% 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_336__3225914_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:36.393

% Could-be-implicit typings (6)
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).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (33)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OIDFT, type,
    fFT_Mirabelle_IDFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
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__Int__Oint, type,
    times_times_int : int > int > int).
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__Int__Oint, type,
    zero_zero_int : int).
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__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__Int__Oint, type,
    numeral_numeral_int : num > int).
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_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__Int__Oint, type,
    divide_divide_int : int > int > int).
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__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_m, type,
    m : nat).

% Relevant facts (245)
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_IDFT__def, axiom,
    ((fFT_Mirabelle_IDFT = (^[N : nat]: (^[A : nat > complex]: (^[I2 : nat]: (groups59700922omplex @ (^[K : nat]: (divide1210191872omplex @ (A @ K) @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ (times_times_nat @ I2 @ K)))) @ (set_or562006527an_nat @ zero_zero_nat @ N)))))))). % IDFT_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_divide__eq__eq__numeral1_I1_J, axiom,
    ((![B : complex, W : num, A2 : complex]: (((divide1210191872omplex @ B @ (numera632737353omplex @ W)) = A2) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => ((B = (times_times_complex @ A2 @ (numera632737353omplex @ W)))))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % divide_eq_eq_numeral1(1)
thf(fact_6_divide__eq__eq__numeral1_I1_J, axiom,
    ((![B : real, W : num, A2 : real]: (((divide_divide_real @ B @ (numeral_numeral_real @ W)) = A2) = (((((~ (((numeral_numeral_real @ W) = zero_zero_real)))) => ((B = (times_times_real @ A2 @ (numeral_numeral_real @ W)))))) & (((((numeral_numeral_real @ W) = zero_zero_real)) => ((A2 = zero_zero_real))))))))). % divide_eq_eq_numeral1(1)
thf(fact_7_eq__divide__eq__numeral1_I1_J, axiom,
    ((![A2 : complex, B : complex, W : num]: ((A2 = (divide1210191872omplex @ B @ (numera632737353omplex @ W))) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => (((times_times_complex @ A2 @ (numera632737353omplex @ W)) = B)))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % eq_divide_eq_numeral1(1)
thf(fact_8_eq__divide__eq__numeral1_I1_J, axiom,
    ((![A2 : real, B : real, W : num]: ((A2 = (divide_divide_real @ B @ (numeral_numeral_real @ W))) = (((((~ (((numeral_numeral_real @ W) = zero_zero_real)))) => (((times_times_real @ A2 @ (numeral_numeral_real @ W)) = B)))) & (((((numeral_numeral_real @ W) = zero_zero_real)) => ((A2 = zero_zero_real))))))))). % eq_divide_eq_numeral1(1)
thf(fact_9_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_10_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_11_power__Suc0__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_12_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_13_power__Suc0__right, axiom,
    ((![A2 : real]: ((power_power_real @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_14_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K2)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_15_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K2)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_16_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ K2)) = zero_zero_real)))). % power_zero_numeral
thf(fact_17_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N2)) = zero_zero_complex)))). % power_0_Suc
thf(fact_18_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N2)) = zero_zero_nat)))). % power_0_Suc
thf(fact_19_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_real @ zero_zero_real @ (suc @ N2)) = zero_zero_real)))). % power_0_Suc
thf(fact_20_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_21_mult__divide__mult__cancel__left__if, axiom,
