% 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_618__5374060_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:35.472

% 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 (44)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Nat__Onat, type,
    fundam170929432ly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1032801442poly_a : poly_poly_a > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_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_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_Ouminus__class_Ouminus_001tf__a, type,
    uminus_uminus_a : a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_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_Polynomial_Ofold__coeffs_001tf__a_001t__Polynomial__Opoly_Itf__a_J, type,
    fold_coeffs_a_poly_a : (a > poly_a > poly_a) > poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_Itf__a_J, type,
    pcompose_poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_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_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).
thf(sy_v_x____, type,
    x : a).
thf(sy_v_y____, type,
    y : a).

% Relevant facts (240)
thf(fact_0__092_060open_062_092_060forall_062z_O_Az_A_092_060noteq_062_A_I0_058_058_Ha_J_A_092_060longrightarrow_062_Apoly_Acs_Az_A_061_A_I0_058_058_Ha_J_092_060close_062, axiom,
    ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ cs @ Z) = zero_zero_a))))). % \<open>\<forall>z. z \<noteq> (0::'a) \<longrightarrow> poly cs z = (0::'a)\<close>
thf(fact_1_nc, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ p)))))). % nc
thf(fact_2_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_3_pCons__eq__iff, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: (((pCons_a @ A @ P) = (pCons_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_4_pCons_Oprems, axiom,
    ((~ ((fundam236050252nt_a_a @ (poly_a2 @ (pCons_a @ c @ cs))))))). % pCons.prems
thf(fact_5_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_6_pderiv_Ocases, axiom,
    ((![X : poly_a]: (~ ((![A2 : a, P2 : poly_a]: (~ ((X = (pCons_a @ A2 @ P2)))))))))). % pderiv.cases
thf(fact_7_synthetic__div__pCons, axiom,
    ((![A : a, P : poly_a, C : a]: ((synthetic_div_a @ (pCons_a @ A @ P) @ C) = (pCons_a @ (poly_a2 @ P @ C) @ (synthetic_div_a @ P @ C)))))). % synthetic_div_pCons
thf(fact_8_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X2 : a]: (![Y : a]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_9_poly__pCons, axiom,
    ((![A : a, P : poly_a, X : a]: ((poly_a2 @ (pCons_a @ A @ P) @ X) = (plus_plus_a @ A @ (times_times_a @ X @ (poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_10_poly__pCons, axiom,
    ((![A : nat, P : poly_nat, X : nat]: ((poly_nat2 @ (pCons_nat @ A @ P) @ X) = (plus_plus_nat @ A @ (times_times_nat @ X @ (poly_nat2 @ P @ X))))))). % poly_pCons
thf(fact_11_poly__offset, axiom,
    ((![P : poly_a, A : a]: (?[Q2 : poly_a]: (((fundam247907092size_a @ Q2) = (fundam247907092size_a @ P)) & (![X3 : a]: ((poly_a2 @ Q2 @ X3) = (poly_a2 @ P @ (plus_plus_a @ A @ X3))))))))). % poly_offset
thf(fact_12_offset__poly__single, axiom,
    ((![A : a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ H) = (pCons_a @ A @ zero_zero_poly_a))))). % offset_poly_single
thf(fact_13_poly__offset__poly, axiom,
    ((![P : poly_nat, H : nat, X : nat]: ((poly_nat2 @ (fundam170929432ly_nat @ P @ H) @ X) = (poly_nat2 @ P @ (plus_plus_nat @ H @ X)))))). % poly_offset_poly
thf(fact_14_poly__offset__poly, axiom,
    ((![P : poly_a, H : a, X : a]: ((poly_a2 @ (fundam1358810038poly_a @ P @ H) @ X) = (poly_a2 @ P @ (plus_plus_a @ H @ X)))))). % poly_offset_poly
thf(fact_15_poly__pcompose, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (pcompose_a @ P @ Q) @ X) = (poly_a2 @ P @ (poly_a2 @ Q @ X)))))). % poly_pcompose
thf(fact_16_add__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: ((plus_plus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q)) = (pCons_nat @ (plus_plus_nat @ A @ B) @ (plus_plus_poly_nat @ P @ Q)))))). % add_pCons
thf(fact_17_add__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((plus_plus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (plus_plus_a @ A @ B) @ (plus_plus_poly_a @ P @ Q)))))). % add_pCons
thf(fact_18_poly__mult, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q) @ X) = (times_times_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_mult
thf(fact_19_poly__mult, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q) @ X) = (times_times_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q @ X)))))). % poly_mult
thf(fact_20_poly__add, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q) @ X) = (plus_plus_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q @ X)))))). % poly_add
thf(fact_21_poly__add, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q) @ X) = (plus_plus_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_add
