% 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/FFT/prob_177__3224480_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:41.692

% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J, type,
    set_complex : $tType).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (29)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    groups443808152omplex : (complex > complex) > set_complex > complex).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal, type,
    semiri2110766477t_real : nat > real).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex, type,
    neg_nu484426047omplex : complex > complex).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal, type,
    neg_nu1973887165c_real : real > real).
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_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex, type,
    collect_complex : (complex > $o) > set_complex).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal, type,
    set_or2075149659n_real : real > real > set_real).
thf(sy_c_member_001t__Complex__Ocomplex, type,
    member_complex : complex > set_complex > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_k, type,
    k : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (240)
thf(fact_0_k_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ k))). % k(1)
thf(fact_1_k_I2_J, axiom,
    ((ord_less_nat @ k @ n))). % k(2)
thf(fact_2_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_3_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_4_divide__eq__1__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = one_one_real) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_5_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_6_div__self, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real))))). % div_self
thf(fact_7_div__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % div_self
thf(fact_8_div__self, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) => ((divide_divide_nat @ A @ A) = one_one_nat))))). % div_self
thf(fact_9_one__eq__divide__iff, axiom,
    ((![A : real, B : real]: ((one_one_real = (divide_divide_real @ A @ B)) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_10_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_11_divide__self, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real))))). % divide_self
thf(fact_12_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_13_divide__self__if, axiom,
    ((![A : real]: (((A = zero_zero_real) => ((divide_divide_real @ A @ A) = zero_zero_real)) & ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real)))))). % divide_self_if
thf(fact_14_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_15_divide__eq__eq__1, axiom,
    ((![B : real, A : real]: (((divide_divide_real @ B @ A) = one_one_real) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % divide_eq_eq_1
thf(fact_16_eq__divide__eq__1, axiom,
    ((![B : real, A : real]: ((one_one_real = (divide_divide_real @ B @ A)) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % eq_divide_eq_1
thf(fact_17_one__divide__eq__0__iff, axiom,
    ((![A : real]: (((divide_divide_real @ one_one_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % one_divide_eq_0_iff
thf(fact_18_zero__eq__1__divide__iff, axiom,
    ((![A : real]: ((zero_zero_real = (divide_divide_real @ one_one_real @ A)) = (A = zero_zero_real))))). % zero_eq_1_divide_iff
thf(fact_19_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A2) = zero_zero_complex)))). % sum.neutral_const
thf(fact_20_sum_Oneutral__const, axiom,
    ((![A2 : set_complex]: ((groups443808152omplex @ (^[Uu : complex]: zero_zero_complex) @ A2) = zero_zero_complex)))). % sum.neutral_const
thf(fact_21_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_22_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_23_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_24_div__by__1, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ one_one_complex) = A)))). % div_by_1
thf(fact_25_div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % div_by_1
thf(fact_26_div__by__1, axiom,
    ((![A : real]: ((divide_divide_real @ A @ one_one_real) = A)))). % div_by_1
thf(fact_27_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_28_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_29_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_30_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_31_division__ring__divide__zero, axiom,
    ((![A : real]: ((divide_divide_real @ A @ zero_zero_real) = zero_zero_real)))). % division_ring_divide_zero
thf(fact_32_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_33_divide__cancel__right, axiom,
    ((![A : real, C : real, B : real]: (((divide_divide_real @ A @ C) = (divide_divide_real @ B @ C)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_right
thf(fact_34_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_35_divide__cancel__left, axiom,
    ((![C : real, A : real, B : real]: (((divide_divide_real @ C @ A) = (divide_divide_real @ C @ B)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_left
thf(fact_36_div__by__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % div_by_0