    ((![C : real, A2 : real, B : real]: (((C = zero_zero_real) => ((divide_divide_real @ (times_times_real @ C @ A2) @ (times_times_real @ C @ B)) = zero_zero_real)) & ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A2) @ (times_times_real @ C @ B)) = (divide_divide_real @ A2 @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_22_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_23_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A2) @ (times_times_real @ C @ B)) = (divide_divide_real @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_24_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_25_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_26_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : real, B : real]: ((~ ((A2 = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_27_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : int, B : int]: ((~ ((A2 = zero_zero_int))) => ((divide_divide_int @ (times_times_int @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_28_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_29_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_30_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_int @ M) = (numeral_numeral_int @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_31_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_32_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_33_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_34_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_35_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_36_mult__cancel__right, axiom,
    ((![A2 : real, C : real, B : real]: (((times_times_real @ A2 @ C) = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_37_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_38_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_39_mult__cancel__left, axiom,
    ((![C : real, A2 : real, B : real]: (((times_times_real @ C @ A2) = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_40_mult__eq__0__iff, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_41_mult__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_42_mult__eq__0__iff, axiom,
    ((![A2 : real, B : real]: (((times_times_real @ A2 @ B) = zero_zero_real) = (((A2 = zero_zero_real)) | ((B = zero_zero_real))))))). % mult_eq_0_iff
thf(fact_43_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_44_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_45_mult__zero__right, axiom,
    ((![A2 : real]: ((times_times_real @ A2 @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_46_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_47_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_48_mult__zero__left, axiom,
    ((![A2 : real]: ((times_times_real @ zero_zero_real @ A2) = zero_zero_real)))). % mult_zero_left
thf(fact_49_division__ring__divide__zero, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_50_division__ring__divide__zero, axiom,
    ((![A2 : real]: ((divide_divide_real @ A2 @ zero_zero_real) = zero_zero_real)))). % division_ring_divide_zero
thf(fact_51_divide__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((divide1210191872omplex @ A2 @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % divide_cancel_right
thf(fact_52_divide__cancel__right, axiom,
    ((![A2 : real, C : real, B : real]: (((divide_divide_real @ A2 @ C) = (divide_divide_real @ B @ C)) = (((C = zero_zero_real)) | ((A2 = B))))))). % divide_cancel_right
thf(fact_53_divide__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((divide1210191872omplex @ C @ A2) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % divide_cancel_left
thf(fact_54_divide__cancel__left, axiom,
    ((![C : real, A2 : real, B : real]: (((divide_divide_real @ C @ A2) = (divide_divide_real @ C @ B)) = (((C = zero_zero_real)) | ((A2 = B))))))). % divide_cancel_left
thf(fact_55_div__by__0, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % div_by_0
thf(fact_56_div__by__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_57_div__by__0, axiom,
    ((![A2 : real]: ((divide_divide_real @ A2 @ zero_zero_real) = zero_zero_real)))). % div_by_0
thf(fact_58_div__by__0, axiom,
    ((![A2 : int]: ((divide_divide_int @ A2 @ zero_zero_int) = zero_zero_int)))). % div_by_0
thf(fact_59_divide__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((divide1210191872omplex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_60_divide__eq__0__iff, axiom,
    ((![A2 : real, B : real]: (((divide_divide_real @ A2 @ B) = zero_zero_real) = (((A2 = zero_zero_real)) | ((B = zero_zero_real))))))). % divide_eq_0_iff
thf(fact_61_div__0, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ zero_zero_complex @ A2) = zero_zero_complex)))). % div_0
thf(fact_62_div__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % div_0
thf(fact_63_div__0, axiom,
    ((![A2 : real]: ((divide_divide_real @ zero_zero_real @ A2) = zero_zero_real)))). % div_0
thf(fact_64_div__0, axiom,
    ((![A2 : int]: ((divide_divide_int @ zero_zero_int @ A2) = zero_zero_int)))). % div_0
thf(fact_65_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_66_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_67_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N2)) = (numeral_numeral_int @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_68_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_69_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_70_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_71_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : int]: ((times_times_int @ (numeral_numeral_int @ V) @ (times_times_int @ (numeral_numeral_int @ W) @ Z)) = (times_times_int @ (numeral_numeral_int @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_72_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_73_times__divide__eq__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A2 @ B) @ C))))). % times_divide_eq_right