thf(fact_22_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_23_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_24_pCons__eq__0__iff, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((pCons_poly_a @ A @ P) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) & ((P = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_25_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_26_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_27_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_28_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_29_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_30_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_31_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_32_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_33_pcompose__const, axiom,
    ((![A : a, Q : poly_a]: ((pcompose_a @ (pCons_a @ A @ zero_zero_poly_a) @ Q) = (pCons_a @ A @ zero_zero_poly_a))))). % pcompose_const
thf(fact_34_pcompose__pCons, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((pcompose_a @ (pCons_a @ A @ P) @ Q) = (plus_plus_poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ (times_times_poly_a @ Q @ (pcompose_a @ P @ Q))))))). % pcompose_pCons
thf(fact_35_mult__poly__0__left, axiom,
    ((![Q : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_36_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_37_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_38_offset__poly__eq__0__iff, axiom,
    ((![P : poly_a, H : a]: (((fundam1358810038poly_a @ P @ H) = zero_zero_poly_a) = (P = zero_zero_poly_a))))). % offset_poly_eq_0_iff
thf(fact_39_pCons__induct, axiom,
    ((![P3 : poly_poly_a > $o, P : poly_poly_a]: ((P3 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P2 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P2 = zero_z2096148049poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_a @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_40_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A2 : nat, P2 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_41_pCons__induct, axiom,
    ((![P3 : poly_a > $o, P : poly_a]: ((P3 @ zero_zero_poly_a) => ((![A2 : a, P2 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P2 = zero_zero_poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_a @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_42_pderiv_Oinduct, axiom,
    ((![P3 : poly_a > $o, A0 : poly_a]: ((![A2 : a, P2 : poly_a]: (((~ ((P2 = zero_zero_poly_a))) => (P3 @ P2)) => (P3 @ (pCons_a @ A2 @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_43_poly__induct2, axiom,
    ((![P3 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P2 : poly_a, B2 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A2 @ P2) @ (pCons_a @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_44_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_45_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_46_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_47_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_48_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_49_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_50_add__cancel__left__left, axiom,
    ((![B : a, A : a]: (((plus_plus_a @ B @ A) = A) = (B = zero_zero_a))))). % add_cancel_left_left
thf(fact_51_add__cancel__left__left, axiom,
    ((![B : poly_a, A : poly_a]: (((plus_plus_poly_a @ B @ A) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_left
thf(fact_52_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_53_add__cancel__left__right, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = A) = (B = zero_zero_a))))). % add_cancel_left_right
thf(fact_54_add__cancel__left__right, axiom,
    ((![A : poly_a, B : poly_a]: (((plus_plus_poly_a @ A @ B) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_right
thf(fact_55_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_56_add__cancel__right__left, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ B @ A)) = (B = zero_zero_a))))). % add_cancel_right_left
thf(fact_57_add__cancel__right__left, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ B @ A)) = (B = zero_zero_poly_a))))). % add_cancel_right_left
thf(fact_58_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_59_add__cancel__right__right, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ A @ B)) = (B = zero_zero_a))))). % add_cancel_right_right
thf(fact_60_add__cancel__right__right, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ A @ B)) = (B = zero_zero_poly_a))))). % add_cancel_right_right
