% 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_280__3225480_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:13.910

% Could-be-implicit typings (5)
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__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (38)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
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_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
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__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > 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__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal, type,
    groups2069495480t_real : (nat > real) > set_nat > real).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > real).
thf(sy_c_Num_Opow, type,
    pow : num > num > num).
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__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat, type,
    semiri2013084963it_nat : nat > nat > nat).
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_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_m, type,
    m : nat).

% Relevant facts (240)
thf(fact_0_root__cancel1, axiom,
    ((![M : nat, I : nat, J : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I @ J)))))). % root_cancel1
thf(fact_1_DFT__def, axiom,
    ((fFT_Mirabelle_DFT = (^[N : nat]: (^[A : nat > complex]: (^[I2 : nat]: (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ (times_times_nat @ I2 @ J2)) @ (A @ J2))) @ (set_or562006527an_nat @ zero_zero_nat @ N)))))))). % DFT_def
thf(fact_2_zero__eq__power2, axiom,
    ((![A2 : complex]: (((power_power_complex @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A2 = zero_zero_complex))))). % zero_eq_power2
thf(fact_3_zero__eq__power2, axiom,
    ((![A2 : nat]: (((power_power_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A2 = zero_zero_nat))))). % zero_eq_power2
thf(fact_4_zero__eq__power2, axiom,
    ((![A2 : real]: (((power_power_real @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_real) = (A2 = zero_zero_real))))). % zero_eq_power2
thf(fact_5_mult__eq__1__iff, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N2 = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_6_one__eq__mult__iff, axiom,
    ((![M : nat, N2 : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N2)) = (((M = (suc @ zero_zero_nat))) & ((N2 = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_7_power__Suc0__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_8_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_9_power__Suc0__right, axiom,
    ((![A2 : real]: ((power_power_real @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_10_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_11_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_12_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ K)) = zero_zero_real)))). % power_zero_numeral
thf(fact_13_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N2)) = zero_zero_complex)))). % power_0_Suc
thf(fact_14_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N2)) = zero_zero_nat)))). % power_0_Suc
thf(fact_15_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_real @ zero_zero_real @ (suc @ N2)) = zero_zero_real)))). % power_0_Suc
thf(fact_16_power__odd__eq, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))) = (times_times_complex @ A2 @ (power_power_complex @ (power_power_complex @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_17_power__odd__eq, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))) = (times_times_nat @ A2 @ (power_power_nat @ (power_power_nat @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_18_power__odd__eq, axiom,
    ((![A2 : real, N2 : nat]: ((power_power_real @ A2 @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))) = (times_times_real @ A2 @ (power_power_real @ (power_power_real @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_19_sum_Oneutral__const, axiom,
    ((![A3 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A3) = zero_zero_complex)))). % sum.neutral_const
thf(fact_20_mult__is__0, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N2 = zero_zero_nat))))))). % mult_is_0
thf(fact_21_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_22_mult__cancel1, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N2)) = (((M = N2)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_23_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_24_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_25_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_26_power__Suc__0, axiom,
    ((![N2 : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N2) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_27_power__mult__numeral, axiom,
    ((![A2 : complex, M : num, N2 : num]: ((power_power_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N2)) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N2))))))). % power_mult_numeral
thf(fact_28_power__mult__numeral, axiom,
