% 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_111__5368006_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:11.995

% Could-be-implicit typings (6)
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 (50)
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_It__Nat__Onat_J, type,
    fundam1276917024ly_nat : poly_poly_nat > poly_nat > poly_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_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Polynomial__Opoly_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_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_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_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Polynomial__Opoly_Itf__a_J, type,
    neg_nu60143741poly_a : poly_a > poly_a).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001tf__a, type,
    neg_numeral_dbl_a : a > a).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat, type,
    monom_nat : nat > nat > poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    monom_poly_nat : poly_nat > nat > poly_poly_nat).
thf(sy_c_Polynomial_Omonom_001t__Polynomial__Opoly_Itf__a_J, type,
    monom_poly_a : poly_a > nat > poly_poly_a).
thf(sy_c_Polynomial_Omonom_001tf__a, type,
    monom_a : a > nat > poly_a).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    coeff_poly_nat : poly_poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_Itf__a_J, type,
    coeff_poly_a : poly_poly_a > nat > poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001tf__a, type,
    coeff_a : poly_a > nat > a).
thf(sy_c_Polynomial_Opos__poly_001t__Nat__Onat, type,
    pos_poly_nat : poly_nat > $o).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
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_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_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (232)
thf(fact_0__092_060open_062psize_A_Ioffset__poly_Ap_Aa_J_A_061_Apsize_Ap_092_060close_062, axiom,
    (((fundam247907092size_a @ (fundam1358810038poly_a @ p @ a2)) = (fundam247907092size_a @ p)))). % \<open>psize (offset_poly p a) = psize p\<close>
thf(fact_1_poly__offset__poly, axiom,
    ((![P : poly_poly_nat, H : poly_nat, X : poly_nat]: ((poly_poly_nat2 @ (fundam1276917024ly_nat @ P @ H) @ X) = (poly_poly_nat2 @ P @ (plus_plus_poly_nat @ H @ X)))))). % poly_offset_poly
thf(fact_2_poly__offset__poly, axiom,
    ((![P : poly_poly_a, H : poly_a, X : poly_a]: ((poly_poly_a2 @ (fundam1343031620poly_a @ P @ H) @ X) = (poly_poly_a2 @ P @ (plus_plus_poly_a @ H @ X)))))). % poly_offset_poly
thf(fact_3_poly__offset__poly, axiom,
    ((![P : poly_nat, H : nat, X : nat]: ((poly_nat2 @ (fundam170929432ly_nat @ P @ H) @ X) = (poly_nat2 @ P @ (plus_plus_nat @ H @ X)))))). % poly_offset_poly
thf(fact_4_poly__offset__poly, axiom,
    ((![P : poly_a, H : a, X : a]: ((poly_a2 @ (fundam1358810038poly_a @ P @ H) @ X) = (poly_a2 @ P @ (plus_plus_a @ H @ X)))))). % poly_offset_poly
thf(fact_5_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_6_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_7_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_8_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_9_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_10_add__left__cancel, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: (((plus_plus_poly_a @ A @ B) = (plus_plus_poly_a @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_11_add__left__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_12_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_13_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_14_add__right__cancel, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: (((plus_plus_poly_a @ B @ A) = (plus_plus_poly_a @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_15_add__right__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_16_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_17_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_18_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_19_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_20_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_21_is__num__normalize_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)))))). % is_num_normalize(1)
thf(fact_22_is__num__normalize_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)))))). % is_num_normalize(1)
thf(fact_23_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_24_group__cancel_Oadd1, axiom,