thf(fact_61_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_62_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y2 : nat]: (((plus_plus_nat @ X @ Y2) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_63_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_64_add__right__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_65_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_66_add__left__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_67_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y2)) = (((X = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_68_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_69_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_70_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_71_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_72_mult_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((times_times_a @ B @ (times_times_a @ A @ C)) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % mult.left_commute
thf(fact_73_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_74_mult_Ocommute, axiom,
    ((times_times_a = (^[A3 : a]: (^[B3 : a]: (times_times_a @ B3 @ A3)))))). % mult.commute
thf(fact_75_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_76_mult_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A @ B) @ C) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % mult.assoc
thf(fact_77_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_78_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (times_times_a @ A @ B) @ C) = (times_times_a @ A @ (times_times_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_79_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_80_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_81_add__right__imp__eq, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_82_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_83_add__left__imp__eq, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_84_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_85_add_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A @ C)) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_86_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_87_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_88_add_Oright__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_89_add_Oleft__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_90_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_91_add_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_92_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_93_group__cancel_Oadd2, axiom,
    ((![B4 : a, K : a, B : a, A : a]: ((B4 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B4) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_94_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_95_group__cancel_Oadd1, axiom,
    ((![A4 : a, K : a, A : a, B : a]: ((A4 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A4 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_96_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_97_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_98_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_99_add_Ogroup__left__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.group_left_neutral
thf(fact_100_add_Ogroup__left__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.group_left_neutral
thf(fact_101_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_102_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_103_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_104_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_105_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_106_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_107_mult__cancel__right, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_108_mult__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_109_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_110_mult__cancel__left, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_111_mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_112_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_113_mult__eq__0__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) = (((A = zero_zero_poly_a)) | ((B = zero_zero_poly_a))))))). % mult_eq_0_iff
thf(fact_114_mult__eq__0__iff, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % mult_eq_0_iff
thf(fact_115_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_116_mult__zero__right, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_117_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_118_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_119_mult__zero__left, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_120_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_121_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_122_pCons_Ohyps_I2_J, axiom,
    (((~ ((fundam236050252nt_a_a @ (poly_a2 @ cs)))) => (?[K2 : nat, A2 : a, Q2 : poly_a]: ((~ ((A2 = zero_zero_a))) & ((~ ((K2 = zero_zero_nat))) & (((plus_plus_nat @ (plus_plus_nat @ (fundam247907092size_a @ Q2) @ K2) @ one_one_nat) = (fundam247907092size_a @ cs)) & (![Z : a]: ((poly_a2 @ cs @ Z) = (plus_plus_a @ (poly_a2 @ cs @ zero_zero_a) @ (times_times_a @ (power_power_a @ Z @ K2) @ (poly_a2 @ (pCons_a @ A2 @ Q2) @ Z)))))))))))). % pCons.hyps(2)
thf(fact_123_mult_Oleft__neutral, axiom,
    ((![A : a]: ((times_times_a @ one_one_a @ A) = A)))). % mult.left_neutral
