% 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_83__5367666_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:26:53.545

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

% Explicit typings (47)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Nat__Onat, type,
    fundam170929432ly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1343031620poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_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_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_It__Nat__Onat_J_J, type,
    plus_p1835221865ly_nat : poly_poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    plus_p672445791poly_a : poly_poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
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_It__Nat__Onat_J_J, type,
    times_1465266917ly_nat : poly_poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    times_1069126883poly_a : poly_poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    times_545135445poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    zero_z2064990175poly_a : poly_poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat, type,
    is_zero_nat : poly_nat > $o).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_Itf__a_J, type,
    is_zero_poly_a : poly_poly_a > $o).
thf(sy_c_Polynomial_Ois__zero_001tf__a, type,
    is_zero_a : poly_a > $o).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    pCons_poly_poly_a : poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat, type,
    pderiv_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pderiv_poly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat, type,
    poly_cutoff_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_cutoff_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    smult_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    smult_poly_poly_a : poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_Itf__a_J, type,
    smult_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osmult_001tf__a, type,
    smult_a : a > poly_a > poly_a).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_h, type,
    h : a).
thf(sy_v_pa, type,
    pa : poly_a).

% Relevant facts (246)
thf(fact_0_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_1_offset__poly__0, axiom,
    ((![H : poly_a]: ((fundam1343031620poly_a @ zero_z2096148049poly_a @ H) = zero_z2096148049poly_a)))). % offset_poly_0
thf(fact_2_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_3_offset__poly__pCons, axiom,
    ((![A : nat, P : poly_nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ P) @ H) = (plus_plus_poly_nat @ (smult_nat @ H @ (fundam170929432ly_nat @ P @ H)) @ (pCons_nat @ A @ (fundam170929432ly_nat @ P @ H))))))). % offset_poly_pCons
thf(fact_4_offset__poly__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, H : poly_a]: ((fundam1343031620poly_a @ (pCons_poly_a @ A @ P) @ H) = (plus_p1976640465poly_a @ (smult_poly_a @ H @ (fundam1343031620poly_a @ P @ H)) @ (pCons_poly_a @ A @ (fundam1343031620poly_a @ P @ H))))))). % offset_poly_pCons
thf(fact_5_offset__poly__pCons, axiom,
    ((![A : a, P : poly_a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ P) @ H) = (plus_plus_poly_a @ (smult_a @ H @ (fundam1358810038poly_a @ P @ H)) @ (pCons_a @ A @ (fundam1358810038poly_a @ P @ H))))))). % offset_poly_pCons
thf(fact_6_offset__poly__single, axiom,
    ((![A : nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ H) = (pCons_nat @ A @ zero_zero_poly_nat))))). % offset_poly_single
thf(fact_7_offset__poly__single, axiom,
    ((![A : poly_a, H : poly_a]: ((fundam1343031620poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ H) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % offset_poly_single
thf(fact_8_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_9_offset__poly__eq__0__lemma, axiom,
    ((![C : nat, P : poly_nat, A : nat]: (((plus_plus_poly_nat @ (smult_nat @ C @ P) @ (pCons_nat @ A @ P)) = zero_zero_poly_nat) => (P = zero_zero_poly_nat))))). % offset_poly_eq_0_lemma
thf(fact_10_offset__poly__eq__0__lemma, axiom,
    ((![C : poly_a, P : poly_poly_a, A : poly_a]: (((plus_p1976640465poly_a @ (smult_poly_a @ C @ P) @ (pCons_poly_a @ A @ P)) = zero_z2096148049poly_a) => (P = zero_z2096148049poly_a))))). % offset_poly_eq_0_lemma
thf(fact_11_offset__poly__eq__0__lemma, axiom,
    ((![C : a, P : poly_a, A : a]: (((plus_plus_poly_a @ (smult_a @ C @ P) @ (pCons_a @ A @ P)) = zero_zero_poly_a) => (P = zero_zero_poly_a))))). % offset_poly_eq_0_lemma
thf(fact_12_add__pCons, axiom,
    ((![A : poly_nat, P : poly_poly_nat, B : poly_nat, Q : poly_poly_nat]: ((plus_p1835221865ly_nat @ (pCons_poly_nat @ A @ P) @ (pCons_poly_nat @ B @ Q)) = (pCons_poly_nat @ (plus_plus_poly_nat @ A @ B) @ (plus_p1835221865ly_nat @ P @ Q)))))). % add_pCons
thf(fact_13_add__pCons, axiom,