thf(fact_74_times__divide__eq__right, axiom,
    ((![A2 : real, B : real, C : real]: ((times_times_real @ A2 @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A2 @ B) @ C))))). % times_divide_eq_right
thf(fact_75_divide__divide__eq__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ A2 @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A2 @ C) @ B))))). % divide_divide_eq_right
thf(fact_76_divide__divide__eq__right, axiom,
    ((![A2 : real, B : real, C : real]: ((divide_divide_real @ A2 @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A2 @ C) @ B))))). % divide_divide_eq_right
thf(fact_77_divide__divide__eq__left, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A2 @ B) @ C) = (divide1210191872omplex @ A2 @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_78_divide__divide__eq__left, axiom,
    ((![A2 : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A2 @ B) @ C) = (divide_divide_real @ A2 @ (times_times_real @ B @ C)))))). % divide_divide_eq_left
thf(fact_79_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A2 : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A2) = (divide1210191872omplex @ (times_times_complex @ B @ A2) @ C))))). % times_divide_eq_left
thf(fact_80_times__divide__eq__left, axiom,
    ((![B : real, C : real, A2 : real]: ((times_times_real @ (divide_divide_real @ B @ C) @ A2) = (divide_divide_real @ (times_times_real @ B @ A2) @ C))))). % times_divide_eq_left
thf(fact_81_num__double, axiom,
    ((![N2 : num]: ((times_times_num @ (bit0 @ one) @ N2) = (bit0 @ N2))))). % num_double
thf(fact_82_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_83_power__Suc__0, axiom,
    ((![N2 : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N2) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_84_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_85_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_86_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_87_mult__cancel2, axiom,
    ((![M : nat, K2 : nat, N2 : nat]: (((times_times_nat @ M @ K2) = (times_times_nat @ N2 @ K2)) = (((M = N2)) | ((K2 = zero_zero_nat))))))). % mult_cancel2
thf(fact_88_mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N2 : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N2)) = (((M = N2)) | ((K2 = zero_zero_nat))))))). % mult_cancel1
thf(fact_89_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_90_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_91_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_92_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A2 @ C) @ (times_times_real @ C @ B)) = (divide_divide_real @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_93_nonzero__mult__div__cancel__right, axiom,
    ((![B : complex, A2 : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_94_nonzero__mult__div__cancel__right, axiom,
    ((![B : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_95_nonzero__mult__div__cancel__right, axiom,
    ((![B : real, A2 : real]: ((~ ((B = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_96_nonzero__mult__div__cancel__right, axiom,
    ((![B : int, A2 : int]: ((~ ((B = zero_zero_int))) => ((divide_divide_int @ (times_times_int @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_97_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_98_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A2 @ C) @ (times_times_real @ B @ C)) = (divide_divide_real @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_99_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_100_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A2) @ (times_times_real @ B @ C)) = (divide_divide_real @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_101_root__nonzero, axiom,
    ((![N2 : nat]: (~ (((fFT_Mirabelle_root @ N2) = zero_zero_complex)))))). % root_nonzero
thf(fact_102_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_103_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_104_mult__right__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_105_mult__right__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_106_mult__right__cancel, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ A2 @ C) = (times_times_real @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_107_mult__left__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_108_mult__left__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_109_mult__left__cancel, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ C @ A2) = (times_times_real @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_110_no__zero__divisors, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A2 @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_111_no__zero__divisors, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A2 @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_112_no__zero__divisors, axiom,
