% 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_442__3227790_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:21.011

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Set__Oset_It__Int__Oint_J, type,
    set_int : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (25)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint, type,
    minus_minus_int : int > int > int).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
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_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__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint, type,
    power_power_int : int > nat > int).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint, type,
    set_or1199280219an_int : int > int > set_int).
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_n, type,
    n : nat).

% Relevant facts (234)
thf(fact_0_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_1_diff__i, axiom,
    ((![K : nat]: ((ord_less_nat @ K @ n) => (ord_less_nat @ (minus_minus_nat @ K @ i) @ n))))). % diff_i
thf(fact_2_i__diff, axiom,
    ((![K : nat]: (ord_less_nat @ (minus_minus_nat @ i @ K) @ n)))). % i_diff
thf(fact_3_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_4_of__nat__0, axiom,
    (((semiri356525583omplex @ zero_zero_nat) = zero_zero_complex))). % of_nat_0
thf(fact_5_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_6_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_7_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_complex = (semiri356525583omplex @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_8_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_9_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_10_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri356525583omplex @ M) = zero_zero_complex) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_11_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_12_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_13_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_14_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_15_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (power_power_nat @ M @ N)) = (power_power_complex @ (semiri356525583omplex @ M) @ N))))). % of_nat_power
thf(fact_16_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (power_power_nat @ M @ N)) = (power_power_int @ (semiri2019852685at_int @ M) @ N))))). % of_nat_power
thf(fact_17_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_18_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_complex @ (semiri356525583omplex @ B) @ W) = (semiri356525583omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_19_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_int @ (semiri2019852685at_int @ B) @ W) = (semiri2019852685at_int @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_20_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri356525583omplex @ X) = (power_power_complex @ (semiri356525583omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_21_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2019852685at_int @ X) = (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_22_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_23_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (times_times_nat @ M @ N)) = (times_times_complex @ (semiri356525583omplex @ M) @ (semiri356525583omplex @ N)))))). % of_nat_mult
thf(fact_24_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (times_times_nat @ M @ N)) = (times_times_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % of_nat_mult
thf(fact_25_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (times_times_nat @ M @ N)) = (times_times_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % of_nat_mult
thf(fact_26_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_27_diff__self, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % diff_self
thf(fact_28_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_29_diff__0__right, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_0_right
thf(fact_30_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_31_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri356525583omplex @ M) = (semiri356525583omplex @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_32_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_33_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri1382578993at_nat @ M) = (semiri1382578993at_nat @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_34_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_35_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_36_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_37_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_38_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_39_diff__zero, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_zero
thf(fact_40_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_41_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_42_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_43_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_44_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_45_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_46_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_47_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_48_diff__gt__0__iff__gt, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ (minus_minus_int @ A @ B)) = (ord_less_int @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_49_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_50_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_51_zero__less__diff, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N @ M)) = (ord_less_nat @ M @ N))))). % zero_less_diff