thf(fact_37_div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_38_div__by__0, axiom,
    ((![A : real]: ((divide_divide_real @ A @ zero_zero_real) = zero_zero_real)))). % div_by_0
thf(fact_39_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_40_divide__eq__0__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = zero_zero_real) = (((A = zero_zero_real)) | ((B = zero_zero_real))))))). % divide_eq_0_iff
thf(fact_41_div__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ zero_zero_complex @ A) = zero_zero_complex)))). % div_0
thf(fact_42_div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % div_0
thf(fact_43_div__0, axiom,
    ((![A : real]: ((divide_divide_real @ zero_zero_real @ A) = zero_zero_real)))). % div_0
thf(fact_44_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_45_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_46_power__inject__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M) = (power_power_real @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_47_zero__less__divide__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_real @ one_one_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_divide_1_iff
thf(fact_48_less__divide__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ A @ B)))))). % less_divide_eq_1_pos
thf(fact_49_less__divide__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ B @ A)))))). % less_divide_eq_1_neg
thf(fact_50_divide__less__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ B @ A)))))). % divide_less_eq_1_pos
thf(fact_51_divide__less__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ A @ B)))))). % divide_less_eq_1_neg
thf(fact_52_divide__less__0__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ (divide_divide_real @ one_one_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % divide_less_0_1_iff
thf(fact_53_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_54_power__strict__increasing__iff, axiom,
    ((![B : real, X : nat, Y : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X) @ (power_power_real @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_55_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_56_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_57_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_58_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_59_power__strict__decreasing__iff, axiom,
    ((![B : real, M : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_60_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_61_linordered__field__no__lb, axiom,
    ((![X2 : real]: (?[Y2 : real]: (ord_less_real @ Y2 @ X2))))). % linordered_field_no_lb
thf(fact_62_linordered__field__no__ub, axiom,
    ((![X2 : real]: (?[X_1 : real]: (ord_less_real @ X2 @ X_1))))). % linordered_field_no_ub
thf(fact_63_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_64_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_65_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_strict_increasing
thf(fact_66_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_67_power__less__imp__less__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_68_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_69_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_strict_decreasing
thf(fact_70_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_71_one__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ one_one_real @ (power_power_real @ A @ N))))))). % one_less_power
thf(fact_72_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_73_not__one__less__zero, axiom,
    ((~ ((ord_less_real @ one_one_real @ zero_zero_real))))). % not_one_less_zero
thf(fact_74_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_75_zero__less__one, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % zero_less_one
thf(fact_76_divide__strict__right__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ zero_zero_real) => (ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono_neg
thf(fact_77_mem__Collect__eq, axiom,
    ((![A : complex, P : complex > $o]: ((member_complex @ A @ (collect_complex @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_78_Collect__mem__eq, axiom,
    ((![A2 : set_complex]: ((collect_complex @ (^[X3 : complex]: (member_complex @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_79_Collect__cong, axiom,
    ((![P : complex > $o, Q : complex > $o]: ((![X4 : complex]: ((P @ X4) = (Q @ X4))) => ((collect_complex @ P) = (collect_complex @ Q)))))). % Collect_cong
thf(fact_80_divide__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono
thf(fact_81_zero__less__divide__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_real @ A @ B)) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ zero_zero_real @ B)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ B @ zero_zero_real))))))))). % zero_less_divide_iff
thf(fact_82_divide__less__cancel, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ B)))) & ((((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ A)))) & ((~ ((C = zero_zero_real))))))))))). % divide_less_cancel
