% 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_72__5367500_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:26:45.007

% 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 (59)
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_Itf__a_J_J, type,
    plus_p1976640465poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J, type,
    plus_plus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    times_545135445poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J_J, type,
    zero_z2064990175poly_a : poly_poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_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_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_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_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_poly_a2 : poly_poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_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_Opoly__shift_001t__Nat__Onat, type,
    poly_shift_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_shift_poly_a : nat > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Opoly__shift_001tf__a, type,
    poly_shift_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    reflec581648976poly_a : poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    reflect_poly_poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001tf__a, type,
    reflect_poly_a : poly_a > poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat, type,
    synthetic_div_nat : poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_Itf__a_J, type,
    synthetic_div_poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > 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).
thf(sy_v_x, type,
    x : 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__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_4_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_5_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_6_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X : poly_poly_a]: ((poly_poly_poly_a2 @ zero_z2064990175poly_a @ X) = zero_z2096148049poly_a)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_10_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_11_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_12_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_13_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_14_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_15_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_16_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_17_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_18_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_19_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_20_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_21_poly__add, axiom,
    ((![P : poly_poly_nat, Q : poly_poly_nat, X : poly_nat]: ((poly_poly_nat2 @ (plus_p1835221865ly_nat @ P @ Q) @ X) = (plus_plus_poly_nat @ (poly_poly_nat2 @ P @ X) @ (poly_poly_nat2 @ Q @ X)))))). % poly_add
thf(fact_22_poly__add, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (plus_p1976640465poly_a @ P @ Q) @ X) = (plus_plus_poly_a @ (poly_poly_a2 @ P @ X) @ (poly_poly_a2 @ Q @ X)))))). % poly_add
thf(fact_23_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_24_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_25_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_26_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_27_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_28_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_29_add_Oleft__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A) = A)))). % add.left_neutral
thf(fact_30_add_Oleft__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A) = A)))). % add.left_neutral
thf(fact_31_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_32_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_33_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_34_add_Oright__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.right_neutral
thf(fact_35_add_Oright__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.right_neutral
thf(fact_36_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_37_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_38_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_57_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_58_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_59_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_60_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_add_Ocommute, axiom,
    ((plus_plus_a = (^[A2 : a]: (^[B2 : a]: (plus_plus_a @ B2 @ A2)))))). % add.commute
thf(fact_70_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_71_add_Ocommute, axiom,
    ((plus_plus_poly_nat = (^[A2 : poly_nat]: (^[B2 : poly_nat]: (plus_plus_poly_nat @ B2 @ A2)))))). % add.commute
thf(fact_72_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A2 : poly_a]: (^[B2 : poly_a]: (plus_plus_poly_a @ B2 @ A2)))))). % add.commute
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A4 : nat, P2 : poly_nat]: (~ ((X = (pCons_nat @ A4 @ P2)))))))))). % pderiv.cases
thf(fact_91_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A4 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_92_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A4 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_93_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A4 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A4 @ Q2)))))))))). % pCons_cases
thf(fact_94_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_95_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_96_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_97_add_Ocomm__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.comm_neutral
thf(fact_98_add_Ocomm__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.comm_neutral
thf(fact_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_120_verit__sum__simplify, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % verit_sum_simplify
