% 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_149__3224268_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:30.111

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J, type,
    set_complex : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (23)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    groups443808152omplex : (complex > complex) > set_complex > complex).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat, type,
    groups1415553210ex_nat : (complex > nat) > set_complex > 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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_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__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex, type,
    collect_complex : (complex > $o) > set_complex).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Complex__Ocomplex, type,
    member_complex : complex > set_complex > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_k, type,
    k : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (181)
thf(fact_0_k_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ k))). % k(1)
thf(fact_1_k_I2_J, axiom,
    ((ord_less_nat @ k @ n))). % k(2)
thf(fact_2_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_3_sum_Oneutral__const, axiom,
    ((![A : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A) = zero_zero_complex)))). % sum.neutral_const
thf(fact_4_sum_Oneutral__const, axiom,
    ((![A : set_complex]: ((groups443808152omplex @ (^[Uu : complex]: zero_zero_complex) @ A) = zero_zero_complex)))). % sum.neutral_const
thf(fact_5_sum_Oneutral, axiom,
    ((![A : set_nat, G : nat > complex]: ((![X : nat]: ((member_nat @ X @ A) => ((G @ X) = zero_zero_complex))) => ((groups59700922omplex @ G @ A) = zero_zero_complex))))). % sum.neutral
thf(fact_6_sum_Oneutral, axiom,
    ((![A : set_complex, G : complex > complex]: ((![X : complex]: ((member_complex @ X @ A) => ((G @ X) = zero_zero_complex))) => ((groups443808152omplex @ G @ A) = zero_zero_complex))))). % sum.neutral
thf(fact_7_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : complex > nat, A : set_complex]: ((~ (((groups1415553210ex_nat @ G @ A) = zero_zero_nat))) => (~ ((![A2 : complex]: ((member_complex @ A2 @ A) => ((G @ A2) = zero_zero_nat))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_8_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > nat, A : set_nat]: ((~ (((groups1842438620at_nat @ G @ A) = zero_zero_nat))) => (~ ((![A2 : nat]: ((member_nat @ A2 @ A) => ((G @ A2) = zero_zero_nat))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_9_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > complex, A : set_nat]: ((~ (((groups59700922omplex @ G @ A) = zero_zero_complex))) => (~ ((![A2 : nat]: ((member_nat @ A2 @ A) => ((G @ A2) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_10_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : complex > complex, A : set_complex]: ((~ (((groups443808152omplex @ G @ A) = zero_zero_complex))) => (~ ((![A2 : complex]: ((member_complex @ A2 @ A) => ((G @ A2) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_11_power__not__zero, axiom,
    ((![A3 : complex, N : nat]: ((~ ((A3 = zero_zero_complex))) => (~ (((power_power_complex @ A3 @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_12_power__not__zero, axiom,
    ((![A3 : nat, N : nat]: ((~ ((A3 = zero_zero_nat))) => (~ (((power_power_nat @ A3 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_13_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B : set_nat, A : set_nat]: ((groups59700922omplex @ (^[I : nat]: (groups59700922omplex @ (G @ I) @ B)) @ A) = (groups59700922omplex @ (^[J : nat]: (groups59700922omplex @ (^[I : nat]: (G @ I @ J)) @ A)) @ B))))). % sum.swap
thf(fact_14_sum_Oswap, axiom,
    ((![G : nat > complex > complex, B : set_complex, A : set_nat]: ((groups59700922omplex @ (^[I : nat]: (groups443808152omplex @ (G @ I) @ B)) @ A) = (groups443808152omplex @ (^[J : complex]: (groups59700922omplex @ (^[I : nat]: (G @ I @ J)) @ A)) @ B))))). % sum.swap
thf(fact_15_sum_Oswap, axiom,
    ((![G : complex > nat > complex, B : set_nat, A : set_complex]: ((groups443808152omplex @ (^[I : complex]: (groups59700922omplex @ (G @ I) @ B)) @ A) = (groups59700922omplex @ (^[J : nat]: (groups443808152omplex @ (^[I : complex]: (G @ I @ J)) @ A)) @ B))))). % sum.swap
