% 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_320__5370712_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:11.689

% Could-be-implicit typings (7)
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__Real__Oreal_J, type,
    poly_real : $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__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (49)
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_001t__Real__Oreal, type,
    fundam1552870388y_real : poly_real > real > poly_real).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Real__Oreal, type,
    fundam1947011094e_real : poly_real > nat).
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_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_It__Real__Oreal_J, type,
    plus_plus_poly_real : poly_real > poly_real > poly_real).
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_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    uminus1736902417poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    uminus1613791741y_real : poly_real > poly_real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_Itf__a_J, type,
    uminus_uminus_poly_a : poly_a > poly_a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_real : real > real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a, type,
    uminus_uminus_a : a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
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_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : 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_001t__Real__Oreal, type,
    degree_real : poly_real > 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_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal, type,
    pCons_real : real > poly_real > poly_real).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > 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_Itf__a_J, type,
    smult_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osmult_001t__Real__Oreal, type,
    smult_real : real > poly_real > poly_real).
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__Real__Oreal, type,
    synthetic_div_real : poly_real > real > poly_real).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_v_e, type,
    e : real).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_z, type,
    z : a).

% Relevant facts (245)
thf(fact_0_degree__offset__poly, axiom,
    ((![P : poly_nat, H : nat]: ((degree_nat @ (fundam170929432ly_nat @ P @ H)) = (degree_nat @ P))))). % degree_offset_poly
thf(fact_1_degree__offset__poly, axiom,
    ((![P : poly_real, H : real]: ((degree_real @ (fundam1552870388y_real @ P @ H)) = (degree_real @ P))))). % degree_offset_poly
thf(fact_2_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_3_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_4_offset__poly__0, axiom,
    ((![H : real]: ((fundam1552870388y_real @ zero_zero_poly_real @ H) = zero_zero_poly_real)))). % offset_poly_0
thf(fact_5_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_6_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_7_offset__poly__eq__0__iff, axiom,
    ((![P : poly_real, H : real]: (((fundam1552870388y_real @ P @ H) = zero_zero_poly_real) = (P = zero_zero_poly_real))))). % offset_poly_eq_0_iff
thf(fact_8_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_9_degree__minus, axiom,
    ((![P : poly_a]: ((degree_a @ (uminus_uminus_poly_a @ P)) = (degree_a @ P))))). % degree_minus
thf(fact_10_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_11_offset__poly__single, axiom,
    ((![A : real, H : real]: ((fundam1552870388y_real @ (pCons_real @ A @ zero_zero_poly_real) @ H) = (pCons_real @ A @ zero_zero_poly_real))))). % offset_poly_single
thf(fact_12_offset__poly__single, axiom,
    ((![A : a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ H) = (pCons_a @ A @ zero_zero_poly_a))))). % offset_poly_single
thf(fact_13_poly__offset__poly, axiom,
    ((![P : poly_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_14_poly__offset__poly, axiom,
    ((![P : poly_real, H : real, X : real]: ((poly_real2 @ (fundam1552870388y_real @ P @ H) @ X) = (poly_real2 @ P @ (plus_plus_real @ H @ X)))))). % poly_offset_poly
thf(fact_15_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_16_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_17_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_18_offset__poly__pCons, axiom,
    ((![A : real, P : poly_real, H : real]: ((fundam1552870388y_real @ (pCons_real @ A @ P) @ H) = (plus_plus_poly_real @ (smult_real @ H @ (fundam1552870388y_real @ P @ H)) @ (pCons_real @ A @ (fundam1552870388y_real @ P @ H))))))). % offset_poly_pCons
thf(fact_19_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_20_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_21_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_22_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_23_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_24_pCons__0__0, axiom,