    ((![A2 : nat, M : num, N2 : num]: ((power_power_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N2)) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N2))))))). % power_mult_numeral
thf(fact_29_power__mult__numeral, axiom,
    ((![A2 : real, M : num, N2 : num]: ((power_power_real @ (power_power_real @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N2)) = (power_power_real @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N2))))))). % power_mult_numeral
thf(fact_30_mult__cancel2, axiom,
    ((![M : nat, K : nat, N2 : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N2 @ K)) = (((M = N2)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_31_root__nonzero, axiom,
    ((![N2 : nat]: (~ (((fFT_Mirabelle_root @ N2) = zero_zero_complex)))))). % root_nonzero
thf(fact_32_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_33_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_34_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > complex, G : nat > complex]: ((![A4 : nat]: ((member_nat @ A4 @ S) => ((I @ (J @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => (member_nat @ (J @ A4) @ T))) => ((![B : nat]: ((member_nat @ B @ T) => ((J @ (I @ B)) = B))) => ((![B : nat]: ((member_nat @ B @ T) => (member_nat @ (I @ B) @ S))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => ((H @ (J @ A4)) = (G @ A4)))) => ((groups59700922omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_35_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A3 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B2) => ((member_nat @ (K @ Y3) @ A3) & ((H @ (K @ Y3)) = Y3)))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H @ X3) @ B2) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_36_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A3 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B2) => (?[X4 : nat]: (((member_nat @ X4 @ A3) & ((H @ X4) = Y3)) & (![Ya : nat]: (((member_nat @ Ya @ A3) & ((H @ Ya) = Y3)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H @ X3) @ B2) & ((Gamma @ (H @ X3)) = (Phi @ X3))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general
thf(fact_37_sum_Ocong, axiom,
    ((![A3 : set_nat, B2 : set_nat, G : nat > complex, H : nat > complex]: ((A3 = B2) => ((![X3 : nat]: ((member_nat @ X3 @ B2) => ((G @ X3) = (H @ X3)))) => ((groups59700922omplex @ G @ A3) = (groups59700922omplex @ H @ B2))))))). % sum.cong
thf(fact_38_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B2 : set_nat, A3 : set_nat]: ((groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (G @ I2) @ B2)) @ A3) = (groups59700922omplex @ (^[J2 : nat]: (groups59700922omplex @ (^[I2 : nat]: (G @ I2 @ J2)) @ A3)) @ B2))))). % sum.swap
thf(fact_39_power__not__zero, axiom,
    ((![A2 : complex, N2 : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N2) = zero_zero_complex))))))). % power_not_zero
thf(fact_40_power__not__zero, axiom,
    ((![A2 : nat, N2 : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N2) = zero_zero_nat))))))). % power_not_zero
thf(fact_41_power__not__zero, axiom,
    ((![A2 : real, N2 : nat]: ((~ ((A2 = zero_zero_real))) => (~ (((power_power_real @ A2 @ N2) = zero_zero_real))))))). % power_not_zero
thf(fact_42_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N2 : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N2) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N2))))))). % power_commuting_commutes
thf(fact_43_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N2 : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N2) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N2))))))). % power_commuting_commutes
thf(fact_44_power__commuting__commutes, axiom,
    ((![X : real, Y : real, N2 : nat]: (((times_times_real @ X @ Y) = (times_times_real @ Y @ X)) => ((times_times_real @ (power_power_real @ X @ N2) @ Y) = (times_times_real @ Y @ (power_power_real @ X @ N2))))))). % power_commuting_commutes
thf(fact_45_power__mult__distrib, axiom,
    ((![A2 : complex, B3 : complex, N2 : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B3) @ N2) = (times_times_complex @ (power_power_complex @ A2 @ N2) @ (power_power_complex @ B3 @ N2)))))). % power_mult_distrib
thf(fact_46_power__mult__distrib, axiom,
    ((![A2 : nat, B3 : nat, N2 : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B3) @ N2) = (times_times_nat @ (power_power_nat @ A2 @ N2) @ (power_power_nat @ B3 @ N2)))))). % power_mult_distrib
thf(fact_47_power__mult__distrib, axiom,
    ((![A2 : real, B3 : real, N2 : nat]: ((power_power_real @ (times_times_real @ A2 @ B3) @ N2) = (times_times_real @ (power_power_real @ A2 @ N2) @ (power_power_real @ B3 @ N2)))))). % power_mult_distrib
thf(fact_48_power__commutes, axiom,
    ((![A2 : complex, N2 : nat]: ((times_times_complex @ (power_power_complex @ A2 @ N2) @ A2) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N2)))))). % power_commutes
thf(fact_49_power__commutes, axiom,
    ((![A2 : nat, N2 : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N2) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N2)))))). % power_commutes