thf(fact_16_sum_Oswap, axiom,
    ((![G : complex > complex > complex, B : set_complex, A : set_complex]: ((groups443808152omplex @ (^[I : complex]: (groups443808152omplex @ (G @ I) @ B)) @ A) = (groups443808152omplex @ (^[J : complex]: (groups443808152omplex @ (^[I : complex]: (G @ I @ J)) @ A)) @ B))))). % sum.swap
thf(fact_17_sum_Ocong, axiom,
    ((![A : set_nat, B : set_nat, G : nat > complex, H : nat > complex]: ((A = B) => ((![X : nat]: ((member_nat @ X @ B) => ((G @ X) = (H @ X)))) => ((groups59700922omplex @ G @ A) = (groups59700922omplex @ H @ B))))))). % sum.cong
thf(fact_18_sum_Ocong, axiom,
    ((![A : set_complex, B : set_complex, G : complex > complex, H : complex > complex]: ((A = B) => ((![X : complex]: ((member_complex @ X @ B) => ((G @ X) = (H @ X)))) => ((groups443808152omplex @ G @ A) = (groups443808152omplex @ H @ B))))))). % sum.cong
thf(fact_19_sum_Oeq__general, axiom,
    ((![B : set_nat, A : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y : nat]: ((member_nat @ Y @ B) => (?[X2 : nat]: (((member_nat @ X2 @ A) & ((H @ X2) = Y)) & (![Ya : nat]: (((member_nat @ Ya @ A) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_nat @ (H @ X) @ B) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups59700922omplex @ Phi @ A) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general
thf(fact_20_sum_Oeq__general, axiom,
    ((![B : set_complex, A : set_nat, H : nat > complex, Gamma : complex > complex, Phi : nat > complex]: ((![Y : complex]: ((member_complex @ Y @ B) => (?[X2 : nat]: (((member_nat @ X2 @ A) & ((H @ X2) = Y)) & (![Ya : nat]: (((member_nat @ Ya @ A) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_complex @ (H @ X) @ B) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups59700922omplex @ Phi @ A) = (groups443808152omplex @ Gamma @ B))))))). % sum.eq_general
thf(fact_21_sum_Oeq__general, axiom,
    ((![B : set_nat, A : set_complex, H : complex > nat, Gamma : nat > complex, Phi : complex > complex]: ((![Y : nat]: ((member_nat @ Y @ B) => (?[X2 : complex]: (((member_complex @ X2 @ A) & ((H @ X2) = Y)) & (![Ya : complex]: (((member_complex @ Ya @ A) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : complex]: ((member_complex @ X @ A) => ((member_nat @ (H @ X) @ B) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups443808152omplex @ Phi @ A) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general
thf(fact_22_sum_Oeq__general, axiom,
    ((![B : set_complex, A : set_complex, H : complex > complex, Gamma : complex > complex, Phi : complex > complex]: ((![Y : complex]: ((member_complex @ Y @ B) => (?[X2 : complex]: (((member_complex @ X2 @ A) & ((H @ X2) = Y)) & (![Ya : complex]: (((member_complex @ Ya @ A) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : complex]: ((member_complex @ X @ A) => ((member_complex @ (H @ X) @ B) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups443808152omplex @ Phi @ A) = (groups443808152omplex @ Gamma @ B))))))). % sum.eq_general
thf(fact_23_sum_Oeq__general__inverses, axiom,
    ((![B : set_nat, K : nat > nat, A : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y : nat]: ((member_nat @ Y @ B) => ((member_nat @ (K @ Y) @ A) & ((H @ (K @ Y)) = Y)))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_nat @ (H @ X) @ B) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups59700922omplex @ Phi @ A) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_24_sum_Oeq__general__inverses, axiom,
    ((![B : set_complex, K : complex > nat, A : set_nat, H : nat > complex, Gamma : complex > complex, Phi : nat > complex]: ((![Y : complex]: ((member_complex @ Y @ B) => ((member_nat @ (K @ Y) @ A) & ((H @ (K @ Y)) = Y)))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_complex @ (H @ X) @ B) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups59700922omplex @ Phi @ A) = (groups443808152omplex @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_25_sum_Oeq__general__inverses, axiom,
    ((![B : set_nat, K : nat > complex, A : set_complex, H : complex > nat, Gamma : nat > complex, Phi : complex > complex]: ((![Y : nat]: ((member_nat @ Y @ B) => ((member_complex @ (K @ Y) @ A) & ((H @ (K @ Y)) = Y)))) => ((![X : complex]: ((member_complex @ X @ A) => ((member_nat @ (H @ X) @ B) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups443808152omplex @ Phi @ A) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_26_sum_Oeq__general__inverses, axiom,
