% 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_628__5374136_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:44.154

% Could-be-implicit typings (5)
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 (37)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1032801442poly_a : poly_poly_a > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_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_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_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
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_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).

% Relevant facts (225)
thf(fact_0_nc, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ p)))))). % nc
thf(fact_1__092_060open_062_092_060not_062_A_I_092_060forall_062z_O_Az_A_092_060noteq_062_A_I0_058_058_Ha_J_A_092_060longrightarrow_062_Apoly_Acs_Az_A_061_A_I0_058_058_Ha_J_J_092_060close_062, axiom,
    ((~ ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ cs @ Z) = zero_zero_a))))))). % \<open>\<not> (\<forall>z. z \<noteq> (0::'a) \<longrightarrow> poly cs z = (0::'a))\<close>
thf(fact_2__092_060open_062_092_060exists_062k_Aa_Aq_O_Aa_A_092_060noteq_062_A_I0_058_058_Ha_J_A_092_060and_062_ASuc_A_Ipsize_Aq_A_L_Ak_J_A_061_Apsize_Acs_A_092_060and_062_A_I_092_060forall_062z_O_Apoly_Acs_Az_A_061_Az_A_094_Ak_A_K_Apoly_A_IpCons_Aa_Aq_J_Az_J_092_060close_062, axiom,
    ((?[K : nat, A : a, Q : poly_a]: ((~ ((A = zero_zero_a))) & (((suc @ (plus_plus_nat @ (fundam247907092size_a @ Q) @ K)) = (fundam247907092size_a @ cs)) & (![Z2 : a]: ((poly_a2 @ cs @ Z2) = (times_times_a @ (power_power_a @ Z2 @ K) @ (poly_a2 @ (pCons_a @ A @ Q) @ Z2))))))))). % \<open>\<exists>k a q. a \<noteq> (0::'a) \<and> Suc (psize q + k) = psize cs \<and> (\<forall>z. poly cs z = z ^ k * poly (pCons a q) z)\<close>
thf(fact_3_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_4_poly__offset, axiom,
    ((![P : poly_poly_a, A2 : poly_a]: (?[Q : poly_poly_a]: (((fundam1032801442poly_a @ Q) = (fundam1032801442poly_a @ P)) & (![X : poly_a]: ((poly_poly_a2 @ Q @ X) = (poly_poly_a2 @ P @ (plus_plus_poly_a @ A2 @ X))))))))). % poly_offset
thf(fact_5_poly__offset, axiom,
    ((![P : poly_a, A2 : a]: (?[Q : poly_a]: (((fundam247907092size_a @ Q) = (fundam247907092size_a @ P)) & (![X : a]: ((poly_a2 @ Q @ X) = (poly_a2 @ P @ (plus_plus_a @ A2 @ X))))))))). % poly_offset
thf(fact_6_poly__decompose__lemma, axiom,
    ((![P : poly_poly_a]: ((~ ((![Z2 : poly_a]: ((~ ((Z2 = zero_zero_poly_a))) => ((poly_poly_a2 @ P @ Z2) = zero_zero_poly_a))))) => (?[K : nat, A : poly_a, Q : poly_poly_a]: ((~ ((A = zero_zero_poly_a))) & (((suc @ (plus_plus_nat @ (fundam1032801442poly_a @ Q) @ K)) = (fundam1032801442poly_a @ P)) & (![Z2 : poly_a]: ((poly_poly_a2 @ P @ Z2) = (times_times_poly_a @ (power_power_poly_a @ Z2 @ K) @ (poly_poly_a2 @ (pCons_poly_a @ A @ Q) @ Z2))))))))))). % poly_decompose_lemma
thf(fact_7_poly__decompose__lemma, axiom,
    ((![P : poly_a]: ((~ ((![Z2 : a]: ((~ ((Z2 = zero_zero_a))) => ((poly_a2 @ P @ Z2) = zero_zero_a))))) => (?[K : nat, A : a, Q : poly_a]: ((~ ((A = zero_zero_a))) & (((suc @ (plus_plus_nat @ (fundam247907092size_a @ Q) @ K)) = (fundam247907092size_a @ P)) & (![Z2 : a]: ((poly_a2 @ P @ Z2) = (times_times_a @ (power_power_a @ Z2 @ K) @ (poly_a2 @ (pCons_a @ A @ Q) @ Z2))))))))))). % poly_decompose_lemma
thf(fact_8_poly__pCons, axiom,
    ((![A2 : poly_a, P : poly_poly_a, X2 : poly_a]: ((poly_poly_a2 @ (pCons_poly_a @ A2 @ P) @ X2) = (plus_plus_poly_a @ A2 @ (times_times_poly_a @ X2 @ (poly_poly_a2 @ P @ X2))))))). % poly_pCons
thf(fact_9_poly__pCons, axiom,
    ((![A2 : a, P : poly_a, X2 : a]: ((poly_a2 @ (pCons_a @ A2 @ P) @ X2) = (plus_plus_a @ A2 @ (times_times_a @ X2 @ (poly_a2 @ P @ X2))))))). % poly_pCons
thf(fact_10_poly__pCons, axiom,
    ((![A2 : nat, P : poly_nat, X2 : nat]: ((poly_nat2 @ (pCons_nat @ A2 @ P) @ X2) = (plus_plus_nat @ A2 @ (times_times_nat @ X2 @ (poly_nat2 @ P @ X2))))))). % poly_pCons
thf(fact_11_power__eq__0__iff, axiom,
    ((![A2 : poly_a, N : nat]: (((power_power_poly_a @ A2 @ N) = zero_zero_poly_a) = (((A2 = zero_zero_poly_a)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_12_power__eq__0__iff, axiom,
    ((![A2 : a, N : nat]: (((power_power_a @ A2 @ N) = zero_zero_a) = (((A2 = zero_zero_a)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_13_power__eq__0__iff, axiom,