thf(fact_50_power__commutes, axiom,
    ((![A2 : real, N2 : nat]: ((times_times_real @ (power_power_real @ A2 @ N2) @ A2) = (times_times_real @ A2 @ (power_power_real @ A2 @ N2)))))). % power_commutes
thf(fact_51_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > complex, A3 : set_nat]: ((~ (((groups59700922omplex @ G @ A3) = zero_zero_complex))) => (~ ((![A4 : nat]: ((member_nat @ A4 @ A3) => ((G @ A4) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_52_sum_Oneutral, axiom,
    ((![A3 : set_nat, G : nat > complex]: ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((G @ X3) = zero_zero_complex))) => ((groups59700922omplex @ G @ A3) = zero_zero_complex))))). % sum.neutral
thf(fact_53_not0__implies__Suc, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (?[M2 : nat]: (N2 = (suc @ M2))))))). % not0_implies_Suc
thf(fact_54_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_55_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_56_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_57_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_58_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_59_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_60_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N2 : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M @ N2))))))). % diff_induct
thf(fact_61_nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N2)))))). % nat_induct
thf(fact_62_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_63_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_64_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_65_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_66_power__mult, axiom,
    ((![A2 : complex, M : nat, N2 : nat]: ((power_power_complex @ A2 @ (times_times_nat @ M @ N2)) = (power_power_complex @ (power_power_complex @ A2 @ M) @ N2))))). % power_mult
thf(fact_67_power__mult, axiom,
    ((![A2 : nat, M : nat, N2 : nat]: ((power_power_nat @ A2 @ (times_times_nat @ M @ N2)) = (power_power_nat @ (power_power_nat @ A2 @ M) @ N2))))). % power_mult
thf(fact_68_power__mult, axiom,
    ((![A2 : real, M : nat, N2 : nat]: ((power_power_real @ A2 @ (times_times_nat @ M @ N2)) = (power_power_real @ (power_power_real @ A2 @ M) @ N2))))). % power_mult
thf(fact_69_mult__0, axiom,
    ((![N2 : nat]: ((times_times_nat @ zero_zero_nat @ N2) = zero_zero_nat)))). % mult_0
thf(fact_70_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N2)) = (M = N2))))). % Suc_mult_cancel1
thf(fact_71_sum__distrib__right, axiom,
    ((![F : nat > complex, A3 : set_nat, R : complex]: ((times_times_complex @ (groups59700922omplex @ F @ A3) @ R) = (groups59700922omplex @ (^[N : nat]: (times_times_complex @ (F @ N) @ R)) @ A3))))). % sum_distrib_right
thf(fact_72_sum__distrib__left, axiom,
    ((![R : complex, F : nat > complex, A3 : set_nat]: ((times_times_complex @ R @ (groups59700922omplex @ F @ A3)) = (groups59700922omplex @ (^[N : nat]: (times_times_complex @ R @ (F @ N))) @ A3))))). % sum_distrib_left
thf(fact_73_sum__product, axiom,
    ((![F : nat > complex, A3 : set_nat, G : nat > complex, B2 : set_nat]: ((times_times_complex @ (groups59700922omplex @ F @ A3) @ (groups59700922omplex @ G @ B2)) = (groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (F @ I2) @ (G @ J2))) @ B2)) @ A3))))). % sum_product
thf(fact_74_power__Suc2, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ N2)) = (times_times_complex @ (power_power_complex @ A2 @ N2) @ A2))))). % power_Suc2
thf(fact_75_power__Suc2, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ N2)) = (times_times_nat @ (power_power_nat @ A2 @ N2) @ A2))))). % power_Suc2
thf(fact_76_power__Suc2, axiom,
    ((![A2 : real, N2 : nat]: ((power_power_real @ A2 @ (suc @ N2)) = (times_times_real @ (power_power_real @ A2 @ N2) @ A2))))). % power_Suc2
thf(fact_77_power__Suc, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ N2)) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N2)))))). % power_Suc
thf(fact_78_power__Suc, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ N2)) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N2)))))). % power_Suc
thf(fact_79_power__Suc, axiom,
    ((![A2 : real, N2 : nat]: ((power_power_real @ A2 @ (suc @ N2)) = (times_times_real @ A2 @ (power_power_real @ A2 @ N2)))))). % power_Suc
thf(fact_80_sum__cong__Suc, axiom,
    ((![A3 : set_nat, F : nat > complex, G : nat > complex]: ((~ ((member_nat @ zero_zero_nat @ A3))) => ((![X3 : nat]: ((member_nat @ (suc @ X3) @ A3) => ((F @ (suc @ X3)) = (G @ (suc @ X3))))) => ((groups59700922omplex @ F @ A3) = (groups59700922omplex @ G @ A3))))))). % sum_cong_Suc
thf(fact_81_power__numeral__even, axiom,
    ((![Z : complex, W : num]: ((power_power_complex @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_complex @ (power_power_complex @ Z @ (numeral_numeral_nat @ W)) @ (power_power_complex @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_82_power__numeral__even, axiom,