thf(fact_83_divide__less__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (divide_divide_real @ A @ B) @ zero_zero_real) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ B @ zero_zero_real)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ zero_zero_real @ B))))))))). % divide_less_0_iff
thf(fact_84_divide__pos__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_pos_pos
thf(fact_85_divide__pos__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_pos_neg
thf(fact_86_divide__neg__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_neg_pos
thf(fact_87_divide__neg__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_neg_neg
thf(fact_88_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_89_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_90_less__divide__eq__1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ A @ B)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ B @ A))))))))). % less_divide_eq_1
thf(fact_91_divide__less__eq__1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ B @ A)))) | ((((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ A @ B)))) | ((A = zero_zero_real))))))))). % divide_less_eq_1
thf(fact_92_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_93_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_94_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_95_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > complex, G : nat > complex]: ((![A3 : nat]: ((member_nat @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => (member_nat @ (J @ A3) @ T))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => ((J @ (I @ B2)) = B2))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => (member_nat @ (I @ B2) @ S))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups59700922omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_96_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : complex > nat, J : nat > complex, T : set_complex, H : complex > complex, G : nat > complex]: ((![A3 : nat]: ((member_nat @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => (member_complex @ (J @ A3) @ T))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => ((J @ (I @ B2)) = B2))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => (member_nat @ (I @ B2) @ S))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups59700922omplex @ G @ S) = (groups443808152omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_97_sum_Oreindex__bij__witness, axiom,
    ((![S : set_complex, I : nat > complex, J : complex > nat, T : set_nat, H : nat > complex, G : complex > complex]: ((![A3 : complex]: ((member_complex @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : complex]: ((member_complex @ A3 @ S) => (member_nat @ (J @ A3) @ T))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => ((J @ (I @ B2)) = B2))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => (member_complex @ (I @ B2) @ S))) => ((![A3 : complex]: ((member_complex @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups443808152omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_98_sum_Oreindex__bij__witness, axiom,
    ((![S : set_complex, I : complex > complex, J : complex > complex, T : set_complex, H : complex > complex, G : complex > complex]: ((![A3 : complex]: ((member_complex @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : complex]: ((member_complex @ A3 @ S) => (member_complex @ (J @ A3) @ T))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => ((J @ (I @ B2)) = B2))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => (member_complex @ (I @ B2) @ S))) => ((![A3 : complex]: ((member_complex @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups443808152omplex @ G @ S) = (groups443808152omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_99_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => ((member_nat @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X4 : nat]: ((member_nat @ X4 @ A2) => ((member_nat @ (H @ X4) @ B3) & (((K @ (H @ X4)) = X4) & ((Gamma @ (H @ X4)) = (Phi @ X4)))))) => ((groups59700922omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B3))))))). % sum.eq_general_inverses
thf(fact_100_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_complex, K : complex > nat, A2 : set_nat, H : nat > complex, Gamma : complex > complex, Phi : nat > complex]: ((![Y2 : complex]: ((member_complex @ Y2 @ B3) => ((member_nat @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X4 : nat]: ((member_nat @ X4 @ A2) => ((member_complex @ (H @ X4) @ B3) & (((K @ (H @ X4)) = X4) & ((Gamma @ (H @ X4)) = (Phi @ X4)))))) => ((groups59700922omplex @ Phi @ A2) = (groups443808152omplex @ Gamma @ B3))))))). % sum.eq_general_inverses
thf(fact_101_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_nat, K : nat > complex, A2 : set_complex, H : complex > nat, Gamma : nat > complex, Phi : complex > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => ((member_complex @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X4 : complex]: ((member_complex @ X4 @ A2) => ((member_nat @ (H @ X4) @ B3) & (((K @ (H @ X4)) = X4) & ((Gamma @ (H @ X4)) = (Phi @ X4)))))) => ((groups443808152omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B3))))))). % sum.eq_general_inverses