thf(fact_52_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_53_mult__less__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_54_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_55_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_56_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_57_power__eq__0__iff, axiom,
    ((![A : int, N : nat]: (((power_power_int @ A @ N) = zero_zero_int) = (((A = zero_zero_int)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_58_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_59_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_60_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_61_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_62_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_int @ (power_power_int @ (semiri2019852685at_int @ B) @ W) @ (semiri2019852685at_int @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_63_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_64_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_int @ zero_zero_int @ (power_power_int @ (semiri2019852685at_int @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_65_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_66_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_67_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_68_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_69_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_70_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_71_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_72_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_73_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_74_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_75_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_76_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_77_power__mult, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A @ M) @ N))))). % power_mult
thf(fact_78_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_79_power__mult, axiom,
    ((![A : int, M : nat, N : nat]: ((power_power_int @ A @ (times_times_nat @ M @ N)) = (power_power_int @ (power_power_int @ A @ M) @ N))))). % power_mult
thf(fact_80_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_81_diff__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (minus_minus_nat @ M @ N)) = (minus_minus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % diff_mult_distrib2
thf(fact_82_diff__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (minus_minus_nat @ M @ N) @ K) = (minus_minus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % diff_mult_distrib
thf(fact_83_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_84_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_85_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_86_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_87_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_88_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_89_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_90_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_91_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_92_diff__strict__right__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ C)))))). % diff_strict_right_mono
thf(fact_93_diff__strict__left__mono, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => (ord_less_int @ (minus_minus_int @ C @ A) @ (minus_minus_int @ C @ B)))))). % diff_strict_left_mono
thf(fact_94_diff__eq__diff__less, axiom,
    ((![A : int, B : int, C : int, D : int]: (((minus_minus_int @ A @ B) = (minus_minus_int @ C @ D)) => ((ord_less_int @ A @ B) = (ord_less_int @ C @ D)))))). % diff_eq_diff_less
thf(fact_95_diff__strict__mono, axiom,
    ((![A : int, B : int, D : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ D @ C) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ D))))))). % diff_strict_mono
thf(fact_96_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_97_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_98_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_99_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_100_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_101_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_102_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_103_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N) @ K))))). % less_imp_diff_less
thf(fact_104_diff__less__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_nat @ M @ N) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_105_root__cancel, axiom,
    ((![D : nat, N : nat, K : nat]: ((ord_less_nat @ zero_zero_nat @ D) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D @ N)) @ (times_times_nat @ D @ K)) = (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)))))). % root_cancel
thf(fact_106_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_107_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_108_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_109_mult_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((times_times_complex @ B @ (times_times_complex @ A @ C)) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.left_commute
thf(fact_110_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_111_mult_Oleft__commute, axiom,
    ((![B : int, A : int, C : int]: ((times_times_int @ B @ (times_times_int @ A @ C)) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % mult.left_commute
thf(fact_112_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_113_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_114_mult_Ocommute, axiom,
    ((times_times_int = (^[A2 : int]: (^[B2 : int]: (times_times_int @ B2 @ A2)))))). % mult.commute
thf(fact_115_mult_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.assoc
thf(fact_116_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_117_mult_Oassoc, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (times_times_int @ A @ B) @ C) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % mult.assoc
