% 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_1083__5378106_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:47.107

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

% Explicit typings (77)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_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_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    uminus1736902417poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_Itf__a_J, type,
    uminus_uminus_poly_a : poly_a > poly_a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a, type,
    uminus_uminus_a : 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_Polynomial_Ocontent_001t__Nat__Onat, type,
    content_nat : poly_nat > 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_Omap__poly_001t__Nat__Onat_001t__Nat__Onat, type,
    map_poly_nat_nat : (nat > nat) > poly_nat > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    map_po495548498ly_nat : (nat > poly_nat) > poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Polynomial__Opoly_Itf__a_J, type,
    map_poly_nat_poly_a : (nat > poly_a) > poly_nat > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001tf__a, type,
    map_poly_nat_a : (nat > a) > poly_nat > poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001t__Nat__Onat, type,
    map_poly_poly_a_nat : (poly_a > nat) > poly_poly_a > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001t__Polynomial__Opoly_Itf__a_J, type,
    map_po495521320poly_a : (poly_a > poly_a) > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001tf__a, type,
    map_poly_poly_a_a : (poly_a > a) > poly_poly_a > poly_a).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001t__Nat__Onat, type,
    map_poly_a_nat : (a > nat) > poly_a > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    map_poly_a_poly_nat : (a > poly_nat) > poly_a > poly_poly_nat).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001t__Polynomial__Opoly_Itf__a_J, type,
    map_poly_a_poly_a : (a > poly_a) > poly_a > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001tf__a, type,
    map_poly_a_a : (a > a) > poly_a > poly_a).
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_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    dvd_dvd_poly_nat : poly_nat > poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    dvd_dv1351350402poly_a : poly_poly_poly_a > poly_poly_poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    dvd_dvd_poly_poly_a : poly_poly_a > poly_poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_Itf__a_J, type,
    dvd_dvd_poly_a : poly_a > poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001tf__a, type,
    dvd_dvd_a : a > a > $o).
thf(sy_v_x, type,
    x : a).

% Relevant facts (246)
thf(fact_0_mpoly__base__conv_I2_J, axiom,
    ((![C : poly_a, X : poly_a]: (C = (poly_poly_a2 @ (pCons_poly_a @ C @ zero_z2096148049poly_a) @ X))))). % mpoly_base_conv(2)
thf(fact_1_mpoly__base__conv_I2_J, axiom,
    ((![C : a, X : a]: (C = (poly_a2 @ (pCons_a @ C @ zero_zero_poly_a) @ X))))). % mpoly_base_conv(2)
thf(fact_2_mpoly__base__conv_I1_J, axiom,
    ((![X : poly_poly_a]: (zero_z2096148049poly_a = (poly_poly_poly_a2 @ zero_z2064990175poly_a @ X))))). % mpoly_base_conv(1)
thf(fact_3_mpoly__base__conv_I1_J, axiom,
    ((![X : a]: (zero_zero_a = (poly_a2 @ zero_zero_poly_a @ X))))). % mpoly_base_conv(1)
thf(fact_4_mpoly__base__conv_I1_J, axiom,
    ((![X : poly_a]: (zero_zero_poly_a = (poly_poly_a2 @ zero_z2096148049poly_a @ X))))). % mpoly_base_conv(1)
thf(fact_5_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_10_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_11_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_12_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_13_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_14_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_15_pCons__eq__0__iff, axiom,
    ((![A : poly_poly_a, P : poly_poly_poly_a]: (((pCons_poly_poly_a @ A @ P) = zero_z2064990175poly_a) = (((A = zero_z2096148049poly_a)) & ((P = zero_z2064990175poly_a))))))). % pCons_eq_0_iff
thf(fact_16_pCons__eq__0__iff, axiom,
    ((![A : poly_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_17_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_18_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_19_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_20_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_21_pCons__induct, axiom,
    ((![P2 : poly_poly_nat > $o, P : poly_poly_nat]: ((P2 @ zero_z1059985641ly_nat) => ((![A2 : poly_nat, P3 : poly_poly_nat]: (((~ ((A2 = zero_zero_poly_nat))) | (~ ((P3 = zero_z1059985641ly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_poly_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_22_pCons__induct, axiom,