    ((![Z : nat, W : num]: ((power_power_nat @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_nat @ (power_power_nat @ Z @ (numeral_numeral_nat @ W)) @ (power_power_nat @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_83_power__numeral__even, axiom,
    ((![Z : real, W : num]: ((power_power_real @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_real @ (power_power_real @ Z @ (numeral_numeral_nat @ W)) @ (power_power_real @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_84_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_85_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_86_zero__power2, axiom,
    (((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_real))). % zero_power2
thf(fact_87_power2__eq__square, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_complex @ A2 @ A2))))). % power2_eq_square
thf(fact_88_power2__eq__square, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A2 @ A2))))). % power2_eq_square
thf(fact_89_power2__eq__square, axiom,
    ((![A2 : real]: ((power_power_real @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_real @ A2 @ A2))))). % power2_eq_square
thf(fact_90_power4__eq__xxxx, axiom,
    ((![X : complex]: ((power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_complex @ (times_times_complex @ (times_times_complex @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_91_power4__eq__xxxx, axiom,
    ((![X : nat]: ((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_nat @ (times_times_nat @ (times_times_nat @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_92_power4__eq__xxxx, axiom,
    ((![X : real]: ((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_real @ (times_times_real @ (times_times_real @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_93_power__even__eq, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)) = (power_power_complex @ (power_power_complex @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_94_power__even__eq, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)) = (power_power_nat @ (power_power_nat @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_95_power__even__eq, axiom,
    ((![A2 : real, N2 : nat]: ((power_power_real @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)) = (power_power_real @ (power_power_real @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_96_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > nat, K : nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((groups1842438620at_nat @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K)) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ zero_zero_nat @ K))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_97_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > real, K : nat]: (((F @ zero_zero_nat) = zero_zero_real) => ((groups2069495480t_real @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K)) = (groups2069495480t_real @ F @ (set_or562006527an_nat @ zero_zero_nat @ K))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_98_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > complex, K : nat]: (((F @ zero_zero_nat) = zero_zero_complex) => ((groups59700922omplex @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K)) = (groups59700922omplex @ F @ (set_or562006527an_nat @ zero_zero_nat @ K))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_99_double__not__eq__Suc__double, axiom,
    ((![M : nat, N2 : nat]: (~ (((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)))))))). % double_not_eq_Suc_double
thf(fact_100_Suc__double__not__eq__double, axiom,
    ((![M : nat, N2 : nat]: (~ (((suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))))))). % Suc_double_not_eq_double
thf(fact_101_numeral__2__eq__2, axiom,
    (((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat))))). % numeral_2_eq_2
thf(fact_102_num__double, axiom,
    ((![N2 : num]: ((times_times_num @ (bit0 @ one) @ N2) = (bit0 @ N2))))). % num_double
thf(fact_103_semiring__norm_I83_J, axiom,
    ((![N2 : num]: (~ ((one = (bit0 @ N2))))))). % semiring_norm(83)
thf(fact_104_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_105_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N2)) = (numera632737353omplex @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_106_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_107_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N2)) = (numeral_numeral_real @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_108_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (times_times_complex @ (numera632737353omplex @ W) @ Z)) = (times_times_complex @ (numera632737353omplex @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_109_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_110_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (times_times_real @ (numeral_numeral_real @ W) @ Z)) = (times_times_real @ (numeral_numeral_real @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_111_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B3 : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B3 @ C)) = (((C = zero_zero_complex)) | ((A2 = B3))))))). % mult_cancel_right