    ((![A2 : real, B : real]: ((~ ((A2 = zero_zero_real))) => ((~ ((B = zero_zero_real))) => (~ (((times_times_real @ A2 @ B) = zero_zero_real)))))))). % no_zero_divisors
thf(fact_113_divisors__zero, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) => ((A2 = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_114_divisors__zero, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) => ((A2 = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_115_divisors__zero, axiom,
    ((![A2 : real, B : real]: (((times_times_real @ A2 @ B) = zero_zero_real) => ((A2 = zero_zero_real) | (B = zero_zero_real)))))). % divisors_zero
thf(fact_116_mult__not__zero, axiom,
    ((![A2 : nat, B : nat]: ((~ (((times_times_nat @ A2 @ B) = zero_zero_nat))) => ((~ ((A2 = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_117_mult__not__zero, axiom,
    ((![A2 : complex, B : complex]: ((~ (((times_times_complex @ A2 @ B) = zero_zero_complex))) => ((~ ((A2 = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_118_mult__not__zero, axiom,
    ((![A2 : real, B : real]: ((~ (((times_times_real @ A2 @ B) = zero_zero_real))) => ((~ ((A2 = zero_zero_real))) & (~ ((B = zero_zero_real)))))))). % mult_not_zero
thf(fact_119_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N2))))))). % zero_neq_numeral
thf(fact_120_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_real = (numeral_numeral_real @ N2))))))). % zero_neq_numeral
thf(fact_121_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_int = (numeral_numeral_int @ N2))))))). % zero_neq_numeral
thf(fact_122_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N2))))))). % zero_neq_numeral
thf(fact_123_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_124_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_125_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_126_divide__divide__eq__left_H, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A2 @ B) @ C) = (divide1210191872omplex @ A2 @ (times_times_complex @ C @ B)))))). % divide_divide_eq_left'
thf(fact_127_divide__divide__eq__left_H, axiom,
    ((![A2 : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A2 @ B) @ C) = (divide_divide_real @ A2 @ (times_times_real @ C @ B)))))). % divide_divide_eq_left'
thf(fact_128_divide__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ W) @ (times_times_complex @ Y @ Z)))))). % divide_divide_times_eq
thf(fact_129_divide__divide__times__eq, axiom,
    ((![X : real, Y : real, Z : real, W : real]: ((divide_divide_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z @ W)) = (divide_divide_real @ (times_times_real @ X @ W) @ (times_times_real @ Y @ Z)))))). % divide_divide_times_eq
thf(fact_130_times__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((times_times_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ Z) @ (times_times_complex @ Y @ W)))))). % times_divide_times_eq
thf(fact_131_times__divide__times__eq, axiom,
    ((![X : real, Y : real, Z : real, W : real]: ((times_times_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z @ W)) = (divide_divide_real @ (times_times_real @ X @ Z) @ (times_times_real @ Y @ W)))))). % times_divide_times_eq
thf(fact_132_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_133_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_134_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_135_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N2 : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N2) = (times_times_nat @ (power_power_nat @ A2 @ N2) @ (power_power_nat @ B @ N2)))))). % power_mult_distrib
thf(fact_136_power__mult__distrib, axiom,
    ((![A2 : complex, B : complex, N2 : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B) @ N2) = (times_times_complex @ (power_power_complex @ A2 @ N2) @ (power_power_complex @ B @ N2)))))). % power_mult_distrib
thf(fact_137_power__mult__distrib, axiom,
    ((![A2 : real, B : real, N2 : nat]: ((power_power_real @ (times_times_real @ A2 @ B) @ N2) = (times_times_real @ (power_power_real @ A2 @ N2) @ (power_power_real @ B @ N2)))))). % power_mult_distrib
thf(fact_138_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_139_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_140_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_141_power__divide, axiom,
    ((![A2 : complex, B : complex, N2 : nat]: ((power_power_complex @ (divide1210191872omplex @ A2 @ B) @ N2) = (divide1210191872omplex @ (power_power_complex @ A2 @ N2) @ (power_power_complex @ B @ N2)))))). % power_divide
thf(fact_142_power__divide, axiom,