    ((![P2 : poly_poly_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_23_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_24_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_25_pCons__eq__iff, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: (((pCons_nat @ A @ P) = (pCons_nat @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_26_pCons__eq__iff, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: (((pCons_poly_a @ A @ P) = (pCons_poly_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_27_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_28_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_29_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_30_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_31_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_32_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_33_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_a > $o, P : poly_nat, Q : poly_a]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A2 : nat, P3 : poly_nat, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_34_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_poly_a > $o, P : poly_nat, Q : poly_poly_a]: ((P2 @ zero_zero_poly_nat @ zero_z2096148049poly_a) => ((![A2 : nat, P3 : poly_nat, B2 : poly_a, Q2 : poly_poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_nat @ A2 @ P3) @ (pCons_poly_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_35_poly__induct2, axiom,
    ((![P2 : poly_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_36_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_37_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_38_mpoly__base__conv_I3_J, axiom,
    ((![X : a]: (X = (poly_a2 @ (pCons_a @ zero_zero_a @ (pCons_a @ one_one_a @ zero_zero_poly_a)) @ X))))). % mpoly_base_conv(3)
thf(fact_39_mpoly__base__conv_I3_J, axiom,
    ((![X : poly_a]: (X = (poly_poly_a2 @ (pCons_poly_a @ zero_zero_poly_a @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)) @ X))))). % mpoly_base_conv(3)
thf(fact_40_mpoly__base__conv_I3_J, axiom,
    ((![X : poly_poly_a]: (X = (poly_poly_poly_a2 @ (pCons_poly_poly_a @ zero_z2096148049poly_a @ (pCons_poly_poly_a @ one_one_poly_poly_a @ zero_z2064990175poly_a)) @ X))))). % mpoly_base_conv(3)
thf(fact_41_offset__poly__single, axiom,
    ((![A : a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ H) = (pCons_a @ A @ zero_zero_poly_a))))). % offset_poly_single
thf(fact_42_offset__poly__single, axiom,
    ((![A : 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_43_offset__poly__single, axiom,
    ((![A : nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ H) = (pCons_nat @ A @ zero_zero_poly_nat))))). % offset_poly_single
thf(fact_44_poly__1, axiom,
    ((![X : a]: ((poly_a2 @ one_one_poly_a @ X) = one_one_a)))). % poly_1
thf(fact_45_poly__1, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ one_one_poly_poly_a @ X) = one_one_poly_a)))). % poly_1
thf(fact_46_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_47_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_48_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_49_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_50_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_51_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_52_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_53_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_54_pCons__one, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % pCons_one
thf(fact_55_pCons__one, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % pCons_one
thf(fact_56_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_57_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_58_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_59_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_60_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_61_offset__poly__0, axiom,
    ((![H : poly_a]: ((fundam1343031620poly_a @ zero_z2096148049poly_a @ H) = zero_z2096148049poly_a)))). % offset_poly_0
thf(fact_62_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_63_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_64_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_65_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_66_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_67_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_68_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A2 : nat, P3 : poly_nat]: (~ ((X = (pCons_nat @ A2 @ P3)))))))))). % pderiv.cases
