% 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_587__5373828_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:20.826

% 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 (49)
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_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    plus_p1976640465poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J, type,
    plus_plus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    times_545135445poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : 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_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
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_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_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_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_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    power_power_poly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    power_276493840poly_a : poly_poly_a > nat > poly_poly_a).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_Itf__a_J, type,
    power_power_poly_a : poly_a > nat > poly_a).
thf(sy_c_Power_Opower__class_Opower_001tf__a, type,
    power_power_a : a > nat > a).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_c____, type,
    c : a).
thf(sy_v_cs____, type,
    cs : poly_a).
thf(sy_v_k, type,
    k : nat).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_q, type,
    q : poly_a).
thf(sy_v_z, type,
    z : a).

% Relevant facts (245)
thf(fact_0_nz, axiom,
    ((~ ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ p @ Z) = zero_zero_a))))))). % nz
thf(fact_1_True, axiom,
    ((c = zero_zero_a))). % True
thf(fact_2_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_3_pCons_Oprems, axiom,
    ((~ ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ (pCons_a @ c @ cs) @ Z) = zero_zero_a))))))). % pCons.prems
thf(fact_4_pCons_Ohyps_I2_J, axiom,
    (((~ ((![Z2 : a]: ((~ ((Z2 = zero_zero_a))) => ((poly_a2 @ cs @ Z2) = zero_zero_a))))) => (?[K : nat, A : a]: ((~ ((A = zero_zero_a))) & (?[Q : poly_a]: ((((cs = zero_zero_poly_a) => ((suc @ (plus_plus_nat @ (if_nat @ (Q = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ Q))) @ K)) = zero_zero_nat)) & ((~ ((cs = zero_zero_poly_a))) => ((suc @ (plus_plus_nat @ (if_nat @ (Q = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ Q))) @ K)) = (suc @ (degree_a @ cs))))) & (![Z2 : a]: ((poly_a2 @ cs @ Z2) = (times_times_a @ (power_power_a @ Z2 @ K) @ (poly_a2 @ (pCons_a @ A @ Q) @ Z2))))))))))). % pCons.hyps(2)
thf(fact_5_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % 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_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_11_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_12_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_13_power__Suc0__right, axiom,
    ((![A2 : poly_a]: ((power_power_poly_a @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_14_power__Suc0__right, axiom,
    ((![A2 : a]: ((power_power_a @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_15_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_16_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_poly_nat @ zero_zero_poly_nat @ (suc @ N)) = zero_zero_poly_nat)))). % power_0_Suc
thf(fact_17_power__0__Suc, axiom,
    ((![N : nat]: ((power_276493840poly_a @ zero_z2096148049poly_a @ (suc @ N)) = zero_z2096148049poly_a)))). % power_0_Suc
thf(fact_18_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_poly_a @ zero_zero_poly_a @ (suc @ N)) = zero_zero_poly_a)))). % power_0_Suc
thf(fact_19_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_a @ zero_zero_a @ (suc @ N)) = zero_zero_a)))). % power_0_Suc
thf(fact_20_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_21_mult__cancel__left1, axiom,
    ((![C : poly_poly_a, B : poly_poly_a]: ((C = (times_545135445poly_a @ C @ B)) = (((C = zero_z2096148049poly_a)) | ((B = one_one_poly_poly_a))))))). % mult_cancel_left1
thf(fact_22_mult__cancel__left1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_left1
thf(fact_23_mult__cancel__left1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_left1
thf(fact_24_mult__cancel__left2, axiom,
    ((![C : poly_poly_a, A2 : poly_poly_a]: (((times_545135445poly_a @ C @ A2) = C) = (((C = zero_z2096148049poly_a)) | ((A2 = one_one_poly_poly_a))))))). % mult_cancel_left2
thf(fact_25_mult__cancel__left2, axiom,
    ((![C : a, A2 : a]: (((times_times_a @ C @ A2) = C) = (((C = zero_zero_a)) | ((A2 = one_one_a))))))). % mult_cancel_left2
thf(fact_26_mult__cancel__left2, axiom,
    ((![C : poly_a, A2 : poly_a]: (((times_times_poly_a @ C @ A2) = C) = (((C = zero_zero_poly_a)) | ((A2 = one_one_poly_a))))))). % mult_cancel_left2
thf(fact_27_mult__cancel__right1, axiom,
    ((![C : poly_poly_a, B : poly_poly_a]: ((C = (times_545135445poly_a @ B @ C)) = (((C = zero_z2096148049poly_a)) | ((B = one_one_poly_poly_a))))))). % mult_cancel_right1
thf(fact_28_mult__cancel__right1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_right1
thf(fact_29_mult__cancel__right1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_right1
thf(fact_30_mult__cancel__right2, axiom,
    ((![A2 : poly_poly_a, C : poly_poly_a]: (((times_545135445poly_a @ A2 @ C) = C) = (((C = zero_z2096148049poly_a)) | ((A2 = one_one_poly_poly_a))))))). % mult_cancel_right2
thf(fact_31_mult__cancel__right2, axiom,
