% 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_318__5370702_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:10.186

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

% Explicit typings (39)
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_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_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_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001tf__a, type,
    neg_numeral_dbl_a : 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_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
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_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_v_e, type,
    e : real).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_thesis____, type,
    thesis : $o).
thf(sy_v_z, type,
    z : a).

% Relevant facts (239)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_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_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_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_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_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_degree__offset__poly, axiom,
    ((![P : poly_nat, H : nat]: ((degree_nat @ (fundam170929432ly_nat @ P @ H)) = (degree_nat @ P))))). % degree_offset_poly
thf(fact_35_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_49_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_50_add_Ocommute, axiom,
    ((plus_plus_poly_nat = (^[A3 : poly_nat]: (^[B3 : poly_nat]: (plus_plus_poly_nat @ B3 @ A3)))))). % add.commute
thf(fact_51_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A3 : poly_a]: (^[B3 : poly_a]: (plus_plus_poly_a @ B3 @ A3)))))). % add.commute
thf(fact_52_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_53_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_54_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_55_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_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_degree__add__le, axiom,
    ((![P : poly_nat, N : nat, Q : poly_nat]: ((ord_less_eq_nat @ (degree_nat @ P) @ N) => ((ord_less_eq_nat @ (degree_nat @ Q) @ N) => (ord_less_eq_nat @ (degree_nat @ (plus_plus_poly_nat @ P @ Q)) @ N)))))). % degree_add_le
thf(fact_65_degree__add__le, axiom,
    ((![P : poly_a, N : nat, Q : poly_a]: ((ord_less_eq_nat @ (degree_a @ P) @ N) => ((ord_less_eq_nat @ (degree_a @ Q) @ N) => (ord_less_eq_nat @ (degree_a @ (plus_plus_poly_a @ P @ Q)) @ N)))))). % degree_add_le
thf(fact_66_degree__add__less, axiom,
    ((![P : poly_nat, N : nat, Q : poly_nat]: ((ord_less_nat @ (degree_nat @ P) @ N) => ((ord_less_nat @ (degree_nat @ Q) @ N) => (ord_less_nat @ (degree_nat @ (plus_plus_poly_nat @ P @ Q)) @ N)))))). % degree_add_less
thf(fact_67_degree__add__less, axiom,
    ((![P : poly_a, N : nat, Q : poly_a]: ((ord_less_nat @ (degree_a @ P) @ N) => ((ord_less_nat @ (degree_a @ Q) @ N) => (ord_less_nat @ (degree_a @ (plus_plus_poly_a @ P @ Q)) @ N)))))). % degree_add_less
thf(fact_68_degree__add__eq__left, axiom,
    ((![Q : poly_nat, P : poly_nat]: ((ord_less_nat @ (degree_nat @ Q) @ (degree_nat @ P)) => ((degree_nat @ (plus_plus_poly_nat @ P @ Q)) = (degree_nat @ P)))))). % degree_add_eq_left
thf(fact_69_degree__add__eq__left, axiom,
    ((![Q : poly_a, P : poly_a]: ((ord_less_nat @ (degree_a @ Q) @ (degree_a @ P)) => ((degree_a @ (plus_plus_poly_a @ P @ Q)) = (degree_a @ P)))))). % degree_add_eq_left
thf(fact_70_degree__add__eq__right, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((ord_less_nat @ (degree_nat @ P) @ (degree_nat @ Q)) => ((degree_nat @ (plus_plus_poly_nat @ P @ Q)) = (degree_nat @ Q)))))). % degree_add_eq_right
thf(fact_71_degree__add__eq__right, axiom,
    ((![P : poly_a, Q : poly_a]: ((ord_less_nat @ (degree_a @ P) @ (degree_a @ Q)) => ((degree_a @ (plus_plus_poly_a @ P @ Q)) = (degree_a @ Q)))))). % degree_add_eq_right
thf(fact_72_dbl__def, axiom,
    ((neg_numeral_dbl_a = (^[X2 : a]: (plus_plus_a @ X2 @ X2))))). % dbl_def
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_82_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_83_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_84_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_85_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_86_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_87_lead__coeff__monom, axiom,
    ((![C : a, N : nat]: ((coeff_a @ (monom_a @ C @ N) @ (degree_a @ (monom_a @ C @ N))) = C)))). % lead_coeff_monom
thf(fact_88_add__mono__thms__linordered__field_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(4)
thf(fact_89_add__mono__thms__linordered__field_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(4)
thf(fact_90_add__mono__thms__linordered__field_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(3)
thf(fact_91_add__mono__thms__linordered__field_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(3)
thf(fact_92_add__le__less__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_le_less_mono
thf(fact_93_add__le__less__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_le_less_mono