    ((![A : poly_poly_a, P : poly_poly_poly_a, B : poly_poly_a, Q : poly_poly_poly_a]: ((plus_p672445791poly_a @ (pCons_poly_poly_a @ A @ P) @ (pCons_poly_poly_a @ B @ Q)) = (pCons_poly_poly_a @ (plus_p1976640465poly_a @ A @ B) @ (plus_p672445791poly_a @ P @ Q)))))). % add_pCons
thf(fact_14_add__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: ((plus_p1976640465poly_a @ (pCons_poly_a @ A @ P) @ (pCons_poly_a @ B @ Q)) = (pCons_poly_a @ (plus_plus_poly_a @ A @ B) @ (plus_p1976640465poly_a @ P @ Q)))))). % add_pCons
thf(fact_15_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_16_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_17_smult__0__left, axiom,
    ((![P : poly_poly_nat]: ((smult_poly_nat @ zero_zero_poly_nat @ P) = zero_z1059985641ly_nat)))). % smult_0_left
thf(fact_18_smult__0__left, axiom,
    ((![P : poly_poly_poly_a]: ((smult_poly_poly_a @ zero_z2096148049poly_a @ P) = zero_z2064990175poly_a)))). % smult_0_left
thf(fact_19_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_20_smult__0__left, axiom,
    ((![P : poly_poly_a]: ((smult_poly_a @ zero_zero_poly_a @ P) = zero_z2096148049poly_a)))). % smult_0_left
thf(fact_21_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_22_smult__eq__0__iff, axiom,
    ((![A : poly_nat, P : poly_poly_nat]: (((smult_poly_nat @ A @ P) = zero_z1059985641ly_nat) = (((A = zero_zero_poly_nat)) | ((P = zero_z1059985641ly_nat))))))). % smult_eq_0_iff
thf(fact_23_smult__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((smult_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) | ((P = zero_zero_poly_nat))))))). % smult_eq_0_iff
thf(fact_24_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_25_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_26_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_27_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_28_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_29_pCons__eq__0__iff, axiom,
    ((![A : poly_nat, P : poly_poly_nat]: (((pCons_poly_nat @ A @ P) = zero_z1059985641ly_nat) = (((A = zero_zero_poly_nat)) & ((P = zero_z1059985641ly_nat))))))). % pCons_eq_0_iff
thf(fact_30_pCons__eq__0__iff, axiom,
    ((![A : poly_poly_a, P : poly_poly_poly_a]: (((pCons_poly_poly_a @ A @ P) = zero_z2064990175poly_a) = (((A = zero_z2096148049poly_a)) & ((P = zero_z2064990175poly_a))))))). % pCons_eq_0_iff
thf(fact_31_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_32_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_33_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_34_smult__0__right, axiom,
    ((![A : nat]: ((smult_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % smult_0_right
thf(fact_35_smult__0__right, axiom,
    ((![A : poly_a]: ((smult_poly_a @ A @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % smult_0_right
thf(fact_36_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_37_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_38_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_39_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_40_add_Oleft__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A) = A)))). % add.left_neutral
thf(fact_41_add_Oleft__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A) = A)))). % add.left_neutral
thf(fact_42_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_43_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_44_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_45_add_Oright__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.right_neutral
thf(fact_46_add_Oright__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.right_neutral
thf(fact_47_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_48_add__cancel__left__left, axiom,
    ((![B : poly_nat, A : poly_nat]: (((plus_plus_poly_nat @ B @ A) = A) = (B = zero_zero_poly_nat))))). % add_cancel_left_left
thf(fact_49_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_50_add__cancel__left__right, axiom,
    ((![A : poly_nat, B : poly_nat]: (((plus_plus_poly_nat @ A @ B) = A) = (B = zero_zero_poly_nat))))). % add_cancel_left_right
thf(fact_51_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_52_add__right__cancel, axiom,
    ((![B : poly_nat, A : poly_nat, C : poly_nat]: (((plus_plus_poly_nat @ B @ A) = (plus_plus_poly_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_53_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_54_add__left__cancel, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: (((plus_plus_poly_nat @ A @ B) = (plus_plus_poly_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_55_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_56_pCons__eq__iff, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: (((pCons_nat @ A @ P) = (pCons_nat @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_57_pCons__eq__iff, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: (((pCons_poly_a @ A @ P) = (pCons_poly_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_58_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_59_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_60_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_61_add__cancel__right__right, axiom,