    ((![A2 : a, C : a]: (((times_times_a @ A2 @ C) = C) = (((C = zero_zero_a)) | ((A2 = one_one_a))))))). % mult_cancel_right2
thf(fact_32_mult__cancel__right2, axiom,
    ((![A2 : poly_a, C : poly_a]: (((times_times_poly_a @ A2 @ C) = C) = (((C = zero_zero_poly_a)) | ((A2 = one_one_poly_a))))))). % mult_cancel_right2
thf(fact_33_poly__power, axiom,
    ((![P : poly_nat, N : nat, X : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X) = (power_power_nat @ (poly_nat2 @ P @ X) @ N))))). % poly_power
thf(fact_34_poly__power, axiom,
    ((![P : poly_poly_a, N : nat, X : poly_a]: ((poly_poly_a2 @ (power_276493840poly_a @ P @ N) @ X) = (power_power_poly_a @ (poly_poly_a2 @ P @ X) @ N))))). % poly_power
thf(fact_35_poly__power, axiom,
    ((![P : poly_a, N : nat, X : a]: ((poly_a2 @ (power_power_poly_a @ P @ N) @ X) = (power_power_a @ (poly_a2 @ P @ X) @ N))))). % poly_power
thf(fact_36_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_37_poly__1, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ one_one_poly_poly_a @ X) = one_one_poly_a)))). % poly_1
thf(fact_38_poly__1, axiom,
    ((![X : a]: ((poly_a2 @ one_one_poly_a @ X) = one_one_a)))). % poly_1
thf(fact_39_poly__add, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (plus_p1976640465poly_a @ P @ Q2) @ X) = (plus_plus_poly_a @ (poly_poly_a2 @ P @ X) @ (poly_poly_a2 @ Q2 @ X)))))). % poly_add
thf(fact_40_poly__add, axiom,
    ((![P : poly_a, Q2 : poly_a, X : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q2) @ X) = (plus_plus_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q2 @ X)))))). % poly_add
thf(fact_41_poly__add, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q2) @ X) = (plus_plus_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q2 @ X)))))). % poly_add
thf(fact_42_pCons__eq__iff, axiom,
    ((![A2 : a, P : poly_a, B : a, Q2 : poly_a]: (((pCons_a @ A2 @ P) = (pCons_a @ B @ Q2)) = (((A2 = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_43_mult__cancel__right, axiom,
    ((![A2 : poly_poly_a, C : poly_poly_a, B : poly_poly_a]: (((times_545135445poly_a @ A2 @ C) = (times_545135445poly_a @ B @ C)) = (((C = zero_z2096148049poly_a)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_44_mult__cancel__right, axiom,
    ((![A2 : a, C : a, B : a]: (((times_times_a @ A2 @ C) = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_45_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_46_mult__cancel__right, axiom,
    ((![A2 : poly_a, C : poly_a, B : poly_a]: (((times_times_poly_a @ A2 @ C) = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_47_mult__cancel__left, axiom,
    ((![C : poly_poly_a, A2 : poly_poly_a, B : poly_poly_a]: (((times_545135445poly_a @ C @ A2) = (times_545135445poly_a @ C @ B)) = (((C = zero_z2096148049poly_a)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_48_mult__cancel__left, axiom,
    ((![C : a, A2 : a, B : a]: (((times_times_a @ C @ A2) = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_49_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_50_mult__cancel__left, axiom,
    ((![C : poly_a, A2 : poly_a, B : poly_a]: (((times_times_poly_a @ C @ A2) = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_51_mult__eq__0__iff, axiom,
    ((![A2 : poly_nat, B : poly_nat]: (((times_times_poly_nat @ A2 @ B) = zero_zero_poly_nat) = (((A2 = zero_zero_poly_nat)) | ((B = zero_zero_poly_nat))))))). % mult_eq_0_iff
thf(fact_52_mult__eq__0__iff, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: (((times_545135445poly_a @ A2 @ B) = zero_z2096148049poly_a) = (((A2 = zero_z2096148049poly_a)) | ((B = zero_z2096148049poly_a))))))). % mult_eq_0_iff
thf(fact_53_mult__eq__0__iff, axiom,
    ((![A2 : a, B : a]: (((times_times_a @ A2 @ B) = zero_zero_a) = (((A2 = zero_zero_a)) | ((B = zero_zero_a))))))). % mult_eq_0_iff
thf(fact_54_mult__eq__0__iff, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_55_mult__eq__0__iff, axiom,
    ((![A2 : poly_a, B : poly_a]: (((times_times_poly_a @ A2 @ B) = zero_zero_poly_a) = (((A2 = zero_zero_poly_a)) | ((B = zero_zero_poly_a))))))). % mult_eq_0_iff
thf(fact_56_mult__zero__right, axiom,
    ((![A2 : poly_nat]: ((times_times_poly_nat @ A2 @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_zero_right
thf(fact_57_mult__zero__right, axiom,
    ((![A2 : poly_poly_a]: ((times_545135445poly_a @ A2 @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % mult_zero_right
thf(fact_58_mult__zero__right, axiom,
    ((![A2 : a]: ((times_times_a @ A2 @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_59_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_60_mult__zero__right, axiom,
    ((![A2 : poly_a]: ((times_times_poly_a @ A2 @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_61_mult__zero__left, axiom,
    ((![A2 : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ A2) = zero_zero_poly_nat)))). % mult_zero_left
thf(fact_62_mult__zero__left, axiom,
    ((![A2 : poly_poly_a]: ((times_545135445poly_a @ zero_z2096148049poly_a @ A2) = zero_z2096148049poly_a)))). % mult_zero_left
thf(fact_63_mult__zero__left, axiom,
    ((![A2 : a]: ((times_times_a @ zero_zero_a @ A2) = zero_zero_a)))). % mult_zero_left
thf(fact_64_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_65_mult__zero__left, axiom,
    ((![A2 : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A2) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_66_power__one, axiom,
