% 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_92__5367760_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:02.038

% 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 (45)
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_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_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    degree_poly_nat : poly_poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_Itf__a_J, type,
    degree_poly_a : poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    pCons_poly_poly_a : poly_poly_a > poly_poly_poly_a > poly_poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_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_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).

% Relevant facts (245)
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__iff, axiom,
    ((![P : poly_nat, H : nat]: (((fundam170929432ly_nat @ P @ H) = zero_zero_poly_nat) = (P = zero_zero_poly_nat))))). % offset_poly_eq_0_iff
thf(fact_10_offset__poly__eq__0__iff, axiom,
    ((![P : poly_poly_a, H : poly_a]: (((fundam1343031620poly_a @ P @ H) = zero_z2096148049poly_a) = (P = zero_z2096148049poly_a))))). % offset_poly_eq_0_iff
thf(fact_11_offset__poly__eq__0__iff, axiom,
    ((![P : poly_a, H : a]: (((fundam1358810038poly_a @ P @ H) = zero_zero_poly_a) = (P = zero_zero_poly_a))))). % offset_poly_eq_0_iff
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_degree__smult__eq, axiom,
    ((![A : poly_nat, P : poly_poly_nat]: (((A = zero_zero_poly_nat) => ((degree_poly_nat @ (smult_poly_nat @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_poly_nat))) => ((degree_poly_nat @ (smult_poly_nat @ A @ P)) = (degree_poly_nat @ P))))))). % degree_smult_eq
thf(fact_21_degree__smult__eq, axiom,
    ((![A : nat, P : poly_nat]: (((A = zero_zero_nat) => ((degree_nat @ (smult_nat @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_nat))) => ((degree_nat @ (smult_nat @ A @ P)) = (degree_nat @ P))))))). % degree_smult_eq
thf(fact_22_smult__0__left, axiom,
    ((![P : poly_poly_nat]: ((smult_poly_nat @ zero_zero_poly_nat @ P) = zero_z1059985641ly_nat)))). % smult_0_left
thf(fact_23_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_24_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_25_smult__0__left, axiom,
    ((![P : poly_poly_a]: ((smult_poly_a @ zero_zero_poly_a @ P) = zero_z2096148049poly_a)))). % smult_0_left
thf(fact_26_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_27_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_28_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_29_pCons__0__0, axiom,
    (((pCons_poly_nat @ zero_zero_poly_nat @ zero_z1059985641ly_nat) = zero_z1059985641ly_nat))). % pCons_0_0
thf(fact_30_pCons__0__0, axiom,
    (((pCons_poly_poly_a @ zero_z2096148049poly_a @ zero_z2064990175poly_a) = zero_z2064990175poly_a))). % pCons_0_0
thf(fact_31_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_32_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_33_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_34_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_35_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_36_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_37_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_38_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_39_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_40_smult__0__right, axiom,
    ((![A : nat]: ((smult_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % smult_0_right
thf(fact_41_smult__0__right, axiom,
    ((![A : poly_a]: ((smult_poly_a @ A @ zero_z2096148049poly_a) = zero_z2096148049poly_a)))). % smult_0_right
thf(fact_42_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_43_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_44_add_Oleft__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A) = A)))). % add.left_neutral
thf(fact_45_add_Oleft__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ zero_z2096148049poly_a @ A) = A)))). % add.left_neutral
thf(fact_46_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_47_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_48_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_49_add_Oright__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.right_neutral
thf(fact_50_add_Oright__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.right_neutral
thf(fact_51_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_52_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_53_add__cancel__left__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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_70_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_71_degree__0, axiom,
    (((degree_poly_a @ zero_z2096148049poly_a) = zero_zero_nat))). % degree_0
thf(fact_72_degree__pCons__eq__if, axiom,
    ((![P : poly_a, A : a]: (((P = zero_zero_poly_a) => ((degree_a @ (pCons_a @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P)) = (suc @ (degree_a @ P)))))))). % degree_pCons_eq_if
thf(fact_73_degree__pCons__eq__if, axiom,
