% 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_596__5373884_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:27.846

% 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 (40)
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_c____, type,
    c : a).
thf(sy_v_cs____, type,
    cs : poly_a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (244)
thf(fact_0_nz, axiom,
    ((~ ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ p @ Z) = zero_zero_a))))))). % nz
thf(fact_1_False, axiom,
    ((~ ((c = zero_zero_a))))). % False
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_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_6_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_7_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_8_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_9_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_10_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_11_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_12_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % poly_0
thf(fact_13_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_14_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_15_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_16_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_17_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_18_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_19_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_20_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_21_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_22_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_23_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_24_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_25_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_26_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_27_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_28_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_29_power__Suc0__right, axiom,
    ((![A2 : a]: ((power_power_a @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_30_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_31_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_32_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_33_power__0__Suc, axiom,
    ((![N : nat]: ((power_276493840poly_a @ zero_z2096148049poly_a @ (suc @ N)) = zero_z2096148049poly_a)))). % power_0_Suc
thf(fact_34_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_a @ zero_zero_a @ (suc @ N)) = zero_zero_a)))). % power_0_Suc
thf(fact_35_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_36_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_37_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_38_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_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_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_41_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_42_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_43_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_44_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_45_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_46_pCons__eq__iff, axiom,
    ((![A2 : nat, P : poly_nat, B : nat, Q2 : poly_nat]: (((pCons_nat @ A2 @ P) = (pCons_nat @ B @ Q2)) = (((A2 = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_47_pCons__eq__iff, axiom,
    ((![A2 : poly_a, P : poly_poly_a, B : poly_a, Q2 : poly_poly_a]: (((pCons_poly_a @ A2 @ P) = (pCons_poly_a @ B @ Q2)) = (((A2 = B)) & ((P = Q2))))))). % pCons_eq_iff
thf(fact_48_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_49_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_50_add__pCons, axiom,
    ((![A2 : poly_a, P : poly_poly_a, B : poly_a, Q2 : poly_poly_a]: ((plus_p1976640465poly_a @ (pCons_poly_a @ A2 @ P) @ (pCons_poly_a @ B @ Q2)) = (pCons_poly_a @ (plus_plus_poly_a @ A2 @ B) @ (plus_p1976640465poly_a @ P @ Q2)))))). % add_pCons
thf(fact_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_pderiv_Ocases, axiom,
    ((![X : poly_a]: (~ ((![A : a, P2 : poly_a]: (~ ((X = (pCons_a @ A @ P2)))))))))). % pderiv.cases
thf(fact_70_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A : nat, P2 : poly_nat]: (~ ((X = (pCons_nat @ A @ P2)))))))))). % pderiv.cases
thf(fact_71_pderiv_Ocases, axiom,
    ((![X : poly_poly_a]: (~ ((![A : poly_a, P2 : poly_poly_a]: (~ ((X = (pCons_poly_a @ A @ P2)))))))))). % pderiv.cases