thf(fact_112_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B3 : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B3 @ C)) = (((C = zero_zero_nat)) | ((A2 = B3))))))). % mult_cancel_right
thf(fact_113_mult__cancel__right, axiom,
    ((![A2 : real, C : real, B3 : real]: (((times_times_real @ A2 @ C) = (times_times_real @ B3 @ C)) = (((C = zero_zero_real)) | ((A2 = B3))))))). % mult_cancel_right
thf(fact_114_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_115_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_116_semiring__norm_I87_J, axiom,
    ((![M : num, N2 : num]: (((bit0 @ M) = (bit0 @ N2)) = (M = N2))))). % semiring_norm(87)
thf(fact_117_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_118_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_119_mult__zero__left, axiom,
    ((![A2 : real]: ((times_times_real @ zero_zero_real @ A2) = zero_zero_real)))). % mult_zero_left
thf(fact_120_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_121_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_122_mult__zero__right, axiom,
    ((![A2 : real]: ((times_times_real @ A2 @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_123_mult__eq__0__iff, axiom,
    ((![A2 : complex, B3 : complex]: (((times_times_complex @ A2 @ B3) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B3 = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_124_mult__eq__0__iff, axiom,
    ((![A2 : nat, B3 : nat]: (((times_times_nat @ A2 @ B3) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B3 = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_125_mult__eq__0__iff, axiom,
    ((![A2 : real, B3 : real]: (((times_times_real @ A2 @ B3) = zero_zero_real) = (((A2 = zero_zero_real)) | ((B3 = zero_zero_real))))))). % mult_eq_0_iff
thf(fact_126_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B3 : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B3)) = (((C = zero_zero_complex)) | ((A2 = B3))))))). % mult_cancel_left
thf(fact_127_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B3 : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B3)) = (((C = zero_zero_nat)) | ((A2 = B3))))))). % mult_cancel_left
thf(fact_128_mult__cancel__left, axiom,
    ((![C : real, A2 : real, B3 : real]: (((times_times_real @ C @ A2) = (times_times_real @ C @ B3)) = (((C = zero_zero_real)) | ((A2 = B3))))))). % mult_cancel_left
thf(fact_129_semiring__norm_I13_J, axiom,
    ((![M : num, N2 : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N2)) = (bit0 @ (bit0 @ (times_times_num @ M @ N2))))))). % semiring_norm(13)
thf(fact_130_semiring__norm_I12_J, axiom,
    ((![N2 : num]: ((times_times_num @ one @ N2) = N2)))). % semiring_norm(12)
thf(fact_131_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_132_mult__not__zero, axiom,
    ((![A2 : complex, B3 : complex]: ((~ (((times_times_complex @ A2 @ B3) = zero_zero_complex))) => ((~ ((A2 = zero_zero_complex))) & (~ ((B3 = zero_zero_complex)))))))). % mult_not_zero
thf(fact_133_mult__not__zero, axiom,
    ((![A2 : nat, B3 : nat]: ((~ (((times_times_nat @ A2 @ B3) = zero_zero_nat))) => ((~ ((A2 = zero_zero_nat))) & (~ ((B3 = zero_zero_nat)))))))). % mult_not_zero
thf(fact_134_mult__not__zero, axiom,
    ((![A2 : real, B3 : real]: ((~ (((times_times_real @ A2 @ B3) = zero_zero_real))) => ((~ ((A2 = zero_zero_real))) & (~ ((B3 = zero_zero_real)))))))). % mult_not_zero
thf(fact_135_divisors__zero, axiom,
    ((![A2 : complex, B3 : complex]: (((times_times_complex @ A2 @ B3) = zero_zero_complex) => ((A2 = zero_zero_complex) | (B3 = zero_zero_complex)))))). % divisors_zero
thf(fact_136_divisors__zero, axiom,
    ((![A2 : nat, B3 : nat]: (((times_times_nat @ A2 @ B3) = zero_zero_nat) => ((A2 = zero_zero_nat) | (B3 = zero_zero_nat)))))). % divisors_zero
thf(fact_137_divisors__zero, axiom,
    ((![A2 : real, B3 : real]: (((times_times_real @ A2 @ B3) = zero_zero_real) => ((A2 = zero_zero_real) | (B3 = zero_zero_real)))))). % divisors_zero
thf(fact_138_no__zero__divisors, axiom,
    ((![A2 : complex, B3 : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B3 = zero_zero_complex))) => (~ (((times_times_complex @ A2 @ B3) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_139_no__zero__divisors, axiom,
    ((![A2 : nat, B3 : nat]: ((~ ((A2 = zero_zero_nat))) => ((~ ((B3 = zero_zero_nat))) => (~ (((times_times_nat @ A2 @ B3) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_140_no__zero__divisors, axiom,
    ((![A2 : real, B3 : real]: ((~ ((A2 = zero_zero_real))) => ((~ ((B3 = zero_zero_real))) => (~ (((times_times_real @ A2 @ B3) = zero_zero_real)))))))). % no_zero_divisors
thf(fact_141_mult__left__cancel, axiom,
    ((![C : complex, A2 : complex, B3 : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B3)) = (A2 = B3)))))). % mult_left_cancel
thf(fact_142_mult__left__cancel, axiom,
    ((![C : nat, A2 : nat, B3 : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B3)) = (A2 = B3)))))). % mult_left_cancel
thf(fact_143_mult__left__cancel, axiom,