    (((pCons_real @ zero_zero_real @ zero_zero_poly_real) = zero_zero_poly_real))). % pCons_0_0
thf(fact_25_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_26_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_27_pCons__eq__0__iff, axiom,
    ((![A : real, P : poly_real]: (((pCons_real @ A @ P) = zero_zero_poly_real) = (((A = zero_zero_real)) & ((P = zero_zero_poly_real))))))). % pCons_eq_0_iff
thf(fact_28_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_29_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_30_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_31_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_32_smult__0__left, axiom,
    ((![P : poly_real]: ((smult_real @ zero_zero_real @ P) = zero_zero_poly_real)))). % smult_0_left
thf(fact_33_smult__0__left, axiom,
    ((![P : poly_poly_a]: ((smult_poly_a @ zero_zero_poly_a @ P) = zero_z2096148049poly_a)))). % smult_0_left
thf(fact_34_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_35_smult__eq__0__iff, axiom,
    ((![A : real, P : poly_real]: (((smult_real @ A @ P) = zero_zero_poly_real) = (((A = zero_zero_real)) | ((P = zero_zero_poly_real))))))). % smult_eq_0_iff
thf(fact_36_smult__eq__0__iff, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((smult_poly_a @ A @ P) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) | ((P = zero_z2096148049poly_a))))))). % smult_eq_0_iff
thf(fact_37_smult__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((smult_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) | ((P = zero_zero_poly_a))))))). % smult_eq_0_iff
thf(fact_38_degree__smult__eq, axiom,
    ((![A : a, P : poly_a]: (((A = zero_zero_a) => ((degree_a @ (smult_a @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_a))) => ((degree_a @ (smult_a @ A @ P)) = (degree_a @ P))))))). % degree_smult_eq
thf(fact_39_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_40_degree__smult__eq, axiom,
    ((![A : real, P : poly_real]: (((A = zero_zero_real) => ((degree_real @ (smult_real @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_real))) => ((degree_real @ (smult_real @ A @ P)) = (degree_real @ P))))))). % degree_smult_eq
thf(fact_41_degree__smult__eq, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((A = zero_zero_poly_a) => ((degree_poly_a @ (smult_poly_a @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_poly_a))) => ((degree_poly_a @ (smult_poly_a @ A @ P)) = (degree_poly_a @ P))))))). % degree_smult_eq
thf(fact_42_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_43_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_44_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_45_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_46_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_47_add__pCons, axiom,
    ((![A : real, P : poly_real, B : real, Q : poly_real]: ((plus_plus_poly_real @ (pCons_real @ A @ P) @ (pCons_real @ B @ Q)) = (pCons_real @ (plus_plus_real @ A @ B) @ (plus_plus_poly_real @ P @ Q)))))). % add_pCons
thf(fact_48_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_49_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_50_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_51_poly__add, axiom,
    ((![P : poly_real, Q : poly_real, X : real]: ((poly_real2 @ (plus_plus_poly_real @ P @ Q) @ X) = (plus_plus_real @ (poly_real2 @ P @ X) @ (poly_real2 @ Q @ X)))))). % poly_add
thf(fact_52_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_53_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_54_minus__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a]: ((uminus1736902417poly_a @ (pCons_poly_a @ A @ P)) = (pCons_poly_a @ (uminus_uminus_poly_a @ A) @ (uminus1736902417poly_a @ P)))))). % minus_pCons
thf(fact_55_minus__pCons, axiom,
    ((![A : a, P : poly_a]: ((uminus_uminus_poly_a @ (pCons_a @ A @ P)) = (pCons_a @ (uminus_uminus_a @ A) @ (uminus_uminus_poly_a @ P)))))). % minus_pCons
thf(fact_56_poly__minus, axiom,
    ((![P : poly_real, X : real]: ((poly_real2 @ (uminus1613791741y_real @ P) @ X) = (uminus_uminus_real @ (poly_real2 @ P @ X)))))). % poly_minus
thf(fact_57_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_58_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_59_pCons__induct, axiom,
    ((![P2 : poly_real > $o, P : poly_real]: ((P2 @ zero_zero_poly_real) => ((![A2 : real, P3 : poly_real]: (((~ ((A2 = zero_zero_real))) | (~ ((P3 = zero_zero_poly_real)))) => ((P2 @ P3) => (P2 @ (pCons_real @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_60_pCons__induct, axiom,