    ((![A2 : real, B : real, N2 : nat]: ((power_power_real @ (divide_divide_real @ A2 @ B) @ N2) = (divide_divide_real @ (power_power_real @ A2 @ N2) @ (power_power_real @ B @ N2)))))). % power_divide
thf(fact_143_not0__implies__Suc, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (?[M2 : nat]: (N2 = (suc @ M2))))))). % not0_implies_Suc
thf(fact_144_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_145_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_146_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_147_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_148_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_149_zero__induct, axiom,
    ((![P : nat > $o, K2 : nat]: ((P @ K2) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_150_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_151_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_152_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_153_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_154_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_155_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_156_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_157_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_158_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_159_mult__0, axiom,
    ((![N2 : nat]: ((times_times_nat @ zero_zero_nat @ N2) = zero_zero_nat)))). % mult_0
thf(fact_160_Suc__mult__cancel1, axiom,
    ((![K2 : nat, M : nat, N2 : nat]: (((times_times_nat @ (suc @ K2) @ M) = (times_times_nat @ (suc @ K2) @ N2)) = (M = N2))))). % Suc_mult_cancel1
thf(fact_161_lambda__zero, axiom,
    (((^[H : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_162_lambda__zero, axiom,
    (((^[H : complex]: zero_zero_complex) = (times_times_complex @ zero_zero_complex)))). % lambda_zero
thf(fact_163_lambda__zero, axiom,
    (((^[H : real]: zero_zero_real) = (times_times_real @ zero_zero_real)))). % lambda_zero
thf(fact_164_nonzero__eq__divide__eq, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((A2 = (divide1210191872omplex @ B @ C)) = ((times_times_complex @ A2 @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_165_nonzero__eq__divide__eq, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => ((A2 = (divide_divide_real @ B @ C)) = ((times_times_real @ A2 @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_166_nonzero__divide__eq__eq, axiom,
    ((![C : complex, B : complex, A2 : complex]: ((~ ((C = zero_zero_complex))) => (((divide1210191872omplex @ B @ C) = A2) = (B = (times_times_complex @ A2 @ C))))))). % nonzero_divide_eq_eq
thf(fact_167_nonzero__divide__eq__eq, axiom,
    ((![C : real, B : real, A2 : real]: ((~ ((C = zero_zero_real))) => (((divide_divide_real @ B @ C) = A2) = (B = (times_times_real @ A2 @ C))))))). % nonzero_divide_eq_eq
thf(fact_168_eq__divide__imp, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = B) => (A2 = (divide1210191872omplex @ B @ C))))))). % eq_divide_imp
thf(fact_169_eq__divide__imp, axiom,
    ((![C : real, A2 : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ A2 @ C) = B) => (A2 = (divide_divide_real @ B @ C))))))). % eq_divide_imp
thf(fact_170_divide__eq__imp, axiom,
    ((![C : complex, B : complex, A2 : complex]: ((~ ((C = zero_zero_complex))) => ((B = (times_times_complex @ A2 @ C)) => ((divide1210191872omplex @ B @ C) = A2)))))). % divide_eq_imp
thf(fact_171_divide__eq__imp, axiom,
    ((![C : real, B : real, A2 : real]: ((~ ((C = zero_zero_real))) => ((B = (times_times_real @ A2 @ C)) => ((divide_divide_real @ B @ C) = A2)))))). % divide_eq_imp
thf(fact_172_eq__divide__eq, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((A2 = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ A2 @ C) = B)))) & ((((C = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % eq_divide_eq
thf(fact_173_eq__divide__eq, axiom,
    ((![A2 : real, B : real, C : real]: ((A2 = (divide_divide_real @ B @ C)) = (((((~ ((C = zero_zero_real)))) => (((times_times_real @ A2 @ C) = B)))) & ((((C = zero_zero_real)) => ((A2 = zero_zero_real))))))))). % eq_divide_eq
thf(fact_174_divide__eq__eq, axiom,
    ((![B : complex, C : complex, A2 : complex]: (((divide1210191872omplex @ B @ C) = A2) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ A2 @ C))))) & ((((C = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % divide_eq_eq
thf(fact_175_divide__eq__eq, axiom,
    ((![B : real, C : real, A2 : real]: (((divide_divide_real @ B @ C) = A2) = (((((~ ((C = zero_zero_real)))) => ((B = (times_times_real @ A2 @ C))))) & ((((C = zero_zero_real)) => ((A2 = zero_zero_real))))))))). % divide_eq_eq
thf(fact_176_frac__eq__eq, axiom,
    ((![Y : complex, Z : complex, X : complex, W : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z = zero_zero_complex))) => (((divide1210191872omplex @ X @ Y) = (divide1210191872omplex @ W @ Z)) = ((times_times_complex @ X @ Z) = (times_times_complex @ W @ Y)))))))). % frac_eq_eq