    ((![A2 : nat, N : nat]: (((power_power_nat @ A2 @ N) = zero_zero_nat) = (((A2 = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_14_add__gr__0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_15_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_16_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_17_power__Suc0__right, axiom,
    ((![A2 : a]: ((power_power_a @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_18_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_19_power__Suc0__right, axiom,
    ((![A2 : poly_a]: ((power_power_poly_a @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_20_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_a @ zero_zero_a @ (suc @ N)) = zero_zero_a)))). % power_0_Suc
thf(fact_21_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_22_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_23_add__less__same__cancel1, axiom,
    ((![B : nat, A2 : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A2) @ B) = (ord_less_nat @ A2 @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_24_add__less__same__cancel2, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ B) @ B) = (ord_less_nat @ A2 @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_25_less__add__same__cancel1, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ (plus_plus_nat @ A2 @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_26_less__add__same__cancel2, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ (plus_plus_nat @ B @ A2)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_27_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_28_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_29_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_30_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_31_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_32_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_33_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_34_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_35_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_36_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_37_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_38_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_39_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_40_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_41_zero__eq__add__iff__both__eq__0, axiom,
    ((![X2 : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X2 @ Y)) = (((X2 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_42_add__eq__0__iff__both__eq__0, axiom,
    ((![X2 : nat, Y : nat]: (((plus_plus_nat @ X2 @ Y) = zero_zero_nat) = (((X2 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_add_Oright__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ A2 @ zero_zero_a) = A2)))). % add.right_neutral
thf(fact_56_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_57_add_Oright__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ A2 @ zero_zero_poly_a) = A2)))). % add.right_neutral
thf(fact_58_add_Oleft__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ zero_zero_a @ A2) = A2)))). % add.left_neutral
thf(fact_59_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_60_add_Oleft__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A2) = A2)))). % add.left_neutral
thf(fact_61_add__less__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A2 @ B))))). % add_less_cancel_right
thf(fact_62_add__less__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A2 @ B))))). % add_less_cancel_left
thf(fact_63_nat__power__eq__Suc__0__iff, axiom,
    ((![X2 : nat, M : nat]: (((power_power_nat @ X2 @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X2 = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_64_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_65_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_66_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_67_nat__zero__less__power__iff, axiom,
    ((![X2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X2 @ N)) = (((ord_less_nat @ zero_zero_nat @ X2)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_68_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_69_mult__less__cancel2, axiom,