    ((![A : poly_nat, B : poly_nat]: ((A = (plus_plus_poly_nat @ A @ B)) = (B = zero_zero_poly_nat))))). % add_cancel_right_right
thf(fact_62_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_63_add__cancel__right__left, axiom,
    ((![A : poly_nat, B : poly_nat]: ((A = (plus_plus_poly_nat @ B @ A)) = (B = zero_zero_poly_nat))))). % add_cancel_right_left
thf(fact_64_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_65_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_66_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_67_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_68_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_69_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_70_add__right__imp__eq, axiom,
    ((![B : poly_nat, A : poly_nat, C : poly_nat]: (((plus_plus_poly_nat @ B @ A) = (plus_plus_poly_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_71_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_72_add__left__imp__eq, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: (((plus_plus_poly_nat @ A @ B) = (plus_plus_poly_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_73_add_Oleft__commute, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((plus_plus_poly_a @ B @ (plus_plus_poly_a @ A @ C)) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % add.left_commute
thf(fact_74_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_75_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_76_add_Oleft__commute, axiom,
    ((![B : poly_nat, A : poly_nat, C : poly_nat]: ((plus_plus_poly_nat @ B @ (plus_plus_poly_nat @ A @ C)) = (plus_plus_poly_nat @ A @ (plus_plus_poly_nat @ B @ C)))))). % add.left_commute
thf(fact_77_add_Oleft__commute, axiom,
    ((![B : poly_poly_a, A : poly_poly_a, C : poly_poly_a]: ((plus_p1976640465poly_a @ B @ (plus_p1976640465poly_a @ A @ C)) = (plus_p1976640465poly_a @ A @ (plus_p1976640465poly_a @ B @ C)))))). % add.left_commute
thf(fact_78_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A2 : poly_a]: (^[B2 : poly_a]: (plus_plus_poly_a @ B2 @ A2)))))). % add.commute
thf(fact_79_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_80_add_Ocommute, axiom,
    ((plus_plus_a = (^[A2 : a]: (^[B2 : a]: (plus_plus_a @ B2 @ A2)))))). % add.commute
thf(fact_81_add_Ocommute, axiom,
    ((plus_plus_poly_nat = (^[A2 : poly_nat]: (^[B2 : poly_nat]: (plus_plus_poly_nat @ B2 @ A2)))))). % add.commute
thf(fact_82_add_Ocommute, axiom,
    ((plus_p1976640465poly_a = (^[A2 : poly_poly_a]: (^[B2 : poly_poly_a]: (plus_p1976640465poly_a @ B2 @ A2)))))). % add.commute
thf(fact_83_add_Oassoc, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % add.assoc
thf(fact_84_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_85_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_86_add_Oassoc, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((plus_plus_poly_nat @ (plus_plus_poly_nat @ A @ B) @ C) = (plus_plus_poly_nat @ A @ (plus_plus_poly_nat @ B @ C)))))). % add.assoc
thf(fact_87_add_Oassoc, axiom,
    ((![A : poly_poly_a, B : poly_poly_a, C : poly_poly_a]: ((plus_p1976640465poly_a @ (plus_p1976640465poly_a @ A @ B) @ C) = (plus_p1976640465poly_a @ A @ (plus_p1976640465poly_a @ B @ C)))))). % add.assoc
thf(fact_88_group__cancel_Oadd2, axiom,
    ((![B3 : poly_a, K : poly_a, B : poly_a, A : poly_a]: ((B3 = (plus_plus_poly_a @ K @ B)) => ((plus_plus_poly_a @ A @ B3) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add2
thf(fact_89_group__cancel_Oadd2, axiom,
    ((![B3 : nat, K : nat, B : nat, A : nat]: ((B3 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_90_group__cancel_Oadd2, axiom,
    ((![B3 : a, K : a, B : a, A : a]: ((B3 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B3) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_91_group__cancel_Oadd2, axiom,
    ((![B3 : poly_nat, K : poly_nat, B : poly_nat, A : poly_nat]: ((B3 = (plus_plus_poly_nat @ K @ B)) => ((plus_plus_poly_nat @ A @ B3) = (plus_plus_poly_nat @ K @ (plus_plus_poly_nat @ A @ B))))))). % group_cancel.add2
thf(fact_92_group__cancel_Oadd2, axiom,
    ((![B3 : poly_poly_a, K : poly_poly_a, B : poly_poly_a, A : poly_poly_a]: ((B3 = (plus_p1976640465poly_a @ K @ B)) => ((plus_p1976640465poly_a @ A @ B3) = (plus_p1976640465poly_a @ K @ (plus_p1976640465poly_a @ A @ B))))))). % group_cancel.add2
thf(fact_93_group__cancel_Oadd1, axiom,
    ((![A3 : poly_a, K : poly_a, A : poly_a, B : poly_a]: ((A3 = (plus_plus_poly_a @ K @ A)) => ((plus_plus_poly_a @ A3 @ B) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_94_group__cancel_Oadd1, axiom,
    ((![A3 : nat, K : nat, A : nat, B : nat]: ((A3 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A3 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_95_group__cancel_Oadd1, axiom,