    ((![N : nat]: ((power_power_a @ one_one_a @ N) = one_one_a)))). % power_one
thf(fact_67_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_68_power__one, axiom,
    ((![N : nat]: ((power_power_poly_a @ one_one_poly_a @ N) = one_one_poly_a)))). % power_one
thf(fact_69_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_70_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_71_power__one__right, axiom,
    ((![A2 : a]: ((power_power_a @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_72_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_73_power__one__right, axiom,
    ((![A2 : poly_a]: ((power_power_poly_a @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_74_poly__mult, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q2) @ X) = (times_times_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q2 @ X)))))). % poly_mult
thf(fact_75_poly__mult, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (times_545135445poly_a @ P @ Q2) @ X) = (times_times_poly_a @ (poly_poly_a2 @ P @ X) @ (poly_poly_a2 @ Q2 @ X)))))). % poly_mult
thf(fact_76_poly__mult, axiom,
    ((![P : poly_a, Q2 : poly_a, X : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q2) @ X) = (times_times_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q2 @ X)))))). % poly_mult
thf(fact_77_pCons__eq__0__iff, axiom,
    ((![A2 : poly_nat, P : poly_poly_nat]: (((pCons_poly_nat @ A2 @ P) = zero_z1059985641ly_nat) = (((A2 = zero_zero_poly_nat)) & ((P = zero_z1059985641ly_nat))))))). % pCons_eq_0_iff
thf(fact_78_pCons__eq__0__iff, axiom,
    ((![A2 : poly_poly_a, P : poly_poly_poly_a]: (((pCons_poly_poly_a @ A2 @ P) = zero_z2064990175poly_a) = (((A2 = zero_z2096148049poly_a)) & ((P = zero_z2064990175poly_a))))))). % pCons_eq_0_iff
thf(fact_79_pCons__eq__0__iff, axiom,
    ((![A2 : a, P : poly_a]: (((pCons_a @ A2 @ P) = zero_zero_poly_a) = (((A2 = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_80_pCons__eq__0__iff, axiom,
    ((![A2 : nat, P : poly_nat]: (((pCons_nat @ A2 @ P) = zero_zero_poly_nat) = (((A2 = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_81_pCons__eq__0__iff, axiom,
    ((![A2 : poly_a, P : poly_poly_a]: (((pCons_poly_a @ A2 @ P) = zero_z2096148049poly_a) = (((A2 = zero_zero_poly_a)) & ((P = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_82_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_83_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_84_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_85_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_86_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_87_add__pCons, axiom,
    ((![A2 : a, P : poly_a, B : a, Q2 : poly_a]: ((plus_plus_poly_a @ (pCons_a @ A2 @ P) @ (pCons_a @ B @ Q2)) = (pCons_a @ (plus_plus_a @ A2 @ B) @ (plus_plus_poly_a @ P @ Q2)))))). % add_pCons
thf(fact_88_add__pCons, axiom,
    ((![A2 : nat, P : poly_nat, B : nat, Q2 : poly_nat]: ((plus_plus_poly_nat @ (pCons_nat @ A2 @ P) @ (pCons_nat @ B @ Q2)) = (pCons_nat @ (plus_plus_nat @ A2 @ B) @ (plus_plus_poly_nat @ P @ Q2)))))). % add_pCons
thf(fact_89_degree__1, axiom,
    (((degree_a @ one_one_poly_a) = zero_zero_nat))). % degree_1
thf(fact_90_poly__pCons, axiom,
    ((![A2 : a, P : poly_a, X : a]: ((poly_a2 @ (pCons_a @ A2 @ P) @ X) = (plus_plus_a @ A2 @ (times_times_a @ X @ (poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_91_poly__pCons, axiom,
    ((![A2 : nat, P : poly_nat, X : nat]: ((poly_nat2 @ (pCons_nat @ A2 @ P) @ X) = (plus_plus_nat @ A2 @ (times_times_nat @ X @ (poly_nat2 @ P @ X))))))). % poly_pCons
thf(fact_92_poly__pCons, axiom,
    ((![A2 : poly_a, P : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (pCons_poly_a @ A2 @ P) @ X) = (plus_plus_poly_a @ A2 @ (times_times_poly_a @ X @ (poly_poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_93_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_94_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_95_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_96_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_97_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_98_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_99_degree__pCons__eq__if, axiom,
    ((![P : poly_a, A2 : a]: (((P = zero_zero_poly_a) => ((degree_a @ (pCons_a @ A2 @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A2 @ P)) = (suc @ (degree_a @ P)))))))). % degree_pCons_eq_if
thf(fact_100_degree__pCons__eq__if, axiom,
    ((![P : poly_nat, A2 : nat]: (((P = zero_zero_poly_nat) => ((degree_nat @ (pCons_nat @ A2 @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A2 @ P)) = (suc @ (degree_nat @ P)))))))). % degree_pCons_eq_if
thf(fact_101_degree__pCons__eq__if, axiom,
    ((![P : poly_poly_a, A2 : poly_a]: (((P = zero_z2096148049poly_a) => ((degree_poly_a @ (pCons_poly_a @ A2 @ P)) = zero_zero_nat)) & ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A2 @ P)) = (suc @ (degree_poly_a @ P)))))))). % degree_pCons_eq_if
thf(fact_102_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A : a, Q : poly_a]: (~ ((P = (pCons_a @ A @ Q)))))))))). % pCons_cases
thf(fact_103_pderiv_Ocases, axiom,
    ((![X : poly_a]: (~ ((![A : a, P2 : poly_a]: (~ ((X = (pCons_a @ A @ P2)))))))))). % pderiv.cases
thf(fact_104_mult__poly__add__left, axiom,
    ((![P : poly_a, Q2 : poly_a, R : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ P @ Q2) @ R) = (plus_plus_poly_a @ (times_times_poly_a @ P @ R) @ (times_times_poly_a @ Q2 @ R)))))). % mult_poly_add_left