    ((![M : nat, K2 : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K2) @ (times_times_nat @ N @ K2)) = (((ord_less_nat @ zero_zero_nat @ K2)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_70_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_71_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_72_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_73_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_74_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_75_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_76_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_77_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_78_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_79_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_80_nat__add__left__cancel__less, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K2 @ M) @ (plus_plus_nat @ K2 @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_81_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_82_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_83_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_84_poly__mult, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X2 : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q2) @ X2) = (times_times_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q2 @ X2)))))). % poly_mult
thf(fact_85_poly__mult, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a, X2 : poly_a]: ((poly_poly_a2 @ (times_545135445poly_a @ P @ Q2) @ X2) = (times_times_poly_a @ (poly_poly_a2 @ P @ X2) @ (poly_poly_a2 @ Q2 @ X2)))))). % poly_mult
thf(fact_86_poly__mult, axiom,
    ((![P : poly_a, Q2 : poly_a, X2 : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q2) @ X2) = (times_times_a @ (poly_a2 @ P @ X2) @ (poly_a2 @ Q2 @ X2)))))). % poly_mult
thf(fact_87_poly__add, axiom,
    ((![P : poly_poly_a, Q2 : poly_poly_a, X2 : poly_a]: ((poly_poly_a2 @ (plus_p1976640465poly_a @ P @ Q2) @ X2) = (plus_plus_poly_a @ (poly_poly_a2 @ P @ X2) @ (poly_poly_a2 @ Q2 @ X2)))))). % poly_add
thf(fact_88_poly__add, axiom,
    ((![P : poly_nat, Q2 : poly_nat, X2 : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q2) @ X2) = (plus_plus_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q2 @ X2)))))). % poly_add
thf(fact_89_poly__add, axiom,
    ((![P : poly_a, Q2 : poly_a, X2 : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q2) @ X2) = (plus_plus_a @ (poly_a2 @ P @ X2) @ (poly_a2 @ Q2 @ X2)))))). % poly_add
thf(fact_90_poly__power, axiom,
    ((![P : poly_nat, N : nat, X2 : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X2) = (power_power_nat @ (poly_nat2 @ P @ X2) @ N))))). % poly_power
thf(fact_91_poly__power, axiom,
    ((![P : poly_poly_a, N : nat, X2 : poly_a]: ((poly_poly_a2 @ (power_276493840poly_a @ P @ N) @ X2) = (power_power_poly_a @ (poly_poly_a2 @ P @ X2) @ N))))). % poly_power
thf(fact_92_poly__power, axiom,
    ((![P : poly_a, N : nat, X2 : a]: ((poly_a2 @ (power_power_poly_a @ P @ N) @ X2) = (power_power_a @ (poly_a2 @ P @ X2) @ N))))). % poly_power
thf(fact_93_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_94_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_95_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_96_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_97_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_98_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_99_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_100_poly__0, axiom,
    ((![X2 : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X2) = zero_zero_poly_a)))). % poly_0
thf(fact_101_poly__0, axiom,
    ((![X2 : a]: ((poly_a2 @ zero_zero_poly_a @ X2) = zero_zero_a)))). % poly_0
thf(fact_102_psize__eq__0__iff, axiom,
    ((![P : poly_a]: (((fundam247907092size_a @ P) = zero_zero_nat) = (P = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_103_psize__eq__0__iff, axiom,
    ((![P : poly_poly_a]: (((fundam1032801442poly_a @ P) = zero_zero_nat) = (P = zero_z2096148049poly_a))))). % psize_eq_0_iff
thf(fact_104_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_105_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_106_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_107_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_108_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_109_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_110_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_111_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_112_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_113_pderiv_Oinduct, axiom,
    ((![P2 : poly_poly_a > $o, A0 : poly_poly_a]: ((![A : poly_a, P3 : poly_poly_a]: (((~ ((P3 = zero_z2096148049poly_a))) => (P2 @ P3)) => (P2 @ (pCons_poly_a @ A @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_114_pderiv_Oinduct, axiom,