    ((![A3 : a, K : a, A : a, B : a]: ((A3 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A3 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_96_group__cancel_Oadd1, axiom,
    ((![A3 : poly_nat, K : poly_nat, A : poly_nat, B : poly_nat]: ((A3 = (plus_plus_poly_nat @ K @ A)) => ((plus_plus_poly_nat @ A3 @ B) = (plus_plus_poly_nat @ K @ (plus_plus_poly_nat @ A @ B))))))). % group_cancel.add1
thf(fact_97_group__cancel_Oadd1, axiom,
    ((![A3 : poly_poly_a, K : poly_poly_a, A : poly_poly_a, B : poly_poly_a]: ((A3 = (plus_p1976640465poly_a @ K @ A)) => ((plus_p1976640465poly_a @ A3 @ B) = (plus_p1976640465poly_a @ K @ (plus_p1976640465poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_98_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_99_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_100_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_101_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_102_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((plus_plus_poly_nat @ (plus_plus_poly_nat @ A @ B) @ C) = (plus_plus_poly_nat @ A @ (plus_plus_poly_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_103_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : poly_poly_a, B : poly_poly_a, C : poly_poly_a]: ((plus_p1976640465poly_a @ (plus_p1976640465poly_a @ A @ B) @ C) = (plus_p1976640465poly_a @ A @ (plus_p1976640465poly_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_104_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A4 : nat, P2 : poly_nat]: (~ ((X = (pCons_nat @ A4 @ P2)))))))))). % pderiv.cases
thf(fact_105_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A4 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_106_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A4 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_107_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A4 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_108_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_109_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_110_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_111_add_Ocomm__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.comm_neutral
thf(fact_112_add_Ocomm__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.comm_neutral
thf(fact_113_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_114_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_115_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_116_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_117_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_118_pderiv_Oinduct, axiom,
    ((![P3 : poly_nat > $o, A0 : poly_nat]: ((![A4 : nat, P2 : poly_nat]: (((~ ((P2 = zero_zero_poly_nat))) => (P3 @ P2)) => (P3 @ (pCons_nat @ A4 @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_119_poly__induct2, axiom,
    ((![P3 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A4 : a, P2 : poly_a, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A4 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_120_poly__induct2, axiom,
    ((![P3 : poly_a > poly_nat > $o, P : poly_a, Q : poly_nat]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A4 : a, P2 : poly_a, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A4 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_121_poly__induct2, axiom,
    ((![P3 : poly_a > poly_poly_a > $o, P : poly_a, Q : poly_poly_a]: ((P3 @ zero_zero_poly_a @ zero_z2096148049poly_a) => ((![A4 : a, P2 : poly_a, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A4 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_122_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_a > $o, P : poly_nat, Q : poly_a]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A4 : nat, P2 : poly_nat, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A4 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_123_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_nat > $o, P : poly_nat, Q : poly_nat]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A4 : nat, P2 : poly_nat, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A4 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_124_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_poly_a > $o, P : poly_nat, Q : poly_poly_a]: ((P3 @ zero_zero_poly_nat @ zero_z2096148049poly_a) => ((![A4 : nat, P2 : poly_nat, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A4 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_125_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_a > $o, P : poly_poly_a, Q : poly_a]: ((P3 @ zero_z2096148049poly_a @ zero_zero_poly_a) => ((![A4 : poly_a, P2 : poly_poly_a, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A4 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_126_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_nat > $o, P : poly_poly_a, Q : poly_nat]: ((P3 @ zero_z2096148049poly_a @ zero_zero_poly_nat) => ((![A4 : poly_a, P2 : poly_poly_a, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A4 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_127_poly__induct2, axiom,