thf(fact_121_verit__sum__simplify, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % verit_sum_simplify
thf(fact_122_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_123_synthetic__div__pCons, axiom,
    ((![A : nat, P : poly_nat, C : nat]: ((synthetic_div_nat @ (pCons_nat @ A @ P) @ C) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_pCons
thf(fact_124_synthetic__div__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, C : poly_a]: ((synthetic_div_poly_a @ (pCons_poly_a @ A @ P) @ C) = (pCons_poly_a @ (poly_poly_a2 @ P @ C) @ (synthetic_div_poly_a @ P @ C)))))). % synthetic_div_pCons
thf(fact_125_is__zero__null, axiom,
    ((is_zero_a = (^[P4 : poly_a]: (P4 = zero_zero_poly_a))))). % is_zero_null
thf(fact_126_is__zero__null, axiom,
    ((is_zero_nat = (^[P4 : poly_nat]: (P4 = zero_zero_poly_nat))))). % is_zero_null
thf(fact_127_is__zero__null, axiom,
    ((is_zero_poly_a = (^[P4 : poly_poly_a]: (P4 = zero_z2096148049poly_a))))). % is_zero_null
thf(fact_128_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_129_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_cutoff_0
thf(fact_130_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_cutoff_0
thf(fact_131_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_a]: (((poly_a2 @ (reflect_poly_a @ P) @ zero_zero_a) = zero_zero_a) = (P = zero_zero_poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_132_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_a]: (((poly_poly_a2 @ (reflect_poly_poly_a @ P) @ zero_zero_poly_a) = zero_zero_poly_a) = (P = zero_z2096148049poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_133_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_134_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_nat]: (((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_135_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_poly_a]: (((poly_poly_poly_a2 @ (reflec581648976poly_a @ P) @ zero_z2096148049poly_a) = zero_z2096148049poly_a) = (P = zero_z2064990175poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_136_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_137_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_138_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_139_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_140_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_141_poly__pCons, axiom,
    ((![A : poly_nat, P : poly_poly_nat, X : poly_nat]: ((poly_poly_nat2 @ (pCons_poly_nat @ A @ P) @ X) = (plus_plus_poly_nat @ A @ (times_times_poly_nat @ X @ (poly_poly_nat2 @ P @ X))))))). % poly_pCons
thf(fact_142_poly__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (pCons_poly_a @ A @ P) @ X) = (plus_plus_poly_a @ A @ (times_times_poly_a @ X @ (poly_poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_143_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_144_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_145_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_nat @ N @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % poly_shift_0
thf(fact_146_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_poly_a @ N @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % poly_shift_0
thf(fact_147_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_148_poly__mult, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (times_545135445poly_a @ P @ Q) @ X) = (times_times_poly_a @ (poly_poly_a2 @ P @ X) @ (poly_poly_a2 @ Q @ X)))))). % poly_mult
thf(fact_149_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_150_pderiv__0, axiom,
    (((pderiv_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pderiv_0
thf(fact_151_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_152_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_153_reflect__poly__0, axiom,
    (((reflect_poly_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % reflect_poly_0
thf(fact_154_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_155_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_156_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_157_pderiv__singleton, axiom,
    ((![A : nat]: ((pderiv_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = zero_zero_poly_nat)))). % pderiv_singleton
thf(fact_158_reflect__poly__const, axiom,
    ((![A : a]: ((reflect_poly_a @ (pCons_a @ A @ zero_zero_poly_a)) = (pCons_a @ A @ zero_zero_poly_a))))). % reflect_poly_const
thf(fact_159_reflect__poly__const, axiom,
    ((![A : nat]: ((reflect_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = (pCons_nat @ A @ zero_zero_poly_nat))))). % reflect_poly_const
thf(fact_160_reflect__poly__const, axiom,
    ((![A : poly_a]: ((reflect_poly_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a)) = (pCons_poly_a @ A @ zero_z2096148049poly_a))))). % reflect_poly_const
thf(fact_161_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_162_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_163_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_182_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_183_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_184_mult__zero__left, axiom,
    ((![A : poly_poly_a]: ((times_545135445poly_a @ zero_z2096148049poly_a @ A) = zero_z2096148049poly_a)))). % mult_zero_left
thf(fact_185_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_186_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_187_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_188_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_189_mult__zero__right, axiom,
    ((![A : poly_poly_a]: ((times_545135445poly_a @ A @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % mult_zero_right
thf(fact_190_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_191_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_192_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_193_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_194_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_195_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_196_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_197_no__zero__divisors, axiom,
