% 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_620__5374076_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:36.969

% 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 (79)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Nat__Onat, type,
    fundam170929432ly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1343031620poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Nat__Onat, type,
    fundam1567013434ze_nat : poly_nat > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1032801442poly_a : poly_poly_a > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    one_one_poly_poly_a : poly_poly_a).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    zero_z2064990175poly_a : poly_poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_If_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    if_poly_nat : $o > poly_nat > poly_nat > poly_nat).
thf(sy_c_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    if_poly_poly_a : $o > poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_If_001t__Polynomial__Opoly_Itf__a_J, type,
    if_poly_a : $o > poly_a > poly_a > poly_a).
thf(sy_c_If_001tf__a, type,
    if_a : $o > a > a > a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    degree_poly_nat : poly_poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    degree_poly_poly_a : poly_poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_Itf__a_J, type,
    degree_poly_a : poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat, type,
    is_zero_nat : poly_nat > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_Itf__a_J, type,
    is_zero_poly_a : poly_poly_a > $o).
thf(sy_c_Polynomial_Ois__zero_001tf__a, type,
    is_zero_a : poly_a > $o).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat, type,
    monom_nat : nat > nat > poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    monom_poly_nat : poly_nat > nat > poly_poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    monom_poly_poly_a : poly_poly_a > nat > poly_poly_poly_a).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_Itf__a_J, type,
    monom_poly_a : poly_a > nat > poly_poly_a).
thf(sy_c_Polynomial_Omonom_001tf__a, type,
    monom_a : a > nat > poly_a).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    order_poly_poly_a : poly_poly_a > poly_poly_poly_a > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_Itf__a_J, type,
    order_poly_a : poly_a > poly_poly_a > nat).
thf(sy_c_Polynomial_Oorder_001tf__a, type,
    order_a : a > poly_a > nat).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    pCons_poly_poly_a : poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pcompose_poly_nat : poly_poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    pcompose_poly_poly_a : poly_poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_Itf__a_J, type,
    pcompose_poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_poly_a2 : poly_poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    coeff_poly_nat : poly_poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    coeff_poly_poly_a : poly_poly_poly_a > nat > poly_poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_Itf__a_J, type,
    coeff_poly_a : poly_poly_a > nat > poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001tf__a, type,
    coeff_a : poly_a > nat > a).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat, type,
    poly_cutoff_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_cutoff_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat, type,
    poly_shift_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_shift_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__shift_001tf__a, type,
    poly_shift_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    reflec581648976poly_a : poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    reflect_poly_poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001tf__a, type,
    reflect_poly_a : poly_a > poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat, type,
    synthetic_div_nat : poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_Itf__a_J, type,
    synthetic_div_poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_v_c____, type,
    c : a).
thf(sy_v_cs____, type,
    cs : poly_a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (242)
thf(fact_0_nc, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ p)))))). % nc
thf(fact_1_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X : a]: (![Y : a]: ((F @ X) = (F @ Y)))))))). % constant_def
thf(fact_2__092_060open_062_092_060forall_062z_O_Az_A_092_060noteq_062_A_I0_058_058_Ha_J_A_092_060longrightarrow_062_Apoly_Acs_Az_A_061_A_I0_058_058_Ha_J_092_060close_062, axiom,
    ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ cs @ Z) = zero_zero_a))))). % \<open>\<forall>z. z \<noteq> (0::'a) \<longrightarrow> poly cs z = (0::'a)\<close>
thf(fact_3__092_060open_062_092_060And_062y_Ax_O_Apoly_A_IpCons_Ac_Acs_J_Ax_A_061_Apoly_A_IpCons_Ac_Acs_J_Ay_092_060close_062, axiom,