    ((![P2 : poly_poly_a > $o, P : poly_poly_a]: ((P2 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P3 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P3 = zero_z2096148049poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_poly_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_61_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_62_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_63_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_64_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_65_smult__add__left, axiom,
    ((![A : real, B : real, P : poly_real]: ((smult_real @ (plus_plus_real @ A @ B) @ P) = (plus_plus_poly_real @ (smult_real @ A @ P) @ (smult_real @ B @ P)))))). % smult_add_left
thf(fact_66_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_67_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_68_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X2 : real]: ((poly_real2 @ P @ X2) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_69_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_70_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_71_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_72_add_Oleft__inverse, axiom,
    ((![A : a]: ((plus_plus_a @ (uminus_uminus_a @ A) @ A) = zero_zero_a)))). % add.left_inverse
thf(fact_73_add_Oleft__inverse, axiom,
    ((![A : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ A) = zero_zero_real)))). % add.left_inverse
thf(fact_74_add_Oleft__inverse, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ (uminus_uminus_poly_a @ A) @ A) = zero_zero_poly_a)))). % add.left_inverse
thf(fact_75_add_Oright__inverse, axiom,
    ((![A : a]: ((plus_plus_a @ A @ (uminus_uminus_a @ A)) = zero_zero_a)))). % add.right_inverse
thf(fact_76_add_Oright__inverse, axiom,
    ((![A : real]: ((plus_plus_real @ A @ (uminus_uminus_real @ A)) = zero_zero_real)))). % add.right_inverse
thf(fact_77_add_Oright__inverse, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ (uminus_uminus_poly_a @ A)) = zero_zero_poly_a)))). % add.right_inverse
thf(fact_78_add__minus__cancel, axiom,
    ((![A : a, B : a]: ((plus_plus_a @ A @ (plus_plus_a @ (uminus_uminus_a @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_79_add__minus__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ (uminus_uminus_real @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_80_add__minus__cancel, axiom,
    ((![A : poly_a, B : poly_a]: ((plus_plus_poly_a @ A @ (plus_plus_poly_a @ (uminus_uminus_poly_a @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_81_minus__add__cancel, axiom,
    ((![A : a, B : a]: ((plus_plus_a @ (uminus_uminus_a @ A) @ (plus_plus_a @ A @ B)) = B)))). % minus_add_cancel
thf(fact_82_minus__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ (plus_plus_real @ A @ B)) = B)))). % minus_add_cancel
thf(fact_83_minus__add__cancel, axiom,
    ((![A : poly_a, B : poly_a]: ((plus_plus_poly_a @ (uminus_uminus_poly_a @ A) @ (plus_plus_poly_a @ A @ B)) = B)))). % minus_add_cancel
thf(fact_84_minus__add__distrib, axiom,
    ((![A : a, B : a]: ((uminus_uminus_a @ (plus_plus_a @ A @ B)) = (plus_plus_a @ (uminus_uminus_a @ A) @ (uminus_uminus_a @ B)))))). % minus_add_distrib
thf(fact_85_minus__add__distrib, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)))))). % minus_add_distrib
thf(fact_86_minus__add__distrib, axiom,
    ((![A : poly_a, B : poly_a]: ((uminus_uminus_poly_a @ (plus_plus_poly_a @ A @ B)) = (plus_plus_poly_a @ (uminus_uminus_poly_a @ A) @ (uminus_uminus_poly_a @ B)))))). % minus_add_distrib
thf(fact_87_add_Oinverse__neutral, axiom,
    (((uminus_uminus_real @ zero_zero_real) = zero_zero_real))). % add.inverse_neutral
thf(fact_88_add_Oinverse__neutral, axiom,
    (((uminus_uminus_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % add.inverse_neutral
thf(fact_89_neg__0__equal__iff__equal, axiom,
    ((![A : real]: ((zero_zero_real = (uminus_uminus_real @ A)) = (zero_zero_real = A))))). % neg_0_equal_iff_equal
thf(fact_90_neg__0__equal__iff__equal, axiom,
    ((![A : poly_a]: ((zero_zero_poly_a = (uminus_uminus_poly_a @ A)) = (zero_zero_poly_a = A))))). % neg_0_equal_iff_equal
thf(fact_91_neg__equal__0__iff__equal, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % neg_equal_0_iff_equal
thf(fact_92_neg__equal__0__iff__equal, axiom,
    ((![A : poly_a]: (((uminus_uminus_poly_a @ A) = zero_zero_poly_a) = (A = zero_zero_poly_a))))). % neg_equal_0_iff_equal
thf(fact_93_equal__neg__zero, axiom,
    ((![A : real]: ((A = (uminus_uminus_real @ A)) = (A = zero_zero_real))))). % equal_neg_zero
thf(fact_94_neg__equal__zero, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = A) = (A = zero_zero_real))))). % neg_equal_zero
thf(fact_95_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_96_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_97_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_98_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_99_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_100_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_101_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_102_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_103_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_104_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_105_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_106_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_107_neg__equal__iff__equal, axiom,