thf(fact_72_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A : a, Q : poly_a]: (~ ((P = (pCons_a @ A @ Q)))))))))). % pCons_cases
thf(fact_73_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A : nat, Q : poly_nat]: (~ ((P = (pCons_nat @ A @ Q)))))))))). % pCons_cases
thf(fact_74_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A : poly_a, Q : poly_poly_a]: (~ ((P = (pCons_poly_a @ A @ Q)))))))))). % pCons_cases
thf(fact_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_power__Suc2, axiom,
    ((![A2 : a, N : nat]: ((power_power_a @ A2 @ (suc @ N)) = (times_times_a @ (power_power_a @ A2 @ N) @ A2))))). % power_Suc2
thf(fact_99_power__Suc2, axiom,
    ((![A2 : nat, N : nat]: ((power_power_nat @ A2 @ (suc @ N)) = (times_times_nat @ (power_power_nat @ A2 @ N) @ A2))))). % power_Suc2
thf(fact_100_power__Suc, axiom,
    ((![A2 : a, N : nat]: ((power_power_a @ A2 @ (suc @ N)) = (times_times_a @ A2 @ (power_power_a @ A2 @ N)))))). % power_Suc
thf(fact_101_power__Suc, axiom,
    ((![A2 : nat, N : nat]: ((power_power_nat @ A2 @ (suc @ N)) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N)))))). % power_Suc
thf(fact_102_power__add, axiom,
    ((![A2 : a, M : nat, N : nat]: ((power_power_a @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_a @ (power_power_a @ A2 @ M) @ (power_power_a @ A2 @ N)))))). % power_add
thf(fact_103_power__add, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A2 @ M) @ (power_power_nat @ A2 @ N)))))). % power_add
thf(fact_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_degree__pCons__0, axiom,
    ((![A2 : a]: ((degree_a @ (pCons_a @ A2 @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_113_degree__pCons__0, axiom,
    ((![A2 : nat]: ((degree_nat @ (pCons_nat @ A2 @ zero_zero_poly_nat)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_114_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_115_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_116_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_117_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_118_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_119_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_120_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_121_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_122_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_123_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_124_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_125_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_126_add__cancel__right__right, axiom,
    ((![A2 : a, B : a]: ((A2 = (plus_plus_a @ A2 @ B)) = (B = zero_zero_a))))). % add_cancel_right_right
thf(fact_127_add__cancel__right__right, axiom,
    ((![A2 : poly_a, B : poly_a]: ((A2 = (plus_plus_poly_a @ A2 @ B)) = (B = zero_zero_poly_a))))). % add_cancel_right_right
thf(fact_128_add__cancel__right__right, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ A2 @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_129_add__cancel__right__right, axiom,
    ((![A2 : poly_nat, B : poly_nat]: ((A2 = (plus_plus_poly_nat @ A2 @ B)) = (B = zero_zero_poly_nat))))). % add_cancel_right_right
thf(fact_130_add__cancel__right__right, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: ((A2 = (plus_p1976640465poly_a @ A2 @ B)) = (B = zero_z2096148049poly_a))))). % add_cancel_right_right
thf(fact_131_add__left__cancel, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_132_add__left__cancel, axiom,
    ((![A2 : a, B : a, C : a]: (((plus_plus_a @ A2 @ B) = (plus_plus_a @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_133_add__right__cancel, axiom,
    ((![B : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B @ A2) = (plus_plus_nat @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_134_add__right__cancel, axiom,
    ((![B : a, A2 : a, C : a]: (((plus_plus_a @ B @ A2) = (plus_plus_a @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_135_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_136_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_137_add_Oleft__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ zero_zero_a @ A2) = A2)))). % add.left_neutral
thf(fact_138_add_Oleft__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A2) = A2)))). % add.left_neutral
thf(fact_139_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_140_add_Oleft__neutral, axiom,
    ((![A2 : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A2) = A2)))). % add.left_neutral