    ((![X2 : a, Y2 : a]: ((poly_a2 @ (pCons_a @ c @ cs) @ X2) = (poly_a2 @ (pCons_a @ c @ cs) @ Y2))))). % \<open>\<And>y x. poly (pCons c cs) x = poly (pCons c cs) y\<close>
thf(fact_4_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_5_poly__0, axiom,
    ((![X2 : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X2) = zero_zero_poly_nat)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X2 : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X2) = zero_z2096148049poly_a)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X2 : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X2) = zero_zero_poly_a)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X2 : a]: ((poly_a2 @ zero_zero_poly_a @ X2) = zero_zero_a)))). % poly_0
thf(fact_10_pCons_Oprems, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ (pCons_a @ c @ cs))))))). % pCons.prems
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 : poly_a]: ((zero_zero_poly_a = X2) = (X2 = zero_zero_poly_a))))). % zero_reorient
thf(fact_13_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_14_zero__reorient, axiom,
    ((![X2 : poly_nat]: ((zero_zero_poly_nat = X2) = (X2 = zero_zero_poly_nat))))). % zero_reorient
thf(fact_15_zero__reorient, axiom,
    ((![X2 : poly_poly_a]: ((zero_z2096148049poly_a = X2) = (X2 = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_16_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_17_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_18_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_19_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_20_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_33_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_34_reflect__poly__0, axiom,
    (((reflect_poly_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % reflect_poly_0
thf(fact_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_pderiv_Ocases, axiom,
    ((![X2 : poly_a]: (~ ((![A2 : a, P3 : poly_a]: (~ ((X2 = (pCons_a @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_51_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_61_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_62_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_63_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_64_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_65_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_66_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_67_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_68_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_69_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_70_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_71_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_shift_0
thf(fact_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_84_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_85_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_86_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_87_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_poly_a @ zero_z2064990175poly_a @ N) = zero_z2096148049poly_a)))). % coeff_0
thf(fact_88_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_89_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_90_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_91_coeff__pCons__0, axiom,
    ((![A : a, P : poly_a]: ((coeff_a @ (pCons_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_92_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_93_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_94_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_95_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_96_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_97_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_98_zero__poly_Orep__eq, axiom,
    (((coeff_poly_poly_a @ zero_z2064990175poly_a) = (^[Uu : nat]: zero_z2096148049poly_a)))). % zero_poly.rep_eq
thf(fact_99_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_100_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_101_zero__poly_Orep__eq, axiom,
    (((coeff_poly_a @ zero_z2096148049poly_a) = (^[Uu : nat]: zero_zero_poly_a)))). % zero_poly.rep_eq
thf(fact_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_111_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_112_offset__poly__0, axiom,
    ((![H : poly_a]: ((fundam1343031620poly_a @ zero_z2096148049poly_a @ H) = zero_z2096148049poly_a)))). % offset_poly_0
thf(fact_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_monom__eq__const__iff, axiom,
    ((![C : poly_nat, N : nat, D : poly_nat]: (((monom_poly_nat @ C @ N) = (pCons_poly_nat @ D @ zero_z1059985641ly_nat)) = (((C = D)) & ((((C = zero_zero_poly_nat)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_133_monom__eq__const__iff, axiom,
    ((![C : poly_poly_a, N : nat, D : poly_poly_a]: (((monom_poly_poly_a @ C @ N) = (pCons_poly_poly_a @ D @ zero_z2064990175poly_a)) = (((C = D)) & ((((C = zero_z2096148049poly_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_134_monom__eq__const__iff, axiom,