    ((![A2 : poly_nat, K : poly_nat, A : poly_nat, B : poly_nat]: ((A2 = (plus_plus_poly_nat @ K @ A)) => ((plus_plus_poly_nat @ A2 @ B) = (plus_plus_poly_nat @ K @ (plus_plus_poly_nat @ A @ B))))))). % group_cancel.add1
thf(fact_25_group__cancel_Oadd1, axiom,
    ((![A2 : poly_a, K : poly_a, A : poly_a, B : poly_a]: ((A2 = (plus_plus_poly_a @ K @ A)) => ((plus_plus_poly_a @ A2 @ B) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_26_group__cancel_Oadd1, axiom,
    ((![A2 : a, K : a, A : a, B : a]: ((A2 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A2 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_27_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_28_group__cancel_Oadd2, axiom,
    ((![B2 : a, K : a, B : a, A : a]: ((B2 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B2) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_29_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_30_group__cancel_Oadd2, axiom,
    ((![B2 : poly_nat, K : poly_nat, B : poly_nat, A : poly_nat]: ((B2 = (plus_plus_poly_nat @ K @ B)) => ((plus_plus_poly_nat @ A @ B2) = (plus_plus_poly_nat @ K @ (plus_plus_poly_nat @ A @ B))))))). % group_cancel.add2
thf(fact_31_group__cancel_Oadd2, axiom,
    ((![B2 : poly_a, K : poly_a, B : poly_a, A : poly_a]: ((B2 = (plus_plus_poly_a @ K @ B)) => ((plus_plus_poly_a @ A @ B2) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add2
thf(fact_32_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_33_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_34_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_35_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_36_add_Oleft__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_37_add_Oleft__cancel, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: (((plus_plus_poly_a @ A @ B) = (plus_plus_poly_a @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_38_add_Oright__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_39_add_Oright__cancel, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: (((plus_plus_poly_a @ B @ A) = (plus_plus_poly_a @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_40_add__right__imp__eq, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_41_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_42_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_43_add__right__imp__eq, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: (((plus_plus_poly_a @ B @ A) = (plus_plus_poly_a @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_44_add__left__imp__eq, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_45_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_46_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_47_add__left__imp__eq, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: (((plus_plus_poly_a @ A @ B) = (plus_plus_poly_a @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_48_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_49_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_50_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_51_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_52_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_53_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_54_add_Ocommute, axiom,
    ((plus_plus_poly_nat = (^[A3 : poly_nat]: (^[B3 : poly_nat]: (plus_plus_poly_nat @ B3 @ A3)))))). % add.commute
thf(fact_55_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A3 : poly_a]: (^[B3 : poly_a]: (plus_plus_poly_a @ B3 @ A3)))))). % add.commute
thf(fact_56_coeff__add, axiom,
    ((![P : poly_poly_nat, Q : poly_poly_nat, N : nat]: ((coeff_poly_nat @ (plus_p1835221865ly_nat @ P @ Q) @ N) = (plus_plus_poly_nat @ (coeff_poly_nat @ P @ N) @ (coeff_poly_nat @ Q @ N)))))). % coeff_add
thf(fact_57_coeff__add, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, N : nat]: ((coeff_poly_a @ (plus_p1976640465poly_a @ P @ Q) @ N) = (plus_plus_poly_a @ (coeff_poly_a @ P @ N) @ (coeff_poly_a @ Q @ N)))))). % coeff_add
thf(fact_58_coeff__add, axiom,
    ((![P : poly_nat, Q : poly_nat, N : nat]: ((coeff_nat @ (plus_plus_poly_nat @ P @ Q) @ N) = (plus_plus_nat @ (coeff_nat @ P @ N) @ (coeff_nat @ Q @ N)))))). % coeff_add
thf(fact_59_coeff__add, axiom,
    ((![P : poly_a, Q : poly_a, N : nat]: ((coeff_a @ (plus_plus_poly_a @ P @ Q) @ N) = (plus_plus_a @ (coeff_a @ P @ N) @ (coeff_a @ Q @ N)))))). % coeff_add
thf(fact_60_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_61_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_62_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_63_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_64_dbl__def, axiom,
    ((neg_numeral_dbl_a = (^[X2 : a]: (plus_plus_a @ X2 @ X2))))). % dbl_def