thf(fact_105_pderiv_Oinduct, axiom,
    ((![P3 : poly_a > $o, A0 : poly_a]: ((![A : a, P2 : poly_a]: (((~ ((P2 = zero_zero_poly_a))) => (P3 @ P2)) => (P3 @ (pCons_a @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_106_pderiv_Oinduct, axiom,
    ((![P3 : poly_nat > $o, A0 : poly_nat]: ((![A : nat, P2 : poly_nat]: (((~ ((P2 = zero_zero_poly_nat))) => (P3 @ P2)) => (P3 @ (pCons_nat @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_107_pderiv_Oinduct, axiom,
    ((![P3 : poly_poly_a > $o, A0 : poly_poly_a]: ((![A : poly_a, P2 : poly_poly_a]: (((~ ((P2 = zero_z2096148049poly_a))) => (P3 @ P2)) => (P3 @ (pCons_poly_a @ A @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_108_poly__induct2, axiom,
    ((![P3 : poly_a > poly_a > $o, P : poly_a, Q2 : poly_a]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A : a, P2 : poly_a, B2 : a, Q : poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_a @ A @ P2) @ (pCons_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_109_poly__induct2, axiom,
    ((![P3 : poly_a > poly_nat > $o, P : poly_a, Q2 : poly_nat]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A : a, P2 : poly_a, B2 : nat, Q : poly_nat]: ((P3 @ P2 @ Q) => (P3 @ (pCons_a @ A @ P2) @ (pCons_nat @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_110_poly__induct2, axiom,
    ((![P3 : poly_a > poly_poly_a > $o, P : poly_a, Q2 : poly_poly_a]: ((P3 @ zero_zero_poly_a @ zero_z2096148049poly_a) => ((![A : a, P2 : poly_a, B2 : poly_a, Q : poly_poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_a @ A @ P2) @ (pCons_poly_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_111_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_a > $o, P : poly_nat, Q2 : poly_a]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A : nat, P2 : poly_nat, B2 : a, Q : poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_112_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_nat > $o, P : poly_nat, Q2 : poly_nat]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A : nat, P2 : poly_nat, B2 : nat, Q : poly_nat]: ((P3 @ P2 @ Q) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_nat @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_113_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_poly_a > $o, P : poly_nat, Q2 : poly_poly_a]: ((P3 @ zero_zero_poly_nat @ zero_z2096148049poly_a) => ((![A : nat, P2 : poly_nat, B2 : poly_a, Q : poly_poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_nat @ A @ P2) @ (pCons_poly_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_114_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_a > $o, P : poly_poly_a, Q2 : poly_a]: ((P3 @ zero_z2096148049poly_a @ zero_zero_poly_a) => ((![A : poly_a, P2 : poly_poly_a, B2 : a, Q : poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_poly_a @ A @ P2) @ (pCons_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_115_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_nat > $o, P : poly_poly_a, Q2 : poly_nat]: ((P3 @ zero_z2096148049poly_a @ zero_zero_poly_nat) => ((![A : poly_a, P2 : poly_poly_a, B2 : nat, Q : poly_nat]: ((P3 @ P2 @ Q) => (P3 @ (pCons_poly_a @ A @ P2) @ (pCons_nat @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_116_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_poly_a > $o, P : poly_poly_a, Q2 : poly_poly_a]: ((P3 @ zero_z2096148049poly_a @ zero_z2096148049poly_a) => ((![A : poly_a, P2 : poly_poly_a, B2 : poly_a, Q : poly_poly_a]: ((P3 @ P2 @ Q) => (P3 @ (pCons_poly_a @ A @ P2) @ (pCons_poly_a @ B2 @ Q)))) => (P3 @ P @ Q2)))))). % poly_induct2
thf(fact_117_power__mult, axiom,
    ((![A2 : a, M : nat, N : nat]: ((power_power_a @ A2 @ (times_times_nat @ M @ N)) = (power_power_a @ (power_power_a @ A2 @ M) @ N))))). % power_mult
thf(fact_118_power__mult, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A2 @ M) @ N))))). % power_mult
thf(fact_119_power__mult, axiom,
    ((![A2 : poly_a, M : nat, N : nat]: ((power_power_poly_a @ A2 @ (times_times_nat @ M @ N)) = (power_power_poly_a @ (power_power_poly_a @ A2 @ M) @ N))))). % power_mult
thf(fact_120_mult__poly__0__right, axiom,
    ((![P : poly_nat]: ((times_times_poly_nat @ P @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_poly_0_right
thf(fact_121_mult__poly__0__right, axiom,
    ((![P : poly_poly_a]: ((times_545135445poly_a @ P @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % mult_poly_0_right
thf(fact_122_mult__poly__0__right, axiom,
    ((![P : poly_a]: ((times_times_poly_a @ P @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_poly_0_right
thf(fact_123_mult__poly__0__left, axiom,
    ((![Q2 : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ Q2) = zero_zero_poly_nat)))). % mult_poly_0_left
thf(fact_124_mult__poly__0__left, axiom,
    ((![Q2 : poly_poly_a]: ((times_545135445poly_a @ zero_z2096148049poly_a @ Q2) = zero_z2096148049poly_a)))). % mult_poly_0_left
thf(fact_125_mult__poly__0__left, axiom,
    ((![Q2 : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q2) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_126_degree__power__eq, axiom,
    ((![P : poly_poly_a, N : nat]: ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (power_276493840poly_a @ P @ N)) = (times_times_nat @ N @ (degree_poly_a @ P))))))). % degree_power_eq
thf(fact_127_degree__power__eq, axiom,
    ((![P : poly_a, N : nat]: ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (power_power_poly_a @ P @ N)) = (times_times_nat @ N @ (degree_a @ P))))))). % degree_power_eq