    ((![A : poly_nat, B : poly_nat]: ((~ ((A = zero_zero_poly_nat))) => ((~ ((B = zero_zero_poly_nat))) => (~ (((times_times_poly_nat @ A @ B) = zero_zero_poly_nat)))))))). % no_zero_divisors
thf(fact_198_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_199_divisors__zero, 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)))))). % divisors_zero
thf(fact_200_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_201_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_202_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_203_mult__not__zero, 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_not_zero
thf(fact_204_mult__not__zero, axiom,
    ((![A : poly_poly_a, B : poly_poly_a]: ((~ (((times_545135445poly_a @ A @ B) = zero_z2096148049poly_a))) => ((~ ((A = zero_z2096148049poly_a))) & (~ ((B = zero_z2096148049poly_a)))))))). % mult_not_zero
thf(fact_205_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_206_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_207_combine__common__factor, axiom,
    ((![A : poly_nat, E : poly_nat, B : poly_nat, C : poly_nat]: ((plus_plus_poly_nat @ (times_times_poly_nat @ A @ E) @ (plus_plus_poly_nat @ (times_times_poly_nat @ B @ E) @ C)) = (plus_plus_poly_nat @ (times_times_poly_nat @ (plus_plus_poly_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_208_combine__common__factor, axiom,
    ((![A : poly_a, E : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (times_times_poly_a @ A @ E) @ (plus_plus_poly_a @ (times_times_poly_a @ B @ E) @ C)) = (plus_plus_poly_a @ (times_times_poly_a @ (plus_plus_poly_a @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_209_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_210_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_211_distrib__right, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((times_times_poly_nat @ (plus_plus_poly_nat @ A @ B) @ C) = (plus_plus_poly_nat @ (times_times_poly_nat @ A @ C) @ (times_times_poly_nat @ B @ C)))))). % distrib_right
thf(fact_212_distrib__right, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ (times_times_poly_a @ A @ C) @ (times_times_poly_a @ B @ C)))))). % distrib_right
thf(fact_213_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_214_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_215_distrib__left, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((times_times_poly_nat @ A @ (plus_plus_poly_nat @ B @ C)) = (plus_plus_poly_nat @ (times_times_poly_nat @ A @ B) @ (times_times_poly_nat @ A @ C)))))). % distrib_left
thf(fact_216_distrib__left, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ A @ (plus_plus_poly_a @ B @ C)) = (plus_plus_poly_a @ (times_times_poly_a @ A @ B) @ (times_times_poly_a @ A @ C)))))). % distrib_left
thf(fact_217_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_218_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_219_comm__semiring__class_Odistrib, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((times_times_poly_nat @ (plus_plus_poly_nat @ A @ B) @ C) = (plus_plus_poly_nat @ (times_times_poly_nat @ A @ C) @ (times_times_poly_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_220_comm__semiring__class_Odistrib, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ (times_times_poly_a @ A @ C) @ (times_times_poly_a @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_221_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_222_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_223_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_224_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_225_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_226_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_227_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_228_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_229_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_230_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_231_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_232_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_233_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_234_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_235_pcompose__pCons, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((pcompose_nat @ (pCons_nat @ A @ P) @ Q) = (plus_plus_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ (times_times_poly_nat @ Q @ (pcompose_nat @ P @ Q))))))). % pcompose_pCons
thf(fact_236_pcompose__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, Q : poly_poly_a]: ((pcompose_poly_a @ (pCons_poly_a @ A @ P) @ Q) = (plus_p1976640465poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ (times_545135445poly_a @ Q @ (pcompose_poly_a @ P @ Q))))))). % pcompose_pCons
thf(fact_237_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_238_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_239_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_240_pcompose__0, axiom,
    ((![Q : poly_nat]: ((pcompose_nat @ zero_zero_poly_nat @ Q) = zero_zero_poly_nat)))). % pcompose_0
thf(fact_241_pcompose__0, axiom,
    ((![Q : poly_poly_a]: ((pcompose_poly_a @ zero_z2096148049poly_a @ Q) = zero_z2096148049poly_a)))). % pcompose_0
thf(fact_242_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_243_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
thf(fact_244_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_245_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

% Conjectures (3)
thf(conj_0, hypothesis,
    (((~ ((a2 = zero_zero_a))) | (~ ((pa = zero_zero_poly_a)))))).
thf(conj_1, hypothesis,
    (((poly_a2 @ (fundam1358810038poly_a @ pa @ h) @ x) = (poly_a2 @ pa @ (plus_plus_a @ h @ x))))).
thf(conj_2, conjecture,
    (((poly_a2 @ (fundam1358810038poly_a @ (pCons_a @ a2 @ pa) @ h) @ x) = (poly_a2 @ (pCons_a @ a2 @ pa) @ (plus_plus_a @ h @ x))))).