    ((![B : set_complex, K : complex > complex, A : set_complex, H : complex > complex, Gamma : complex > complex, Phi : complex > complex]: ((![Y : complex]: ((member_complex @ Y @ B) => ((member_complex @ (K @ Y) @ A) & ((H @ (K @ Y)) = Y)))) => ((![X : complex]: ((member_complex @ X @ A) => ((member_complex @ (H @ X) @ B) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups443808152omplex @ Phi @ A) = (groups443808152omplex @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_27_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I2 : nat > nat, J2 : nat > nat, T : set_nat, H : nat > complex, G : nat > complex]: ((![A2 : nat]: ((member_nat @ A2 @ S) => ((I2 @ (J2 @ A2)) = A2))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => (member_nat @ (J2 @ A2) @ T))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => ((J2 @ (I2 @ B2)) = B2))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => (member_nat @ (I2 @ B2) @ S))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => ((H @ (J2 @ A2)) = (G @ A2)))) => ((groups59700922omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_28_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I2 : complex > nat, J2 : nat > complex, T : set_complex, H : complex > complex, G : nat > complex]: ((![A2 : nat]: ((member_nat @ A2 @ S) => ((I2 @ (J2 @ A2)) = A2))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => (member_complex @ (J2 @ A2) @ T))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => ((J2 @ (I2 @ B2)) = B2))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => (member_nat @ (I2 @ B2) @ S))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => ((H @ (J2 @ A2)) = (G @ A2)))) => ((groups59700922omplex @ G @ S) = (groups443808152omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_29_sum_Oreindex__bij__witness, axiom,
    ((![S : set_complex, I2 : nat > complex, J2 : complex > nat, T : set_nat, H : nat > complex, G : complex > complex]: ((![A2 : complex]: ((member_complex @ A2 @ S) => ((I2 @ (J2 @ A2)) = A2))) => ((![A2 : complex]: ((member_complex @ A2 @ S) => (member_nat @ (J2 @ A2) @ T))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => ((J2 @ (I2 @ B2)) = B2))) => ((![B2 : nat]: ((member_nat @ B2 @ T) => (member_complex @ (I2 @ B2) @ S))) => ((![A2 : complex]: ((member_complex @ A2 @ S) => ((H @ (J2 @ A2)) = (G @ A2)))) => ((groups443808152omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_30_sum_Oreindex__bij__witness, axiom,
    ((![S : set_complex, I2 : complex > complex, J2 : complex > complex, T : set_complex, H : complex > complex, G : complex > complex]: ((![A2 : complex]: ((member_complex @ A2 @ S) => ((I2 @ (J2 @ A2)) = A2))) => ((![A2 : complex]: ((member_complex @ A2 @ S) => (member_complex @ (J2 @ A2) @ T))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => ((J2 @ (I2 @ B2)) = B2))) => ((![B2 : complex]: ((member_complex @ B2 @ T) => (member_complex @ (I2 @ B2) @ S))) => ((![A2 : complex]: ((member_complex @ A2 @ S) => ((H @ (J2 @ A2)) = (G @ A2)))) => ((groups443808152omplex @ G @ S) = (groups443808152omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_31_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_32_nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X3 @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_33_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A3 : nat]: ((~ ((A3 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A3))))). % bot_nat_0.not_eq_extremum
thf(fact_34_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_35_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_36_power__eq__0__iff, axiom,
    ((![A3 : complex, N : nat]: (((power_power_complex @ A3 @ N) = zero_zero_complex) = (((A3 = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_37_power__eq__0__iff, axiom,
    ((![A3 : nat, N : nat]: (((power_power_nat @ A3 @ N) = zero_zero_nat) = (((A3 = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_38_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_39_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_40_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_41_less__not__refl3, axiom,