thf(fact_65_dbl__def, axiom,
    ((neg_nu60143741poly_a = (^[X2 : poly_a]: (plus_plus_poly_a @ X2 @ X2))))). % dbl_def
thf(fact_66_add__monom, axiom,
    ((![A : poly_nat, N : nat, B : poly_nat]: ((plus_p1835221865ly_nat @ (monom_poly_nat @ A @ N) @ (monom_poly_nat @ B @ N)) = (monom_poly_nat @ (plus_plus_poly_nat @ A @ B) @ N))))). % add_monom
thf(fact_67_add__monom, axiom,
    ((![A : poly_a, N : nat, B : poly_a]: ((plus_p1976640465poly_a @ (monom_poly_a @ A @ N) @ (monom_poly_a @ B @ N)) = (monom_poly_a @ (plus_plus_poly_a @ A @ B) @ N))))). % add_monom
thf(fact_68_add__monom, axiom,
    ((![A : nat, N : nat, B : nat]: ((plus_plus_poly_nat @ (monom_nat @ A @ N) @ (monom_nat @ B @ N)) = (monom_nat @ (plus_plus_nat @ A @ B) @ N))))). % add_monom
thf(fact_69_add__monom, axiom,
    ((![A : a, N : nat, B : a]: ((plus_plus_poly_a @ (monom_a @ A @ N) @ (monom_a @ B @ N)) = (monom_a @ (plus_plus_a @ A @ B) @ N))))). % add_monom
thf(fact_70_plus__poly_Orep__eq, axiom,
    ((![X : poly_poly_nat, Xa : poly_poly_nat]: ((coeff_poly_nat @ (plus_p1835221865ly_nat @ X @ Xa)) = (^[N2 : nat]: (plus_plus_poly_nat @ (coeff_poly_nat @ X @ N2) @ (coeff_poly_nat @ Xa @ N2))))))). % plus_poly.rep_eq
thf(fact_71_plus__poly_Orep__eq, axiom,
    ((![X : poly_poly_a, Xa : poly_poly_a]: ((coeff_poly_a @ (plus_p1976640465poly_a @ X @ Xa)) = (^[N2 : nat]: (plus_plus_poly_a @ (coeff_poly_a @ X @ N2) @ (coeff_poly_a @ Xa @ N2))))))). % plus_poly.rep_eq
thf(fact_72_plus__poly_Orep__eq, axiom,
    ((![X : poly_nat, Xa : poly_nat]: ((coeff_nat @ (plus_plus_poly_nat @ X @ Xa)) = (^[N2 : nat]: (plus_plus_nat @ (coeff_nat @ X @ N2) @ (coeff_nat @ Xa @ N2))))))). % plus_poly.rep_eq
thf(fact_73_plus__poly_Orep__eq, axiom,
    ((![X : poly_a, Xa : poly_a]: ((coeff_a @ (plus_plus_poly_a @ X @ Xa)) = (^[N2 : nat]: (plus_plus_a @ (coeff_a @ X @ N2) @ (coeff_a @ Xa @ N2))))))). % plus_poly.rep_eq
thf(fact_74_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_75_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_76_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_77_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_78_pos__poly__add, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((pos_poly_nat @ P) => ((pos_poly_nat @ Q) => (pos_poly_nat @ (plus_plus_poly_nat @ P @ Q))))))). % pos_poly_add
thf(fact_79_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_80_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_81_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_82_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_83_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_84_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_85_add_Oleft__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ zero_zero_poly_nat @ A) = A)))). % add.left_neutral
thf(fact_86_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_87_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_88_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_89_add_Oright__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.right_neutral
thf(fact_90_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_91_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_92_add__cancel__left__left, axiom,
    ((![B : a, A : a]: (((plus_plus_a @ B @ A) = A) = (B = zero_zero_a))))). % add_cancel_left_left
thf(fact_93_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_94_add__cancel__left__left, axiom,
    ((![B : poly_a, A : poly_a]: (((plus_plus_poly_a @ B @ A) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_left
thf(fact_95_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_96_add__cancel__left__right, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = A) = (B = zero_zero_a))))). % add_cancel_left_right
thf(fact_97_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_98_add__cancel__left__right, axiom,
    ((![A : poly_a, B : poly_a]: (((plus_plus_poly_a @ A @ B) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_right
thf(fact_99_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_100_add__cancel__right__left, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ B @ A)) = (B = zero_zero_a))))). % add_cancel_right_left
thf(fact_101_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_102_add__cancel__right__left, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ B @ A)) = (B = zero_zero_poly_a))))). % add_cancel_right_left
thf(fact_103_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_104_add__cancel__right__right, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ A @ B)) = (B = zero_zero_a))))). % add_cancel_right_right