    ((![C : a, N : nat, D : a]: (((monom_a @ C @ N) = (pCons_a @ D @ zero_zero_poly_a)) = (((C = D)) & ((((C = zero_zero_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_135_monom__eq__const__iff, axiom,
    ((![C : nat, N : nat, D : nat]: (((monom_nat @ C @ N) = (pCons_nat @ D @ zero_zero_poly_nat)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_136_monom__eq__const__iff, axiom,
    ((![C : poly_a, N : nat, D : poly_a]: (((monom_poly_a @ C @ N) = (pCons_poly_a @ D @ zero_z2096148049poly_a)) = (((C = D)) & ((((C = zero_zero_poly_a)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_137_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_138_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_139_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_poly_a @ N @ one_one_poly_poly_a) = one_one_poly_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_poly_a @ N @ one_one_poly_poly_a) = zero_z2096148049poly_a)))))). % poly_shift_1
thf(fact_140_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_141_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_142_pcompose__0, axiom,
    ((![Q : poly_poly_a]: ((pcompose_poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % pcompose_0
thf(fact_143_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_144_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_145_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_146_monom__eq__0__iff, axiom,
    ((![A : poly_nat, N : nat]: (((monom_poly_nat @ A @ N) = zero_z1059985641ly_nat) = (A = zero_zero_poly_nat))))). % monom_eq_0_iff
thf(fact_147_monom__eq__0__iff, axiom,
    ((![A : poly_poly_a, N : nat]: (((monom_poly_poly_a @ A @ N) = zero_z2064990175poly_a) = (A = zero_z2096148049poly_a))))). % monom_eq_0_iff
thf(fact_148_monom__eq__0__iff, axiom,
    ((![A : a, N : nat]: (((monom_a @ A @ N) = zero_zero_poly_a) = (A = zero_zero_a))))). % monom_eq_0_iff
thf(fact_149_monom__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((monom_nat @ A @ N) = zero_zero_poly_nat) = (A = zero_zero_nat))))). % monom_eq_0_iff
thf(fact_150_monom__eq__0__iff, axiom,
    ((![A : poly_a, N : nat]: (((monom_poly_a @ A @ N) = zero_z2096148049poly_a) = (A = zero_zero_poly_a))))). % monom_eq_0_iff
thf(fact_151_monom__eq__0, axiom,
    ((![N : nat]: ((monom_a @ zero_zero_a @ N) = zero_zero_poly_a)))). % monom_eq_0
thf(fact_152_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_a @ zero_zero_poly_a @ N) = zero_z2096148049poly_a)))). % monom_eq_0
thf(fact_153_monom__eq__0, axiom,
    ((![N : nat]: ((monom_nat @ zero_zero_nat @ N) = zero_zero_poly_nat)))). % monom_eq_0
thf(fact_154_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_nat @ zero_zero_poly_nat @ N) = zero_z1059985641ly_nat)))). % monom_eq_0
thf(fact_155_monom__eq__0, axiom,
    ((![N : nat]: ((monom_poly_poly_a @ zero_z2096148049poly_a @ N) = zero_z2064990175poly_a)))). % monom_eq_0
thf(fact_156_coeff__monom, axiom,
    ((![M : nat, N : nat, A : a]: (((M = N) => ((coeff_a @ (monom_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_a @ (monom_a @ A @ M) @ N) = zero_zero_a)))))). % coeff_monom
thf(fact_157_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_a]: (((M = N) => ((coeff_poly_a @ (monom_poly_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_a @ (monom_poly_a @ A @ M) @ N) = zero_zero_poly_a)))))). % coeff_monom
thf(fact_158_coeff__monom, axiom,
    ((![M : nat, N : nat, A : nat]: (((M = N) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = zero_zero_nat)))))). % coeff_monom
thf(fact_159_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_nat]: (((M = N) => ((coeff_poly_nat @ (monom_poly_nat @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_nat @ (monom_poly_nat @ A @ M) @ N) = zero_zero_poly_nat)))))). % coeff_monom
thf(fact_160_coeff__monom, axiom,
    ((![M : nat, N : nat, A : poly_poly_a]: (((M = N) => ((coeff_poly_poly_a @ (monom_poly_poly_a @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_poly_poly_a @ (monom_poly_poly_a @ A @ M) @ N) = zero_z2096148049poly_a)))))). % coeff_monom
thf(fact_161_poly__1, axiom,
    ((![X2 : a]: ((poly_a2 @ one_one_poly_a @ X2) = one_one_a)))). % poly_1
thf(fact_162_poly__1, axiom,
    ((![X2 : poly_a]: ((poly_poly_a2 @ one_one_poly_poly_a @ X2) = one_one_poly_a)))). % poly_1
thf(fact_163_poly__1, axiom,
    ((![X2 : nat]: ((poly_nat2 @ one_one_poly_nat @ X2) = one_one_nat)))). % poly_1
thf(fact_164_order__0__monom, axiom,
    ((![C : a, N : nat]: ((~ ((C = zero_zero_a))) => ((order_a @ zero_zero_a @ (monom_a @ C @ N)) = N))))). % order_0_monom
thf(fact_165_order__0__monom, axiom,
    ((![C : poly_a, N : nat]: ((~ ((C = zero_zero_poly_a))) => ((order_poly_a @ zero_zero_poly_a @ (monom_poly_a @ C @ N)) = N))))). % order_0_monom
thf(fact_166_order__0__monom, axiom,
    ((![C : poly_poly_a, N : nat]: ((~ ((C = zero_z2096148049poly_a))) => ((order_poly_poly_a @ zero_z2096148049poly_a @ (monom_poly_poly_a @ C @ N)) = N))))). % order_0_monom