    ((![A : poly_a, B : poly_a]: (((uminus_uminus_poly_a @ A) = (uminus_uminus_poly_a @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_108_add_Oinverse__inverse, axiom,
    ((![A : poly_a]: ((uminus_uminus_poly_a @ (uminus_uminus_poly_a @ A)) = A)))). % add.inverse_inverse
thf(fact_109_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_110_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_111_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_112_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_113_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_114_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_115_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_116_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_117_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_118_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_119_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_120_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_121_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_122_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_123_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_124_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_125_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_126_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_127_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_128_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_129_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_130_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_131_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_132_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_133_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_134_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_135_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_136_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_137_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_138_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_139_add__right__imp__eq, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_140_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_141_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_142_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_143_add__left__imp__eq, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_144_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_145_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_146_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_147_add_Oleft__commute, axiom,
    ((![B : real, A : real, C : real]: ((plus_plus_real @ B @ (plus_plus_real @ A @ C)) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.left_commute
thf(fact_148_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_149_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_150_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_151_add_Ocommute, axiom,
    ((plus_plus_real = (^[A3 : real]: (^[B3 : real]: (plus_plus_real @ B3 @ A3)))))). % add.commute
thf(fact_152_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A3 : poly_a]: (^[B3 : poly_a]: (plus_plus_poly_a @ B3 @ A3)))))). % add.commute
thf(fact_153_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_154_add_Oright__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_155_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_156_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_157_add_Oleft__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_158_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_159_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_160_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_161_add_Oassoc, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.assoc
thf(fact_162_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_163_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_164_group__cancel_Oadd2, axiom,
    ((![B4 : a, K : a, B : a, A : a]: ((B4 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B4) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_165_group__cancel_Oadd2, axiom,
    ((![B4 : real, K : real, B : real, A : real]: ((B4 = (plus_plus_real @ K @ B)) => ((plus_plus_real @ A @ B4) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add2
thf(fact_166_group__cancel_Oadd2, axiom,
    ((![B4 : poly_a, K : poly_a, B : poly_a, A : poly_a]: ((B4 = (plus_plus_poly_a @ K @ B)) => ((plus_plus_poly_a @ A @ B4) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add2