thf(fact_128_degree__linear__power, axiom,
    ((![A2 : nat, N : nat]: ((degree_nat @ (power_power_poly_nat @ (pCons_nat @ A2 @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat)) @ N)) = N)))). % degree_linear_power
thf(fact_129_degree__linear__power, axiom,
    ((![A2 : poly_a, N : nat]: ((degree_poly_a @ (power_276493840poly_a @ (pCons_poly_a @ A2 @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a)) @ N)) = N)))). % degree_linear_power
thf(fact_130_degree__linear__power, axiom,
    ((![A2 : a, N : nat]: ((degree_a @ (power_power_poly_a @ (pCons_a @ A2 @ (pCons_a @ one_one_a @ zero_zero_poly_a)) @ N)) = N)))). % degree_linear_power
thf(fact_131_pCons__induct, axiom,
    ((![P3 : poly_poly_nat > $o, P : poly_poly_nat]: ((P3 @ zero_z1059985641ly_nat) => ((![A : poly_nat, P2 : poly_poly_nat]: (((~ ((A = zero_zero_poly_nat))) | (~ ((P2 = zero_z1059985641ly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_poly_nat @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_132_pCons__induct, axiom,
    ((![P3 : poly_poly_poly_a > $o, P : poly_poly_poly_a]: ((P3 @ zero_z2064990175poly_a) => ((![A : poly_poly_a, P2 : poly_poly_poly_a]: (((~ ((A = zero_z2096148049poly_a))) | (~ ((P2 = zero_z2064990175poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_poly_a @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_133_pCons__induct, axiom,
    ((![P3 : poly_a > $o, P : poly_a]: ((P3 @ zero_zero_poly_a) => ((![A : a, P2 : poly_a]: (((~ ((A = zero_zero_a))) | (~ ((P2 = zero_zero_poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_a @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_134_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A : nat, P2 : poly_nat]: (((~ ((A = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_135_pCons__induct, axiom,
    ((![P3 : poly_poly_a > $o, P : poly_poly_a]: ((P3 @ zero_z2096148049poly_a) => ((![A : poly_a, P2 : poly_poly_a]: (((~ ((A = zero_zero_poly_a))) | (~ ((P2 = zero_z2096148049poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_a @ A @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_136_pCons__one, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % pCons_one
thf(fact_137_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_138_pCons__one, axiom,
    (((pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a) = one_one_poly_poly_a))). % pCons_one
thf(fact_139_degree__eq__zeroE, axiom,
    ((![P : poly_a]: (((degree_a @ P) = zero_zero_nat) => (~ ((![A : a]: (~ ((P = (pCons_a @ A @ zero_zero_poly_a))))))))))). % degree_eq_zeroE
thf(fact_140_degree__eq__zeroE, axiom,
    ((![P : poly_nat]: (((degree_nat @ P) = zero_zero_nat) => (~ ((![A : nat]: (~ ((P = (pCons_nat @ A @ zero_zero_poly_nat))))))))))). % degree_eq_zeroE
thf(fact_141_degree__eq__zeroE, axiom,
    ((![P : poly_poly_a]: (((degree_poly_a @ P) = zero_zero_nat) => (~ ((![A : poly_a]: (~ ((P = (pCons_poly_a @ A @ zero_z2096148049poly_a))))))))))). % degree_eq_zeroE
thf(fact_142_degree__pCons__0, axiom,
    ((![A2 : a]: ((degree_a @ (pCons_a @ A2 @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_143_degree__pCons__0, axiom,
    ((![A2 : nat]: ((degree_nat @ (pCons_nat @ A2 @ zero_zero_poly_nat)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_144_degree__pCons__0, axiom,
    ((![A2 : poly_a]: ((degree_poly_a @ (pCons_poly_a @ A2 @ zero_z2096148049poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_145_degree__pCons__eq, axiom,
    ((![P : poly_a, A2 : a]: ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A2 @ P)) = (suc @ (degree_a @ P))))))). % degree_pCons_eq
thf(fact_146_degree__pCons__eq, axiom,
    ((![P : poly_nat, A2 : nat]: ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A2 @ P)) = (suc @ (degree_nat @ P))))))). % degree_pCons_eq
thf(fact_147_degree__pCons__eq, axiom,
    ((![P : poly_poly_a, A2 : poly_a]: ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A2 @ P)) = (suc @ (degree_poly_a @ P))))))). % degree_pCons_eq
thf(fact_148_degree__mult__eq__0, axiom,
    ((![P : poly_nat, Q2 : poly_nat]: (((degree_nat @ (times_times_poly_nat @ P @ Q2)) = zero_zero_nat) = (((P = zero_zero_poly_nat)) | ((((Q2 = zero_zero_poly_nat)) | ((((~ ((P = zero_zero_poly_nat)))) & ((((~ ((Q2 = zero_zero_poly_nat)))) & (((((degree_nat @ P) = zero_zero_nat)) & (((degree_nat @ Q2) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_149_degree__mult__eq__0, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a]: (((degree_poly_a @ (times_545135445poly_a @ P @ Q2)) = zero_zero_nat) = (((P = zero_z2096148049poly_a)) | ((((Q2 = zero_z2096148049poly_a)) | ((((~ ((P = zero_z2096148049poly_a)))) & ((((~ ((Q2 = zero_z2096148049poly_a)))) & (((((degree_poly_a @ P) = zero_zero_nat)) & (((degree_poly_a @ Q2) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_150_degree__mult__eq__0, axiom,
    ((![P : poly_a, Q2 : poly_a]: (((degree_a @ (times_times_poly_a @ P @ Q2)) = zero_zero_nat) = (((P = zero_zero_poly_a)) | ((((Q2 = zero_zero_poly_a)) | ((((~ ((P = zero_zero_poly_a)))) & ((((~ ((Q2 = zero_zero_poly_a)))) & (((((degree_a @ P) = zero_zero_nat)) & (((degree_a @ Q2) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_151_degree__mult__eq, axiom,
    ((![P : poly_nat, Q2 : poly_nat]: ((~ ((P = zero_zero_poly_nat))) => ((~ ((Q2 = zero_zero_poly_nat))) => ((degree_nat @ (times_times_poly_nat @ P @ Q2)) = (plus_plus_nat @ (degree_nat @ P) @ (degree_nat @ Q2)))))))). % degree_mult_eq