    ((![P3 : poly_poly_a > poly_poly_a > $o, P : poly_poly_a, Q : poly_poly_a]: ((P3 @ zero_z2096148049poly_a @ zero_z2096148049poly_a) => ((![A4 : poly_a, P2 : poly_poly_a, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A4 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_128_smult__add__right, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((smult_a @ A @ (plus_plus_poly_a @ P @ Q)) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (smult_a @ A @ Q)))))). % smult_add_right
thf(fact_129_smult__add__right, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((smult_nat @ A @ (plus_plus_poly_nat @ P @ Q)) = (plus_plus_poly_nat @ (smult_nat @ A @ P) @ (smult_nat @ A @ Q)))))). % smult_add_right
thf(fact_130_smult__add__right, axiom,
    ((![A : poly_a, P : poly_poly_a, Q : poly_poly_a]: ((smult_poly_a @ A @ (plus_p1976640465poly_a @ P @ Q)) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ P) @ (smult_poly_a @ A @ Q)))))). % smult_add_right
thf(fact_131_pCons__induct, axiom,
    ((![P3 : poly_poly_nat > $o, P : poly_poly_nat]: ((P3 @ zero_z1059985641ly_nat) => ((![A4 : poly_nat, P2 : poly_poly_nat]: (((~ ((A4 = zero_zero_poly_nat))) | (~ ((P2 = zero_z1059985641ly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_poly_nat @ A4 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_132_pCons__induct, axiom,
    ((![P3 : poly_poly_poly_a > $o, P : poly_poly_poly_a]: ((P3 @ zero_z2064990175poly_a) => ((![A4 : poly_poly_a, P2 : poly_poly_poly_a]: (((~ ((A4 = zero_z2096148049poly_a))) | (~ ((P2 = zero_z2064990175poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_poly_a @ A4 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_133_pCons__induct, axiom,
    ((![P3 : poly_a > $o, P : poly_a]: ((P3 @ zero_zero_poly_a) => ((![A4 : a, P2 : poly_a]: (((~ ((A4 = zero_zero_a))) | (~ ((P2 = zero_zero_poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_a @ A4 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_134_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A4 : nat, P2 : poly_nat]: (((~ ((A4 = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A4 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_135_pCons__induct, axiom,
    ((![P3 : poly_poly_a > $o, P : poly_poly_a]: ((P3 @ zero_z2096148049poly_a) => ((![A4 : poly_a, P2 : poly_poly_a]: (((~ ((A4 = zero_zero_poly_a))) | (~ ((P2 = zero_z2096148049poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_a @ A4 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_136_smult__add__left, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ (plus_plus_poly_a @ A @ B) @ P) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ P) @ (smult_poly_a @ B @ P)))))). % smult_add_left
thf(fact_137_smult__add__left, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ (plus_plus_nat @ A @ B) @ P) = (plus_plus_poly_nat @ (smult_nat @ A @ P) @ (smult_nat @ B @ P)))))). % smult_add_left
thf(fact_138_smult__add__left, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ (plus_plus_a @ A @ B) @ P) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (smult_a @ B @ P)))))). % smult_add_left
thf(fact_139_smult__add__left, axiom,
    ((![A : poly_nat, B : poly_nat, P : poly_poly_nat]: ((smult_poly_nat @ (plus_plus_poly_nat @ A @ B) @ P) = (plus_p1835221865ly_nat @ (smult_poly_nat @ A @ P) @ (smult_poly_nat @ B @ P)))))). % smult_add_left
thf(fact_140_smult__add__left, axiom,
    ((![A : poly_poly_a, B : poly_poly_a, P : poly_poly_poly_a]: ((smult_poly_poly_a @ (plus_p1976640465poly_a @ A @ B) @ P) = (plus_p672445791poly_a @ (smult_poly_poly_a @ A @ P) @ (smult_poly_poly_a @ B @ P)))))). % smult_add_left
thf(fact_141_synthetic__div__unique__lemma, axiom,
    ((![C : a, P : poly_a, A : a]: (((smult_a @ C @ P) = (pCons_a @ A @ P)) => (P = zero_zero_poly_a))))). % synthetic_div_unique_lemma
thf(fact_142_synthetic__div__unique__lemma, axiom,
    ((![C : nat, P : poly_nat, A : nat]: (((smult_nat @ C @ P) = (pCons_nat @ A @ P)) => (P = zero_zero_poly_nat))))). % synthetic_div_unique_lemma
thf(fact_143_synthetic__div__unique__lemma, axiom,
    ((![C : poly_a, P : poly_poly_a, A : poly_a]: (((smult_poly_a @ C @ P) = (pCons_poly_a @ A @ P)) => (P = zero_z2096148049poly_a))))). % synthetic_div_unique_lemma
thf(fact_144_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_145_verit__sum__simplify, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % verit_sum_simplify
thf(fact_146_verit__sum__simplify, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % verit_sum_simplify
thf(fact_147_mult__pCons__right, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((times_times_poly_a @ P @ (pCons_a @ A @ Q)) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (pCons_a @ zero_zero_a @ (times_times_poly_a @ P @ Q))))))). % mult_pCons_right