thf(fact_105_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_106_add__cancel__right__right, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ A @ B)) = (B = zero_zero_poly_a))))). % add_cancel_right_right
thf(fact_107_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_108_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_109_psize__eq__0__iff, axiom,
    ((![P : poly_a]: (((fundam247907092size_a @ P) = zero_zero_nat) = (P = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_110_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_111_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_112_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_113_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_114_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_115_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_116_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_117_monom__eq__0, axiom,
    ((![N : nat]: ((monom_nat @ zero_zero_nat @ N) = zero_zero_poly_nat)))). % monom_eq_0
thf(fact_118_monom__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((monom_nat @ A @ N) = zero_zero_poly_nat) = (A = zero_zero_nat))))). % monom_eq_0_iff
thf(fact_119_coeff__monom, axiom,
    ((![M : nat, N : nat, A : nat]: (((M = N) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = A)) & ((~ ((M = N))) => ((coeff_nat @ (monom_nat @ A @ M) @ N) = zero_zero_nat)))))). % coeff_monom
thf(fact_120_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_121_monom_Orep__eq, axiom,
    ((![X : nat, Xa : nat]: ((coeff_nat @ (monom_nat @ X @ Xa)) = (^[N2 : nat]: (if_nat @ (Xa = N2) @ X @ zero_zero_nat)))))). % monom.rep_eq
thf(fact_122_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A4 : nat, P3 : poly_nat]: (((~ ((A4 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A4 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_123_monom__eq__iff_H, axiom,
    ((![C : nat, N : nat, D : nat, M : nat]: (((monom_nat @ C @ N) = (monom_nat @ D @ M)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = M))))))))). % monom_eq_iff'
thf(fact_124_poly__0__coeff__0, axiom,
    ((![P : poly_a]: ((poly_a2 @ P @ zero_zero_a) = (coeff_a @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_125_poly__0__coeff__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ P @ zero_zero_nat) = (coeff_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_126_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_127_monom__eq__const__iff, axiom,
    ((![C : nat, N : nat, D : nat]: (((monom_nat @ C @ N) = (pCons_nat @ D @ zero_zero_poly_nat)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = zero_zero_nat))))))))). % monom_eq_const_iff
thf(fact_128_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_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_141_add_Ocomm__neutral, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % add.comm_neutral
thf(fact_142_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_143_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_144_add_Ogroup__left__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.group_left_neutral
thf(fact_145_add_Ogroup__left__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.group_left_neutral
thf(fact_146_synthetic__div__correct, axiom,
    ((![P : poly_nat, C : nat]: ((plus_plus_poly_nat @ P @ (smult_nat @ C @ (synthetic_div_nat @ P @ C))) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_correct
thf(fact_147_synthetic__div__correct, axiom,
    ((![P : poly_a, C : a]: ((plus_plus_poly_a @ P @ (smult_a @ C @ (synthetic_div_a @ P @ C))) = (pCons_a @ (poly_a2 @ P @ C) @ (synthetic_div_a @ P @ C)))))). % synthetic_div_correct
thf(fact_148_synthetic__div__unique, axiom,
    ((![P : poly_nat, C : nat, Q : poly_nat, R : nat]: (((plus_plus_poly_nat @ P @ (smult_nat @ C @ Q)) = (pCons_nat @ R @ Q)) => ((R = (poly_nat2 @ P @ C)) & (Q = (synthetic_div_nat @ P @ C))))))). % synthetic_div_unique
thf(fact_149_synthetic__div__unique, axiom,
    ((![P : poly_a, C : a, Q : poly_a, R : a]: (((plus_plus_poly_a @ P @ (smult_a @ C @ Q)) = (pCons_a @ R @ Q)) => ((R = (poly_a2 @ P @ C)) & (Q = (synthetic_div_a @ P @ C))))))). % synthetic_div_unique
thf(fact_150_verit__sum__simplify, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % verit_sum_simplify
thf(fact_151_verit__sum__simplify, axiom,
    ((![A : poly_nat]: ((plus_plus_poly_nat @ A @ zero_zero_poly_nat) = A)))). % verit_sum_simplify
thf(fact_152_verit__sum__simplify, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % verit_sum_simplify