    ((![P2 : poly_nat > $o, A0 : poly_nat]: ((![A : nat, P3 : poly_nat]: (((~ ((P3 = zero_zero_poly_nat))) => (P2 @ P3)) => (P2 @ (pCons_nat @ A @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_115_pderiv_Oinduct, axiom,
    ((![P2 : poly_a > $o, A0 : poly_a]: ((![A : a, P3 : poly_a]: (((~ ((P3 = zero_zero_poly_a))) => (P2 @ P3)) => (P2 @ (pCons_a @ A @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_116_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_poly_a > $o, P : poly_poly_a, Q2 : poly_poly_a]: ((P2 @ zero_z2096148049poly_a @ zero_z2096148049poly_a) => ((![A : poly_a, P3 : poly_poly_a, B2 : poly_a, Q : poly_poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_poly_a @ A @ P3) @ (pCons_poly_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_117_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_nat > $o, P : poly_poly_a, Q2 : poly_nat]: ((P2 @ zero_z2096148049poly_a @ zero_zero_poly_nat) => ((![A : poly_a, P3 : poly_poly_a, B2 : nat, Q : poly_nat]: ((P2 @ P3 @ Q) => (P2 @ (pCons_poly_a @ A @ P3) @ (pCons_nat @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_118_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_poly_a > $o, P : poly_nat, Q2 : poly_poly_a]: ((P2 @ zero_zero_poly_nat @ zero_z2096148049poly_a) => ((![A : nat, P3 : poly_nat, B2 : poly_a, Q : poly_poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_nat @ A @ P3) @ (pCons_poly_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_119_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_nat > $o, P : poly_nat, Q2 : poly_nat]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A : nat, P3 : poly_nat, B2 : nat, Q : poly_nat]: ((P2 @ P3 @ Q) => (P2 @ (pCons_nat @ A @ P3) @ (pCons_nat @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_120_poly__induct2, axiom,
    ((![P2 : poly_poly_a > poly_a > $o, P : poly_poly_a, Q2 : poly_a]: ((P2 @ zero_z2096148049poly_a @ zero_zero_poly_a) => ((![A : poly_a, P3 : poly_poly_a, B2 : a, Q : poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_poly_a @ A @ P3) @ (pCons_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_121_poly__induct2, axiom,
    ((![P2 : poly_nat > poly_a > $o, P : poly_nat, Q2 : poly_a]: ((P2 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A : nat, P3 : poly_nat, B2 : a, Q : poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_nat @ A @ P3) @ (pCons_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_122_poly__induct2, axiom,
    ((![P2 : poly_a > poly_poly_a > $o, P : poly_a, Q2 : poly_poly_a]: ((P2 @ zero_zero_poly_a @ zero_z2096148049poly_a) => ((![A : a, P3 : poly_a, B2 : poly_a, Q : poly_poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_a @ A @ P3) @ (pCons_poly_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_123_poly__induct2, axiom,
    ((![P2 : poly_a > poly_nat > $o, P : poly_a, Q2 : poly_nat]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A : a, P3 : poly_a, B2 : nat, Q : poly_nat]: ((P2 @ P3 @ Q) => (P2 @ (pCons_a @ A @ P3) @ (pCons_nat @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_124_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q2 : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A : a, P3 : poly_a, B2 : a, Q : poly_a]: ((P2 @ P3 @ Q) => (P2 @ (pCons_a @ A @ P3) @ (pCons_a @ B2 @ Q)))) => (P2 @ P @ Q2)))))). % poly_induct2
thf(fact_125_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X3 : a]: (![Y3 : a]: ((F @ X3) = (F @ Y3)))))))). % constant_def
thf(fact_126_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_127_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K2 : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K2) => (ord_less_nat @ (times_times_nat @ K2 @ I) @ (times_times_nat @ K2 @ J))))))). % mult_less_mono2
thf(fact_128_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K2 : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K2) => (ord_less_nat @ (times_times_nat @ I @ K2) @ (times_times_nat @ J @ K2))))))). % mult_less_mono1
thf(fact_129_Suc__mult__less__cancel1, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ (suc @ K2) @ M) @ (times_times_nat @ (suc @ K2) @ N)) = (ord_less_nat @ M @ N))))). % Suc_mult_less_cancel1
thf(fact_130_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_131_power__gt__expt, axiom,
    ((![N : nat, K2 : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => (ord_less_nat @ K2 @ (power_power_nat @ N @ K2)))))). % power_gt_expt