thf(fact_167_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_168_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_169_group__cancel_Oadd1, axiom,
    ((![A4 : real, K : real, A : real, B : real]: ((A4 = (plus_plus_real @ K @ A)) => ((plus_plus_real @ A4 @ B) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add1
thf(fact_170_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_171_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_172_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (K = L)) => ((plus_plus_real @ I @ K) = (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_173_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_174_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_175_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_176_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_177_minus__equation__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((uminus_uminus_poly_a @ A) = B) = ((uminus_uminus_poly_a @ B) = A))))). % minus_equation_iff
thf(fact_178_equation__minus__iff, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (uminus_uminus_poly_a @ B)) = (B = (uminus_uminus_poly_a @ A)))))). % equation_minus_iff
thf(fact_179_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_180_degree__pCons__0, axiom,
    ((![A : a]: ((degree_a @ (pCons_a @ A @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_181_add_Ogroup__left__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.group_left_neutral
thf(fact_182_add_Ogroup__left__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.group_left_neutral
thf(fact_183_add_Ogroup__left__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.group_left_neutral
thf(fact_184_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_185_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_186_add_Ocomm__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.comm_neutral
thf(fact_187_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_188_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_189_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_190_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_191_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_192_add_Oinverse__distrib__swap, axiom,
    ((![A : a, B : a]: ((uminus_uminus_a @ (plus_plus_a @ A @ B)) = (plus_plus_a @ (uminus_uminus_a @ B) @ (uminus_uminus_a @ A)))))). % add.inverse_distrib_swap
thf(fact_193_add_Oinverse__distrib__swap, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)))))). % add.inverse_distrib_swap
thf(fact_194_add_Oinverse__distrib__swap, axiom,
    ((![A : poly_a, B : poly_a]: ((uminus_uminus_poly_a @ (plus_plus_poly_a @ A @ B)) = (plus_plus_poly_a @ (uminus_uminus_poly_a @ B) @ (uminus_uminus_poly_a @ A)))))). % add.inverse_distrib_swap
thf(fact_195_group__cancel_Oneg1, axiom,
    ((![A4 : a, K : a, A : a]: ((A4 = (plus_plus_a @ K @ A)) => ((uminus_uminus_a @ A4) = (plus_plus_a @ (uminus_uminus_a @ K) @ (uminus_uminus_a @ A))))))). % group_cancel.neg1
thf(fact_196_group__cancel_Oneg1, axiom,
    ((![A4 : real, K : real, A : real]: ((A4 = (plus_plus_real @ K @ A)) => ((uminus_uminus_real @ A4) = (plus_plus_real @ (uminus_uminus_real @ K) @ (uminus_uminus_real @ A))))))). % group_cancel.neg1
thf(fact_197_group__cancel_Oneg1, axiom,
    ((![A4 : poly_a, K : poly_a, A : poly_a]: ((A4 = (plus_plus_poly_a @ K @ A)) => ((uminus_uminus_poly_a @ A4) = (plus_plus_poly_a @ (uminus_uminus_poly_a @ K) @ (uminus_uminus_poly_a @ A))))))). % group_cancel.neg1
thf(fact_198_neg__eq__iff__add__eq__0, axiom,
    ((![A : a, B : a]: (((uminus_uminus_a @ A) = B) = ((plus_plus_a @ A @ B) = zero_zero_a))))). % neg_eq_iff_add_eq_0
thf(fact_199_neg__eq__iff__add__eq__0, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = B) = ((plus_plus_real @ A @ B) = zero_zero_real))))). % neg_eq_iff_add_eq_0
thf(fact_200_neg__eq__iff__add__eq__0, axiom,
    ((![A : poly_a, B : poly_a]: (((uminus_uminus_poly_a @ A) = B) = ((plus_plus_poly_a @ A @ B) = zero_zero_poly_a))))). % neg_eq_iff_add_eq_0
thf(fact_201_eq__neg__iff__add__eq__0, axiom,
    ((![A : a, B : a]: ((A = (uminus_uminus_a @ B)) = ((plus_plus_a @ A @ B) = zero_zero_a))))). % eq_neg_iff_add_eq_0
thf(fact_202_eq__neg__iff__add__eq__0, axiom,
    ((![A : real, B : real]: ((A = (uminus_uminus_real @ B)) = ((plus_plus_real @ A @ B) = zero_zero_real))))). % eq_neg_iff_add_eq_0
thf(fact_203_eq__neg__iff__add__eq__0, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (uminus_uminus_poly_a @ B)) = ((plus_plus_poly_a @ A @ B) = zero_zero_poly_a))))). % eq_neg_iff_add_eq_0
thf(fact_204_add_Oinverse__unique, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = zero_zero_a) => ((uminus_uminus_a @ A) = B))))). % add.inverse_unique
thf(fact_205_add_Oinverse__unique, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = zero_zero_real) => ((uminus_uminus_real @ A) = B))))). % add.inverse_unique
thf(fact_206_add_Oinverse__unique, axiom,