thf(fact_167_pcompose__const, axiom,
    ((![A : a, Q : poly_a]: ((pcompose_a @ (pCons_a @ A @ zero_zero_poly_a) @ Q) = (pCons_a @ A @ zero_zero_poly_a))))). % pcompose_const
thf(fact_168_pcompose__const, axiom,
    ((![A : nat, Q : poly_nat]: ((pcompose_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ Q) = (pCons_nat @ A @ zero_zero_poly_nat))))). % pcompose_const
thf(fact_169_pcompose__const, axiom,
    ((![A : poly_a, Q : poly_poly_a]: ((pcompose_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ Q) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % pcompose_const
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_one__poly__eq__simps_I2_J, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % one_poly_eq_simps(2)
thf(fact_182_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_183_one__poly__eq__simps_I2_J, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % one_poly_eq_simps(2)
thf(fact_184_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_a = (pCons_a @ one_one_a @ zero_zero_poly_a)))). % one_poly_eq_simps(1)
thf(fact_185_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_186_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_poly_a = (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)))). % one_poly_eq_simps(1)
thf(fact_187_monom__eq__1, axiom,
    (((monom_nat @ one_one_nat @ zero_zero_nat) = one_one_poly_nat))). % monom_eq_1
thf(fact_188_lead__coeff__1, axiom,
    (((coeff_nat @ one_one_poly_nat @ (degree_nat @ one_one_poly_nat)) = one_one_nat))). % lead_coeff_1
thf(fact_189_pcompose__idR, axiom,
    ((![P : poly_a]: ((pcompose_a @ P @ (pCons_a @ zero_zero_a @ (pCons_a @ one_one_a @ zero_zero_poly_a))) = P)))). % pcompose_idR
thf(fact_190_pcompose__idR, axiom,
    ((![P : poly_poly_a]: ((pcompose_poly_a @ P @ (pCons_poly_a @ zero_zero_poly_a @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a))) = P)))). % pcompose_idR
thf(fact_191_pcompose__idR, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P)))). % pcompose_idR
thf(fact_192_pcompose__idR, axiom,
    ((![P : poly_poly_nat]: ((pcompose_poly_nat @ P @ (pCons_poly_nat @ zero_zero_poly_nat @ (pCons_poly_nat @ one_one_poly_nat @ zero_z1059985641ly_nat))) = P)))). % pcompose_idR
thf(fact_193_pcompose__idR, axiom,
    ((![P : poly_poly_poly_a]: ((pcompose_poly_poly_a @ P @ (pCons_poly_poly_a @ zero_z2096148049poly_a @ (pCons_poly_poly_a @ one_one_poly_poly_a @ zero_z2064990175poly_a))) = P)))). % pcompose_idR
thf(fact_194_monom__eq__1__iff, axiom,
    ((![C : nat, N : nat]: (((monom_nat @ C @ N) = one_one_poly_nat) = (((C = one_one_nat)) & ((N = zero_zero_nat))))))). % monom_eq_1_iff
thf(fact_195_degree__monom__eq, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => ((degree_a @ (monom_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_196_degree__monom__eq, axiom,
    ((![A : poly_a, N : nat]: ((~ ((A = zero_zero_poly_a))) => ((degree_poly_a @ (monom_poly_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_197_degree__monom__eq, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => ((degree_nat @ (monom_nat @ A @ N)) = N))))). % degree_monom_eq
thf(fact_198_degree__monom__eq, axiom,
    ((![A : poly_nat, N : nat]: ((~ ((A = zero_zero_poly_nat))) => ((degree_poly_nat @ (monom_poly_nat @ A @ N)) = N))))). % degree_monom_eq
thf(fact_199_degree__monom__eq, axiom,
    ((![A : poly_poly_a, N : nat]: ((~ ((A = zero_z2096148049poly_a))) => ((degree_poly_poly_a @ (monom_poly_poly_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_200_one__reorient, axiom,
    ((![X2 : nat]: ((one_one_nat = X2) = (X2 = one_one_nat))))). % one_reorient
thf(fact_201_monom__eq__iff_H, axiom,
    ((![C : a, N : nat, D : a, M : nat]: (((monom_a @ C @ N) = (monom_a @ D @ M)) = (((C = D)) & ((((C = zero_zero_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_202_monom__eq__iff_H, axiom,
    ((![C : poly_a, N : nat, D : poly_a, M : nat]: (((monom_poly_a @ C @ N) = (monom_poly_a @ D @ M)) = (((C = D)) & ((((C = zero_zero_poly_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_203_monom__eq__iff_H, axiom,
    ((![C : nat, N : nat, D : nat, M : nat]: (((monom_nat @ C @ N) = (monom_nat @ D @ M)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_204_monom__eq__iff_H, axiom,
    ((![C : poly_nat, N : nat, D : poly_nat, M : nat]: (((monom_poly_nat @ C @ N) = (monom_poly_nat @ D @ M)) = (((C = D)) & ((((C = zero_zero_poly_nat)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_205_monom__eq__iff_H, axiom,
    ((![C : poly_poly_a, N : nat, D : poly_poly_a, M : nat]: (((monom_poly_poly_a @ C @ N) = (monom_poly_poly_a @ D @ M)) = (((C = D)) & ((((C = zero_z2096148049poly_a)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_206_pCons__one, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % pCons_one
thf(fact_207_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_208_pCons__one, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % pCons_one
thf(fact_209_poly__pcompose, axiom,