thf(fact_177_frac__eq__eq, axiom,
    ((![Y : real, Z : real, X : real, W : real]: ((~ ((Y = zero_zero_real))) => ((~ ((Z = zero_zero_real))) => (((divide_divide_real @ X @ Y) = (divide_divide_real @ W @ Z)) = ((times_times_real @ X @ Z) = (times_times_real @ W @ Y)))))))). % frac_eq_eq
thf(fact_178_mult__numeral__1__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ (numeral_numeral_nat @ one)) = A2)))). % mult_numeral_1_right
thf(fact_179_mult__numeral__1__right, axiom,
    ((![A2 : real]: ((times_times_real @ A2 @ (numeral_numeral_real @ one)) = A2)))). % mult_numeral_1_right
thf(fact_180_mult__numeral__1__right, axiom,
    ((![A2 : int]: ((times_times_int @ A2 @ (numeral_numeral_int @ one)) = A2)))). % mult_numeral_1_right
thf(fact_181_mult__numeral__1__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ (numera632737353omplex @ one)) = A2)))). % mult_numeral_1_right
thf(fact_182_mult__numeral__1, axiom,
    ((![A2 : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_183_mult__numeral__1, axiom,
    ((![A2 : real]: ((times_times_real @ (numeral_numeral_real @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_184_mult__numeral__1, axiom,
    ((![A2 : int]: ((times_times_int @ (numeral_numeral_int @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_185_mult__numeral__1, axiom,
    ((![A2 : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_186_divide__numeral__1, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ (numera632737353omplex @ one)) = A2)))). % divide_numeral_1
thf(fact_187_divide__numeral__1, axiom,
    ((![A2 : real]: ((divide_divide_real @ A2 @ (numeral_numeral_real @ one)) = A2)))). % divide_numeral_1
thf(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_eq__divide__eq__numeral_I1_J, axiom,
    ((![W : num, B : complex, C : complex]: (((numera632737353omplex @ W) = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ (numera632737353omplex @ W) @ C) = B)))) & ((((C = zero_zero_complex)) => (((numera632737353omplex @ W) = zero_zero_complex))))))))). % eq_divide_eq_numeral(1)
thf(fact_195_eq__divide__eq__numeral_I1_J, axiom,
    ((![W : num, B : real, C : real]: (((numeral_numeral_real @ W) = (divide_divide_real @ B @ C)) = (((((~ ((C = zero_zero_real)))) => (((times_times_real @ (numeral_numeral_real @ W) @ C) = B)))) & ((((C = zero_zero_real)) => (((numeral_numeral_real @ W) = zero_zero_real))))))))). % eq_divide_eq_numeral(1)
thf(fact_196_divide__eq__eq__numeral_I1_J, axiom,
    ((![B : complex, C : complex, W : num]: (((divide1210191872omplex @ B @ C) = (numera632737353omplex @ W)) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ (numera632737353omplex @ W) @ C))))) & ((((C = zero_zero_complex)) => (((numera632737353omplex @ W) = zero_zero_complex))))))))). % divide_eq_eq_numeral(1)
thf(fact_197_divide__eq__eq__numeral_I1_J, axiom,
    ((![B : real, C : real, W : num]: (((divide_divide_real @ B @ C) = (numeral_numeral_real @ W)) = (((((~ ((C = zero_zero_real)))) => ((B = (times_times_real @ (numeral_numeral_real @ W) @ C))))) & ((((C = zero_zero_real)) => (((numeral_numeral_real @ W) = zero_zero_real))))))))). % divide_eq_eq_numeral(1)
thf(fact_198_numeral__1__eq__Suc__0, axiom,
    (((numeral_numeral_nat @ one) = (suc @ zero_zero_nat)))). % numeral_1_eq_Suc_0
thf(fact_199_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_200_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_201_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_202_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_203_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_204_zero__power2, axiom,
    (((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_real))). % zero_power2
thf(fact_205_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_206_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_207_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_208_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_209_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_210_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_211_numeral__2__eq__2, axiom,
    (((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat))))). % numeral_2_eq_2
thf(fact_212_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_213_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_214_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_215_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_216_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_217_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_218_div2__Suc__Suc, axiom,
    ((![M : nat]: ((divide_divide_nat @ (suc @ (suc @ M)) @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (divide_divide_nat @ M @ (numeral_numeral_nat @ (bit0 @ one)))))))). % div2_Suc_Suc