thf(fact_132_n__less__n__mult__m, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M) => (ord_less_nat @ N @ (times_times_nat @ N @ M))))))). % n_less_n_mult_m
thf(fact_133_n__less__m__mult__n, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M) => (ord_less_nat @ N @ (times_times_nat @ M @ N))))))). % n_less_m_mult_n
thf(fact_134_one__less__mult, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M) => (ord_less_nat @ (suc @ zero_zero_nat) @ (times_times_nat @ M @ N))))))). % one_less_mult
thf(fact_135_zero__reorient, axiom,
    ((![X2 : a]: ((zero_zero_a = X2) = (X2 = zero_zero_a))))). % zero_reorient
thf(fact_136_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_137_zero__reorient, axiom,
    ((![X2 : poly_a]: ((zero_zero_poly_a = X2) = (X2 = zero_zero_poly_a))))). % zero_reorient
thf(fact_138_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_139_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_140_mult_Oleft__commute, axiom,
    ((![B : poly_a, A2 : poly_a, C : poly_a]: ((times_times_poly_a @ B @ (times_times_poly_a @ A2 @ C)) = (times_times_poly_a @ A2 @ (times_times_poly_a @ B @ C)))))). % mult.left_commute
thf(fact_141_mult_Ocommute, axiom,
    ((times_times_a = (^[A3 : a]: (^[B3 : a]: (times_times_a @ B3 @ A3)))))). % mult.commute
thf(fact_142_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_143_mult_Ocommute, axiom,
    ((times_times_poly_a = (^[A3 : poly_a]: (^[B3 : poly_a]: (times_times_poly_a @ B3 @ A3)))))). % mult.commute
thf(fact_144_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_145_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_146_mult_Oassoc, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A2 @ B) @ C) = (times_times_poly_a @ A2 @ (times_times_poly_a @ B @ C)))))). % mult.assoc
thf(fact_147_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_148_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_149_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A2 : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A2 @ B) @ C) = (times_times_poly_a @ A2 @ (times_times_poly_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_150_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_151_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_152_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_153_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_154_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_155_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_156_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_157_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_170_Suc__inject, axiom,
    ((![X2 : nat, Y : nat]: (((suc @ X2) = (suc @ Y)) => (X2 = Y))))). % Suc_inject
thf(fact_171_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_neqE_nat
thf(fact_172_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P2 @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P2 @ M2))))))) => (P2 @ N))))). % infinite_descent
thf(fact_173_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P2 @ M2))) => (P2 @ N2))) => (P2 @ N))))). % nat_less_induct
thf(fact_174_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_175_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_176_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_177_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_178_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_179_pderiv_Ocases, axiom,
    ((![X2 : poly_a]: (~ ((![A : a, P3 : poly_a]: (~ ((X2 = (pCons_a @ A @ P3)))))))))). % pderiv.cases
thf(fact_180_pderiv_Ocases, axiom,
    ((![X2 : poly_poly_a]: (~ ((![A : poly_a, P3 : poly_poly_a]: (~ ((X2 = (pCons_poly_a @ A @ P3)))))))))). % pderiv.cases
thf(fact_181_pderiv_Ocases, axiom,
    ((![X2 : poly_nat]: (~ ((![A : nat, P3 : poly_nat]: (~ ((X2 = (pCons_nat @ A @ P3)))))))))). % pderiv.cases
thf(fact_182_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A : a, Q : poly_a]: (~ ((P = (pCons_a @ A @ Q)))))))))). % pCons_cases
thf(fact_183_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A : poly_a, Q : poly_poly_a]: (~ ((P = (pCons_poly_a @ A @ Q)))))))))). % pCons_cases
thf(fact_184_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A : nat, Q : poly_nat]: (~ ((P = (pCons_nat @ A @ Q)))))))))). % pCons_cases
thf(fact_185_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_186_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_187_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_188_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_189_add_Ogroup__left__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ zero_zero_a @ A2) = A2)))). % add.group_left_neutral