    ((![S2 : nat, T2 : nat]: ((ord_less_nat @ S2 @ T2) => (~ ((S2 = T2))))))). % less_not_refl3
thf(fact_42_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_43_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_44_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_45_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) => ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_nat @ Y2 @ X3)))))). % linorder_neqE_nat
thf(fact_46_nat__power__less__imp__less, axiom,
    ((![I2 : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I2) => ((ord_less_nat @ (power_power_nat @ I2 @ M) @ (power_power_nat @ I2 @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_47_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_48_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_49_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_50_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_51_bot__nat__0_Oextremum__strict, axiom,
    ((![A3 : nat]: (~ ((ord_less_nat @ A3 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_52_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_53_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_54_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_55_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_56_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_57_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_58_mem__Collect__eq, axiom,
    ((![A3 : nat, P : nat > $o]: ((member_nat @ A3 @ (collect_nat @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_59_mem__Collect__eq, axiom,
    ((![A3 : complex, P : complex > $o]: ((member_complex @ A3 @ (collect_complex @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_60_Collect__mem__eq, axiom,
    ((![A : set_nat]: ((collect_nat @ (^[X4 : nat]: (member_nat @ X4 @ A))) = A)))). % Collect_mem_eq
thf(fact_61_Collect__mem__eq, axiom,
    ((![A : set_complex]: ((collect_complex @ (^[X4 : complex]: (member_complex @ X4 @ A))) = A)))). % Collect_mem_eq
thf(fact_62_Collect__cong, axiom,
    ((![P : complex > $o, Q : complex > $o]: ((![X : complex]: ((P @ X) = (Q @ X))) => ((collect_complex @ P) = (collect_complex @ Q)))))). % Collect_cong
thf(fact_63_zero__less__power, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A3) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A3 @ N)))))). % zero_less_power
thf(fact_64_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_65_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_66_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_67_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_68_all__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((![M3 : nat]: (((ord_less_nat @ M3 @ N)) => ((P @ M3)))) = (![X4 : nat]: (((member_nat @ X4 @ (set_or562006527an_nat @ zero_zero_nat @ N))) => ((P @ X4)))))))). % all_nat_less_eq
thf(fact_69_ex__nat__less__eq, axiom,
    ((![N : nat, P : nat > $o]: ((?[M3 : nat]: (((ord_less_nat @ M3 @ N)) & ((P @ M3)))) = (?[X4 : nat]: (((member_nat @ X4 @ (set_or562006527an_nat @ zero_zero_nat @ N))) & ((P @ X4)))))))). % ex_nat_less_eq
thf(fact_70_atLeastLessThan__eq__iff, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: ((ord_less_nat @ A3 @ B3) => ((ord_less_nat @ C @ D) => (((set_or562006527an_nat @ A3 @ B3) = (set_or562006527an_nat @ C @ D)) = (((A3 = C)) & ((B3 = D))))))))). % atLeastLessThan_eq_iff
thf(fact_71_atLeastLessThan__inj_I1_J, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A3 @ B3) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A3 @ B3) => ((ord_less_nat @ C @ D) => (A3 = C))))))). % atLeastLessThan_inj(1)
thf(fact_72_atLeastLessThan__inj_I2_J, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A3 @ B3) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A3 @ B3) => ((ord_less_nat @ C @ D) => (B3 = D))))))). % atLeastLessThan_inj(2)
thf(fact_73_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_74_of__nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X3) @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_75_power__strict__decreasing__iff, axiom,
    ((![B3 : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B3) => ((ord_less_nat @ B3 @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B3 @ M) @ (power_power_nat @ B3 @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_76_power__strict__mono, axiom,