thf(fact_102_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_complex, K : complex > complex, A2 : set_complex, H : complex > complex, Gamma : complex > complex, Phi : complex > complex]: ((![Y2 : complex]: ((member_complex @ Y2 @ B3) => ((member_complex @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X4 : complex]: ((member_complex @ X4 @ A2) => ((member_complex @ (H @ X4) @ B3) & (((K @ (H @ X4)) = X4) & ((Gamma @ (H @ X4)) = (Phi @ X4)))))) => ((groups443808152omplex @ Phi @ A2) = (groups443808152omplex @ Gamma @ B3))))))). % sum.eq_general_inverses
thf(fact_103_sum_Oeq__general, axiom,
    ((![B3 : set_nat, A2 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => (?[X2 : nat]: (((member_nat @ X2 @ A2) & ((H @ X2) = Y2)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y2)) => (Ya = X2))))))) => ((![X4 : nat]: ((member_nat @ X4 @ A2) => ((member_nat @ (H @ X4) @ B3) & ((Gamma @ (H @ X4)) = (Phi @ X4))))) => ((groups59700922omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B3))))))). % sum.eq_general
thf(fact_104_sum_Oeq__general, axiom,
    ((![B3 : set_complex, A2 : set_nat, H : nat > complex, Gamma : complex > complex, Phi : nat > complex]: ((![Y2 : complex]: ((member_complex @ Y2 @ B3) => (?[X2 : nat]: (((member_nat @ X2 @ A2) & ((H @ X2) = Y2)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y2)) => (Ya = X2))))))) => ((![X4 : nat]: ((member_nat @ X4 @ A2) => ((member_complex @ (H @ X4) @ B3) & ((Gamma @ (H @ X4)) = (Phi @ X4))))) => ((groups59700922omplex @ Phi @ A2) = (groups443808152omplex @ Gamma @ B3))))))). % sum.eq_general
thf(fact_105_sum_Oeq__general, axiom,
    ((![B3 : set_nat, A2 : set_complex, H : complex > nat, Gamma : nat > complex, Phi : complex > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => (?[X2 : complex]: (((member_complex @ X2 @ A2) & ((H @ X2) = Y2)) & (![Ya : complex]: (((member_complex @ Ya @ A2) & ((H @ Ya) = Y2)) => (Ya = X2))))))) => ((![X4 : complex]: ((member_complex @ X4 @ A2) => ((member_nat @ (H @ X4) @ B3) & ((Gamma @ (H @ X4)) = (Phi @ X4))))) => ((groups443808152omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B3))))))). % sum.eq_general
thf(fact_106_sum_Oeq__general, axiom,
    ((![B3 : set_complex, A2 : set_complex, H : complex > complex, Gamma : complex > complex, Phi : complex > complex]: ((![Y2 : complex]: ((member_complex @ Y2 @ B3) => (?[X2 : complex]: (((member_complex @ X2 @ A2) & ((H @ X2) = Y2)) & (![Ya : complex]: (((member_complex @ Ya @ A2) & ((H @ Ya) = Y2)) => (Ya = X2))))))) => ((![X4 : complex]: ((member_complex @ X4 @ A2) => ((member_complex @ (H @ X4) @ B3) & ((Gamma @ (H @ X4)) = (Phi @ X4))))) => ((groups443808152omplex @ Phi @ A2) = (groups443808152omplex @ Gamma @ B3))))))). % sum.eq_general
thf(fact_107_sum_Ocong, axiom,
    ((![A2 : set_nat, B3 : set_nat, G : nat > complex, H : nat > complex]: ((A2 = B3) => ((![X4 : nat]: ((member_nat @ X4 @ B3) => ((G @ X4) = (H @ X4)))) => ((groups59700922omplex @ G @ A2) = (groups59700922omplex @ H @ B3))))))). % sum.cong
thf(fact_108_sum_Ocong, axiom,
    ((![A2 : set_complex, B3 : set_complex, G : complex > complex, H : complex > complex]: ((A2 = B3) => ((![X4 : complex]: ((member_complex @ X4 @ B3) => ((G @ X4) = (H @ X4)))) => ((groups443808152omplex @ G @ A2) = (groups443808152omplex @ H @ B3))))))). % sum.cong
thf(fact_109_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B3 : set_nat, A2 : set_nat]: ((groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (G @ I2) @ B3)) @ A2) = (groups59700922omplex @ (^[J2 : nat]: (groups59700922omplex @ (^[I2 : nat]: (G @ I2 @ J2)) @ A2)) @ B3))))). % sum.swap
thf(fact_110_sum_Oswap, axiom,
    ((![G : nat > complex > complex, B3 : set_complex, A2 : set_nat]: ((groups59700922omplex @ (^[I2 : nat]: (groups443808152omplex @ (G @ I2) @ B3)) @ A2) = (groups443808152omplex @ (^[J2 : complex]: (groups59700922omplex @ (^[I2 : nat]: (G @ I2 @ J2)) @ A2)) @ B3))))). % sum.swap
thf(fact_111_sum_Oswap, axiom,
    ((![G : complex > nat > complex, B3 : set_nat, A2 : set_complex]: ((groups443808152omplex @ (^[I2 : complex]: (groups59700922omplex @ (G @ I2) @ B3)) @ A2) = (groups59700922omplex @ (^[J2 : nat]: (groups443808152omplex @ (^[I2 : complex]: (G @ I2 @ J2)) @ A2)) @ B3))))). % sum.swap
thf(fact_112_sum_Oswap, axiom,
    ((![G : complex > complex > complex, B3 : set_complex, A2 : set_complex]: ((groups443808152omplex @ (^[I2 : complex]: (groups443808152omplex @ (G @ I2) @ B3)) @ A2) = (groups443808152omplex @ (^[J2 : complex]: (groups443808152omplex @ (^[I2 : complex]: (G @ I2 @ J2)) @ A2)) @ B3))))). % sum.swap
thf(fact_113_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_114_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_115_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_116_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_117_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_118_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_119_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > complex, A2 : set_nat]: ((~ (((groups59700922omplex @ G @ A2) = zero_zero_complex))) => (~ ((![A3 : nat]: ((member_nat @ A3 @ A2) => ((G @ A3) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_120_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : complex > complex, A2 : set_complex]: ((~ (((groups443808152omplex @ G @ A2) = zero_zero_complex))) => (~ ((![A3 : complex]: ((member_complex @ A3 @ A2) => ((G @ A3) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_121_sum_Oneutral, axiom,