thf(fact_148_mult__pCons__right, axiom,
    ((![P : poly_poly_a, A : poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ P @ (pCons_poly_a @ A @ Q)) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ P) @ (pCons_poly_a @ zero_zero_poly_a @ (times_545135445poly_a @ P @ Q))))))). % mult_pCons_right
thf(fact_149_mult__pCons__right, axiom,
    ((![P : poly_nat, A : nat, Q : poly_nat]: ((times_times_poly_nat @ P @ (pCons_nat @ A @ Q)) = (plus_plus_poly_nat @ (smult_nat @ A @ P) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q))))))). % mult_pCons_right
thf(fact_150_mult__pCons__right, axiom,
    ((![P : poly_poly_nat, A : poly_nat, Q : poly_poly_nat]: ((times_1465266917ly_nat @ P @ (pCons_poly_nat @ A @ Q)) = (plus_p1835221865ly_nat @ (smult_poly_nat @ A @ P) @ (pCons_poly_nat @ zero_zero_poly_nat @ (times_1465266917ly_nat @ P @ Q))))))). % mult_pCons_right
thf(fact_151_mult__pCons__right, axiom,
    ((![P : poly_poly_poly_a, A : poly_poly_a, Q : poly_poly_poly_a]: ((times_1069126883poly_a @ P @ (pCons_poly_poly_a @ A @ Q)) = (plus_p672445791poly_a @ (smult_poly_poly_a @ A @ P) @ (pCons_poly_poly_a @ zero_z2096148049poly_a @ (times_1069126883poly_a @ P @ Q))))))). % mult_pCons_right
thf(fact_152_mult__pCons__left, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((times_times_poly_a @ (pCons_a @ A @ P) @ Q) = (plus_plus_poly_a @ (smult_a @ A @ Q) @ (pCons_a @ zero_zero_a @ (times_times_poly_a @ P @ Q))))))). % mult_pCons_left
thf(fact_153_mult__pCons__left, axiom,
    ((![A : poly_a, P : poly_poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ (pCons_poly_a @ A @ P) @ Q) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ Q) @ (pCons_poly_a @ zero_zero_poly_a @ (times_545135445poly_a @ P @ Q))))))). % mult_pCons_left
thf(fact_154_mult__pCons__left, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((times_times_poly_nat @ (pCons_nat @ A @ P) @ Q) = (plus_plus_poly_nat @ (smult_nat @ A @ Q) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q))))))). % mult_pCons_left
thf(fact_155_mult__pCons__left, axiom,
    ((![A : poly_nat, P : poly_poly_nat, Q : poly_poly_nat]: ((times_1465266917ly_nat @ (pCons_poly_nat @ A @ P) @ Q) = (plus_p1835221865ly_nat @ (smult_poly_nat @ A @ Q) @ (pCons_poly_nat @ zero_zero_poly_nat @ (times_1465266917ly_nat @ P @ Q))))))). % mult_pCons_left
thf(fact_156_mult__pCons__left, axiom,
    ((![A : poly_poly_a, P : poly_poly_poly_a, Q : poly_poly_poly_a]: ((times_1069126883poly_a @ (pCons_poly_poly_a @ A @ P) @ Q) = (plus_p672445791poly_a @ (smult_poly_poly_a @ A @ Q) @ (pCons_poly_poly_a @ zero_z2096148049poly_a @ (times_1069126883poly_a @ P @ Q))))))). % mult_pCons_left
thf(fact_157_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_158_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_159_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_160_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_161_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_162_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_163_smult__one, axiom,
    ((![C : nat]: ((smult_nat @ C @ one_one_poly_nat) = (pCons_nat @ C @ zero_zero_poly_nat))))). % smult_one
thf(fact_164_pderiv_Osimps, axiom,
    ((![P : poly_poly_nat, A : poly_nat]: (((P = zero_z1059985641ly_nat) => ((pderiv_poly_nat @ (pCons_poly_nat @ A @ P)) = zero_z1059985641ly_nat)) & ((~ ((P = zero_z1059985641ly_nat))) => ((pderiv_poly_nat @ (pCons_poly_nat @ A @ P)) = (plus_p1835221865ly_nat @ P @ (pCons_poly_nat @ zero_zero_poly_nat @ (pderiv_poly_nat @ P))))))))). % pderiv.simps
thf(fact_165_pderiv_Osimps, axiom,
    ((![P : poly_nat, A : nat]: (((P = zero_zero_poly_nat) => ((pderiv_nat @ (pCons_nat @ A @ P)) = zero_zero_poly_nat)) & ((~ ((P = zero_zero_poly_nat))) => ((pderiv_nat @ (pCons_nat @ A @ P)) = (plus_plus_poly_nat @ P @ (pCons_nat @ zero_zero_nat @ (pderiv_nat @ P))))))))). % pderiv.simps
thf(fact_166_pderiv_Oelims, axiom,
    ((![X : poly_poly_nat, Y : poly_poly_nat]: (((pderiv_poly_nat @ X) = Y) => (~ ((![A4 : poly_nat, P2 : poly_poly_nat]: ((X = (pCons_poly_nat @ A4 @ P2)) => (~ ((((P2 = zero_z1059985641ly_nat) => (Y = zero_z1059985641ly_nat)) & ((~ ((P2 = zero_z1059985641ly_nat))) => (Y = (plus_p1835221865ly_nat @ P2 @ (pCons_poly_nat @ zero_zero_poly_nat @ (pderiv_poly_nat @ P2)))))))))))))))). % pderiv.elims
thf(fact_167_pderiv_Oelims, axiom,
    ((![X : poly_nat, Y : poly_nat]: (((pderiv_nat @ X) = Y) => (~ ((![A4 : nat, P2 : poly_nat]: ((X = (pCons_nat @ A4 @ P2)) => (~ ((((P2 = zero_zero_poly_nat) => (Y = zero_zero_poly_nat)) & ((~ ((P2 = zero_zero_poly_nat))) => (Y = (plus_plus_poly_nat @ P2 @ (pCons_nat @ zero_zero_nat @ (pderiv_nat @ P2)))))))))))))))). % pderiv.elims
thf(fact_168_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_169_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_170_smult__smult, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (smult_a @ B @ P)) = (smult_a @ (times_times_a @ A @ B) @ P))))). % smult_smult