    ((![C : real, A2 : real, B3 : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ C @ A2) = (times_times_real @ C @ B3)) = (A2 = B3)))))). % mult_left_cancel
thf(fact_144_mult__right__cancel, axiom,
    ((![C : complex, A2 : complex, B3 : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = (times_times_complex @ B3 @ C)) = (A2 = B3)))))). % mult_right_cancel
thf(fact_145_mult__right__cancel, axiom,
    ((![C : nat, A2 : nat, B3 : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A2 @ C) = (times_times_nat @ B3 @ C)) = (A2 = B3)))))). % mult_right_cancel
thf(fact_146_mult__right__cancel, axiom,
    ((![C : real, A2 : real, B3 : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ A2 @ C) = (times_times_real @ B3 @ C)) = (A2 = B3)))))). % mult_right_cancel
thf(fact_147_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N2))))))). % zero_neq_numeral
thf(fact_148_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N2))))))). % zero_neq_numeral
thf(fact_149_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_real = (numeral_numeral_real @ N2))))))). % zero_neq_numeral
thf(fact_150_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N2)) = (((K = zero_zero_nat)) | ((M = N2))))))). % nat_mult_eq_cancel_disj
thf(fact_151_lambda__zero, axiom,
    (((^[H2 : complex]: zero_zero_complex) = (times_times_complex @ zero_zero_complex)))). % lambda_zero
thf(fact_152_lambda__zero, axiom,
    (((^[H2 : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_153_lambda__zero, axiom,
    (((^[H2 : real]: zero_zero_real) = (times_times_real @ zero_zero_real)))). % lambda_zero
thf(fact_154_mult__numeral__1, axiom,
    ((![A2 : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_155_mult__numeral__1, axiom,
    ((![A2 : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_156_mult__numeral__1, axiom,
    ((![A2 : real]: ((times_times_real @ (numeral_numeral_real @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_157_mult__numeral__1__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ (numera632737353omplex @ one)) = A2)))). % mult_numeral_1_right
thf(fact_158_mult__numeral__1__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ (numeral_numeral_nat @ one)) = A2)))). % mult_numeral_1_right
thf(fact_159_mult__numeral__1__right, axiom,
    ((![A2 : real]: ((times_times_real @ A2 @ (numeral_numeral_real @ one)) = A2)))). % mult_numeral_1_right
thf(fact_160_sum_Oshift__bounds__Suc__ivl, axiom,
    ((![G : nat > complex, M : nat, N2 : nat]: ((groups59700922omplex @ G @ (set_or562006527an_nat @ (suc @ M) @ (suc @ N2))) = (groups59700922omplex @ (^[I2 : nat]: (G @ (suc @ I2))) @ (set_or562006527an_nat @ M @ N2)))))). % sum.shift_bounds_Suc_ivl
thf(fact_161_numeral__1__eq__Suc__0, axiom,
    (((numeral_numeral_nat @ one) = (suc @ zero_zero_nat)))). % numeral_1_eq_Suc_0
thf(fact_162_four__x__squared, axiom,
    ((![X : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ (bit0 @ one))) @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one)))) = (power_power_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % four_x_squared
thf(fact_163_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_complex @ (numera632737353omplex @ K) @ (numeral_numeral_nat @ L)) = (numera632737353omplex @ (pow @ K @ L)))))). % power_numeral
thf(fact_164_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_nat @ (numeral_numeral_nat @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_nat @ (pow @ K @ L)))))). % power_numeral
thf(fact_165_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_real @ (numeral_numeral_real @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_real @ (pow @ K @ L)))))). % power_numeral
thf(fact_166_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_167_push__bit__of__Suc__0, axiom,
    ((![N2 : nat]: ((semiri2013084963it_nat @ N2 @ (suc @ zero_zero_nat)) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))))). % push_bit_of_Suc_0
thf(fact_168_push__bit__eq__0__iff, axiom,
    ((![N2 : nat, A2 : nat]: (((semiri2013084963it_nat @ N2 @ A2) = zero_zero_nat) = (A2 = zero_zero_nat))))). % push_bit_eq_0_iff
thf(fact_169_push__bit__of__0, axiom,
    ((![N2 : nat]: ((semiri2013084963it_nat @ N2 @ zero_zero_nat) = zero_zero_nat)))). % push_bit_of_0
thf(fact_170_push__bit__Suc, axiom,
    ((![N2 : nat, A2 : nat]: ((semiri2013084963it_nat @ (suc @ N2) @ A2) = (semiri2013084963it_nat @ N2 @ (times_times_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one)))))))). % push_bit_Suc
thf(fact_171_pow_Osimps_I1_J, axiom,
    ((![X : num]: ((pow @ X @ one) = X)))). % pow.simps(1)
thf(fact_172_push__bit__double, axiom,
    ((![N2 : nat, A2 : nat]: ((semiri2013084963it_nat @ N2 @ (times_times_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one)))) = (times_times_nat @ (semiri2013084963it_nat @ N2 @ A2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % push_bit_double
thf(fact_173_push__bit__nat__def, axiom,
    ((semiri2013084963it_nat = (^[N : nat]: (^[M3 : nat]: (times_times_nat @ M3 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % push_bit_nat_def
thf(fact_174_push__bit__eq__mult, axiom,