    ((![P : poly_nat, A : nat]: (((P = zero_zero_poly_nat) => ((degree_nat @ (pCons_nat @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A @ P)) = (suc @ (degree_nat @ P)))))))). % degree_pCons_eq_if
thf(fact_74_degree__pCons__eq__if, axiom,
    ((![P : poly_poly_a, A : poly_a]: (((P = zero_z2096148049poly_a) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = zero_zero_nat)) & ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = (suc @ (degree_poly_a @ P)))))))). % degree_pCons_eq_if
thf(fact_75_degree__eq__zeroE, axiom,
    ((![P : poly_a]: (((degree_a @ P) = zero_zero_nat) => (~ ((![A2 : a]: (~ ((P = (pCons_a @ A2 @ zero_zero_poly_a))))))))))). % degree_eq_zeroE
thf(fact_76_degree__eq__zeroE, axiom,
    ((![P : poly_nat]: (((degree_nat @ P) = zero_zero_nat) => (~ ((![A2 : nat]: (~ ((P = (pCons_nat @ A2 @ zero_zero_poly_nat))))))))))). % degree_eq_zeroE
thf(fact_77_degree__eq__zeroE, axiom,
    ((![P : poly_poly_a]: (((degree_poly_a @ P) = zero_zero_nat) => (~ ((![A2 : poly_a]: (~ ((P = (pCons_poly_a @ A2 @ zero_z2096148049poly_a))))))))))). % degree_eq_zeroE
thf(fact_78_degree__pCons__0, axiom,
    ((![A : a]: ((degree_a @ (pCons_a @ A @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_79_degree__pCons__0, axiom,
    ((![A : nat]: ((degree_nat @ (pCons_nat @ A @ zero_zero_poly_nat)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_80_degree__pCons__0, axiom,
    ((![A : poly_a]: ((degree_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_81_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_82_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_83_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_84_zero__reorient, axiom,
    ((![X : poly_poly_a]: ((zero_z2096148049poly_a = X) = (X = zero_z2096148049poly_a))))). % zero_reorient
thf(fact_85_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A3 : poly_a]: (^[B2 : poly_a]: (plus_plus_poly_a @ B2 @ A3)))))). % add.commute
thf(fact_96_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A3)))))). % add.commute
thf(fact_97_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B2 : a]: (plus_plus_a @ B2 @ A3)))))). % add.commute
thf(fact_98_add_Ocommute, axiom,
    ((plus_plus_poly_nat = (^[A3 : poly_nat]: (^[B2 : poly_nat]: (plus_plus_poly_nat @ B2 @ A3)))))). % add.commute
thf(fact_99_add_Ocommute, axiom,
    ((plus_p1976640465poly_a = (^[A3 : poly_poly_a]: (^[B2 : poly_poly_a]: (plus_p1976640465poly_a @ B2 @ A3)))))). % add.commute
thf(fact_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_group__cancel_Oadd1, axiom,
    ((![A4 : poly_a, K : poly_a, A : poly_a, B : poly_a]: ((A4 = (plus_plus_poly_a @ K @ A)) => ((plus_plus_poly_a @ A4 @ B) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_111_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_112_group__cancel_Oadd1, axiom,
    ((![A4 : a, K : a, A : a, B : a]: ((A4 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A4 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_113_group__cancel_Oadd1, axiom,
    ((![A4 : poly_nat, K : poly_nat, A : poly_nat, B : poly_nat]: ((A4 = (plus_plus_poly_nat @ K @ A)) => ((plus_plus_poly_nat @ A4 @ B) = (plus_plus_poly_nat @ K @ (plus_plus_poly_nat @ A @ B))))))). % group_cancel.add1
thf(fact_114_group__cancel_Oadd1, axiom,
    ((![A4 : poly_poly_a, K : poly_poly_a, A : poly_poly_a, B : poly_poly_a]: ((A4 = (plus_p1976640465poly_a @ K @ A)) => ((plus_p1976640465poly_a @ A4 @ B) = (plus_p1976640465poly_a @ K @ (plus_p1976640465poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_115_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_116_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_117_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_118_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_119_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_120_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_121_pderiv_Ocases, axiom,
    ((![X : poly_nat]: (~ ((![A2 : nat, P2 : poly_nat]: (~ ((X = (pCons_nat @ A2 @ P2)))))))))). % pderiv.cases
thf(fact_122_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_123_pCons__cases, axiom,
    ((![P : poly_nat]: (~ ((![A2 : nat, Q2 : poly_nat]: (~ ((P = (pCons_nat @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_124_pCons__cases, axiom,
    ((![P : poly_poly_a]: (~ ((![A2 : poly_a, Q2 : poly_poly_a]: (~ ((P = (pCons_poly_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_125_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_126_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_127_add_Ocomm__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.comm_neutral
thf(fact_128_add_Ocomm__neutral, axiom,
    ((![A : poly_poly_a]: ((plus_p1976640465poly_a @ A @ zero_z2096148049poly_a) = A)))). % add.comm_neutral
thf(fact_129_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_130_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_131_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_132_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_133_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_134_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_135_pderiv_Oinduct, axiom,