    ((![A2 : set_nat, G : nat > complex]: ((![X4 : nat]: ((member_nat @ X4 @ A2) => ((G @ X4) = zero_zero_complex))) => ((groups59700922omplex @ G @ A2) = zero_zero_complex))))). % sum.neutral
thf(fact_122_sum_Oneutral, axiom,
    ((![A2 : set_complex, G : complex > complex]: ((![X4 : complex]: ((member_complex @ X4 @ A2) => ((G @ X4) = zero_zero_complex))) => ((groups443808152omplex @ G @ A2) = zero_zero_complex))))). % sum.neutral
thf(fact_123_power__divide, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_124_power__divide, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (divide_divide_real @ A @ B) @ N) = (divide_divide_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_divide
thf(fact_125_sum__divide__distrib, axiom,
    ((![F : nat > complex, A2 : set_nat, R : complex]: ((divide1210191872omplex @ (groups59700922omplex @ F @ A2) @ R) = (groups59700922omplex @ (^[N3 : nat]: (divide1210191872omplex @ (F @ N3) @ R)) @ A2))))). % sum_divide_distrib
thf(fact_126_sum__divide__distrib, axiom,
    ((![F : complex > complex, A2 : set_complex, R : complex]: ((divide1210191872omplex @ (groups443808152omplex @ F @ A2) @ R) = (groups443808152omplex @ (^[N3 : complex]: (divide1210191872omplex @ (F @ N3) @ R)) @ A2))))). % sum_divide_distrib
thf(fact_127_root__summation, axiom,
    ((![K : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ K @ N) => ((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)) @ (set_or562006527an_nat @ zero_zero_nat @ N)) = zero_zero_complex)))))). % root_summation
thf(fact_128_right__inverse__eq, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => (((divide1210191872omplex @ A @ B) = one_one_complex) = (A = B)))))). % right_inverse_eq
thf(fact_129_right__inverse__eq, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => (((divide_divide_real @ A @ B) = one_one_real) = (A = B)))))). % right_inverse_eq
thf(fact_130_power__one__over, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ one_one_complex @ A) @ N) = (divide1210191872omplex @ one_one_complex @ (power_power_complex @ A @ N)))))). % power_one_over
thf(fact_131_power__one__over, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (divide_divide_real @ one_one_real @ A) @ N) = (divide_divide_real @ one_one_real @ (power_power_real @ A @ N)))))). % power_one_over
thf(fact_132_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_133_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_134_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_135_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_136_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_137_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_real @ zero_zero_real @ N) = one_one_real)) & ((~ ((N = zero_zero_nat))) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real)))))). % power_0_left
thf(fact_138_sum__roots__unity, axiom,
    ((![N : nat]: ((ord_less_nat @ one_one_nat @ N) => ((groups443808152omplex @ (^[X3 : complex]: X3) @ (collect_complex @ (^[Z : complex]: ((power_power_complex @ Z @ N) = one_one_complex)))) = zero_zero_complex))))). % sum_roots_unity
thf(fact_139_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_140_div__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((divide_divide_nat @ M @ N) = zero_zero_nat))))). % div_less
thf(fact_141_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_142_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_143_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_144_bits__div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % bits_div_by_1
thf(fact_145_bits__div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_146_bits__div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % bits_div_0
thf(fact_147_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_148_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N4 : nat]: ((~ ((P @ N4))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N4) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_149_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N4 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N4) => (P @ M2))) => (P @ N4))) => (P @ N))))). % nat_less_induct
thf(fact_150_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_151_less__not__refl3, axiom,
    ((![S2 : nat, T2 : nat]: ((ord_less_nat @ S2 @ T2) => (~ ((S2 = T2))))))). % less_not_refl3