    ((semiri2013084963it_nat = (^[N : nat]: (^[A : nat]: (times_times_nat @ A @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % push_bit_eq_mult
thf(fact_175_verit__eq__simplify_I10_J, axiom,
    ((![X2 : num]: (~ ((one = (bit0 @ X2))))))). % verit_eq_simplify(10)
thf(fact_176_calculation, axiom,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J2)) @ (a @ J2))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat))) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))). % calculation
thf(fact_177_root__summation, axiom,
    ((![K : nat, N2 : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ K @ N2) => ((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N2) @ K)) @ (set_or562006527an_nat @ zero_zero_nat @ N2)) = zero_zero_complex)))))). % root_summation
thf(fact_178_add__right__cancel, axiom,
    ((![B3 : complex, A2 : complex, C : complex]: (((plus_plus_complex @ B3 @ A2) = (plus_plus_complex @ C @ A2)) = (B3 = C))))). % add_right_cancel
thf(fact_179_add__right__cancel, axiom,
    ((![B3 : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B3 @ A2) = (plus_plus_nat @ C @ A2)) = (B3 = C))))). % add_right_cancel
thf(fact_180_add__left__cancel, axiom,
    ((![A2 : complex, B3 : complex, C : complex]: (((plus_plus_complex @ A2 @ B3) = (plus_plus_complex @ A2 @ C)) = (B3 = C))))). % add_left_cancel
thf(fact_181_add__left__cancel, axiom,
    ((![A2 : nat, B3 : nat, C : nat]: (((plus_plus_nat @ A2 @ B3) = (plus_plus_nat @ A2 @ C)) = (B3 = C))))). % add_left_cancel
thf(fact_182_not__gr__zero, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr_zero
thf(fact_183_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_184_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_185_add__cancel__right__right, axiom,
    ((![A2 : nat, B3 : nat]: ((A2 = (plus_plus_nat @ A2 @ B3)) = (B3 = zero_zero_nat))))). % add_cancel_right_right
thf(fact_186_add__cancel__right__right, axiom,
    ((![A2 : complex, B3 : complex]: ((A2 = (plus_plus_complex @ A2 @ B3)) = (B3 = zero_zero_complex))))). % add_cancel_right_right
thf(fact_187_add__cancel__right__right, axiom,
    ((![A2 : real, B3 : real]: ((A2 = (plus_plus_real @ A2 @ B3)) = (B3 = zero_zero_real))))). % add_cancel_right_right
thf(fact_188_add__cancel__right__left, axiom,
    ((![A2 : nat, B3 : nat]: ((A2 = (plus_plus_nat @ B3 @ A2)) = (B3 = zero_zero_nat))))). % add_cancel_right_left
thf(fact_189_add__cancel__right__left, axiom,
    ((![A2 : complex, B3 : complex]: ((A2 = (plus_plus_complex @ B3 @ A2)) = (B3 = zero_zero_complex))))). % add_cancel_right_left
thf(fact_190_add__cancel__right__left, axiom,
    ((![A2 : real, B3 : real]: ((A2 = (plus_plus_real @ B3 @ A2)) = (B3 = zero_zero_real))))). % add_cancel_right_left
thf(fact_191_add__cancel__left__right, axiom,
    ((![A2 : nat, B3 : nat]: (((plus_plus_nat @ A2 @ B3) = A2) = (B3 = zero_zero_nat))))). % add_cancel_left_right
thf(fact_192_add__cancel__left__right, axiom,
    ((![A2 : complex, B3 : complex]: (((plus_plus_complex @ A2 @ B3) = A2) = (B3 = zero_zero_complex))))). % add_cancel_left_right
thf(fact_193_add__cancel__left__right, axiom,
    ((![A2 : real, B3 : real]: (((plus_plus_real @ A2 @ B3) = A2) = (B3 = zero_zero_real))))). % add_cancel_left_right
thf(fact_194_add__cancel__left__left, axiom,
    ((![B3 : nat, A2 : nat]: (((plus_plus_nat @ B3 @ A2) = A2) = (B3 = zero_zero_nat))))). % add_cancel_left_left
thf(fact_195_add__cancel__left__left, axiom,
    ((![B3 : complex, A2 : complex]: (((plus_plus_complex @ B3 @ A2) = A2) = (B3 = zero_zero_complex))))). % add_cancel_left_left
thf(fact_196_add__cancel__left__left, axiom,
    ((![B3 : real, A2 : real]: (((plus_plus_real @ B3 @ A2) = A2) = (B3 = zero_zero_real))))). % add_cancel_left_left
thf(fact_197_double__zero__sym, axiom,
    ((![A2 : real]: ((zero_zero_real = (plus_plus_real @ A2 @ A2)) = (A2 = zero_zero_real))))). % double_zero_sym
thf(fact_198_double__zero, axiom,
    ((![A2 : real]: (((plus_plus_real @ A2 @ A2) = zero_zero_real) = (A2 = zero_zero_real))))). % double_zero
thf(fact_199_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_200_add_Oright__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % add.right_neutral
thf(fact_201_add_Oright__neutral, axiom,
    ((![A2 : real]: ((plus_plus_real @ A2 @ zero_zero_real) = A2)))). % add.right_neutral
thf(fact_202_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_203_add_Oleft__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % add.left_neutral
thf(fact_204_add_Oleft__neutral, axiom,
    ((![A2 : real]: ((plus_plus_real @ zero_zero_real @ A2) = A2)))). % add.left_neutral
thf(fact_205_numeral__less__iff, axiom,
    ((![M : num, N2 : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N2)) = (ord_less_num @ M @ N2))))). % numeral_less_iff
thf(fact_206_numeral__less__iff, axiom,
    ((![M : num, N2 : num]: ((ord_less_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N2)) = (ord_less_num @ M @ N2))))). % numeral_less_iff
thf(fact_207_add__less__cancel__right, axiom,