    ((![A3 : nat, B3 : nat, N : nat]: ((ord_less_nat @ A3 @ B3) => ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ B3 @ N)))))))). % power_strict_mono
thf(fact_77_power__one__right, axiom,
    ((![A3 : complex]: ((power_power_complex @ A3 @ one_one_nat) = A3)))). % power_one_right
thf(fact_78_power__one__right, axiom,
    ((![A3 : nat]: ((power_power_nat @ A3 @ one_one_nat) = A3)))). % power_one_right
thf(fact_79_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_80_bot__nat__0_Oextremum, axiom,
    ((![A3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ A3)))). % bot_nat_0.extremum
thf(fact_81_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_82_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_83_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_84_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_85_of__nat__1, axiom,
    (((semiri1382578993at_nat @ one_one_nat) = one_one_nat))). % of_nat_1
thf(fact_86_of__nat__1, axiom,
    (((semiri356525583omplex @ one_one_nat) = one_one_complex))). % of_nat_1
thf(fact_87_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_nat = (semiri1382578993at_nat @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_88_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_complex = (semiri356525583omplex @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_89_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri1382578993at_nat @ N) = one_one_nat) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_90_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri356525583omplex @ N) = one_one_complex) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_91_ivl__subset, axiom,
    ((![I2 : nat, J2 : nat, M : nat, N : nat]: ((ord_less_eq_set_nat @ (set_or562006527an_nat @ I2 @ J2) @ (set_or562006527an_nat @ M @ N)) = (((ord_less_eq_nat @ J2 @ I2)) | ((((ord_less_eq_nat @ M @ I2)) & ((ord_less_eq_nat @ J2 @ N))))))))). % ivl_subset
thf(fact_92_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_93_power__inject__exp, axiom,
    ((![A3 : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A3) => (((power_power_nat @ A3 @ M) = (power_power_nat @ A3 @ N)) = (M = N)))))). % power_inject_exp
thf(fact_94_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_95_of__nat__0, axiom,
    (((semiri356525583omplex @ zero_zero_nat) = zero_zero_complex))). % of_nat_0
thf(fact_96_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_97_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_complex = (semiri356525583omplex @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_98_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_99_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri356525583omplex @ M) = zero_zero_complex) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_100_atLeastLessThan__iff, axiom,
    ((![I2 : nat, L : nat, U : nat]: ((member_nat @ I2 @ (set_or562006527an_nat @ L @ U)) = (((ord_less_eq_nat @ L @ I2)) & ((ord_less_nat @ I2 @ U))))))). % atLeastLessThan_iff
thf(fact_101_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_102_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_103_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_104_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B3 : nat, W : nat, X3 : nat]: (((power_power_complex @ (semiri356525583omplex @ B3) @ W) = (semiri356525583omplex @ X3)) = ((power_power_nat @ B3 @ W) = X3))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_105_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B3 : nat, W : nat, X3 : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B3) @ W) = (semiri1382578993at_nat @ X3)) = ((power_power_nat @ B3 @ W) = X3))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_106_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X3 : nat, B3 : nat, W : nat]: (((semiri356525583omplex @ X3) = (power_power_complex @ (semiri356525583omplex @ B3) @ W)) = (X3 = (power_power_nat @ B3 @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_107_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X3 : nat, B3 : nat, W : nat]: (((semiri1382578993at_nat @ X3) = (power_power_nat @ (semiri1382578993at_nat @ B3) @ W)) = (X3 = (power_power_nat @ B3 @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_108_power__increasing__iff, axiom,