    ((![A : poly_a, B : poly_a]: (((plus_plus_poly_a @ A @ B) = zero_zero_poly_a) => ((uminus_uminus_poly_a @ A) = B))))). % add.inverse_unique
thf(fact_207_ab__group__add__class_Oab__left__minus, axiom,
    ((![A : a]: ((plus_plus_a @ (uminus_uminus_a @ A) @ A) = zero_zero_a)))). % ab_group_add_class.ab_left_minus
thf(fact_208_ab__group__add__class_Oab__left__minus, axiom,
    ((![A : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ A) = zero_zero_real)))). % ab_group_add_class.ab_left_minus
thf(fact_209_ab__group__add__class_Oab__left__minus, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ (uminus_uminus_poly_a @ A) @ A) = zero_zero_poly_a)))). % ab_group_add_class.ab_left_minus
thf(fact_210_add__eq__0__iff, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = zero_zero_a) = (B = (uminus_uminus_a @ A)))))). % add_eq_0_iff
thf(fact_211_add__eq__0__iff, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = zero_zero_real) = (B = (uminus_uminus_real @ A)))))). % add_eq_0_iff
thf(fact_212_add__eq__0__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((plus_plus_poly_a @ A @ B) = zero_zero_poly_a) = (B = (uminus_uminus_poly_a @ A)))))). % add_eq_0_iff
thf(fact_213_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_214_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_215_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_216_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_217_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_218_poly__offset, axiom,
    ((![P : poly_real, A : real]: (?[Q2 : poly_real]: (((fundam1947011094e_real @ Q2) = (fundam1947011094e_real @ P)) & (![X3 : real]: ((poly_real2 @ Q2 @ X3) = (poly_real2 @ P @ (plus_plus_real @ A @ X3))))))))). % poly_offset
thf(fact_219_verit__minus__simplify_I4_J, axiom,
    ((![B : poly_a]: ((uminus_uminus_poly_a @ (uminus_uminus_poly_a @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_220_ep, axiom,
    ((ord_less_real @ zero_zero_real @ e))). % ep
thf(fact_221_synthetic__div__unique, axiom,
    ((![P : poly_real, C : real, Q : poly_real, R : real]: (((plus_plus_poly_real @ P @ (smult_real @ C @ Q)) = (pCons_real @ R @ Q)) => ((R = (poly_real2 @ P @ C)) & (Q = (synthetic_div_real @ P @ C))))))). % synthetic_div_unique
thf(fact_222_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_223_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_224_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_225_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_226_add__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_real @ A @ B))))). % add_less_cancel_left
thf(fact_227_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_228_add__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_real @ A @ B))))). % add_less_cancel_right
thf(fact_229_neg__less__iff__less, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ B))))). % neg_less_iff_less
thf(fact_230_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_231_zero__less__double__add__iff__zero__less__single__add, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_double_add_iff_zero_less_single_add
thf(fact_232_double__add__less__zero__iff__single__add__less__zero, axiom,
    ((![A : real]: ((ord_less_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % double_add_less_zero_iff_single_add_less_zero
thf(fact_233_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_234_less__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel2
thf(fact_235_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_236_less__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel1
thf(fact_237_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_238_add__less__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel2
thf(fact_239_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_240_add__less__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel1
thf(fact_241_neg__less__0__iff__less, axiom,
    ((![A : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ zero_zero_real) = (ord_less_real @ zero_zero_real @ A))))). % neg_less_0_iff_less
thf(fact_242_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_1 : real]: (P2 @ X_1)) => ((?[Z : real]: (![X4 : real]: ((P2 @ X4) => (ord_less_real @ X4 @ Z)))) => (?[S : real]: (![Y2 : real]: ((?[X2 : real]: (((P2 @ X2)) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ S))))))))). % real_sup_exists
thf(fact_243_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X4 : real]: ((ord_less_real @ A @ X4) & ((ord_less_real @ X4 @ B) & ((poly_real2 @ P @ X4) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_244_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X4 : real]: ((ord_less_real @ A @ X4) & ((ord_less_real @ X4 @ B) & ((poly_real2 @ P @ X4) = zero_zero_real)))))))))). % poly_IVT_neg

% Conjectures (1)
thf(conj_0, conjecture,
    (((degree_a @ (fundam1358810038poly_a @ p @ z)) = (degree_a @ p)))).