    ((![P : poly_a, Q : poly_a, X2 : a]: ((poly_a2 @ (pcompose_a @ P @ Q) @ X2) = (poly_a2 @ P @ (poly_a2 @ Q @ X2)))))). % poly_pcompose
thf(fact_210_poly__pcompose, axiom,
    ((![P : poly_nat, Q : poly_nat, X2 : nat]: ((poly_nat2 @ (pcompose_nat @ P @ Q) @ X2) = (poly_nat2 @ P @ (poly_nat2 @ Q @ X2)))))). % poly_pcompose
thf(fact_211_poly__pcompose, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X2 : poly_a]: ((poly_poly_a2 @ (pcompose_poly_a @ P @ Q) @ X2) = (poly_poly_a2 @ P @ (poly_poly_a2 @ Q @ X2)))))). % poly_pcompose
thf(fact_212_monom_Orep__eq, axiom,
    ((![X2 : a, Xa : nat]: ((coeff_a @ (monom_a @ X2 @ Xa)) = (^[N2 : nat]: (if_a @ (Xa = N2) @ X2 @ zero_zero_a)))))). % monom.rep_eq
thf(fact_213_monom_Orep__eq, axiom,
    ((![X2 : poly_a, Xa : nat]: ((coeff_poly_a @ (monom_poly_a @ X2 @ Xa)) = (^[N2 : nat]: (if_poly_a @ (Xa = N2) @ X2 @ zero_zero_poly_a)))))). % monom.rep_eq
thf(fact_214_monom_Orep__eq, axiom,
    ((![X2 : nat, Xa : nat]: ((coeff_nat @ (monom_nat @ X2 @ Xa)) = (^[N2 : nat]: (if_nat @ (Xa = N2) @ X2 @ zero_zero_nat)))))). % monom.rep_eq
thf(fact_215_monom_Orep__eq, axiom,
    ((![X2 : poly_nat, Xa : nat]: ((coeff_poly_nat @ (monom_poly_nat @ X2 @ Xa)) = (^[N2 : nat]: (if_poly_nat @ (Xa = N2) @ X2 @ zero_zero_poly_nat)))))). % monom.rep_eq
thf(fact_216_monom_Orep__eq, axiom,
    ((![X2 : poly_poly_a, Xa : nat]: ((coeff_poly_poly_a @ (monom_poly_poly_a @ X2 @ Xa)) = (^[N2 : nat]: (if_poly_poly_a @ (Xa = N2) @ X2 @ zero_z2096148049poly_a)))))). % monom.rep_eq
thf(fact_217_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_218_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_219_nat__dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ one_one_nat) = (M = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_220_dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ (suc @ zero_zero_nat)) = (M = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_221_dvd__1__left, axiom,
    ((![K : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K)))). % dvd_1_left
thf(fact_222_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_223_Suc__inject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))). % Suc_inject
thf(fact_224_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_225_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_226_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_227_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_228_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N3 : nat]: ((P2 @ N3) => (P2 @ (suc @ N3)))) => (P2 @ N)))))). % nat_induct
thf(fact_229_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P2 @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P2 @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P2 @ X3 @ Y3) => (P2 @ (suc @ X3) @ (suc @ Y3)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_230_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N3 : nat]: ((P2 @ (suc @ N3)) => (P2 @ N3))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_231_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_232_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_233_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_234_old_Onat_Oexhaust, axiom,
    ((![Y2 : nat]: ((~ ((Y2 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y2 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_235_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_236_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_237_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_238_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_239_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_240_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_241_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ B) & (~ ((A = B))))))))). % gcd_nat.not_eq_order_implies_strict

% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T, axiom,
    ((![X2 : a, Y2 : a]: ((if_a @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001tf__a_T, axiom,
    ((![X2 : a, Y2 : a]: ((if_a @ $true @ X2 @ Y2) = X2)))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $true @ X2 @ Y2) = X2)))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_Itf__a_J_T, axiom,
    ((![X2 : poly_a, Y2 : poly_a]: ((if_poly_a @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_Itf__a_J_T, axiom,
    ((![X2 : poly_a, Y2 : poly_a]: ((if_poly_a @ $true @ X2 @ Y2) = X2)))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T, axiom,
    ((![X2 : poly_nat, Y2 : poly_nat]: ((if_poly_nat @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T, axiom,
    ((![X2 : poly_nat, Y2 : poly_nat]: ((if_poly_nat @ $true @ X2 @ Y2) = X2)))).
thf(help_If_3_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![X2 : poly_poly_a, Y2 : poly_poly_a]: ((if_poly_poly_a @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_T, axiom,
    ((![X2 : poly_poly_a, Y2 : poly_poly_a]: ((if_poly_poly_a @ $true @ X2 @ Y2) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ($false)).