thf(fact_152_degree__mult__eq, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a]: ((~ ((P = zero_z2096148049poly_a))) => ((~ ((Q2 = zero_z2096148049poly_a))) => ((degree_poly_a @ (times_545135445poly_a @ P @ Q2)) = (plus_plus_nat @ (degree_poly_a @ P) @ (degree_poly_a @ Q2)))))))). % degree_mult_eq
thf(fact_153_degree__mult__eq, axiom,
    ((![P : poly_a, Q2 : poly_a]: ((~ ((P = zero_zero_poly_a))) => ((~ ((Q2 = zero_zero_poly_a))) => ((degree_a @ (times_times_poly_a @ P @ Q2)) = (plus_plus_nat @ (degree_a @ P) @ (degree_a @ Q2)))))))). % degree_mult_eq
thf(fact_154_mult__right__cancel, axiom,
    ((![C : poly_poly_a, A2 : poly_poly_a, B : poly_poly_a]: ((~ ((C = zero_z2096148049poly_a))) => (((times_545135445poly_a @ A2 @ C) = (times_545135445poly_a @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_155_mult__right__cancel, axiom,
    ((![C : a, A2 : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ A2 @ C) = (times_times_a @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_156_mult__right__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_157_mult__right__cancel, axiom,
    ((![C : poly_a, A2 : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ A2 @ C) = (times_times_poly_a @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_158_mult__left__cancel, axiom,
    ((![C : poly_poly_a, A2 : poly_poly_a, B : poly_poly_a]: ((~ ((C = zero_z2096148049poly_a))) => (((times_545135445poly_a @ C @ A2) = (times_545135445poly_a @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_159_mult__left__cancel, axiom,
    ((![C : a, A2 : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ C @ A2) = (times_times_a @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_160_mult__left__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_161_mult__left__cancel, axiom,
    ((![C : poly_a, A2 : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ C @ A2) = (times_times_poly_a @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_162_no__zero__divisors, axiom,
    ((![A2 : poly_nat, B : poly_nat]: ((~ ((A2 = zero_zero_poly_nat))) => ((~ ((B = zero_zero_poly_nat))) => (~ (((times_times_poly_nat @ A2 @ B) = zero_zero_poly_nat)))))))). % no_zero_divisors
thf(fact_163_no__zero__divisors, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: ((~ ((A2 = zero_z2096148049poly_a))) => ((~ ((B = zero_z2096148049poly_a))) => (~ (((times_545135445poly_a @ A2 @ B) = zero_z2096148049poly_a)))))))). % no_zero_divisors
thf(fact_164_no__zero__divisors, axiom,
    ((![A2 : a, B : a]: ((~ ((A2 = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (~ (((times_times_a @ A2 @ B) = zero_zero_a)))))))). % no_zero_divisors
thf(fact_165_no__zero__divisors, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A2 @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_166_no__zero__divisors, axiom,
    ((![A2 : poly_a, B : poly_a]: ((~ ((A2 = zero_zero_poly_a))) => ((~ ((B = zero_zero_poly_a))) => (~ (((times_times_poly_a @ A2 @ B) = zero_zero_poly_a)))))))). % no_zero_divisors
thf(fact_167_divisors__zero, axiom,
    ((![A2 : poly_nat, B : poly_nat]: (((times_times_poly_nat @ A2 @ B) = zero_zero_poly_nat) => ((A2 = zero_zero_poly_nat) | (B = zero_zero_poly_nat)))))). % divisors_zero
thf(fact_168_divisors__zero, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: (((times_545135445poly_a @ A2 @ B) = zero_z2096148049poly_a) => ((A2 = zero_z2096148049poly_a) | (B = zero_z2096148049poly_a)))))). % divisors_zero
thf(fact_169_divisors__zero, axiom,
    ((![A2 : a, B : a]: (((times_times_a @ A2 @ B) = zero_zero_a) => ((A2 = zero_zero_a) | (B = zero_zero_a)))))). % divisors_zero
thf(fact_170_divisors__zero, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) => ((A2 = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_171_divisors__zero, axiom,
    ((![A2 : poly_a, B : poly_a]: (((times_times_poly_a @ A2 @ B) = zero_zero_poly_a) => ((A2 = zero_zero_poly_a) | (B = zero_zero_poly_a)))))). % divisors_zero
thf(fact_172_mult__not__zero, axiom,
    ((![A2 : poly_nat, B : poly_nat]: ((~ (((times_times_poly_nat @ A2 @ B) = zero_zero_poly_nat))) => ((~ ((A2 = zero_zero_poly_nat))) & (~ ((B = zero_zero_poly_nat)))))))). % mult_not_zero
thf(fact_173_mult__not__zero, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: ((~ (((times_545135445poly_a @ A2 @ B) = zero_z2096148049poly_a))) => ((~ ((A2 = zero_z2096148049poly_a))) & (~ ((B = zero_z2096148049poly_a)))))))). % mult_not_zero
thf(fact_174_mult__not__zero, axiom,
    ((![A2 : a, B : a]: ((~ (((times_times_a @ A2 @ B) = zero_zero_a))) => ((~ ((A2 = zero_zero_a))) & (~ ((B = zero_zero_a)))))))). % mult_not_zero
thf(fact_175_mult__not__zero, axiom,
    ((![A2 : nat, B : nat]: ((~ (((times_times_nat @ A2 @ B) = zero_zero_nat))) => ((~ ((A2 = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_176_mult__not__zero, axiom,
    ((![A2 : poly_a, B : poly_a]: ((~ (((times_times_poly_a @ A2 @ B) = zero_zero_poly_a))) => ((~ ((A2 = zero_zero_poly_a))) & (~ ((B = zero_zero_poly_a)))))))). % mult_not_zero
thf(fact_177_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_a = one_one_poly_a))))). % zero_neq_one
thf(fact_178_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_179_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_180_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_nat = one_one_poly_nat))))). % zero_neq_one
thf(fact_181_zero__neq__one, axiom,