thf(fact_69_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_70_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A2 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_71_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A2 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_72_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_73_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_a = one_one_poly_a))))). % zero_neq_one
thf(fact_74_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_75_zero__neq__one, axiom,
    ((~ ((zero_z2096148049poly_a = one_one_poly_poly_a))))). % zero_neq_one
thf(fact_76_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_nat = one_one_poly_nat))))). % zero_neq_one
thf(fact_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_86_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_87_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_88_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_89_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_90_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_91_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_92_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_93_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_94_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_95_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_96_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_97_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_98_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_99_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_100_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_shift_0
thf(fact_101_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_102_map__poly__1, axiom,
    ((![F : nat > a]: ((map_poly_nat_a @ F @ one_one_poly_nat) = (pCons_a @ (F @ one_one_nat) @ zero_zero_poly_a))))). % map_poly_1
thf(fact_103_map__poly__1, axiom,
    ((![F : nat > poly_a]: ((map_poly_nat_poly_a @ F @ one_one_poly_nat) = (pCons_poly_a @ (F @ one_one_nat) @ zero_z2096148049poly_a))))). % map_poly_1
thf(fact_104_map__poly__1, axiom,
    ((![F : nat > nat]: ((map_poly_nat_nat @ F @ one_one_poly_nat) = (pCons_nat @ (F @ one_one_nat) @ zero_zero_poly_nat))))). % map_poly_1
thf(fact_105_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_106_pcompose__0, axiom,
    ((![Q : poly_poly_a]: ((pcompose_poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % pcompose_0
thf(fact_107_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_108_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_109_reflect__poly__0, axiom,
    (((reflect_poly_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % reflect_poly_0
thf(fact_110_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_111_map__poly__0, axiom,
    ((![F : a > a]: ((map_poly_a_a @ F @ zero_zero_poly_a) = zero_zero_poly_a)))). % map_poly_0
thf(fact_112_map__poly__0, axiom,
    ((![F : a > poly_a]: ((map_poly_a_poly_a @ F @ zero_zero_poly_a) = zero_z2096148049poly_a)))). % map_poly_0
thf(fact_113_map__poly__0, axiom,
    ((![F : a > nat]: ((map_poly_a_nat @ F @ zero_zero_poly_a) = zero_zero_poly_nat)))). % map_poly_0
thf(fact_114_map__poly__0, axiom,
    ((![F : poly_a > a]: ((map_poly_poly_a_a @ F @ zero_z2096148049poly_a) = zero_zero_poly_a)))). % map_poly_0
thf(fact_115_map__poly__0, axiom,
    ((![F : poly_a > poly_a]: ((map_po495521320poly_a @ F @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % map_poly_0
thf(fact_116_map__poly__0, axiom,
    ((![F : poly_a > nat]: ((map_poly_poly_a_nat @ F @ zero_z2096148049poly_a) = zero_zero_poly_nat)))). % map_poly_0
thf(fact_117_map__poly__0, axiom,
    ((![F : nat > a]: ((map_poly_nat_a @ F @ zero_zero_poly_nat) = zero_zero_poly_a)))). % map_poly_0
thf(fact_118_map__poly__0, axiom,
    ((![F : nat > poly_a]: ((map_poly_nat_poly_a @ F @ zero_zero_poly_nat) = zero_z2096148049poly_a)))). % map_poly_0
thf(fact_119_map__poly__0, axiom,
    ((![F : nat > nat]: ((map_poly_nat_nat @ F @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % map_poly_0
thf(fact_120_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_121_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_122_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_123_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_124_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_125_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_126_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_127_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_128_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_129_map__poly__1_H, axiom,
    ((![F : nat > nat]: (((F @ one_one_nat) = one_one_nat) => ((map_poly_nat_nat @ F @ one_one_poly_nat) = one_one_poly_nat))))). % map_poly_1'