thf(fact_219_div__mult__mult1__if, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((C = zero_zero_nat) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = zero_zero_nat)) & ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A2 @ B))))))). % div_mult_mult1_if
thf(fact_220_div__mult__mult1__if, axiom,
    ((![C : int, A2 : int, B : int]: (((C = zero_zero_int) => ((divide_divide_int @ (times_times_int @ C @ A2) @ (times_times_int @ C @ B)) = zero_zero_int)) & ((~ ((C = zero_zero_int))) => ((divide_divide_int @ (times_times_int @ C @ A2) @ (times_times_int @ C @ B)) = (divide_divide_int @ A2 @ B))))))). % div_mult_mult1_if
thf(fact_221_div__mult__mult2, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)) = (divide_divide_nat @ A2 @ B)))))). % div_mult_mult2
thf(fact_222_div__mult__mult2, axiom,
    ((![C : int, A2 : int, B : int]: ((~ ((C = zero_zero_int))) => ((divide_divide_int @ (times_times_int @ A2 @ C) @ (times_times_int @ B @ C)) = (divide_divide_int @ A2 @ B)))))). % div_mult_mult2
thf(fact_223_div__mult__mult1, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A2 @ B)))))). % div_mult_mult1
thf(fact_224_div__mult__mult1, axiom,
    ((![C : int, A2 : int, B : int]: ((~ ((C = zero_zero_int))) => ((divide_divide_int @ (times_times_int @ C @ A2) @ (times_times_int @ C @ B)) = (divide_divide_int @ A2 @ B)))))). % div_mult_mult1
thf(fact_225_sum_Oneutral__const, axiom,
    ((![A3 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A3) = zero_zero_complex)))). % sum.neutral_const
thf(fact_226_nat__mult__div__cancel__disj, axiom,
    ((![K2 : nat, M : nat, N2 : nat]: (((K2 = zero_zero_nat) => ((divide_divide_nat @ (times_times_nat @ K2 @ M) @ (times_times_nat @ K2 @ N2)) = zero_zero_nat)) & ((~ ((K2 = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ K2 @ M) @ (times_times_nat @ K2 @ N2)) = (divide_divide_nat @ M @ N2))))))). % nat_mult_div_cancel_disj
thf(fact_227_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > real, K2 : nat]: (((F @ zero_zero_nat) = zero_zero_real) => ((groups2069495480t_real @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K2)) = (groups2069495480t_real @ F @ (set_or562006527an_nat @ zero_zero_nat @ K2))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_228_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > complex, K2 : nat]: (((F @ zero_zero_nat) = zero_zero_complex) => ((groups59700922omplex @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K2)) = (groups59700922omplex @ F @ (set_or562006527an_nat @ zero_zero_nat @ K2))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_229_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_230_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_231_div__by__Suc__0, axiom,
    ((![M : nat]: ((divide_divide_nat @ M @ (suc @ zero_zero_nat)) = M)))). % div_by_Suc_0
thf(fact_232_semiring__norm_I87_J, axiom,
    ((![M : num, N2 : num]: (((bit0 @ M) = (bit0 @ N2)) = (M = N2))))). % semiring_norm(87)
thf(fact_233_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_234_semiring__norm_I83_J, axiom,
    ((![N2 : num]: (~ ((one = (bit0 @ N2))))))). % semiring_norm(83)
thf(fact_235_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_236_semiring__norm_I12_J, axiom,
    ((![N2 : num]: ((times_times_num @ one @ N2) = N2)))). % semiring_norm(12)
thf(fact_237_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_238_div__mult2__eq, axiom,
    ((![M : nat, N2 : nat, Q : nat]: ((divide_divide_nat @ M @ (times_times_nat @ N2 @ Q)) = (divide_divide_nat @ (divide_divide_nat @ M @ N2) @ Q))))). % div_mult2_eq
thf(fact_239_nat__mult__eq__cancel__disj, axiom,
    ((![K2 : nat, M : nat, N2 : nat]: (((times_times_nat @ K2 @ M) = (times_times_nat @ K2 @ N2)) = (((K2 = zero_zero_nat)) | ((M = N2))))))). % nat_mult_eq_cancel_disj
thf(fact_240_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_241_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_242_real__divide__square__eq, axiom,
    ((![R : real, A2 : real]: ((divide_divide_real @ (times_times_real @ R @ A2) @ (times_times_real @ R @ R)) = (divide_divide_real @ A2 @ R))))). % real_divide_square_eq
thf(fact_243_zdiv__numeral__Bit0, axiom,
    ((![V : num, W : num]: ((divide_divide_int @ (numeral_numeral_int @ (bit0 @ V)) @ (numeral_numeral_int @ (bit0 @ W))) = (divide_divide_int @ (numeral_numeral_int @ V) @ (numeral_numeral_int @ W)))))). % zdiv_numeral_Bit0
thf(fact_244_pow_Osimps_I1_J, axiom,
    ((![X : num]: ((pow @ X @ one) = X)))). % pow.simps(1)

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (divide1210191872omplex @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))) @ (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)))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) = (groups59700922omplex @ (^[N : nat]: (divide1210191872omplex @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) @ (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 @ m) @ (times_times_nat @ i @ N))))) @ (set_or562006527an_nat @ zero_zero_nat @ m))))).