thf(fact_171_smult__smult, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ A @ (smult_poly_a @ B @ P)) = (smult_poly_a @ (times_times_poly_a @ A @ B) @ P))))). % smult_smult
thf(fact_172_smult__smult, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (smult_nat @ B @ P)) = (smult_nat @ (times_times_nat @ A @ B) @ P))))). % smult_smult
thf(fact_173_smult__1__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ one_one_nat @ P) = P)))). % smult_1_left
thf(fact_174_pderiv__0, axiom,
    (((pderiv_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pderiv_0
thf(fact_175_mult__smult__left, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((times_times_poly_a @ (smult_a @ A @ P) @ Q) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_left
thf(fact_176_mult__smult__left, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((times_times_poly_nat @ (smult_nat @ A @ P) @ Q) = (smult_nat @ A @ (times_times_poly_nat @ P @ Q)))))). % mult_smult_left
thf(fact_177_mult__smult__left, axiom,
    ((![A : poly_a, P : poly_poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ (smult_poly_a @ A @ P) @ Q) = (smult_poly_a @ A @ (times_545135445poly_a @ P @ Q)))))). % mult_smult_left
thf(fact_178_mult__smult__right, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((times_times_poly_a @ P @ (smult_a @ A @ Q)) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_right
thf(fact_179_mult__smult__right, axiom,
    ((![P : poly_nat, A : nat, Q : poly_nat]: ((times_times_poly_nat @ P @ (smult_nat @ A @ Q)) = (smult_nat @ A @ (times_times_poly_nat @ P @ Q)))))). % mult_smult_right
thf(fact_180_mult__smult__right, axiom,
    ((![P : poly_poly_a, A : poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ P @ (smult_poly_a @ A @ Q)) = (smult_poly_a @ A @ (times_545135445poly_a @ P @ Q)))))). % mult_smult_right
thf(fact_181_smult__pCons, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (pCons_a @ B @ P)) = (pCons_a @ (times_times_a @ A @ B) @ (smult_a @ A @ P)))))). % smult_pCons
thf(fact_182_smult__pCons, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ A @ (pCons_poly_a @ B @ P)) = (pCons_poly_a @ (times_times_poly_a @ A @ B) @ (smult_poly_a @ A @ P)))))). % smult_pCons
thf(fact_183_smult__pCons, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (pCons_nat @ B @ P)) = (pCons_nat @ (times_times_nat @ A @ B) @ (smult_nat @ A @ P)))))). % smult_pCons
thf(fact_184_pderiv__singleton, axiom,
    ((![A : nat]: ((pderiv_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = zero_zero_poly_nat)))). % pderiv_singleton
thf(fact_185_pderiv__1, axiom,
    (((pderiv_nat @ one_one_poly_nat) = zero_zero_poly_nat))). % pderiv_1
thf(fact_186_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = zero_zero_poly_nat)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_nat @ N @ one_one_poly_nat) = one_one_poly_nat)))))). % poly_cutoff_1
thf(fact_187_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_188_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_189_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_190_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_191_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_192_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_193_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_194_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_195_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_196_pderiv__mult, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((pderiv_nat @ (times_times_poly_nat @ P @ Q)) = (plus_plus_poly_nat @ (times_times_poly_nat @ P @ (pderiv_nat @ Q)) @ (times_times_poly_nat @ Q @ (pderiv_nat @ P))))))). % pderiv_mult
thf(fact_197_crossproduct__eq, axiom,
    ((![W : nat, Y : nat, X : nat, Z : nat]: (((plus_plus_nat @ (times_times_nat @ W @ Y) @ (times_times_nat @ X @ Z)) = (plus_plus_nat @ (times_times_nat @ W @ Z) @ (times_times_nat @ X @ Y))) = (((W = X)) | ((Y = Z))))))). % crossproduct_eq
thf(fact_198_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_199_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_200_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_201_mult__poly__0__left, axiom,
    ((![Q : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % mult_poly_0_left
thf(fact_202_mult__poly__0__left, axiom,
    ((![Q : poly_poly_a]: ((times_545135445poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % mult_poly_0_left
thf(fact_203_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_204_mult__poly__0__right, axiom,
    ((![P : poly_nat]: ((times_times_poly_nat @ P @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_poly_0_right
thf(fact_205_mult__poly__0__right, axiom,
    ((![P : poly_poly_a]: ((times_545135445poly_a @ P @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % mult_poly_0_right
thf(fact_206_pderiv__smult, axiom,
