, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % le_imp_less_Suc
thf(fact_145_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_146_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_147_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_148_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_nat @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_149_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_150_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_151_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_152_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_153_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_154_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_155_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_156_not__le__imp__less, axiom,
    ((![Y4 : nat, X2 : nat]: ((~ ((ord_less_eq_nat @ Y4 @ X2))) => (ord_less_nat @ X2 @ Y4))))). % not_le_imp_less
thf(fact_157_less__le__not__le, axiom,
    ((ord_less_nat = (^[X : nat]: (^[Y : nat]: (((ord_less_eq_nat @ X @ Y)) & ((~ ((ord_less_eq_nat @ Y @ X)))))))))). % less_le_not_le
thf(fact_158_le__imp__less__or__eq, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_eq_nat @ X2 @ Y4) => ((ord_less_nat @ X2 @ Y4) | (X2 = Y4)))))). % le_imp_less_or_eq
thf(fact_159_le__less__linear, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_eq_nat @ X2 @ Y4) | (ord_less_nat @ Y4 @ X2))))). % le_less_linear
thf(fact_160_less__le__trans, axiom,
    ((![X2 : nat, Y4 : nat, Z : nat]: ((ord_less_nat @ X2 @ Y4) => ((ord_less_eq_nat @ Y4 @ Z) => (ord_less_nat @ X2 @ Z)))))). % less_le_trans
thf(fact_161_le__less__trans, axiom,
    ((![X2 : nat, Y4 : nat, Z : nat]: ((ord_less_eq_nat @ X2 @ Y4) => ((ord_less_nat @ Y4 @ Z) => (ord_less_nat @ X2 @ Z)))))). % le_less_trans
thf(fact_162_less__imp__le, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_nat @ X2 @ Y4) => (ord_less_eq_nat @ X2 @ Y4))))). % less_imp_le
thf(fact_163_antisym__conv2, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_eq_nat @ X2 @ Y4) => ((~ ((ord_less_nat @ X2 @ Y4))) = (X2 = Y4)))))). % antisym_conv2
thf(fact_164_antisym__conv1, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X2 @ Y4))) => ((ord_less_eq_nat @ X2 @ Y4) = (X2 = Y4)))))). % antisym_conv1
thf(fact_165_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_166_not__less, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X2 @ Y4))) = (ord_less_eq_nat @ Y4 @ X2))))). % not_less
thf(fact_167_not__le, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X2 @ Y4))) = (ord_less_nat @ Y4 @ X2))))). % not_le
thf(fact_168_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_le_subst2
thf(fact_169_order__less__le__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_le_subst1
thf(fact_170_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_le_less_subst2
thf(fact_171_order__le__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_le_less_subst1
thf(fact_172_less__le, axiom,
    ((ord_less_nat = (^[X : nat]: (^[Y : nat]: (((ord_less_eq_nat @ X @ Y)) & ((~ ((X = Y)))))))))). % less_le
thf(fact_173_le__less, axiom,
    ((ord_less_eq_nat = (^[X : nat]: (^[Y : nat]: (((ord_less_nat @ X @ Y)) | ((X = Y)))))))). % le_less
thf(fact_174_leI, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X2 @ Y4))) => (ord_less_eq_nat @ Y4 @ X2))))). % leI
thf(fact_175_leD, axiom,
    ((![Y4 : nat, X2 : nat]: ((ord_less_eq_nat @ Y4 @ X2) => (~ ((ord_less_nat @ X2 @ Y4))))))). % leD
thf(fact_176_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_177_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_178_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_179_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_180_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_181_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_182_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y2)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_183_gt__ex, axiom,
    ((![X2 : nat]: (?[X_1 : nat]: (ord_less_nat @ X2 @ X_1))))). % gt_ex
thf(fact_184_neqE, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((X2 = Y4))) => ((~ ((ord_less_nat @ X2 @ Y4))) => (ord_less_nat @ Y4 @ X2)))))). % neqE
thf(fact_185_neq__iff, axiom,
    ((![X2 : nat, Y4 : nat]: ((~ ((X2 = Y4))) = (((ord_less_nat @ X2 @ Y4)) | ((ord_less_nat @ Y4 @ X2))))))). % neq_iff
thf(fact_186_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_187_less__imp__neq, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_nat @ X2 @ Y4) => (~ ((X2 = Y4))))))). % less_imp_neq
thf(fact_188_less__asym, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_nat @ X2 @ Y4) => (~ ((ord_less_nat @ Y4 @ X2))))))). % less_asym
thf(fact_189_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_190_less__trans, axiom,
    ((![X2 : nat, Y4 : nat, Z : nat]: ((ord_less_nat @ X2 @ Y4) => ((ord_less_nat @ Y4 @ Z) => (ord_less_nat @ X2 @ Z)))))). % less_trans
thf(fact_191_less__linear, axiom,
    ((![X2 : nat, Y4 : nat]: ((ord_less_nat @ X2 @ Y4) | ((X2 = Y4) | (ord_less_nat @ Y4 @ X2)))))). % less_linear
thf(fact_192_less__irrefl, axiom,
    ((![X2 : nat]: (~ ((ord_less_nat @ X2 @ X2)))))). % less_irrefl
thf(fact_193_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_194_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P3 : $o]: ((P3 = $true) | (P3 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y4 : nat]: ((if_nat @ $false @ X2 @ Y4) = Y4)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y4 : nat]: ((if_nat @ $true @ X2 @ Y4) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((fundam236050252nt_a_a @ (poly_a2 @ p)) = ((degree_a @ p) = zero_zero_nat)))).