thf(fact_94_add__less__le__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_less_le_mono
thf(fact_95_add__less__le__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_less_le_mono
thf(fact_96_degree__monom__le, axiom,
    ((![A : a, N : nat]: (ord_less_eq_nat @ (degree_a @ (monom_a @ A @ N)) @ N)))). % degree_monom_le
thf(fact_97_lead__coeff__add__le, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((ord_less_nat @ (degree_nat @ P) @ (degree_nat @ Q)) => ((coeff_nat @ (plus_plus_poly_nat @ P @ Q) @ (degree_nat @ (plus_plus_poly_nat @ P @ Q))) = (coeff_nat @ Q @ (degree_nat @ Q))))))). % lead_coeff_add_le
thf(fact_98_lead__coeff__add__le, axiom,
    ((![P : poly_a, Q : poly_a]: ((ord_less_nat @ (degree_a @ P) @ (degree_a @ Q)) => ((coeff_a @ (plus_plus_poly_a @ P @ Q) @ (degree_a @ (plus_plus_poly_a @ P @ Q))) = (coeff_a @ Q @ (degree_a @ Q))))))). % lead_coeff_add_le
thf(fact_99_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_100_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_101_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_102_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_103_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_104_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (K = L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_105_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_106_add__strict__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_strict_mono
thf(fact_107_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_108_add__strict__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_strict_left_mono
thf(fact_109_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_110_add__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_strict_right_mono
thf(fact_111_add__less__imp__less__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_imp_less_left
thf(fact_112_add__less__imp__less__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_imp_less_left
thf(fact_113_add__less__imp__less__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_imp_less_right
thf(fact_114_add__less__imp__less__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_imp_less_right
thf(fact_115_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (K = L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_116_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_117_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_118_add__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_mono
thf(fact_119_add__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_left_mono
thf(fact_120_less__eqE, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => (~ ((![C2 : nat]: (~ ((B = (plus_plus_nat @ A @ C2))))))))))). % less_eqE
thf(fact_121_add__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_right_mono
thf(fact_122_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A3 : nat]: (^[B3 : nat]: (?[C3 : nat]: (B3 = (plus_plus_nat @ A3 @ C3)))))))). % le_iff_add
thf(fact_123_add__le__imp__le__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_left
thf(fact_124_add__le__imp__le__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_right
thf(fact_125_nat__add__left__cancel__le, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_eq_nat @ M @ N))))). % nat_add_left_cancel_le
thf(fact_126_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_127_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_128_mono__nat__linear__lb, axiom,
    ((![F : nat > nat, M : nat, K : nat]: ((![M2 : nat, N3 : nat]: ((ord_less_nat @ M2 @ N3) => (ord_less_nat @ (F @ M2) @ (F @ N3)))) => (ord_less_eq_nat @ (plus_plus_nat @ (F @ M) @ K) @ (F @ (plus_plus_nat @ M @ K))))))). % mono_nat_linear_lb
thf(fact_129_nat__descend__induct, axiom,
    ((![N : nat, P2 : nat > $o, M : nat]: ((![K2 : nat]: ((ord_less_nat @ N @ K2) => (P2 @ K2))) => ((![K2 : nat]: ((ord_less_eq_nat @ K2 @ N) => ((![I2 : nat]: ((ord_less_nat @ K2 @ I2) => (P2 @ I2))) => (P2 @ K2)))) => (P2 @ M)))))). % nat_descend_induct
thf(fact_130_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I3 : nat, J2 : nat]: ((ord_less_nat @ I3 @ J2) => (ord_less_nat @ (F @ I3) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_131_ep, axiom,
    ((ord_less_real @ zero_zero_real @ e))). % ep
thf(fact_132_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_133_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_134_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_135_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_136_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_137_antisym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ X) => (X = Y4)))))). % antisym
thf(fact_138_linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) | (ord_less_eq_nat @ Y4 @ X))))). % linear
thf(fact_139_eq__refl, axiom,
    ((![X : nat, Y4 : nat]: ((X = Y4) => (ord_less_eq_nat @ X @ Y4))))). % eq_refl
thf(fact_140_le__cases, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X @ Y4))) => (ord_less_eq_nat @ Y4 @ X))))). % le_cases
thf(fact_141_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_142_le__cases3, axiom,