thf(fact_124_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_125_mult_Oright__neutral, axiom,
    ((![A : a]: ((times_times_a @ A @ one_one_a) = A)))). % mult.right_neutral
thf(fact_126_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_127_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_128_power__one, axiom,
    ((![N : nat]: ((power_power_a @ one_one_a @ N) = one_one_a)))). % power_one
thf(fact_129_power__one__right, axiom,
    ((![A : a]: ((power_power_a @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_130_poly__1, axiom,
    ((![X : a]: ((poly_a2 @ one_one_poly_a @ X) = one_one_a)))). % poly_1
thf(fact_131_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_132_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_133_mult__cancel__left1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_left1
thf(fact_134_mult__cancel__left1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_left1
thf(fact_135_mult__cancel__left2, axiom,
    ((![C : poly_a, A : poly_a]: (((times_times_poly_a @ C @ A) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_left2
thf(fact_136_mult__cancel__left2, axiom,
    ((![C : a, A : a]: (((times_times_a @ C @ A) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_left2
thf(fact_137_mult__cancel__right1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_right1
thf(fact_138_mult__cancel__right1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_right1
thf(fact_139_mult__cancel__right2, axiom,
    ((![A : poly_a, C : poly_a]: (((times_times_poly_a @ A @ C) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_right2
thf(fact_140_mult__cancel__right2, axiom,
    ((![A : a, C : a]: (((times_times_a @ A @ C) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_right2
thf(fact_141_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_142_one__poly__eq__simps_I2_J, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % one_poly_eq_simps(2)
thf(fact_143_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_144_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_a = (pCons_a @ one_one_a @ zero_zero_poly_a)))). % one_poly_eq_simps(1)
thf(fact_145_pcompose__idR, axiom,
    ((![P : poly_a]: ((pcompose_a @ P @ (pCons_a @ zero_zero_a @ (pCons_a @ one_one_a @ zero_zero_poly_a))) = P)))). % pcompose_idR
thf(fact_146_pcompose__idR, axiom,
    ((![P : poly_poly_a]: ((pcompose_poly_a @ P @ (pCons_poly_a @ zero_zero_poly_a @ (pCons_poly_a @ one_one_poly_a @ zero_z2096148049poly_a))) = P)))). % pcompose_idR
thf(fact_147_pcompose__idR, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P)))). % pcompose_idR
thf(fact_148_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_149_power__0, axiom,
    ((![A : a]: ((power_power_a @ A @ zero_zero_nat) = one_one_a)))). % power_0
thf(fact_150_power__mult, axiom,
    ((![A : a, M : nat, N : nat]: ((power_power_a @ A @ (times_times_nat @ M @ N)) = (power_power_a @ (power_power_a @ A @ M) @ N))))). % power_mult
thf(fact_151_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_poly_a @ zero_zero_poly_a @ N) = one_one_poly_a)) & ((~ ((N = zero_zero_nat))) => ((power_power_poly_a @ zero_zero_poly_a @ N) = zero_zero_poly_a)))))). % power_0_left
thf(fact_152_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_153_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_a @ zero_zero_a @ N) = one_one_a)) & ((~ ((N = zero_zero_nat))) => ((power_power_a @ zero_zero_a @ N) = zero_zero_a)))))). % power_0_left
thf(fact_154_left__right__inverse__power, axiom,
    ((![X : a, Y2 : a, N : nat]: (((times_times_a @ X @ Y2) = one_one_a) => ((times_times_a @ (power_power_a @ X @ N) @ (power_power_a @ Y2 @ N)) = one_one_a))))). % left_right_inverse_power
thf(fact_155_left__right__inverse__power, axiom,
    ((![X : nat, Y2 : nat, N : nat]: (((times_times_nat @ X @ Y2) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y2 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_156_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_157_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_158_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_a = one_one_poly_a))))). % zero_neq_one
thf(fact_159_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_160_lambda__one, axiom,
    (((^[X2 : a]: X2) = (times_times_a @ one_one_a)))). % lambda_one
thf(fact_161_lambda__one, axiom,
    (((^[X2 : nat]: X2) = (times_times_nat @ one_one_nat)))). % lambda_one
thf(fact_162_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_163_pCons__one, axiom,
    (((pCons_a @ one_one_a @ zero_zero_poly_a) = one_one_poly_a))). % pCons_one
thf(fact_164_power__not__zero, axiom,
    ((![A : poly_a, N : nat]: ((~ ((A = zero_zero_poly_a))) => (~ (((power_power_poly_a @ A @ N) = zero_zero_poly_a))))))). % power_not_zero
thf(fact_165_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_166_power__not__zero, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => (~ (((power_power_a @ A @ N) = zero_zero_a))))))). % power_not_zero
thf(fact_167_power__add, axiom,