thf(fact_153_verit__sum__simplify, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % verit_sum_simplify
thf(fact_154_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_coeff__smult, axiom,
    ((![A : nat, P : poly_nat, N : nat]: ((coeff_nat @ (smult_nat @ A @ P) @ N) = (times_times_nat @ A @ (coeff_nat @ P @ N)))))). % coeff_smult
thf(fact_163_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_164_poly__smult, axiom,
    ((![A : a, P : poly_a, X : a]: ((poly_a2 @ (smult_a @ A @ P) @ X) = (times_times_a @ A @ (poly_a2 @ P @ X)))))). % poly_smult
thf(fact_165_poly__smult, axiom,
    ((![A : nat, P : poly_nat, X : nat]: ((poly_nat2 @ (smult_nat @ A @ P) @ X) = (times_times_nat @ A @ (poly_nat2 @ P @ X)))))). % poly_smult
thf(fact_166_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_167_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_168_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_169_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_170_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_171_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_172_mult__monom, axiom,
    ((![A : nat, M : nat, B : nat, N : nat]: ((times_times_poly_nat @ (monom_nat @ A @ M) @ (monom_nat @ B @ N)) = (monom_nat @ (times_times_nat @ A @ B) @ (plus_plus_nat @ M @ N)))))). % mult_monom
thf(fact_173_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_174_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_175_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_176_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_177_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_178_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_179_smult__monom__mult, axiom,
    ((![A : nat, B : nat, N : nat, F : poly_nat]: ((smult_nat @ A @ (times_times_poly_nat @ (monom_nat @ B @ N) @ F)) = (times_times_poly_nat @ (monom_nat @ (times_times_nat @ A @ B) @ N) @ F))))). % smult_monom_mult
thf(fact_180_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_181_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_182_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_183_smult_Orep__eq, axiom,
    ((![X : nat, Xa : poly_nat]: ((coeff_nat @ (smult_nat @ X @ Xa)) = (^[N2 : nat]: (times_times_nat @ X @ (coeff_nat @ Xa @ N2))))))). % smult.rep_eq
thf(fact_184_smult__monom, axiom,
    ((![A : nat, B : nat, N : nat]: ((smult_nat @ A @ (monom_nat @ B @ N)) = (monom_nat @ (times_times_nat @ A @ B) @ N))))). % smult_monom
thf(fact_185_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_186_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_187_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_188_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (plus_plus_a @ B @ C)) = (plus_plus_a @ (times_times_a @ A @ B) @ (times_times_a @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_212_ring__class_Oring__distribs_I1_J, 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)))))). % ring_class.ring_distribs(1)
thf(fact_213_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_214_ring__class_Oring__distribs_I2_J, 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)))))). % ring_class.ring_distribs(2)
thf(fact_215_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_216_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_217_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_218_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_219_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_220_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_221_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_222_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_223_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_224_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_225_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_226_Euclid__induct, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B4 : nat]: ((P2 @ A4 @ B4) = (P2 @ B4 @ A4))) => ((![A4 : nat]: (P2 @ A4 @ zero_zero_nat)) => ((![A4 : nat, B4 : nat]: ((P2 @ A4 @ B4) => (P2 @ A4 @ (plus_plus_nat @ A4 @ B4)))) => (P2 @ A @ B))))))). % Euclid_induct
thf(fact_227_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_228_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_229_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_230_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_231_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

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((![X3 : a]: ((poly_a2 @ (fundam1358810038poly_a @ p @ a2) @ X3) = (poly_a2 @ p @ (plus_plus_a @ a2 @ X3)))))).