thf(fact_152_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_153_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_154_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_155_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_156_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_157_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_158_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_159_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_160_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N4 : nat]: ((ord_less_nat @ zero_zero_nat @ N4) => ((~ ((P @ N4))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N4) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_161_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_162_div__eq__dividend__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ M) => (((divide_divide_nat @ M @ N) = M) = (N = one_one_nat)))))). % div_eq_dividend_iff
thf(fact_163_div__less__dividend, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ one_one_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (divide_divide_nat @ M @ N) @ M)))))). % div_less_dividend
thf(fact_164_Euclidean__Division_Odiv__eq__0__iff, axiom,
    ((![M : nat, N : nat]: (((divide_divide_nat @ M @ N) = zero_zero_nat) = (((ord_less_nat @ M @ N)) | ((N = zero_zero_nat))))))). % Euclidean_Division.div_eq_0_iff
thf(fact_165_sum__nth__roots, axiom,
    ((![N : nat, C : complex]: ((ord_less_nat @ one_one_nat @ N) => ((groups443808152omplex @ (^[X3 : complex]: X3) @ (collect_complex @ (^[Z : complex]: ((power_power_complex @ Z @ N) = C)))) = zero_zero_complex))))). % sum_nth_roots
thf(fact_166_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_167_ex__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((?[M3 : nat]: (((ord_less_nat @ M3 @ N)) & ((P @ M3)))) = (?[X3 : nat]: (((member_nat @ X3 @ (set_or562006527an_nat @ zero_zero_nat @ N))) & ((P @ X3)))))))). % ex_nat_less_eq
thf(fact_168_all__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((![M3 : nat]: (((ord_less_nat @ M3 @ N)) => ((P @ M3)))) = (![X3 : nat]: (((member_nat @ X3 @ (set_or562006527an_nat @ zero_zero_nat @ N))) => ((P @ X3)))))))). % all_nat_less_eq
thf(fact_169_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_170_less__numeral__extra_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % less_numeral_extra(1)
thf(fact_171_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_172_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_173_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_174_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_175_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_176_one__reorient, axiom,
    ((![X : real]: ((one_one_real = X) = (X = one_one_real))))). % one_reorient
thf(fact_177_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_178_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_179_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_180_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_181_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_182_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_183_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_184_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_real @ one_one_real @ one_one_real))))). % less_numeral_extra(4)
thf(fact_185_atLeastLessThan__eq__iff, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C @ D)) = (((A = C)) & ((B = D))))))))). % atLeastLessThan_eq_iff
thf(fact_186_atLeastLessThan__eq__iff, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) = (((A = C)) & ((B = D))))))))). % atLeastLessThan_eq_iff
thf(fact_187_atLeastLessThan__inj_I1_J, axiom,
    ((![A : real, B : real, C : real, D : real]: (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C @ D)) => ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (A = C))))))). % atLeastLessThan_inj(1)
thf(fact_188_atLeastLessThan__inj_I1_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (A = C))))))). % atLeastLessThan_inj(1)
thf(fact_189_atLeastLessThan__inj_I2_J, axiom,
    ((![A : real, B : real, C : real, D : real]: (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C @ D)) => ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (B = D))))))). % atLeastLessThan_inj(2)
thf(fact_190_atLeastLessThan__inj_I2_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (B = D))))))). % atLeastLessThan_inj(2)
thf(fact_191_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X4 : real]: (((ord_less_real @ zero_zero_real @ X4) & ((power_power_real @ X4 @ N) = A)) & (![Y3 : real]: (((ord_less_real @ zero_zero_real @ Y3) & ((power_power_real @ Y3 @ N) = A)) => (Y3 = X4)))))))))). % realpow_pos_nth_unique
thf(fact_192_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R2 : real]: ((ord_less_real @ zero_zero_real @ R2) & ((power_power_real @ R2 @ N) = A)))))))). % realpow_pos_nth
thf(fact_193_real__arch__pow__inv, axiom,
    ((![Y : real, X : real]: ((ord_less_real @ zero_zero_real @ Y) => ((ord_less_real @ X @ one_one_real) => (?[N4 : nat]: (ord_less_real @ (power_power_real @ X @ N4) @ Y))))))). % real_arch_pow_inv
thf(fact_194_real__arch__pow, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ one_one_real @ X) => (?[N4 : nat]: (ord_less_real @ Y @ (power_power_real @ X @ N4))))))). % real_arch_pow
thf(fact_195_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_196_dbl__inc__simps_I2_J, axiom,
    (((neg_nu484426047omplex @ zero_zero_complex) = one_one_complex))). % dbl_inc_simps(2)
thf(fact_197_dbl__inc__simps_I2_J, axiom,