    ((![B3 : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B3) => ((ord_less_eq_nat @ (power_power_nat @ B3 @ X3) @ (power_power_nat @ B3 @ Y2)) = (ord_less_eq_nat @ X3 @ Y2)))))). % power_increasing_iff
thf(fact_109_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_110_power__strict__increasing__iff, axiom,
    ((![B3 : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B3) => ((ord_less_nat @ (power_power_nat @ B3 @ X3) @ (power_power_nat @ B3 @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_111_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B3 : nat, W : nat, X3 : nat]: ((ord_less_eq_nat @ (power_power_nat @ (semiri1382578993at_nat @ B3) @ W) @ (semiri1382578993at_nat @ X3)) = (ord_less_eq_nat @ (power_power_nat @ B3 @ W) @ X3))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_112_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X3 : nat, B3 : nat, W : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ X3) @ (power_power_nat @ (semiri1382578993at_nat @ B3) @ W)) = (ord_less_eq_nat @ X3 @ (power_power_nat @ B3 @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_113_power__decreasing__iff, axiom,
    ((![B3 : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B3) => ((ord_less_nat @ B3 @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B3 @ M) @ (power_power_nat @ B3 @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_114_power__mono__iff, axiom,
    ((![A3 : nat, B3 : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_eq_nat @ zero_zero_nat @ B3) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ B3 @ N)) = (ord_less_eq_nat @ A3 @ B3)))))))). % power_mono_iff
thf(fact_115_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_116_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X3 : nat, B3 : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X3) @ (power_power_nat @ (semiri1382578993at_nat @ B3) @ W)) = (ord_less_nat @ X3 @ (power_power_nat @ B3 @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_117_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B3 : nat, W : nat, X3 : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B3) @ W) @ (semiri1382578993at_nat @ X3)) = (ord_less_nat @ (power_power_nat @ B3 @ W) @ X3))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_118_one__reorient, axiom,
    ((![X3 : nat]: ((one_one_nat = X3) = (X3 = one_one_nat))))). % one_reorient
thf(fact_119_one__reorient, axiom,
    ((![X3 : complex]: ((one_one_complex = X3) = (X3 = one_one_complex))))). % one_reorient
thf(fact_120_one__le__power, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A3) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A3 @ N)))))). % one_le_power
thf(fact_121_power__increasing, axiom,
    ((![N : nat, N3 : nat, A3 : nat]: ((ord_less_eq_nat @ N @ N3) => ((ord_less_eq_nat @ one_one_nat @ A3) => (ord_less_eq_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ A3 @ N3))))))). % power_increasing
thf(fact_122_of__nat__mono, axiom,
    ((![I2 : nat, J2 : nat]: ((ord_less_eq_nat @ I2 @ J2) => (ord_less_eq_nat @ (semiri1382578993at_nat @ I2) @ (semiri1382578993at_nat @ J2)))))). % of_nat_mono
thf(fact_123_of__nat__0__le__iff, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N))))). % of_nat_0_le_iff
thf(fact_124_atLeastLessThan__subset__iff, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: ((ord_less_eq_set_nat @ (set_or562006527an_nat @ A3 @ B3) @ (set_or562006527an_nat @ C @ D)) => ((ord_less_eq_nat @ B3 @ A3) | ((ord_less_eq_nat @ C @ A3) & (ord_less_eq_nat @ B3 @ D))))))). % atLeastLessThan_subset_iff
thf(fact_125_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_126_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_127_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_128_power__decreasing, axiom,
    ((![N : nat, N3 : nat, A3 : nat]: ((ord_less_eq_nat @ N @ N3) => ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_eq_nat @ A3 @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A3 @ N3) @ (power_power_nat @ A3 @ N)))))))). % power_decreasing
thf(fact_129_power__le__imp__le__exp, axiom,