thf(fact_130_poly__pcompose, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (pcompose_a @ P @ Q) @ X) = (poly_a2 @ P @ (poly_a2 @ Q @ X)))))). % poly_pcompose
thf(fact_131_poly__pcompose, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (pcompose_poly_a @ P @ Q) @ X) = (poly_poly_a2 @ P @ (poly_poly_a2 @ Q @ X)))))). % poly_pcompose
thf(fact_132_poly__pcompose, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (pcompose_nat @ P @ Q) @ X) = (poly_nat2 @ P @ (poly_nat2 @ Q @ X)))))). % poly_pcompose
thf(fact_133_map__poly__pCons, axiom,
    ((![F : a > a, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_a) => ((map_poly_a_a @ F @ (pCons_a @ C @ P)) = (pCons_a @ (F @ C) @ (map_poly_a_a @ F @ P))))))). % map_poly_pCons
thf(fact_134_map__poly__pCons, axiom,
    ((![F : a > nat, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_nat) => ((map_poly_a_nat @ F @ (pCons_a @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_a_nat @ F @ P))))))). % map_poly_pCons
thf(fact_135_map__poly__pCons, axiom,
    ((![F : nat > a, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_a) => ((map_poly_nat_a @ F @ (pCons_nat @ C @ P)) = (pCons_a @ (F @ C) @ (map_poly_nat_a @ F @ P))))))). % map_poly_pCons
thf(fact_136_map__poly__pCons, axiom,
    ((![F : nat > nat, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((map_poly_nat_nat @ F @ (pCons_nat @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_nat_nat @ F @ P))))))). % map_poly_pCons
thf(fact_137_map__poly__pCons, axiom,
    ((![F : a > poly_a, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_poly_a) => ((map_poly_a_poly_a @ F @ (pCons_a @ C @ P)) = (pCons_poly_a @ (F @ C) @ (map_poly_a_poly_a @ F @ P))))))). % map_poly_pCons
thf(fact_138_map__poly__pCons, axiom,
    ((![F : a > poly_nat, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_poly_nat) => ((map_poly_a_poly_nat @ F @ (pCons_a @ C @ P)) = (pCons_poly_nat @ (F @ C) @ (map_poly_a_poly_nat @ F @ P))))))). % map_poly_pCons
thf(fact_139_map__poly__pCons, axiom,
    ((![F : poly_a > a, C : poly_a, P : poly_poly_a]: (((F @ zero_zero_poly_a) = zero_zero_a) => ((map_poly_poly_a_a @ F @ (pCons_poly_a @ C @ P)) = (pCons_a @ (F @ C) @ (map_poly_poly_a_a @ F @ P))))))). % map_poly_pCons
thf(fact_140_map__poly__pCons, axiom,
    ((![F : poly_a > nat, C : poly_a, P : poly_poly_a]: (((F @ zero_zero_poly_a) = zero_zero_nat) => ((map_poly_poly_a_nat @ F @ (pCons_poly_a @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_poly_a_nat @ F @ P))))))). % map_poly_pCons
thf(fact_141_map__poly__pCons, axiom,
    ((![F : nat > poly_a, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_poly_a) => ((map_poly_nat_poly_a @ F @ (pCons_nat @ C @ P)) = (pCons_poly_a @ (F @ C) @ (map_poly_nat_poly_a @ F @ P))))))). % map_poly_pCons
thf(fact_142_map__poly__pCons, axiom,
    ((![F : nat > poly_nat, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_poly_nat) => ((map_po495548498ly_nat @ F @ (pCons_nat @ C @ P)) = (pCons_poly_nat @ (F @ C) @ (map_po495548498ly_nat @ F @ P))))))). % map_poly_pCons
thf(fact_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_dvd__0__left__iff, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) = (A = zero_zero_a))))). % dvd_0_left_iff
thf(fact_155_dvd__0__left__iff, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) = (A = zero_zero_poly_a))))). % dvd_0_left_iff
thf(fact_156_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_157_dvd__0__left__iff, axiom,
    ((![A : poly_poly_a]: ((dvd_dvd_poly_poly_a @ zero_z2096148049poly_a @ A) = (A = zero_z2096148049poly_a))))). % dvd_0_left_iff
thf(fact_158_dvd__0__left__iff, axiom,
    ((![A : poly_nat]: ((dvd_dvd_poly_nat @ zero_zero_poly_nat @ A) = (A = zero_zero_poly_nat))))). % dvd_0_left_iff
thf(fact_159_dvd__0__right, axiom,
    ((![A : a]: (dvd_dvd_a @ A @ zero_zero_a)))). % dvd_0_right
thf(fact_160_dvd__0__right, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ zero_zero_poly_a)))). % dvd_0_right
thf(fact_161_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_162_dvd__0__right, axiom,
    ((![A : poly_poly_a]: (dvd_dvd_poly_poly_a @ A @ zero_z2096148049poly_a)))). % dvd_0_right
thf(fact_163_dvd__0__right, axiom,
    ((![A : poly_nat]: (dvd_dvd_poly_nat @ A @ zero_zero_poly_nat)))). % dvd_0_right
thf(fact_164_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_poly_a @ zero_z2064990175poly_a @ N) = zero_z2096148049poly_a)))). % coeff_0
thf(fact_165_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_nat @ zero_z1059985641ly_nat @ N) = zero_zero_poly_nat)))). % coeff_0
thf(fact_166_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_167_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_168_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_169_coeff__pCons__0, axiom,