thf(fact_141_add_Oleft__neutral, axiom,
    ((![A2 : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A2) = A2)))). % add.left_neutral
thf(fact_142_add_Oright__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ A2 @ zero_zero_a) = A2)))). % add.right_neutral
thf(fact_143_add_Oright__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ A2 @ zero_zero_poly_a) = A2)))). % add.right_neutral
thf(fact_144_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_145_add_Oright__neutral, axiom,
    ((![A2 : poly_nat]: ((plus_plus_poly_nat @ A2 @ zero_zero_poly_nat) = A2)))). % add.right_neutral
thf(fact_146_add_Oright__neutral, axiom,
    ((![A2 : poly_poly_a]: ((plus_p1976640465poly_a @ A2 @ zero_z2096148049poly_a) = A2)))). % add.right_neutral
thf(fact_147_add__cancel__left__left, axiom,
    ((![B : a, A2 : a]: (((plus_plus_a @ B @ A2) = A2) = (B = zero_zero_a))))). % add_cancel_left_left
thf(fact_148_add__cancel__left__left, axiom,
    ((![B : poly_a, A2 : poly_a]: (((plus_plus_poly_a @ B @ A2) = A2) = (B = zero_zero_poly_a))))). % add_cancel_left_left
thf(fact_149_add__cancel__left__left, axiom,
    ((![B : nat, A2 : nat]: (((plus_plus_nat @ B @ A2) = A2) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_150_add__cancel__left__left, axiom,
    ((![B : poly_nat, A2 : poly_nat]: (((plus_plus_poly_nat @ B @ A2) = A2) = (B = zero_zero_poly_nat))))). % add_cancel_left_left
thf(fact_151_add__cancel__left__left, axiom,
    ((![B : poly_poly_a, A2 : poly_poly_a]: (((plus_p1976640465poly_a @ B @ A2) = A2) = (B = zero_z2096148049poly_a))))). % add_cancel_left_left
thf(fact_152_add__cancel__left__right, axiom,
    ((![A2 : a, B : a]: (((plus_plus_a @ A2 @ B) = A2) = (B = zero_zero_a))))). % add_cancel_left_right
thf(fact_153_add__cancel__left__right, axiom,
    ((![A2 : poly_a, B : poly_a]: (((plus_plus_poly_a @ A2 @ B) = A2) = (B = zero_zero_poly_a))))). % add_cancel_left_right
thf(fact_154_add__cancel__left__right, axiom,
    ((![A2 : nat, B : nat]: (((plus_plus_nat @ A2 @ B) = A2) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_155_add__cancel__left__right, axiom,
    ((![A2 : poly_nat, B : poly_nat]: (((plus_plus_poly_nat @ A2 @ B) = A2) = (B = zero_zero_poly_nat))))). % add_cancel_left_right
thf(fact_156_add__cancel__left__right, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: (((plus_p1976640465poly_a @ A2 @ B) = A2) = (B = zero_z2096148049poly_a))))). % add_cancel_left_right
thf(fact_157_add__cancel__right__left, axiom,
    ((![A2 : a, B : a]: ((A2 = (plus_plus_a @ B @ A2)) = (B = zero_zero_a))))). % add_cancel_right_left
thf(fact_158_add__cancel__right__left, axiom,
    ((![A2 : poly_a, B : poly_a]: ((A2 = (plus_plus_poly_a @ B @ A2)) = (B = zero_zero_poly_a))))). % add_cancel_right_left
thf(fact_159_add__cancel__right__left, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ B @ A2)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_160_add__cancel__right__left, axiom,
    ((![A2 : poly_nat, B : poly_nat]: ((A2 = (plus_plus_poly_nat @ B @ A2)) = (B = zero_zero_poly_nat))))). % add_cancel_right_left
thf(fact_161_add__cancel__right__left, axiom,
    ((![A2 : poly_poly_a, B : poly_poly_a]: ((A2 = (plus_p1976640465poly_a @ B @ A2)) = (B = zero_z2096148049poly_a))))). % add_cancel_right_left
thf(fact_162_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_163_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_164_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_165_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_166_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_167_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_168_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_169_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_170_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_171_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A2 @ B) @ C) = (times_times_a @ A2 @ (times_times_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_172_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A2 @ B) @ C) = (times_times_nat @ A2 @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_173_mult_Oassoc, axiom,
    ((![A2 : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A2 @ B) @ C) = (times_times_a @ A2 @ (times_times_a @ B @ C)))))). % mult.assoc
thf(fact_174_mult_Oassoc, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A2 @ B) @ C) = (times_times_nat @ A2 @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_175_mult_Ocommute, axiom,