    ((![A3 : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A3) => ((ord_less_eq_nat @ (power_power_nat @ A3 @ M) @ (power_power_nat @ A3 @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_130_power__le__one, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_eq_nat @ A3 @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A3 @ N) @ one_one_nat)))))). % power_le_one
thf(fact_131_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_132_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_133_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_134_bot__nat__0_Oextremum__unique, axiom,
    ((![A3 : nat]: ((ord_less_eq_nat @ A3 @ zero_zero_nat) = (A3 = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_135_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A3 : nat]: ((ord_less_eq_nat @ A3 @ zero_zero_nat) => (A3 = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_136_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I2 : nat, J2 : nat]: ((![I3 : nat, J3 : nat]: ((ord_less_nat @ I3 @ J3) => (ord_less_nat @ (F @ I3) @ (F @ J3)))) => ((ord_less_eq_nat @ I2 @ J2) => (ord_less_eq_nat @ (F @ I2) @ (F @ J2))))))). % less_mono_imp_le_mono
thf(fact_137_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_138_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_139_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N4 : nat]: (((ord_less_nat @ M3 @ N4)) | ((M3 = N4)))))))). % le_eq_less_or_eq
thf(fact_140_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_141_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N4 : nat]: (((ord_less_eq_nat @ M3 @ N4)) & ((~ ((M3 = N4)))))))))). % nat_less_le
thf(fact_142_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_143_self__le__power, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A3) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ A3 @ (power_power_nat @ A3 @ N))))))). % self_le_power
thf(fact_144_sum__mono, axiom,
    ((![K2 : set_complex, F : complex > nat, G : complex > nat]: ((![I3 : complex]: ((member_complex @ I3 @ K2) => (ord_less_eq_nat @ (F @ I3) @ (G @ I3)))) => (ord_less_eq_nat @ (groups1415553210ex_nat @ F @ K2) @ (groups1415553210ex_nat @ G @ K2)))))). % sum_mono
thf(fact_145_sum__mono, axiom,
    ((![K2 : set_nat, F : nat > nat, G : nat > nat]: ((![I3 : nat]: ((member_nat @ I3 @ K2) => (ord_less_eq_nat @ (F @ I3) @ (G @ I3)))) => (ord_less_eq_nat @ (groups1842438620at_nat @ F @ K2) @ (groups1842438620at_nat @ G @ K2)))))). % sum_mono
thf(fact_146_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_147_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_148_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_149_power__mono, axiom,
    ((![A3 : nat, B3 : nat, N : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ zero_zero_nat @ A3) => (ord_less_eq_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ B3 @ N))))))). % power_mono
thf(fact_150_zero__le__power, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A3) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A3 @ N)))))). % zero_le_power
thf(fact_151_sum_Oivl__cong, axiom,
    ((![A3 : nat, C : nat, B3 : nat, D : nat, G : nat > complex, H : nat > complex]: ((A3 = C) => ((B3 = D) => ((![X : nat]: ((ord_less_eq_nat @ C @ X) => ((ord_less_nat @ X @ D) => ((G @ X) = (H @ X))))) => ((groups59700922omplex @ G @ (set_or562006527an_nat @ A3 @ B3)) = (groups59700922omplex @ H @ (set_or562006527an_nat @ C @ D))))))))). % sum.ivl_cong
thf(fact_152_sum__nonneg, axiom,
    ((![A : set_complex, F : complex > nat]: ((![X : complex]: ((member_complex @ X @ A) => (ord_less_eq_nat @ zero_zero_nat @ (F @ X)))) => (ord_less_eq_nat @ zero_zero_nat @ (groups1415553210ex_nat @ F @ A)))))). % sum_nonneg
thf(fact_153_sum__nonneg, axiom,
    ((![A : set_nat, F : nat > nat]: ((![X : nat]: ((member_nat @ X @ A) => (ord_less_eq_nat @ zero_zero_nat @ (F @ X)))) => (ord_less_eq_nat @ zero_zero_nat @ (groups1842438620at_nat @ F @ A)))))). % sum_nonneg
thf(fact_154_sum__nonpos, axiom,
    ((![A : set_complex, F : complex > nat]: ((![X : complex]: ((member_complex @ X @ A) => (ord_less_eq_nat @ (F @ X) @ zero_zero_nat))) => (ord_less_eq_nat @ (groups1415553210ex_nat @ F @ A) @ zero_zero_nat))))). % sum_nonpos
thf(fact_155_sum__nonpos, axiom,
    ((![A : set_nat, F : nat > nat]: ((![X : nat]: ((member_nat @ X @ A) => (ord_less_eq_nat @ (F @ X) @ zero_zero_nat))) => (ord_less_eq_nat @ (groups1842438620at_nat @ F @ A) @ zero_zero_nat))))). % sum_nonpos