    ((~ ((zero_z2096148049poly_a = one_one_poly_poly_a))))). % zero_neq_one
thf(fact_182_combine__common__factor, axiom,
    ((![A2 : a, E : a, B : a, C : a]: ((plus_plus_a @ (times_times_a @ A2 @ E) @ (plus_plus_a @ (times_times_a @ B @ E) @ C)) = (plus_plus_a @ (times_times_a @ (plus_plus_a @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_183_combine__common__factor, axiom,
    ((![A2 : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A2 @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_184_combine__common__factor, axiom,
    ((![A2 : poly_a, E : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (times_times_poly_a @ A2 @ E) @ (plus_plus_poly_a @ (times_times_poly_a @ B @ E) @ C)) = (plus_plus_poly_a @ (times_times_poly_a @ (plus_plus_poly_a @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_185_distrib__right, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A2 @ B) @ C) = (plus_plus_a @ (times_times_a @ A2 @ C) @ (times_times_a @ B @ C)))))). % distrib_right
thf(fact_186_distrib__right, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_187_distrib__right, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ A2 @ B) @ C) = (plus_plus_poly_a @ (times_times_poly_a @ A2 @ C) @ (times_times_poly_a @ B @ C)))))). % distrib_right
thf(fact_188_distrib__left, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ A2 @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A2 @ B) @ (times_times_a @ A2 @ C)))))). % distrib_left
thf(fact_189_distrib__left, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ A2 @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A2 @ B) @ (times_times_nat @ A2 @ C)))))). % distrib_left
thf(fact_190_distrib__left, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ A2 @ (plus_plus_poly_a @ B @ C)) = (plus_plus_poly_a @ (times_times_poly_a @ A2 @ B) @ (times_times_poly_a @ A2 @ C)))))). % distrib_left
thf(fact_191_comm__semiring__class_Odistrib, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A2 @ B) @ C) = (plus_plus_a @ (times_times_a @ A2 @ C) @ (times_times_a @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_192_comm__semiring__class_Odistrib, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_193_comm__semiring__class_Odistrib, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ A2 @ B) @ C) = (plus_plus_poly_a @ (times_times_poly_a @ A2 @ C) @ (times_times_poly_a @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_194_ring__class_Oring__distribs_I1_J, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ A2 @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A2 @ B) @ (times_times_a @ A2 @ C)))))). % ring_class.ring_distribs(1)
thf(fact_195_ring__class_Oring__distribs_I1_J, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ A2 @ (plus_plus_poly_a @ B @ C)) = (plus_plus_poly_a @ (times_times_poly_a @ A2 @ B) @ (times_times_poly_a @ A2 @ C)))))). % ring_class.ring_distribs(1)
thf(fact_196_ring__class_Oring__distribs_I2_J, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A2 @ B) @ C) = (plus_plus_a @ (times_times_a @ A2 @ C) @ (times_times_a @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_197_ring__class_Oring__distribs_I2_J, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ A2 @ B) @ C) = (plus_plus_poly_a @ (times_times_poly_a @ A2 @ C) @ (times_times_poly_a @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_198_power__not__zero, axiom,
    ((![A2 : poly_nat, N : nat]: ((~ ((A2 = zero_zero_poly_nat))) => (~ (((power_power_poly_nat @ A2 @ N) = zero_zero_poly_nat))))))). % power_not_zero
thf(fact_199_power__not__zero, axiom,
    ((![A2 : poly_poly_a, N : nat]: ((~ ((A2 = zero_z2096148049poly_a))) => (~ (((power_276493840poly_a @ A2 @ N) = zero_z2096148049poly_a))))))). % power_not_zero
thf(fact_200_power__not__zero, axiom,
    ((![A2 : a, N : nat]: ((~ ((A2 = zero_zero_a))) => (~ (((power_power_a @ A2 @ N) = zero_zero_a))))))). % power_not_zero
thf(fact_201_power__not__zero, axiom,
    ((![A2 : nat, N : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_202_power__not__zero, axiom,
    ((![A2 : poly_a, N : nat]: ((~ ((A2 = zero_zero_poly_a))) => (~ (((power_power_poly_a @ A2 @ N) = zero_zero_poly_a))))))). % power_not_zero
thf(fact_203_power__commuting__commutes, axiom,
    ((![X : a, Y : a, N : nat]: (((times_times_a @ X @ Y) = (times_times_a @ Y @ X)) => ((times_times_a @ (power_power_a @ X @ N) @ Y) = (times_times_a @ Y @ (power_power_a @ X @ N))))))). % power_commuting_commutes
thf(fact_204_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_205_power__commuting__commutes, axiom,
    ((![X : poly_a, Y : poly_a, N : nat]: (((times_times_poly_a @ X @ Y) = (times_times_poly_a @ Y @ X)) => ((times_times_poly_a @ (power_power_poly_a @ X @ N) @ Y) = (times_times_poly_a @ Y @ (power_power_poly_a @ X @ N))))))). % power_commuting_commutes
thf(fact_206_power__mult__distrib, axiom,
    ((![A2 : a, B : a, N : nat]: ((power_power_a @ (times_times_a @ A2 @ B) @ N) = (times_times_a @ (power_power_a @ A2 @ N) @ (power_power_a @ B @ N)))))). % power_mult_distrib
thf(fact_207_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N) = (times_times_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_208_power__mult__distrib, axiom,
    ((![A2 : poly_a, B : poly_a, N : nat]: ((power_power_poly_a @ (times_times_poly_a @ A2 @ B) @ N) = (times_times_poly_a @ (power_power_poly_a @ A2 @ N) @ (power_power_poly_a @ B @ N)))))). % power_mult_distrib