    (((neg_nu1973887165c_real @ zero_zero_real) = one_one_real))). % dbl_inc_simps(2)
thf(fact_198_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_199_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_real @ zero_zero_real @ (power_power_real @ (semiri2110766477t_real @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_200_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_201_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri356525583omplex @ M) = zero_zero_complex) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_202_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2110766477t_real @ M) = zero_zero_real) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_203_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_204_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_complex = (semiri356525583omplex @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_205_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_real = (semiri2110766477t_real @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_206_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_207_of__nat__0, axiom,
    (((semiri356525583omplex @ zero_zero_nat) = zero_zero_complex))). % of_nat_0
thf(fact_208_of__nat__0, axiom,
    (((semiri2110766477t_real @ zero_zero_nat) = zero_zero_real))). % of_nat_0
thf(fact_209_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_210_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_211_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri356525583omplex @ N) = one_one_complex) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_212_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri1382578993at_nat @ N) = one_one_nat) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_213_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri2110766477t_real @ N) = one_one_real) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_214_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_complex = (semiri356525583omplex @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_215_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_nat = (semiri1382578993at_nat @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_216_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_real = (semiri2110766477t_real @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_217_of__nat__1, axiom,
    (((semiri356525583omplex @ one_one_nat) = one_one_complex))). % of_nat_1
thf(fact_218_of__nat__1, axiom,
    (((semiri1382578993at_nat @ one_one_nat) = one_one_nat))). % of_nat_1
thf(fact_219_of__nat__1, axiom,
    (((semiri2110766477t_real @ one_one_nat) = one_one_real))). % of_nat_1
thf(fact_220_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri356525583omplex @ X) = (power_power_complex @ (semiri356525583omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_221_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_222_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2110766477t_real @ X) = (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_223_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_complex @ (semiri356525583omplex @ B) @ W) = (semiri356525583omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_224_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_225_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_real @ (semiri2110766477t_real @ B) @ W) = (semiri2110766477t_real @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_226_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (power_power_nat @ M @ N)) = (power_power_complex @ (semiri356525583omplex @ M) @ N))))). % of_nat_power
thf(fact_227_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_228_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2110766477t_real @ (power_power_nat @ M @ N)) = (power_power_real @ (semiri2110766477t_real @ M) @ N))))). % of_nat_power
thf(fact_229_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_230_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_real @ zero_zero_real @ (semiri2110766477t_real @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_231_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_232_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_real @ (power_power_real @ (semiri2110766477t_real @ B) @ W) @ (semiri2110766477t_real @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_233_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_234_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_235_unique__euclidean__semiring__with__nat__class_Oof__nat__div, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (divide_divide_nat @ M @ N)) = (divide_divide_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_236_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_237_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_real @ (semiri2110766477t_real @ M) @ zero_zero_real)))))). % of_nat_less_0_iff
thf(fact_238_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_239_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)))))). % less_imp_of_nat_less

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ n)) @ k)) @ (set_or562006527an_nat @ zero_zero_nat @ n)) = zero_zero_complex))).