    ((![P3 : poly_nat > $o, A0 : poly_nat]: ((![A2 : nat, P2 : poly_nat]: (((~ ((P2 = zero_zero_poly_nat))) => (P3 @ P2)) => (P3 @ (pCons_nat @ A2 @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_136_poly__induct2, axiom,
    ((![P3 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P2 : poly_a, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A2 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_137_poly__induct2, axiom,
    ((![P3 : poly_a > poly_nat > $o, P : poly_a, Q : poly_nat]: ((P3 @ zero_zero_poly_a @ zero_zero_poly_nat) => ((![A2 : a, P2 : poly_a, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A2 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_138_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) => ((![A2 : a, P2 : poly_a, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_a @ A2 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_139_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_a > $o, P : poly_nat, Q : poly_a]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_a) => ((![A2 : nat, P2 : poly_nat, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A2 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_140_poly__induct2, axiom,
    ((![P3 : poly_nat > poly_nat > $o, P : poly_nat, Q : poly_nat]: ((P3 @ zero_zero_poly_nat @ zero_zero_poly_nat) => ((![A2 : nat, P2 : poly_nat, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A2 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_141_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) => ((![A2 : nat, P2 : poly_nat, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_nat @ A2 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_142_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) => ((![A2 : poly_a, P2 : poly_poly_a, B4 : a, Q2 : poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A2 @ P2) @ (pCons_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_143_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) => ((![A2 : poly_a, P2 : poly_poly_a, B4 : nat, Q2 : poly_nat]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A2 @ P2) @ (pCons_nat @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_144_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) => ((![A2 : poly_a, P2 : poly_poly_a, B4 : poly_a, Q2 : poly_poly_a]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_poly_a @ A2 @ P2) @ (pCons_poly_a @ B4 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_145_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_146_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_147_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_148_pCons__induct, axiom,
    ((![P3 : poly_poly_nat > $o, P : poly_poly_nat]: ((P3 @ zero_z1059985641ly_nat) => ((![A2 : poly_nat, P2 : poly_poly_nat]: (((~ ((A2 = zero_zero_poly_nat))) | (~ ((P2 = zero_z1059985641ly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_poly_nat @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_149_pCons__induct, axiom,
    ((![P3 : poly_poly_poly_a > $o, P : poly_poly_poly_a]: ((P3 @ zero_z2064990175poly_a) => ((![A2 : poly_poly_a, P2 : poly_poly_poly_a]: (((~ ((A2 = zero_z2096148049poly_a))) | (~ ((P2 = zero_z2064990175poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_poly_a @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_150_pCons__induct, axiom,
    ((![P3 : poly_a > $o, P : poly_a]: ((P3 @ zero_zero_poly_a) => ((![A2 : a, P2 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P2 = zero_zero_poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_a @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_151_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A2 : nat, P2 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_152_pCons__induct, axiom,
    ((![P3 : poly_poly_a > $o, P : poly_poly_a]: ((P3 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P2 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P2 = zero_z2096148049poly_a)))) => ((P3 @ P2) => (P3 @ (pCons_poly_a @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_degree__pCons__eq, axiom,
    ((![P : poly_a, A : a]: ((~ ((P = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P)) = (suc @ (degree_a @ P))))))). % degree_pCons_eq
thf(fact_162_degree__pCons__eq, axiom,
    ((![P : poly_nat, A : nat]: ((~ ((P = zero_zero_poly_nat))) => ((degree_nat @ (pCons_nat @ A @ P)) = (suc @ (degree_nat @ P))))))). % degree_pCons_eq
thf(fact_163_degree__pCons__eq, axiom,
    ((![P : poly_poly_a, A : poly_a]: ((~ ((P = zero_z2096148049poly_a))) => ((degree_poly_a @ (pCons_poly_a @ A @ P)) = (suc @ (degree_poly_a @ P))))))). % degree_pCons_eq
thf(fact_164_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_165_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_166_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_167_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_168_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_169_add__is__1, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = (suc @ zero_zero_nat)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_170_one__is__add, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M @ N)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_171_exists__least__lemma, axiom,