thf(fact_209_power__commutes, axiom,
    ((![A2 : a, N : nat]: ((times_times_a @ (power_power_a @ A2 @ N) @ A2) = (times_times_a @ A2 @ (power_power_a @ A2 @ N)))))). % power_commutes
thf(fact_210_power__commutes, axiom,
    ((![A2 : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N)))))). % power_commutes
thf(fact_211_power__commutes, axiom,
    ((![A2 : poly_a, N : nat]: ((times_times_poly_a @ (power_power_poly_a @ A2 @ N) @ A2) = (times_times_poly_a @ A2 @ (power_power_poly_a @ A2 @ N)))))). % power_commutes
thf(fact_212_left__right__inverse__power, axiom,
    ((![X : poly_a, Y : poly_a, N : nat]: (((times_times_poly_a @ X @ Y) = one_one_poly_a) => ((times_times_poly_a @ (power_power_poly_a @ X @ N) @ (power_power_poly_a @ Y @ N)) = one_one_poly_a))))). % left_right_inverse_power
thf(fact_213_mult__Suc__right, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ M @ (suc @ N)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N)))))). % mult_Suc_right
thf(fact_214_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_215_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_216_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_217_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_218_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_219_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_220_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_221_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_222_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_223_mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel1
thf(fact_224_mult__cancel2, axiom,
    ((![M : nat, K2 : nat, N : nat]: (((times_times_nat @ M @ K2) = (times_times_nat @ N @ K2)) = (((M = N)) | ((K2 = zero_zero_nat))))))). % mult_cancel2
thf(fact_225_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_226_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_227_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_228_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_229_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_230_Suc__mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K2) @ M) = (times_times_nat @ (suc @ K2) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_231_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K2 : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K2) = (plus_plus_nat @ (times_times_nat @ M @ K2) @ (times_times_nat @ N @ K2)))))). % add_mult_distrib
thf(fact_232_add__mult__distrib2, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((times_times_nat @ K2 @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K2 @ M) @ (times_times_nat @ K2 @ N)))))). % add_mult_distrib2
thf(fact_233_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_234_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_235_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_236_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_237_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_238_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_239_nat__induct, axiom,
    ((![P3 : nat > $o, N : nat]: ((P3 @ zero_zero_nat) => ((![N2 : nat]: ((P3 @ N2) => (P3 @ (suc @ N2)))) => (P3 @ N)))))). % nat_induct
thf(fact_240_diff__induct, axiom,
    ((![P3 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P3 @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P3 @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P3 @ X3 @ Y3) => (P3 @ (suc @ X3) @ (suc @ Y3)))) => (P3 @ M @ N))))))). % diff_induct
thf(fact_241_zero__induct, axiom,
    ((![P3 : nat > $o, K2 : nat]: ((P3 @ K2) => ((![N2 : nat]: ((P3 @ (suc @ N2)) => (P3 @ N2))) => (P3 @ zero_zero_nat)))))). % zero_induct
thf(fact_242_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_243_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_244_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc

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

% Conjectures (8)
thf(conj_0, hypothesis,
    ((~ ((cs = zero_zero_poly_a))))).
thf(conj_1, hypothesis,
    ($true)).
thf(conj_2, hypothesis,
    ((~ ((z = zero_zero_a))))).
thf(conj_3, hypothesis,
    ((~ (((plus_plus_a @ a2 @ (times_times_a @ z @ (poly_a2 @ q @ z))) = zero_zero_a))))).
thf(conj_4, hypothesis,
    ((~ ((a2 = zero_zero_a))))).
thf(conj_5, hypothesis,
    (((plus_plus_nat @ (if_nat @ (q = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ q))) @ k) = (degree_a @ cs)))).
thf(conj_6, hypothesis,
    ((![Z2 : a]: ((poly_a2 @ cs @ Z2) = (times_times_a @ (power_power_a @ Z2 @ k) @ (plus_plus_a @ a2 @ (times_times_a @ Z2 @ (poly_a2 @ q @ Z2)))))))).
thf(conj_7, conjecture,
    ((?[A3 : a]: ((~ ((A3 = zero_zero_a))) & (?[Q3 : poly_a]: (((~ ((Q3 = zero_zero_poly_a))) | (((plus_plus_nat @ k @ one_one_nat) = (suc @ (degree_a @ cs))) & (![Z : a]: ((times_times_a @ Z @ (times_times_a @ (power_power_a @ Z @ k) @ (plus_plus_a @ a2 @ (times_times_a @ Z @ (poly_a2 @ q @ Z))))) = (times_times_a @ (power_power_a @ Z @ (plus_plus_nat @ k @ one_one_nat)) @ A3))))) & ((Q3 = zero_zero_poly_a) | (((plus_plus_nat @ (degree_a @ Q3) @ (plus_plus_nat @ k @ one_one_nat)) = (degree_a @ cs)) & (![Z : a]: ((times_times_a @ Z @ (times_times_a @ (power_power_a @ Z @ k) @ (plus_plus_a @ a2 @ (times_times_a @ Z @ (poly_a2 @ q @ Z))))) = (times_times_a @ (power_power_a @ Z @ (plus_plus_nat @ k @ one_one_nat)) @ (plus_plus_a @ A3 @ (times_times_a @ Z @ (poly_a2 @ Q3 @ Z)))))))))))))).