thf(fact_118_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_119_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_120_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (times_times_int @ A @ B) @ C) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_121_diff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % diff_right_commute
thf(fact_122_less__iff__diff__less__0, axiom,
    ((ord_less_int = (^[A2 : int]: (^[B2 : int]: (ord_less_int @ (minus_minus_int @ A2 @ B2) @ zero_zero_int)))))). % less_iff_diff_less_0
thf(fact_123_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_124_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_125_zero__less__power, axiom,
    ((![A : int, N : nat]: ((ord_less_int @ zero_zero_int @ A) => (ord_less_int @ zero_zero_int @ (power_power_int @ A @ N)))))). % zero_less_power
thf(fact_126_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_127_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_128_diff__less, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N) @ M)))))). % diff_less
thf(fact_129_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_130_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_131_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_int @ zero_zero_int @ N) = zero_zero_int))))). % zero_power
thf(fact_132_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z : complex]: (Y2 = Z))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_133_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : int]: (^[Z : int]: (Y2 = Z))) = (^[A2 : int]: (^[B2 : int]: ((minus_minus_int @ A2 @ B2) = zero_zero_int)))))). % eq_iff_diff_eq_0
thf(fact_134_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_135_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_136_power__not__zero, axiom,
    ((![A : int, N : nat]: ((~ ((A = zero_zero_int))) => (~ (((power_power_int @ A @ N) = zero_zero_int))))))). % power_not_zero
thf(fact_137_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N))))))). % power_commuting_commutes
thf(fact_138_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_139_power__commuting__commutes, axiom,
    ((![X : int, Y : int, N : nat]: (((times_times_int @ X @ Y) = (times_times_int @ Y @ X)) => ((times_times_int @ (power_power_int @ X @ N) @ Y) = (times_times_int @ Y @ (power_power_int @ X @ N))))))). % power_commuting_commutes
thf(fact_140_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_141_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_142_power__mult__distrib, axiom,
    ((![A : int, B : int, N : nat]: ((power_power_int @ (times_times_int @ A @ B) @ N) = (times_times_int @ (power_power_int @ A @ N) @ (power_power_int @ B @ N)))))). % power_mult_distrib
thf(fact_143_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_144_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_145_power__commutes, axiom,
    ((![A : int, N : nat]: ((times_times_int @ (power_power_int @ A @ N) @ A) = (times_times_int @ A @ (power_power_int @ A @ N)))))). % power_commutes
thf(fact_146_mult__of__nat__commute, axiom,
    ((![X : nat, Y : complex]: ((times_times_complex @ (semiri356525583omplex @ X) @ Y) = (times_times_complex @ Y @ (semiri356525583omplex @ X)))))). % mult_of_nat_commute
thf(fact_147_mult__of__nat__commute, axiom,
    ((![X : nat, Y : int]: ((times_times_int @ (semiri2019852685at_int @ X) @ Y) = (times_times_int @ Y @ (semiri2019852685at_int @ X)))))). % mult_of_nat_commute
thf(fact_148_mult__of__nat__commute, axiom,
    ((![X : nat, Y : nat]: ((times_times_nat @ (semiri1382578993at_nat @ X) @ Y) = (times_times_nat @ Y @ (semiri1382578993at_nat @ X)))))). % mult_of_nat_commute
thf(fact_149_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_150_diffs0__imp__equal, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) => (((minus_minus_nat @ N @ M) = zero_zero_nat) => (M = N)))))). % diffs0_imp_equal
thf(fact_151_nat__mult__less__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % nat_mult_less_cancel_disj
thf(fact_152_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_153_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_154_mult__cancel__right, axiom,
    ((![A : int, C : int, B : int]: (((times_times_int @ A @ C) = (times_times_int @ B @ C)) = (((C = zero_zero_int)) | ((A = B))))))). % mult_cancel_right
thf(fact_155_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_156_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_157_mult__cancel__left, axiom,
    ((![C : int, A : int, B : int]: (((times_times_int @ C @ A) = (times_times_int @ C @ B)) = (((C = zero_zero_int)) | ((A = B))))))). % mult_cancel_left
thf(fact_158_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_159_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_160_mult__eq__0__iff, axiom,
    ((![A : int, B : int]: (((times_times_int @ A @ B) = zero_zero_int) = (((A = zero_zero_int)) | ((B = zero_zero_int))))))). % mult_eq_0_iff
thf(fact_161_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_162_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_163_mult__zero__right, axiom,
    ((![A : int]: ((times_times_int @ A @ zero_zero_int) = zero_zero_int)))). % mult_zero_right
thf(fact_164_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_165_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_166_mult__zero__left, axiom,
    ((![A : int]: ((times_times_int @ zero_zero_int @ A) = zero_zero_int)))). % mult_zero_left
thf(fact_167_all__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((![M3 : nat]: (((ord_less_nat @ M3 @ N)) => ((P @ M3)))) = (![X2 : nat]: (((member_nat @ X2 @ (set_or562006527an_nat @ zero_zero_nat @ N))) => ((P @ X2)))))))). % all_nat_less_eq