thf(fact_156_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K3 : nat]: ((ord_less_eq_nat @ K3 @ N) & ((![I4 : nat]: ((ord_less_nat @ I4 @ K3) => (~ ((P @ I4))))) & (P @ K3))))))))). % ex_least_nat_le
thf(fact_157_power__0, axiom,
    ((![A3 : complex]: ((power_power_complex @ A3 @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_158_power__0, axiom,
    ((![A3 : nat]: ((power_power_nat @ A3 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_159_power__less__imp__less__base, axiom,
    ((![A3 : nat, N : nat, B3 : nat]: ((ord_less_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ B3 @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B3) => (ord_less_nat @ A3 @ B3)))))). % power_less_imp_less_base
thf(fact_160_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_161_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_162_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_163_power__strict__increasing, axiom,
    ((![N : nat, N3 : nat, A3 : nat]: ((ord_less_nat @ N @ N3) => ((ord_less_nat @ one_one_nat @ A3) => (ord_less_nat @ (power_power_nat @ A3 @ N) @ (power_power_nat @ A3 @ N3))))))). % power_strict_increasing
thf(fact_164_power__less__imp__less__exp, axiom,
    ((![A3 : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A3) => ((ord_less_nat @ (power_power_nat @ A3 @ M) @ (power_power_nat @ A3 @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_165_power__eq__iff__eq__base, axiom,
    ((![N : nat, A3 : nat, B3 : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_eq_nat @ zero_zero_nat @ B3) => (((power_power_nat @ A3 @ N) = (power_power_nat @ B3 @ N)) = (A3 = B3)))))))). % power_eq_iff_eq_base
thf(fact_166_power__eq__imp__eq__base, axiom,
    ((![A3 : nat, N : nat, B3 : nat]: (((power_power_nat @ A3 @ N) = (power_power_nat @ B3 @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A3) => ((ord_less_eq_nat @ zero_zero_nat @ B3) => ((ord_less_nat @ zero_zero_nat @ N) => (A3 = B3)))))))). % power_eq_imp_eq_base
thf(fact_167_power__strict__decreasing, axiom,
    ((![N : nat, N3 : nat, A3 : nat]: ((ord_less_nat @ N @ N3) => ((ord_less_nat @ zero_zero_nat @ A3) => ((ord_less_nat @ A3 @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A3 @ N3) @ (power_power_nat @ A3 @ N)))))))). % power_strict_decreasing
thf(fact_168_one__less__power, axiom,
    ((![A3 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A3) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A3 @ N))))))). % one_less_power
thf(fact_169_of__nat__sum, axiom,
    ((![F : nat > nat, A : set_nat]: ((semiri356525583omplex @ (groups1842438620at_nat @ F @ A)) = (groups59700922omplex @ (^[X4 : nat]: (semiri356525583omplex @ (F @ X4))) @ A))))). % of_nat_sum
thf(fact_170_of__nat__sum, axiom,
    ((![F : complex > nat, A : set_complex]: ((semiri356525583omplex @ (groups1415553210ex_nat @ F @ A)) = (groups443808152omplex @ (^[X4 : complex]: (semiri356525583omplex @ (F @ X4))) @ A))))). % of_nat_sum
thf(fact_171_sum__roots__unity, axiom,
    ((![N : nat]: ((ord_less_nat @ one_one_nat @ N) => ((groups443808152omplex @ (^[X4 : complex]: X4) @ (collect_complex @ (^[Z : complex]: ((power_power_complex @ Z @ N) = one_one_complex)))) = zero_zero_complex))))). % sum_roots_unity
thf(fact_172_sum__nth__roots, axiom,
    ((![N : nat, C : complex]: ((ord_less_nat @ one_one_nat @ N) => ((groups443808152omplex @ (^[X4 : complex]: X4) @ (collect_complex @ (^[Z : complex]: ((power_power_complex @ Z @ N) = C)))) = zero_zero_complex))))). % sum_nth_roots
thf(fact_173_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_174_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_175_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_176_le__trans, axiom,
    ((![I2 : nat, J2 : nat, K : nat]: ((ord_less_eq_nat @ I2 @ J2) => ((ord_less_eq_nat @ J2 @ K) => (ord_less_eq_nat @ I2 @ K)))))). % le_trans
thf(fact_177_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_178_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_179_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_180_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (?[X : nat]: ((P @ X) & (![Y3 : nat]: ((P @ Y3) => (ord_less_eq_nat @ Y3 @ X)))))))))). % Nat.ex_has_greatest_nat

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