    ((![A2 : nat, C : nat, B3 : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B3 @ C)) = (ord_less_nat @ A2 @ B3))))). % add_less_cancel_right
thf(fact_208_add__less__cancel__left, axiom,
    ((![C : nat, A2 : nat, B3 : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B3)) = (ord_less_nat @ A2 @ B3))))). % add_less_cancel_left
thf(fact_209_mult_Oleft__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ one_one_complex @ A2) = A2)))). % mult.left_neutral
thf(fact_210_mult_Oleft__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ one_one_nat @ A2) = A2)))). % mult.left_neutral
thf(fact_211_mult_Oleft__neutral, axiom,
    ((![A2 : real]: ((times_times_real @ one_one_real @ A2) = A2)))). % mult.left_neutral
thf(fact_212_mult_Oright__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ one_one_complex) = A2)))). % mult.right_neutral
thf(fact_213_mult_Oright__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ one_one_nat) = A2)))). % mult.right_neutral
thf(fact_214_mult_Oright__neutral, axiom,
    ((![A2 : real]: ((times_times_real @ A2 @ one_one_real) = A2)))). % mult.right_neutral
thf(fact_215_numeral__plus__numeral, axiom,
    ((![M : num, N2 : num]: ((plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N2)) = (numera632737353omplex @ (plus_plus_num @ M @ N2)))))). % numeral_plus_numeral
thf(fact_216_numeral__plus__numeral, axiom,
    ((![M : num, N2 : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N2)))))). % numeral_plus_numeral
thf(fact_217_numeral__plus__numeral, axiom,
    ((![M : num, N2 : num]: ((plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N2)) = (numeral_numeral_real @ (plus_plus_num @ M @ N2)))))). % numeral_plus_numeral
thf(fact_218_add__numeral__left, axiom,
    ((![V : num, W : num, Z : complex]: ((plus_plus_complex @ (numera632737353omplex @ V) @ (plus_plus_complex @ (numera632737353omplex @ W) @ Z)) = (plus_plus_complex @ (numera632737353omplex @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_219_add__numeral__left, axiom,
    ((![V : num, W : num, Z : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_220_add__numeral__left, axiom,
    ((![V : num, W : num, Z : real]: ((plus_plus_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ (numeral_numeral_real @ W) @ Z)) = (plus_plus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_221_power__one, axiom,
    ((![N2 : nat]: ((power_power_complex @ one_one_complex @ N2) = one_one_complex)))). % power_one
thf(fact_222_power__one, axiom,
    ((![N2 : nat]: ((power_power_nat @ one_one_nat @ N2) = one_one_nat)))). % power_one
thf(fact_223_power__one, axiom,
    ((![N2 : nat]: ((power_power_real @ one_one_real @ N2) = one_one_real)))). % power_one
thf(fact_224_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_225_less__nat__zero__code, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_226_neq0__conv, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N2))))). % neq0_conv
thf(fact_227_Suc__less__eq, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N2)) = (ord_less_nat @ M @ N2))))). % Suc_less_eq
thf(fact_228_Suc__mono, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (ord_less_nat @ (suc @ M) @ (suc @ N2)))))). % Suc_mono
thf(fact_229_lessI, axiom,
    ((![N2 : nat]: (ord_less_nat @ N2 @ (suc @ N2))))). % lessI
thf(fact_230_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_231_add__is__0, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N2 = zero_zero_nat))))))). % add_is_0
thf(fact_232_add__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ M @ (suc @ N2)) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc_right
thf(fact_233_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N2 : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N2)) = (ord_less_nat @ M @ N2))))). % nat_add_left_cancel_less
thf(fact_234_power__one__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_235_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_236_power__one__right, axiom,
    ((![A2 : real]: ((power_power_real @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_237_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = one_one_nat) = (((M = one_one_nat)) & ((N2 = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_238_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N2 : nat]: ((one_one_nat = (times_times_nat @ M @ N2)) = (((M = one_one_nat)) & ((N2 = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_239_push__bit__push__bit, axiom,
    ((![M : nat, N2 : nat, A2 : nat]: ((semiri2013084963it_nat @ M @ (semiri2013084963it_nat @ N2 @ A2)) = (semiri2013084963it_nat @ (plus_plus_nat @ M @ N2) @ A2))))). % push_bit_push_bit

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ i) @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)))) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) = (groups59700922omplex @ (^[N : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ i) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ i @ N)) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))) @ (set_or562006527an_nat @ zero_zero_nat @ m))))).