thf(fact_168_ex__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((?[M3 : nat]: (((ord_less_nat @ M3 @ N)) & ((P @ M3)))) = (?[X2 : nat]: (((member_nat @ X2 @ (set_or562006527an_nat @ zero_zero_nat @ N))) & ((P @ X2)))))))). % ex_nat_less_eq
thf(fact_169_nat__mult__less__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (ord_less_nat @ M @ N)))))). % nat_mult_less_cancel1
thf(fact_170_linorder__neqE__linordered__idom, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) => ((~ ((ord_less_int @ X @ Y))) => (ord_less_int @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_171_mult__not__zero, axiom,
    ((![A : complex, B : complex]: ((~ (((times_times_complex @ A @ B) = zero_zero_complex))) => ((~ ((A = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_172_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_173_mult__not__zero, axiom,
    ((![A : int, B : int]: ((~ (((times_times_int @ A @ B) = zero_zero_int))) => ((~ ((A = zero_zero_int))) & (~ ((B = zero_zero_int)))))))). % mult_not_zero
thf(fact_174_divisors__zero, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) => ((A = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_175_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_176_divisors__zero, axiom,
    ((![A : int, B : int]: (((times_times_int @ A @ B) = zero_zero_int) => ((A = zero_zero_int) | (B = zero_zero_int)))))). % divisors_zero
thf(fact_177_no__zero__divisors, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_178_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_179_no__zero__divisors, axiom,
    ((![A : int, B : int]: ((~ ((A = zero_zero_int))) => ((~ ((B = zero_zero_int))) => (~ (((times_times_int @ A @ B) = zero_zero_int)))))))). % no_zero_divisors
thf(fact_180_mult__left__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_181_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_182_mult__left__cancel, axiom,
    ((![C : int, A : int, B : int]: ((~ ((C = zero_zero_int))) => (((times_times_int @ C @ A) = (times_times_int @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_183_mult__right__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_184_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_185_mult__right__cancel, axiom,
    ((![C : int, A : int, B : int]: ((~ ((C = zero_zero_int))) => (((times_times_int @ A @ C) = (times_times_int @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_186_left__diff__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % left_diff_distrib
thf(fact_187_left__diff__distrib, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (minus_minus_int @ A @ B) @ C) = (minus_minus_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)))))). % left_diff_distrib
thf(fact_188_right__diff__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % right_diff_distrib
thf(fact_189_right__diff__distrib, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ A @ (minus_minus_int @ B @ C)) = (minus_minus_int @ (times_times_int @ A @ B) @ (times_times_int @ A @ C)))))). % right_diff_distrib
thf(fact_190_left__diff__distrib_H, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (minus_minus_complex @ B @ C) @ A) = (minus_minus_complex @ (times_times_complex @ B @ A) @ (times_times_complex @ C @ A)))))). % left_diff_distrib'
thf(fact_191_left__diff__distrib_H, axiom,
    ((![B : nat, C : nat, A : nat]: ((times_times_nat @ (minus_minus_nat @ B @ C) @ A) = (minus_minus_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)))))). % left_diff_distrib'
thf(fact_192_left__diff__distrib_H, axiom,
    ((![B : int, C : int, A : int]: ((times_times_int @ (minus_minus_int @ B @ C) @ A) = (minus_minus_int @ (times_times_int @ B @ A) @ (times_times_int @ C @ A)))))). % left_diff_distrib'
thf(fact_193_right__diff__distrib_H, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % right_diff_distrib'
thf(fact_194_right__diff__distrib_H, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (minus_minus_nat @ B @ C)) = (minus_minus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % right_diff_distrib'
thf(fact_195_right__diff__distrib_H, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ A @ (minus_minus_int @ B @ C)) = (minus_minus_int @ (times_times_int @ A @ B) @ (times_times_int @ A @ C)))))). % right_diff_distrib'
thf(fact_196_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_197_mult__neg__neg, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ zero_zero_int) => ((ord_less_int @ B @ zero_zero_int) => (ord_less_int @ zero_zero_int @ (times_times_int @ A @ B))))))). % mult_neg_neg
thf(fact_198_not__square__less__zero, axiom,
    ((![A : int]: (~ ((ord_less_int @ (times_times_int @ A @ A) @ zero_zero_int)))))). % not_square_less_zero
thf(fact_199_mult__less__0__iff, axiom,
    ((![A : int, B : int]: ((ord_less_int @ (times_times_int @ A @ B) @ zero_zero_int) = (((((ord_less_int @ zero_zero_int @ A)) & ((ord_less_int @ B @ zero_zero_int)))) | ((((ord_less_int @ A @ zero_zero_int)) & ((ord_less_int @ zero_zero_int @ B))))))))). % mult_less_0_iff
thf(fact_200_mult__neg__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_neg_pos
thf(fact_201_mult__neg__pos, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ zero_zero_int) => ((ord_less_int @ zero_zero_int @ B) => (ord_less_int @ (times_times_int @ A @ B) @ zero_zero_int)))))). % mult_neg_pos
thf(fact_202_mult__pos__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_pos_neg
thf(fact_203_mult__pos__neg, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ A) => ((ord_less_int @ B @ zero_zero_int) => (ord_less_int @ (times_times_int @ A @ B) @ zero_zero_int)))))). % mult_pos_neg
thf(fact_204_mult__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_pos_pos
thf(fact_205_mult__pos__pos, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ A) => ((ord_less_int @ zero_zero_int @ B) => (ord_less_int @ zero_zero_int @ (times_times_int @ A @ B))))))). % mult_pos_pos