    ((![X : nat, Y4 : nat, Z2 : nat]: (((ord_less_eq_nat @ X @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ Z2)))) => (((ord_less_eq_nat @ Y4 @ X) => (~ ((ord_less_eq_nat @ X @ Z2)))) => (((ord_less_eq_nat @ X @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y4)))) => (((ord_less_eq_nat @ Z2 @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ X)))) => (((ord_less_eq_nat @ Y4 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X)))) => (~ (((ord_less_eq_nat @ Z2 @ X) => (~ ((ord_less_eq_nat @ X @ Y4)))))))))))))). % le_cases3
thf(fact_143_antisym__conv, axiom,
    ((![Y4 : nat, X : nat]: ((ord_less_eq_nat @ Y4 @ X) => ((ord_less_eq_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv
thf(fact_144_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((ord_less_eq_nat @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_145_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_146_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_147_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_148_order__trans, axiom,
    ((![X : nat, Y4 : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ Z2) => (ord_less_eq_nat @ X @ Z2)))))). % order_trans
thf(fact_149_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_150_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B4 : nat]: ((ord_less_eq_nat @ A4 @ B4) => (P2 @ A4 @ B4))) => ((![A4 : nat, B4 : nat]: ((P2 @ B4 @ A4) => (P2 @ A4 @ B4))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_151_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_152_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((ord_less_eq_nat @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_153_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_154_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_155_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_156_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_157_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_158_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_159_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_160_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_161_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_162_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_163_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_164_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_165_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_166_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_167_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_168_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_169_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_170_lt__ex, axiom,
    ((![X : real]: (?[Y : real]: (ord_less_real @ Y @ X))))). % lt_ex
thf(fact_171_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_172_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_173_neqE, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((X = Y4))) => ((~ ((ord_less_nat @ X @ Y4))) => (ord_less_nat @ Y4 @ X)))))). % neqE
thf(fact_174_neqE, axiom,
    ((![X : real, Y4 : real]: ((~ ((X = Y4))) => ((~ ((ord_less_real @ X @ Y4))) => (ord_less_real @ Y4 @ X)))))). % neqE
thf(fact_175_neq__iff, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((X = Y4))) = (((ord_less_nat @ X @ Y4)) | ((ord_less_nat @ Y4 @ X))))))). % neq_iff
thf(fact_176_neq__iff, axiom,
    ((![X : real, Y4 : real]: ((~ ((X = Y4))) = (((ord_less_real @ X @ Y4)) | ((ord_less_real @ Y4 @ X))))))). % neq_iff
thf(fact_177_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_178_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_179_dense, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (?[Z3 : real]: ((ord_less_real @ X @ Z3) & (ord_less_real @ Z3 @ Y4))))))). % dense
thf(fact_180_less__imp__neq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_neq
thf(fact_181_less__imp__neq, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_neq
thf(fact_182_less__asym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_asym
thf(fact_183_less__asym, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_asym
thf(fact_184_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_185_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_186_less__trans, axiom,
    ((![X : nat, Y4 : nat, Z2 : nat]: ((ord_less_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ Z2) => (ord_less_nat @ X @ Z2)))))). % less_trans
thf(fact_187_less__trans, axiom,
    ((![X : real, Y4 : real, Z2 : real]: ((ord_less_real @ X @ Y4) => ((ord_less_real @ Y4 @ Z2) => (ord_less_real @ X @ Z2)))))). % less_trans
thf(fact_188_less__linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) | ((X = Y4) | (ord_less_nat @ Y4 @ X)))))). % less_linear
thf(fact_189_less__linear, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) | ((X = Y4) | (ord_less_real @ Y4 @ X)))))). % less_linear