    ((![A : a, P : poly_a]: ((coeff_a @ (pCons_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_170_coeff__pCons__0, axiom,
    ((![A : nat, P : poly_nat]: ((coeff_nat @ (pCons_nat @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_171_coeff__pCons__0, axiom,
    ((![A : poly_a, P : poly_poly_a]: ((coeff_poly_a @ (pCons_poly_a @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_172_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_173_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_174_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_175_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_176_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_177_const__poly__dvd__const__poly__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ (pCons_nat @ B @ zero_zero_poly_nat)) = (dvd_dvd_nat @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_178_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_179_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_180_const__poly__dvd__iff, axiom,
    ((![C : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ P) = (![N2 : nat]: (dvd_dvd_nat @ C @ (coeff_nat @ P @ N2))))))). % const_poly_dvd_iff
thf(fact_181_dvd__0__left, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) => (A = zero_zero_a))))). % dvd_0_left
thf(fact_182_dvd__0__left, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) => (A = zero_zero_poly_a))))). % dvd_0_left
thf(fact_183_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_184_dvd__0__left, axiom,
    ((![A : poly_poly_a]: ((dvd_dvd_poly_poly_a @ zero_z2096148049poly_a @ A) => (A = zero_z2096148049poly_a))))). % dvd_0_left
thf(fact_185_dvd__0__left, axiom,
    ((![A : poly_nat]: ((dvd_dvd_poly_nat @ zero_zero_poly_nat @ A) => (A = zero_zero_poly_nat))))). % dvd_0_left
thf(fact_186_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_187_unit__imp__dvd, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ B @ A))))). % unit_imp_dvd
thf(fact_188_dvd__unit__imp__unit, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ A @ one_one_nat)))))). % dvd_unit_imp_unit
thf(fact_189_zero__poly_Orep__eq, axiom,
    (((coeff_poly_poly_a @ zero_z2064990175poly_a) = (^[Uu : nat]: zero_z2096148049poly_a)))). % zero_poly.rep_eq
thf(fact_190_zero__poly_Orep__eq, axiom,
    (((coeff_poly_nat @ zero_z1059985641ly_nat) = (^[Uu : nat]: zero_zero_poly_nat)))). % zero_poly.rep_eq
thf(fact_191_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_192_zero__poly_Orep__eq, axiom,
    (((coeff_poly_a @ zero_z2096148049poly_a) = (^[Uu : nat]: zero_zero_poly_a)))). % zero_poly.rep_eq
thf(fact_193_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_194_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_195_is__unit__polyE, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) => (~ ((![C2 : nat]: ((P = (pCons_nat @ C2 @ zero_zero_poly_nat)) => (~ ((dvd_dvd_nat @ C2 @ one_one_nat))))))))))). % is_unit_polyE
thf(fact_196_is__unit__poly__iff, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) = (?[C3 : nat]: (((P = (pCons_nat @ C3 @ zero_zero_poly_nat))) & ((dvd_dvd_nat @ C3 @ one_one_nat)))))))). % is_unit_poly_iff
thf(fact_197_is__unit__const__poly__iff, axiom,
    ((![C : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ one_one_poly_nat) = (dvd_dvd_nat @ C @ one_one_nat))))). % is_unit_const_poly_iff
thf(fact_198_coeff__map__poly, axiom,
    ((![F : a > a, P : poly_a, N : nat]: (((F @ zero_zero_a) = zero_zero_a) => ((coeff_a @ (map_poly_a_a @ F @ P) @ N) = (F @ (coeff_a @ P @ N))))))). % coeff_map_poly
thf(fact_199_coeff__map__poly, axiom,
    ((![F : a > nat, P : poly_a, N : nat]: (((F @ zero_zero_a) = zero_zero_nat) => ((coeff_nat @ (map_poly_a_nat @ F @ P) @ N) = (F @ (coeff_a @ P @ N))))))). % coeff_map_poly
thf(fact_200_coeff__map__poly, axiom,
    ((![F : nat > a, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_a) => ((coeff_a @ (map_poly_nat_a @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_201_coeff__map__poly, axiom,
    ((![F : nat > nat, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((coeff_nat @ (map_poly_nat_nat @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_202_coeff__map__poly, axiom,
    ((![F : a > poly_a, P : poly_a, N : nat]: (((F @ zero_zero_a) = zero_zero_poly_a) => ((coeff_poly_a @ (map_poly_a_poly_a @ F @ P) @ N) = (F @ (coeff_a @ P @ N))))))). % coeff_map_poly
thf(fact_203_coeff__map__poly, axiom,
    ((![F : a > poly_nat, P : poly_a, N : nat]: (((F @ zero_zero_a) = zero_zero_poly_nat) => ((coeff_poly_nat @ (map_poly_a_poly_nat @ F @ P) @ N) = (F @ (coeff_a @ P @ N))))))). % coeff_map_poly