    ((![P3 : nat > $o]: ((~ ((P3 @ zero_zero_nat))) => ((?[X_1 : nat]: (P3 @ X_1)) => (?[N2 : nat]: ((~ ((P3 @ N2))) & (P3 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_172_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_173_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_174_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_175_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_176_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_177_nat__arith_Osuc1, axiom,
    ((![A4 : nat, K : nat, A : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((suc @ A4) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1
thf(fact_178_add__Suc, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc
thf(fact_179_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_180_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_181_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_182_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_183_nat__induct, axiom,
    ((![P3 : nat > $o, N : nat]: ((P3 @ zero_zero_nat) => ((![N2 : nat]: ((P3 @ N2) => (P3 @ (suc @ N2)))) => (P3 @ N)))))). % nat_induct
thf(fact_184_diff__induct, axiom,
    ((![P3 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P3 @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P3 @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P3 @ X3 @ Y3) => (P3 @ (suc @ X3) @ (suc @ Y3)))) => (P3 @ M @ N))))))). % diff_induct
thf(fact_185_zero__induct, axiom,
    ((![P3 : nat > $o, K : nat]: ((P3 @ K) => ((![N2 : nat]: ((P3 @ (suc @ N2)) => (P3 @ N2))) => (P3 @ zero_zero_nat)))))). % zero_induct
thf(fact_186_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_187_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_188_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_189_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_190_old_Onat_Oinducts, axiom,
    ((![P3 : nat > $o, Nat : nat]: ((P3 @ zero_zero_nat) => ((![Nat3 : nat]: ((P3 @ Nat3) => (P3 @ (suc @ Nat3)))) => (P3 @ Nat)))))). % old.nat.inducts
thf(fact_191_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_192_Euclid__induct, axiom,
    ((![P3 : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B4 : nat]: ((P3 @ A2 @ B4) = (P3 @ B4 @ A2))) => ((![A2 : nat]: (P3 @ A2 @ zero_zero_nat)) => ((![A2 : nat, B4 : nat]: ((P3 @ A2 @ B4) => (P3 @ A2 @ (plus_plus_nat @ A2 @ B4)))) => (P3 @ A @ B))))))). % Euclid_induct
thf(fact_193_verit__sum__simplify, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % verit_sum_simplify
thf(fact_194_verit__sum__simplify, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % verit_sum_simplify
thf(fact_195_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_196_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_a, C : a]: (((synthetic_div_a @ P @ C) = zero_zero_poly_a) = ((degree_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_197_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_nat, C : nat]: (((synthetic_div_nat @ P @ C) = zero_zero_poly_nat) = ((degree_nat @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_198_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_poly_a, C : poly_a]: (((synthetic_div_poly_a @ P @ C) = zero_z2096148049poly_a) = ((degree_poly_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_219_synthetic__div__0, axiom,
    ((![C : nat]: ((synthetic_div_nat @ zero_zero_poly_nat @ C) = zero_zero_poly_nat)))). % synthetic_div_0
thf(fact_220_synthetic__div__0, axiom,
    ((![C : poly_a]: ((synthetic_div_poly_a @ zero_z2096148049poly_a @ C) = zero_z2096148049poly_a)))). % synthetic_div_0
thf(fact_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_233_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_234_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_235_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_236_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_237_mult__Suc__right, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ M @ (suc @ N)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N)))))). % mult_Suc_right
thf(fact_238_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_239_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_240_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_241_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_242_mult__Suc, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ (suc @ M) @ N) = (plus_plus_nat @ N @ (times_times_nat @ M @ N)))))). % mult_Suc
thf(fact_243_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_244_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

% Conjectures (3)
thf(conj_0, hypothesis,
    (((degree_a @ (fundam1358810038poly_a @ pa @ h)) = (degree_a @ pa)))).
thf(conj_1, hypothesis,
    ((~ ((pa = zero_zero_poly_a))))).
thf(conj_2, conjecture,
    (((degree_a @ (plus_plus_poly_a @ (smult_a @ h @ (fundam1358810038poly_a @ pa @ h)) @ (pCons_a @ a2 @ (fundam1358810038poly_a @ pa @ h)))) = (suc @ (degree_a @ pa))))).