thf(fact_190_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_191_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_192_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_193_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_194_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_195_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_196_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_197_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_198_less__imp__not__eq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_not_eq
thf(fact_199_less__imp__not__eq, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_not_eq
thf(fact_200_less__not__sym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_not_sym
thf(fact_201_less__not__sym, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_not_sym
thf(fact_202_less__induct, axiom,
    ((![P2 : nat > $o, A : nat]: ((![X3 : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X3) => (P2 @ Y5))) => (P2 @ X3))) => (P2 @ A))))). % less_induct
thf(fact_203_antisym__conv3, axiom,
    ((![Y4 : nat, X : nat]: ((~ ((ord_less_nat @ Y4 @ X))) => ((~ ((ord_less_nat @ X @ Y4))) = (X = Y4)))))). % antisym_conv3
thf(fact_204_antisym__conv3, axiom,
    ((![Y4 : real, X : real]: ((~ ((ord_less_real @ Y4 @ X))) => ((~ ((ord_less_real @ X @ Y4))) = (X = Y4)))))). % antisym_conv3
thf(fact_205_less__imp__not__eq2, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((Y4 = X))))))). % less_imp_not_eq2
thf(fact_206_less__imp__not__eq2, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((Y4 = X))))))). % less_imp_not_eq2
thf(fact_207_less__imp__triv, axiom,
    ((![X : nat, Y4 : nat, P2 : $o]: ((ord_less_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ X) => P2))))). % less_imp_triv
thf(fact_208_less__imp__triv, axiom,
    ((![X : real, Y4 : real, P2 : $o]: ((ord_less_real @ X @ Y4) => ((ord_less_real @ Y4 @ X) => P2))))). % less_imp_triv
thf(fact_209_linorder__cases, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) => ((~ ((X = Y4))) => (ord_less_nat @ Y4 @ X)))))). % linorder_cases
thf(fact_210_linorder__cases, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) => ((~ ((X = Y4))) => (ord_less_real @ Y4 @ X)))))). % linorder_cases
thf(fact_211_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_212_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_213_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_214_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_215_less__imp__not__less, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_imp_not_less
thf(fact_216_less__imp__not__less, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_imp_not_less
thf(fact_217_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X4 : nat]: (P3 @ X4))) = (^[P4 : nat > $o]: (?[N2 : nat]: (((P4 @ N2)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N2)) => ((~ ((P4 @ M3))))))))))))). % exists_least_iff
thf(fact_218_linorder__less__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B4 : nat]: ((ord_less_nat @ A4 @ B4) => (P2 @ A4 @ B4))) => ((![A4 : nat]: (P2 @ A4 @ A4)) => ((![A4 : nat, B4 : nat]: ((P2 @ B4 @ A4) => (P2 @ A4 @ B4))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_219_linorder__less__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A4 : real, B4 : real]: ((ord_less_real @ A4 @ B4) => (P2 @ A4 @ B4))) => ((![A4 : real]: (P2 @ A4 @ A4)) => ((![A4 : real, B4 : real]: ((P2 @ B4 @ A4) => (P2 @ A4 @ B4))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_220_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_221_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_222_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) = (((ord_less_nat @ Y4 @ X)) | ((X = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_223_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) = (((ord_less_real @ Y4 @ X)) | ((X = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_224_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_225_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_226_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_227_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_228_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_229_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_230_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_231_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_232_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_233_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((![M4 : nat]: ((ord_less_nat @ M4 @ N3) => (P2 @ M4))) => (P2 @ N3))) => (P2 @ N))))). % nat_less_induct
thf(fact_234_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P2 @ N3))) => (?[M4 : nat]: ((ord_less_nat @ M4 @ N3) & (~ ((P2 @ M4))))))) => (P2 @ N))))). % infinite_descent
thf(fact_235_linorder__neqE__nat, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((X = Y4))) => ((~ ((ord_less_nat @ X @ Y4))) => (ord_less_nat @ Y4 @ X)))))). % linorder_neqE_nat
thf(fact_236_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_237_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_238_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Q2 : poly_a]: (((degree_a @ Q2) = (degree_a @ p)) => ((![X3 : a]: ((poly_a2 @ Q2 @ X3) = (poly_a2 @ p @ (plus_plus_a @ z @ X3)))) => thesis))))).
thf(conj_1, conjecture,
    (thesis)).