thf(fact_204_coeff__map__poly, axiom,
    ((![F : poly_a > a, P : poly_poly_a, N : nat]: (((F @ zero_zero_poly_a) = zero_zero_a) => ((coeff_a @ (map_poly_poly_a_a @ F @ P) @ N) = (F @ (coeff_poly_a @ P @ N))))))). % coeff_map_poly
thf(fact_205_coeff__map__poly, axiom,
    ((![F : poly_a > nat, P : poly_poly_a, N : nat]: (((F @ zero_zero_poly_a) = zero_zero_nat) => ((coeff_nat @ (map_poly_poly_a_nat @ F @ P) @ N) = (F @ (coeff_poly_a @ P @ N))))))). % coeff_map_poly
thf(fact_206_coeff__map__poly, axiom,
    ((![F : nat > poly_a, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_poly_a) => ((coeff_poly_a @ (map_poly_nat_poly_a @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_207_coeff__map__poly, axiom,
    ((![F : nat > poly_nat, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_poly_nat) => ((coeff_poly_nat @ (map_po495548498ly_nat @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_poly__eq__0__iff__dvd, axiom,
    ((![P : poly_a, C : a]: (((poly_a2 @ P @ C) = zero_zero_a) = (dvd_dvd_poly_a @ (pCons_a @ (uminus_uminus_a @ C) @ (pCons_a @ one_one_a @ zero_zero_poly_a)) @ P))))). % poly_eq_0_iff_dvd
thf(fact_217_poly__eq__0__iff__dvd, axiom,
    ((![P : poly_poly_a, C : poly_a]: (((poly_poly_a2 @ P @ C) = zero_zero_poly_a) = (dvd_dvd_poly_poly_a @ (pCons_poly_a @ (uminus_uminus_poly_a @ C) @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)) @ P))))). % poly_eq_0_iff_dvd
thf(fact_218_poly__eq__0__iff__dvd, axiom,
    ((![P : poly_poly_poly_a, C : poly_poly_a]: (((poly_poly_poly_a2 @ P @ C) = zero_z2096148049poly_a) = (dvd_dv1351350402poly_a @ (pCons_poly_poly_a @ (uminus1736902417poly_a @ C) @ (pCons_poly_poly_a @ one_one_poly_poly_a @ zero_z2064990175poly_a)) @ P))))). % poly_eq_0_iff_dvd
thf(fact_219_dvd__iff__poly__eq__0, axiom,
    ((![C : poly_poly_a, P : poly_poly_poly_a]: ((dvd_dv1351350402poly_a @ (pCons_poly_poly_a @ C @ (pCons_poly_poly_a @ one_one_poly_poly_a @ zero_z2064990175poly_a)) @ P) = ((poly_poly_poly_a2 @ P @ (uminus1736902417poly_a @ C)) = zero_z2096148049poly_a))))). % dvd_iff_poly_eq_0
thf(fact_220_dvd__iff__poly__eq__0, axiom,
    ((![C : a, P : poly_a]: ((dvd_dvd_poly_a @ (pCons_a @ C @ (pCons_a @ one_one_a @ zero_zero_poly_a)) @ P) = ((poly_a2 @ P @ (uminus_uminus_a @ C)) = zero_zero_a))))). % dvd_iff_poly_eq_0
thf(fact_221_dvd__iff__poly__eq__0, axiom,
    ((![C : poly_a, P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ (pCons_poly_a @ C @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)) @ P) = ((poly_poly_a2 @ P @ (uminus_uminus_poly_a @ C)) = zero_zero_poly_a))))). % dvd_iff_poly_eq_0
thf(fact_222_const__poly__dvd__iff__dvd__content, axiom,
    ((![C : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ P) = (dvd_dvd_nat @ C @ (content_nat @ P)))))). % const_poly_dvd_iff_dvd_content
thf(fact_223_add_Oinverse__neutral, axiom,
    (((uminus_uminus_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % add.inverse_neutral
thf(fact_224_add_Oinverse__neutral, axiom,
    (((uminus1736902417poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % add.inverse_neutral
thf(fact_225_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_226_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_227_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_228_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_229_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_230_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_231_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_232_gcd__nat_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A3 : nat]: (^[B3 : nat]: (((dvd_dvd_nat @ A3 @ B3)) & ((dvd_dvd_nat @ B3 @ A3)))))))). % gcd_nat.eq_iff
thf(fact_233_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_234_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_235_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_236_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_237_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_238_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A3 : nat]: (^[B3 : nat]: (((((dvd_dvd_nat @ A3 @ B3)) & ((~ ((A3 = B3)))))) | ((A3 = B3)))))))). % gcd_nat.order_iff_strict
thf(fact_239_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_240_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_241_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_242_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
thf(fact_243_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_244_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_245_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique

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