    ((![A : nat, P : poly_nat]: ((pderiv_nat @ (smult_nat @ A @ P)) = (smult_nat @ A @ (pderiv_nat @ P)))))). % pderiv_smult
thf(fact_207_pderiv__add, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((pderiv_nat @ (plus_plus_poly_nat @ P @ Q)) = (plus_plus_poly_nat @ (pderiv_nat @ P) @ (pderiv_nat @ Q)))))). % pderiv_add
thf(fact_208_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_209_mult__poly__add__left, axiom,
    ((![P : poly_a, Q : poly_a, R : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ P @ Q) @ R) = (plus_plus_poly_a @ (times_times_poly_a @ P @ R) @ (times_times_poly_a @ Q @ R)))))). % mult_poly_add_left
thf(fact_210_mult__poly__add__left, axiom,
    ((![P : poly_nat, Q : poly_nat, R : poly_nat]: ((times_times_poly_nat @ (plus_plus_poly_nat @ P @ Q) @ R) = (plus_plus_poly_nat @ (times_times_poly_nat @ P @ R) @ (times_times_poly_nat @ Q @ R)))))). % mult_poly_add_left
thf(fact_211_mult__poly__add__left, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, R : poly_poly_a]: ((times_545135445poly_a @ (plus_p1976640465poly_a @ P @ Q) @ R) = (plus_p1976640465poly_a @ (times_545135445poly_a @ P @ R) @ (times_545135445poly_a @ Q @ R)))))). % mult_poly_add_left
thf(fact_212_pderiv__iszero, axiom,
    ((![P : poly_nat]: (((pderiv_nat @ P) = zero_zero_poly_nat) => (?[H2 : nat]: (P = (pCons_nat @ H2 @ zero_zero_poly_nat))))))). % pderiv_iszero
thf(fact_213_pderiv__pCons, axiom,
    ((![A : nat, P : poly_nat]: ((pderiv_nat @ (pCons_nat @ A @ P)) = (plus_plus_poly_nat @ P @ (pCons_nat @ zero_zero_nat @ (pderiv_nat @ P))))))). % pderiv_pCons
thf(fact_214_pderiv__pCons, axiom,
    ((![A : poly_nat, P : poly_poly_nat]: ((pderiv_poly_nat @ (pCons_poly_nat @ A @ P)) = (plus_p1835221865ly_nat @ P @ (pCons_poly_nat @ zero_zero_poly_nat @ (pderiv_poly_nat @ P))))))). % pderiv_pCons
thf(fact_215_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_216_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_217_mult__eq__0__iff, axiom,
    ((![A : poly_nat, B : poly_nat]: (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat) = (((A = zero_zero_poly_nat)) | ((B = zero_zero_poly_nat))))))). % mult_eq_0_iff
thf(fact_218_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_219_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_220_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_221_mult__zero__right, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % mult_zero_right
thf(fact_222_mult__zero__right, axiom,
    ((![A : poly_poly_a]: ((times_545135445poly_a @ A @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % mult_zero_right
thf(fact_223_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_224_mult__zero__left, axiom,
    ((![A : poly_nat]: ((times_times_poly_nat @ zero_zero_poly_nat @ A) = zero_zero_poly_nat)))). % mult_zero_left
thf(fact_225_mult__zero__left, axiom,
    ((![A : poly_poly_a]: ((times_545135445poly_a @ zero_z2096148049poly_a @ A) = zero_z2096148049poly_a)))). % mult_zero_left
thf(fact_226_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_227_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_228_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_229_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_230_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_231_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_232_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_233_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_234_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_235_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_236_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_237_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_238_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_239_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_240_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_241_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_242_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_243_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_244_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K))))). % left_add_mult_distrib
thf(fact_245_Euclid__induct, axiom,
    ((![P3 : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B4 : nat]: ((P3 @ A4 @ B4) = (P3 @ B4 @ A4))) => ((![A4 : nat]: (P3 @ A4 @ zero_zero_nat)) => ((![A4 : nat, B4 : nat]: ((P3 @ A4 @ B4) => (P3 @ A4 @ (plus_plus_nat @ A4 @ B4)))) => (P3 @ A @ B))))))). % Euclid_induct

% Conjectures (4)
thf(conj_0, hypothesis,
    (((a2 = zero_zero_a) => (~ ((pa = zero_zero_poly_a)))))).
thf(conj_1, hypothesis,
    ((((fundam1358810038poly_a @ pa @ h) = zero_zero_poly_a) => (pa = zero_zero_poly_a)))).
thf(conj_2, hypothesis,
    (((plus_plus_poly_a @ (smult_a @ h @ (fundam1358810038poly_a @ pa @ h)) @ (pCons_a @ a2 @ (fundam1358810038poly_a @ pa @ h))) = zero_zero_poly_a))).
thf(conj_3, conjecture,
    (((a2 = zero_zero_a) & (pa = zero_zero_poly_a)))).