    ((![A : a, M : nat, N : nat]: ((power_power_a @ A @ (plus_plus_nat @ M @ N)) = (times_times_a @ (power_power_a @ A @ M) @ (power_power_a @ A @ N)))))). % power_add
thf(fact_168_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_169_power__commutes, axiom,
    ((![A : a, N : nat]: ((times_times_a @ (power_power_a @ A @ N) @ A) = (times_times_a @ A @ (power_power_a @ A @ N)))))). % power_commutes
thf(fact_170_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_171_power__mult__distrib, axiom,
    ((![A : a, B : a, N : nat]: ((power_power_a @ (times_times_a @ A @ B) @ N) = (times_times_a @ (power_power_a @ A @ N) @ (power_power_a @ B @ N)))))). % power_mult_distrib
thf(fact_172_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_173_power__commuting__commutes, axiom,
    ((![X : a, Y2 : a, N : nat]: (((times_times_a @ X @ Y2) = (times_times_a @ Y2 @ X)) => ((times_times_a @ (power_power_a @ X @ N) @ Y2) = (times_times_a @ Y2 @ (power_power_a @ X @ N))))))). % power_commuting_commutes
thf(fact_174_power__commuting__commutes, axiom,
    ((![X : nat, Y2 : nat, N : nat]: (((times_times_nat @ X @ Y2) = (times_times_nat @ Y2 @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y2) = (times_times_nat @ Y2 @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_175_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : a]: ((times_times_a @ one_one_a @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_176_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_177_mult_Ocomm__neutral, axiom,
    ((![A : a]: ((times_times_a @ A @ one_one_a) = A)))). % mult.comm_neutral
thf(fact_178_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_179_mult__not__zero, axiom,
    ((![A : poly_a, B : poly_a]: ((~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a))) => ((~ ((A = zero_zero_poly_a))) & (~ ((B = zero_zero_poly_a)))))))). % mult_not_zero
thf(fact_180_mult__not__zero, axiom,
    ((![A : a, B : a]: ((~ (((times_times_a @ A @ B) = zero_zero_a))) => ((~ ((A = zero_zero_a))) & (~ ((B = zero_zero_a)))))))). % mult_not_zero
thf(fact_181_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_182_divisors__zero, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) => ((A = zero_zero_poly_a) | (B = zero_zero_poly_a)))))). % divisors_zero
thf(fact_183_divisors__zero, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) => ((A = zero_zero_a) | (B = zero_zero_a)))))). % divisors_zero
thf(fact_184_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_185_no__zero__divisors, axiom,
    ((![A : poly_a, B : poly_a]: ((~ ((A = zero_zero_poly_a))) => ((~ ((B = zero_zero_poly_a))) => (~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a)))))))). % no_zero_divisors
thf(fact_186_no__zero__divisors, axiom,
    ((![A : a, B : a]: ((~ ((A = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (~ (((times_times_a @ A @ B) = zero_zero_a)))))))). % no_zero_divisors
thf(fact_187_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_188_mult__left__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_189_mult__left__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_190_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_191_mult__right__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_192_mult__right__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_193_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_194_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_195_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A @ B) @ (times_times_a @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_196_comm__semiring__class_Odistrib, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_197_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_198_distrib__left, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A @ B) @ (times_times_a @ A @ C)))))). % distrib_left
thf(fact_199_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_200_distrib__right, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % distrib_right
thf(fact_201_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_202_combine__common__factor, axiom,
    ((![A : a, E : a, B : a, C : a]: ((plus_plus_a @ (times_times_a @ A @ E) @ (plus_plus_a @ (times_times_a @ B @ E) @ C)) = (plus_plus_a @ (times_times_a @ (plus_plus_a @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_203_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_204_lambda__zero, axiom,
    (((^[H2 : poly_a]: zero_zero_poly_a) = (times_times_poly_a @ zero_zero_poly_a)))). % lambda_zero
thf(fact_205_lambda__zero, axiom,
    (((^[H2 : a]: zero_zero_a) = (times_times_a @ zero_zero_a)))). % lambda_zero
thf(fact_206_lambda__zero, axiom,
    (((^[H2 : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_207_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_208_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_209_add__scale__eq__noteq, axiom,