    ((times_times_a = (^[A3 : a]: (^[B3 : a]: (times_times_a @ B3 @ A3)))))). % mult.commute
thf(fact_176_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_177_mult_Oleft__commute, axiom,
    ((![B : a, A2 : a, C : a]: ((times_times_a @ B @ (times_times_a @ A2 @ C)) = (times_times_a @ A2 @ (times_times_a @ B @ C)))))). % mult.left_commute
thf(fact_178_mult_Oleft__commute, axiom,
    ((![B : nat, A2 : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A2 @ C)) = (times_times_nat @ A2 @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_179_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_180_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A2 : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A2 @ B) @ C) = (plus_plus_a @ A2 @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_181_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K2 : nat, L : nat]: (((I = J) & (K2 = L)) => ((plus_plus_nat @ I @ K2) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_182_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K2 : nat, A2 : nat, B : nat]: ((A4 = (plus_plus_nat @ K2 @ A2)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K2 @ (plus_plus_nat @ A2 @ B))))))). % group_cancel.add1
thf(fact_183_group__cancel_Oadd1, axiom,
    ((![A4 : a, K2 : a, A2 : a, B : a]: ((A4 = (plus_plus_a @ K2 @ A2)) => ((plus_plus_a @ A4 @ B) = (plus_plus_a @ K2 @ (plus_plus_a @ A2 @ B))))))). % group_cancel.add1
thf(fact_184_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K2 : nat, B : nat, A2 : nat]: ((B4 = (plus_plus_nat @ K2 @ B)) => ((plus_plus_nat @ A2 @ B4) = (plus_plus_nat @ K2 @ (plus_plus_nat @ A2 @ B))))))). % group_cancel.add2
thf(fact_185_group__cancel_Oadd2, axiom,
    ((![B4 : a, K2 : a, B : a, A2 : a]: ((B4 = (plus_plus_a @ K2 @ B)) => ((plus_plus_a @ A2 @ B4) = (plus_plus_a @ K2 @ (plus_plus_a @ A2 @ B))))))). % group_cancel.add2
thf(fact_186_add_Oassoc, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_187_add_Oassoc, axiom,
    ((![A2 : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A2 @ B) @ C) = (plus_plus_a @ A2 @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_188_add_Oleft__cancel, axiom,
    ((![A2 : a, B : a, C : a]: (((plus_plus_a @ A2 @ B) = (plus_plus_a @ A2 @ C)) = (B = C))))). % add.left_cancel
thf(fact_189_add_Oright__cancel, axiom,
    ((![B : a, A2 : a, C : a]: (((plus_plus_a @ B @ A2) = (plus_plus_a @ C @ A2)) = (B = C))))). % add.right_cancel
thf(fact_190_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_191_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_192_add_Oleft__commute, axiom,
    ((![B : nat, A2 : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A2 @ C)) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_193_add_Oleft__commute, axiom,
    ((![B : a, A2 : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A2 @ C)) = (plus_plus_a @ A2 @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_194_add__left__imp__eq, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ A2 @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_195_add__left__imp__eq, axiom,
    ((![A2 : a, B : a, C : a]: (((plus_plus_a @ A2 @ B) = (plus_plus_a @ A2 @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_196_add__right__imp__eq, axiom,
    ((![B : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B @ A2) = (plus_plus_nat @ C @ A2)) => (B = C))))). % add_right_imp_eq
thf(fact_197_add__right__imp__eq, axiom,
    ((![B : a, A2 : a, C : a]: (((plus_plus_a @ B @ A2) = (plus_plus_a @ C @ A2)) => (B = C))))). % add_right_imp_eq
thf(fact_198_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_199_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_200_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_201_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_202_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_203_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_204_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : a]: ((plus_plus_a @ zero_zero_a @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_205_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_206_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_207_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_208_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_209_add_Ocomm__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ A2 @ zero_zero_a) = A2)))). % add.comm_neutral