thf(fact_190_add_Ogroup__left__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A2) = A2)))). % add.group_left_neutral
thf(fact_191_add_Ocomm__neutral, axiom,
    ((![A2 : a]: ((plus_plus_a @ A2 @ zero_zero_a) = A2)))). % add.comm_neutral
thf(fact_192_add_Ocomm__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.comm_neutral
thf(fact_193_add_Ocomm__neutral, axiom,
    ((![A2 : poly_a]: ((plus_plus_poly_a @ A2 @ zero_zero_poly_a) = A2)))). % add.comm_neutral
thf(fact_194_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_195_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_196_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_197_add__less__imp__less__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A2 @ B))))). % add_less_imp_less_right
thf(fact_198_add__less__imp__less__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A2 @ B))))). % add_less_imp_less_left
thf(fact_199_add__strict__right__mono, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((ord_less_nat @ A2 @ B) => (ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_200_add__strict__left__mono, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((ord_less_nat @ A2 @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_201_add__strict__mono, axiom,
    ((![A2 : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_202_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K2 : nat, L : nat]: (((ord_less_nat @ I @ J) & (K2 = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K2) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_203_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K2 : nat, L : nat]: (((I = J) & (ord_less_nat @ K2 @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K2) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_204_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K2 : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K2 @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K2) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_205_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_206_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_207_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_208_power__commuting__commutes, axiom,
    ((![X2 : a, Y : a, N : nat]: (((times_times_a @ X2 @ Y) = (times_times_a @ Y @ X2)) => ((times_times_a @ (power_power_a @ X2 @ N) @ Y) = (times_times_a @ Y @ (power_power_a @ X2 @ N))))))). % power_commuting_commutes
thf(fact_209_power__commuting__commutes, axiom,
    ((![X2 : nat, Y : nat, N : nat]: (((times_times_nat @ X2 @ Y) = (times_times_nat @ Y @ X2)) => ((times_times_nat @ (power_power_nat @ X2 @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X2 @ N))))))). % power_commuting_commutes
thf(fact_210_power__commuting__commutes, axiom,
    ((![X2 : poly_a, Y : poly_a, N : nat]: (((times_times_poly_a @ X2 @ Y) = (times_times_poly_a @ Y @ X2)) => ((times_times_poly_a @ (power_power_poly_a @ X2 @ N) @ Y) = (times_times_poly_a @ Y @ (power_power_poly_a @ X2 @ N))))))). % power_commuting_commutes
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M3 : nat]: (N = (suc @ M3))))))). % not0_implies_Suc
thf(fact_218_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_219_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_220_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_221_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_222_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_223_zero__induct, axiom,
    ((![P2 : nat > $o, K2 : nat]: ((P2 @ K2) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_224_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X4 : nat]: (P2 @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P2 @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P2 @ X4 @ Y4) => (P2 @ (suc @ X4) @ (suc @ Y4)))) => (P2 @ M @ N))))))). % diff_induct

% Conjectures (4)
thf(conj_0, hypothesis,
    ((~ ((a2 = zero_zero_a))))).
thf(conj_1, hypothesis,
    (((suc @ (plus_plus_nat @ (fundam247907092size_a @ q) @ k)) = (fundam247907092size_a @ cs)))).
thf(conj_2, 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_3, conjecture,
    ((?[Q3 : poly_a]: (((suc @ (suc @ (plus_plus_nat @ (fundam247907092size_a @ Q3) @ k))) = (fundam247907092size_a @ (pCons_a @ c @ cs))) & (![Z : a]: ((Z = zero_zero_a) | (((Z = zero_zero_a) & (ord_less_nat @ zero_zero_nat @ k)) | ((Z = zero_zero_a) | ((poly_a2 @ q @ Z) = (poly_a2 @ Q3 @ Z)))))))))).