    ((![R : poly_a, A : poly_a, B : poly_a, C : poly_a, D : poly_a]: ((~ ((R = zero_zero_poly_a))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_poly_a @ A @ (times_times_poly_a @ R @ C)) = (plus_plus_poly_a @ B @ (times_times_poly_a @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_210_add__scale__eq__noteq, axiom,
    ((![R : a, A : a, B : a, C : a, D : a]: ((~ ((R = zero_zero_a))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_a @ A @ (times_times_a @ R @ C)) = (plus_plus_a @ B @ (times_times_a @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_211_add__scale__eq__noteq, axiom,
    ((![R : nat, A : nat, B : nat, C : nat, D : nat]: ((~ ((R = zero_zero_nat))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_nat @ A @ (times_times_nat @ R @ C)) = (plus_plus_nat @ B @ (times_times_nat @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_212_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_213_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_214_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_215_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_216_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_217_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_218_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_219_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_220_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_221_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_222_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_223_add__0__iff, axiom,
    ((![B : a, A : a]: ((B = (plus_plus_a @ B @ A)) = (A = zero_zero_a))))). % add_0_iff
thf(fact_224_add__0__iff, axiom,
    ((![B : poly_a, A : poly_a]: ((B = (plus_plus_poly_a @ B @ A)) = (A = zero_zero_poly_a))))). % add_0_iff
thf(fact_225_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_226_crossproduct__noteq, axiom,
    ((![A : a, B : a, C : a, D : a]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ D)) = (plus_plus_a @ (times_times_a @ A @ D) @ (times_times_a @ B @ C))))))))). % crossproduct_noteq
thf(fact_227_crossproduct__noteq, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D)) = (plus_plus_nat @ (times_times_nat @ A @ D) @ (times_times_nat @ B @ C))))))))). % crossproduct_noteq
thf(fact_228_crossproduct__eq, axiom,
    ((![W : a, Y2 : a, X : a, Z2 : a]: (((plus_plus_a @ (times_times_a @ W @ Y2) @ (times_times_a @ X @ Z2)) = (plus_plus_a @ (times_times_a @ W @ Z2) @ (times_times_a @ X @ Y2))) = (((W = X)) | ((Y2 = Z2))))))). % crossproduct_eq
thf(fact_229_crossproduct__eq, axiom,
    ((![W : nat, Y2 : nat, X : nat, Z2 : nat]: (((plus_plus_nat @ (times_times_nat @ W @ Y2) @ (times_times_nat @ X @ Z2)) = (plus_plus_nat @ (times_times_nat @ W @ Z2) @ (times_times_nat @ X @ Y2))) = (((W = X)) | ((Y2 = Z2))))))). % crossproduct_eq
thf(fact_230_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_231_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_232_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_233_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_a @ N @ one_one_poly_a) = zero_zero_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_a @ N @ one_one_poly_a) = one_one_poly_a)))))). % poly_cutoff_1
thf(fact_234_pcompose__def, axiom,
    ((pcompose_a = (^[P4 : poly_a]: (^[Q3 : poly_a]: (fold_coeffs_a_poly_a @ (^[A3 : a]: (^[C2 : poly_a]: (plus_plus_poly_a @ (pCons_a @ A3 @ zero_zero_poly_a) @ (times_times_poly_a @ Q3 @ C2)))) @ P4 @ zero_zero_poly_a)))))). % pcompose_def
thf(fact_235_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_236_synthetic__div__correct_H, axiom,
    ((![C : a, P : poly_a]: ((plus_plus_poly_a @ (times_times_poly_a @ (pCons_a @ (uminus_uminus_a @ C) @ (pCons_a @ one_one_a @ zero_zero_poly_a)) @ (synthetic_div_a @ P @ C)) @ (pCons_a @ (poly_a2 @ P @ C) @ zero_zero_poly_a)) = P)))). % synthetic_div_correct'
thf(fact_237_poly__decompose__lemma, axiom,
    ((![P : poly_poly_a]: ((~ ((![Z : poly_a]: ((~ ((Z = zero_zero_poly_a))) => ((poly_poly_a2 @ P @ Z) = zero_zero_poly_a))))) => (?[K2 : nat, A2 : poly_a, Q2 : poly_poly_a]: ((~ ((A2 = zero_zero_poly_a))) & (((suc @ (plus_plus_nat @ (fundam1032801442poly_a @ Q2) @ K2)) = (fundam1032801442poly_a @ P)) & (![Z : poly_a]: ((poly_poly_a2 @ P @ Z) = (times_times_poly_a @ (power_power_poly_a @ Z @ K2) @ (poly_poly_a2 @ (pCons_poly_a @ A2 @ Q2) @ Z))))))))))). % poly_decompose_lemma
thf(fact_238_poly__decompose__lemma, axiom,
    ((![P : poly_a]: ((~ ((![Z : a]: ((~ ((Z = zero_zero_a))) => ((poly_a2 @ P @ Z) = zero_zero_a))))) => (?[K2 : nat, A2 : a, Q2 : poly_a]: ((~ ((A2 = zero_zero_a))) & (((suc @ (plus_plus_nat @ (fundam247907092size_a @ Q2) @ K2)) = (fundam247907092size_a @ P)) & (![Z : a]: ((poly_a2 @ P @ Z) = (times_times_a @ (power_power_a @ Z @ K2) @ (poly_a2 @ (pCons_a @ A2 @ Q2) @ Z))))))))))). % poly_decompose_lemma
thf(fact_239_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject

% Conjectures (1)
thf(conj_0, conjecture,
    (((poly_a2 @ (pCons_a @ c @ cs) @ x) = (poly_a2 @ (pCons_a @ c @ cs) @ y)))).