thf(fact_210_add_Ocomm__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ A2 @ zero_zero_poly_a) = A2)))). % add.comm_neutral
thf(fact_211_add_Ocomm__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.comm_neutral
thf(fact_212_add_Ocomm__neutral, axiom,
    ((![A2 : poly_nat]: ((plus_plus_poly_nat @ A2 @ zero_zero_poly_nat) = A2)))). % add.comm_neutral
thf(fact_213_add_Ocomm__neutral, axiom,
    ((![A2 : poly_poly_a]: ((plus_p1976640465poly_a @ A2 @ zero_z2096148049poly_a) = A2)))). % add.comm_neutral
thf(fact_214_add_Ogroup__left__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ zero_zero_a @ A2) = A2)))). % add.group_left_neutral
thf(fact_215_add_Ogroup__left__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A2) = A2)))). % add.group_left_neutral
thf(fact_216_add_Ogroup__left__neutral, axiom,
    ((![A2 : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A2) = A2)))). % add.group_left_neutral
thf(fact_217_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_218_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_219_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_220_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_221_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_222_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_223_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_224_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_225_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_226_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_227_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_228_old_Onat_Oinducts, axiom,
    ((![P3 : nat > $o, Nat : nat]: ((P3 @ zero_zero_nat) => ((![Nat3 : nat]: ((P3 @ Nat3) => (P3 @ (suc @ Nat3)))) => (P3 @ Nat)))))). % old.nat.inducts
thf(fact_229_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_230_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_231_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_232_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_233_nat__arith_Osuc1, axiom,
    ((![A4 : nat, K2 : nat, A2 : nat]: ((A4 = (plus_plus_nat @ K2 @ A2)) => ((suc @ A4) = (plus_plus_nat @ K2 @ (suc @ A2))))))). % nat_arith.suc1
thf(fact_234_add__Suc, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc
thf(fact_235_mult__Suc, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ (suc @ M) @ N) = (plus_plus_nat @ N @ (times_times_nat @ M @ N)))))). % mult_Suc
thf(fact_236_add__is__1, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = (suc @ zero_zero_nat)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_237_one__is__add, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M @ N)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_238_mult__zero__left, axiom,
    ((![A2 : a]: ((times_times_a @ zero_zero_a @ A2) = zero_zero_a)))). % mult_zero_left
thf(fact_239_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_240_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K2 : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K2)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K2))))). % left_add_mult_distrib
thf(fact_241_nat__mult__eq__cancel__disj, axiom,
    ((![K2 : nat, M : nat, N : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N)) = (((K2 = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_242_Euclid__induct, axiom,
    ((![P3 : nat > nat > $o, A2 : nat, B : nat]: ((![A : nat, B2 : nat]: ((P3 @ A @ B2) = (P3 @ B2 @ A))) => ((![A : nat]: (P3 @ A @ zero_zero_nat)) => ((![A : nat, B2 : nat]: ((P3 @ A @ B2) => (P3 @ A @ (plus_plus_nat @ A @ B2)))) => (P3 @ A2 @ B))))))). % Euclid_induct
thf(fact_243_exists__least__lemma, axiom,
    ((![P3 : nat > $o]: ((~ ((P3 @ zero_zero_nat))) => ((?[X_1 : nat]: (P3 @ X_1)) => (?[N2 : nat]: ((~ ((P3 @ N2))) & (P3 @ (suc @ N2))))))))). % exists_least_lemma

% 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 (1)
thf(conj_0, conjecture,
    ((?[A5 : a]: ((~ ((A5 = zero_zero_a))) & (?[Q3 : poly_a]: (((((pCons_a @ c @ cs) = zero_zero_poly_a) => ((suc @ (plus_plus_nat @ (if_nat @ (Q3 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ Q3))) @ zero_zero_nat)) = zero_zero_nat)) & ((~ (((pCons_a @ c @ cs) = zero_zero_poly_a))) => ((suc @ (plus_plus_nat @ (if_nat @ (Q3 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ Q3))) @ zero_zero_nat)) = (suc @ (degree_a @ (pCons_a @ c @ cs)))))) & (![Z : a]: ((poly_a2 @ (pCons_a @ c @ cs) @ Z) = (times_times_a @ (power_power_a @ Z @ zero_zero_nat) @ (poly_a2 @ (pCons_a @ A5 @ Q3) @ Z)))))))))).