thf(fact_206_mult__pos__neg2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_pos_neg2
thf(fact_207_mult__pos__neg2, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ A) => ((ord_less_int @ B @ zero_zero_int) => (ord_less_int @ (times_times_int @ B @ A) @ zero_zero_int)))))). % mult_pos_neg2
thf(fact_208_zero__less__mult__iff, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ (times_times_int @ A @ B)) = (((((ord_less_int @ zero_zero_int @ A)) & ((ord_less_int @ zero_zero_int @ B)))) | ((((ord_less_int @ A @ zero_zero_int)) & ((ord_less_int @ B @ zero_zero_int))))))))). % zero_less_mult_iff
thf(fact_209_zero__less__mult__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos
thf(fact_210_zero__less__mult__pos, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ (times_times_int @ A @ B)) => ((ord_less_int @ zero_zero_int @ A) => (ord_less_int @ zero_zero_int @ B)))))). % zero_less_mult_pos
thf(fact_211_zero__less__mult__pos2, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ B @ A)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos2
thf(fact_212_zero__less__mult__pos2, axiom,
    ((![B : int, A : int]: ((ord_less_int @ zero_zero_int @ (times_times_int @ B @ A)) => ((ord_less_int @ zero_zero_int @ A) => (ord_less_int @ zero_zero_int @ B)))))). % zero_less_mult_pos2
thf(fact_213_mult__less__cancel__left__neg, axiom,
    ((![C : int, A : int, B : int]: ((ord_less_int @ C @ zero_zero_int) => ((ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B)) = (ord_less_int @ B @ A)))))). % mult_less_cancel_left_neg
thf(fact_214_mult__less__cancel__left__pos, axiom,
    ((![C : int, A : int, B : int]: ((ord_less_int @ zero_zero_int @ C) => ((ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B)) = (ord_less_int @ A @ B)))))). % mult_less_cancel_left_pos
thf(fact_215_mult__strict__left__mono__neg, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => ((ord_less_int @ C @ zero_zero_int) => (ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B))))))). % mult_strict_left_mono_neg
thf(fact_216_mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_strict_left_mono
thf(fact_217_mult__strict__left__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ zero_zero_int @ C) => (ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B))))))). % mult_strict_left_mono
thf(fact_218_mult__less__cancel__left__disj, axiom,
    ((![C : int, A : int, B : int]: ((ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B)) = (((((ord_less_int @ zero_zero_int @ C)) & ((ord_less_int @ A @ B)))) | ((((ord_less_int @ C @ zero_zero_int)) & ((ord_less_int @ B @ A))))))))). % mult_less_cancel_left_disj
thf(fact_219_mult__strict__right__mono__neg, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => ((ord_less_int @ C @ zero_zero_int) => (ord_less_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C))))))). % mult_strict_right_mono_neg
thf(fact_220_mult__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_strict_right_mono
thf(fact_221_mult__strict__right__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ zero_zero_int @ C) => (ord_less_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C))))))). % mult_strict_right_mono
thf(fact_222_mult__less__cancel__right__disj, axiom,
    ((![A : int, C : int, B : int]: ((ord_less_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)) = (((((ord_less_int @ zero_zero_int @ C)) & ((ord_less_int @ A @ B)))) | ((((ord_less_int @ C @ zero_zero_int)) & ((ord_less_int @ B @ A))))))))). % mult_less_cancel_right_disj
thf(fact_223_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_224_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ zero_zero_int @ C) => (ord_less_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_225_nat__mult__eq__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (M = N)))))). % nat_mult_eq_cancel1
thf(fact_226_mult__less__iff1, axiom,
    ((![Z2 : int, X : int, Y : int]: ((ord_less_int @ zero_zero_int @ Z2) => ((ord_less_int @ (times_times_int @ X @ Z2) @ (times_times_int @ Y @ Z2)) = (ord_less_int @ X @ Y)))))). % mult_less_iff1
thf(fact_227_atLeastLessThan__eq__iff, axiom,
    ((![A : int, B : int, C : int, D : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ C @ D) => (((set_or1199280219an_int @ A @ B) = (set_or1199280219an_int @ C @ D)) = (((A = C)) & ((B = D))))))))). % atLeastLessThan_eq_iff
thf(fact_228_atLeastLessThan__eq__iff, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) = (((A = C)) & ((B = D))))))))). % atLeastLessThan_eq_iff
thf(fact_229_atLeastLessThan__inj_I1_J, axiom,
    ((![A : int, B : int, C : int, D : int]: (((set_or1199280219an_int @ A @ B) = (set_or1199280219an_int @ C @ D)) => ((ord_less_int @ A @ B) => ((ord_less_int @ C @ D) => (A = C))))))). % atLeastLessThan_inj(1)
thf(fact_230_atLeastLessThan__inj_I1_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (A = C))))))). % atLeastLessThan_inj(1)
thf(fact_231_atLeastLessThan__inj_I2_J, axiom,
    ((![A : int, B : int, C : int, D : int]: (((set_or1199280219an_int @ A @ B) = (set_or1199280219an_int @ C @ D)) => ((ord_less_int @ A @ B) => ((ord_less_int @ C @ D) => (B = D))))))). % atLeastLessThan_inj(2)
thf(fact_232_atLeastLessThan__inj_I2_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (B = D))))))). % atLeastLessThan_inj(2)
thf(fact_233_zmult__zless__mono2__lemma, axiom,
    ((![I : int, J : int, K : nat]: ((ord_less_int @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_int @ (times_times_int @ (semiri2019852685at_int @ K) @ I) @ (times_times_int @ (semiri2019852685at_int @ K) @ J))))))). % zmult_zless_mono2_lemma

% Conjectures (1)
thf(conj_0, conjecture,
    (((times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ i @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ i)) = (times_times_complex @ (semiri356525583omplex @ n) @ (a @ i))))).
