% 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_435__3227496_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:17.271

% Could-be-implicit typings (3)
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 (27)
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__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__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__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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J, type,
    sup_sup_nat_o : (nat > $o) > (nat > $o) > nat > $o).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat, type,
    sup_sup_nat : nat > nat > nat).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J, type,
    sup_sup_set_nat : set_nat > set_nat > set_nat).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J, type,
    bot_bot_nat_o : nat > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_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_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_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__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_Oinsert_001t__Nat__Onat, type,
    insert_nat : nat > set_nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat, type,
    set_or1544565540an_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 (159)
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_sum_Oempty, axiom,
    ((![G : nat > nat]: ((groups1842438620at_nat @ G @ bot_bot_set_nat) = zero_zero_nat)))). % sum.empty
thf(fact_4_sum_Oempty, axiom,
    ((![G : nat > complex]: ((groups59700922omplex @ G @ bot_bot_set_nat) = zero_zero_complex)))). % sum.empty
thf(fact_5_singleton__conv, axiom,
    ((![A : nat]: ((collect_nat @ (^[X : nat]: (X = A))) = (insert_nat @ A @ bot_bot_set_nat))))). % singleton_conv
thf(fact_6_singleton__conv2, axiom,
    ((![A : nat]: ((collect_nat @ ((^[Y : nat]: (^[Z : nat]: (Y = Z))) @ A)) = (insert_nat @ A @ bot_bot_set_nat))))). % singleton_conv2
thf(fact_7_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups1842438620at_nat @ (^[Uu : nat]: zero_zero_nat) @ A2) = zero_zero_nat)))). % sum.neutral_const
thf(fact_8_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A2) = zero_zero_complex)))). % sum.neutral_const
thf(fact_9_Un__insert__left, axiom,
    ((![A : nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ (insert_nat @ A @ B) @ C) = (insert_nat @ A @ (sup_sup_set_nat @ B @ C)))))). % Un_insert_left
thf(fact_10_Un__insert__right, axiom,
    ((![A2 : set_nat, A : nat, B : set_nat]: ((sup_sup_set_nat @ A2 @ (insert_nat @ A @ B)) = (insert_nat @ A @ (sup_sup_set_nat @ A2 @ B)))))). % Un_insert_right
thf(fact_11_Un__empty, axiom,
    ((![A2 : set_nat, B : set_nat]: (((sup_sup_set_nat @ A2 @ B) = bot_bot_set_nat) = (((A2 = bot_bot_set_nat)) & ((B = bot_bot_set_nat))))))). % Un_empty
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_singletonI, axiom,
    ((![A : nat]: (member_nat @ A @ (insert_nat @ A @ bot_bot_set_nat))))). % singletonI
thf(fact_15_sup__bot__left, axiom,
    ((![X2 : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ X2) = X2)))). % sup_bot_left
thf(fact_16_sup__bot__right, axiom,
    ((![X2 : set_nat]: ((sup_sup_set_nat @ X2 @ bot_bot_set_nat) = X2)))). % sup_bot_right
thf(fact_17_empty__Collect__eq, axiom,
    ((![P : nat > $o]: ((bot_bot_set_nat = (collect_nat @ P)) = (![X : nat]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_18_Collect__empty__eq, axiom,
    ((![P : nat > $o]: (((collect_nat @ P) = bot_bot_set_nat) = (![X : nat]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_19_all__not__in__conv, axiom,
    ((![A2 : set_nat]: ((![X : nat]: (~ ((member_nat @ X @ A2)))) = (A2 = bot_bot_set_nat))))). % all_not_in_conv
thf(fact_20_empty__iff, axiom,
    ((![C2 : nat]: (~ ((member_nat @ C2 @ bot_bot_set_nat)))))). % empty_iff
thf(fact_21_insert__absorb2, axiom,
    ((![X2 : nat, A2 : set_nat]: ((insert_nat @ X2 @ (insert_nat @ X2 @ A2)) = (insert_nat @ X2 @ A2))))). % insert_absorb2
thf(fact_22_insert__iff, axiom,
    ((![A : nat, B2 : nat, A2 : set_nat]: ((member_nat @ A @ (insert_nat @ B2 @ A2)) = (((A = B2)) | ((member_nat @ A @ A2))))))). % insert_iff
thf(fact_23_insertCI, axiom,
    ((![A : nat, B : set_nat, B2 : nat]: (((~ ((member_nat @ A @ B))) => (A = B2)) => (member_nat @ A @ (insert_nat @ B2 @ B)))))). % insertCI
thf(fact_24_sup_Oright__idem, axiom,
    ((![A : set_nat, B2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A @ B2) @ B2) = (sup_sup_set_nat @ A @ B2))))). % sup.right_idem
thf(fact_25_sup__left__idem, axiom,
    ((![X2 : set_nat, Y2 : set_nat]: ((sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ X2 @ Y2)) = (sup_sup_set_nat @ X2 @ Y2))))). % sup_left_idem
thf(fact_26_sup_Oleft__idem, axiom,
    ((![A : set_nat, B2 : set_nat]: ((sup_sup_set_nat @ A @ (sup_sup_set_nat @ A @ B2)) = (sup_sup_set_nat @ A @ B2))))). % sup.left_idem
thf(fact_27_sup__idem, axiom,
    ((![X2 : set_nat]: ((sup_sup_set_nat @ X2 @ X2) = X2)))). % sup_idem
thf(fact_28_sup_Oidem, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ A @ A) = A)))). % sup.idem
thf(fact_29_Un__iff, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)) = (((member_nat @ C2 @ A2)) | ((member_nat @ C2 @ B))))))). % Un_iff
thf(fact_30_UnCI, axiom,
    ((![C2 : nat, B : set_nat, A2 : set_nat]: (((~ ((member_nat @ C2 @ B))) => (member_nat @ C2 @ A2)) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnCI
thf(fact_31_sup__bot_Oright__neutral, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ A @ bot_bot_set_nat) = A)))). % sup_bot.right_neutral
thf(fact_32_sup__bot_Oneutr__eq__iff, axiom,
    ((![A : set_nat, B2 : set_nat]: ((bot_bot_set_nat = (sup_sup_set_nat @ A @ B2)) = (((A = bot_bot_set_nat)) & ((B2 = bot_bot_set_nat))))))). % sup_bot.neutr_eq_iff
thf(fact_33_sup__bot_Oleft__neutral, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ A) = A)))). % sup_bot.left_neutral
thf(fact_34_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_nat, B2 : set_nat]: (((sup_sup_set_nat @ A @ B2) = bot_bot_set_nat) = (((A = bot_bot_set_nat)) & ((B2 = bot_bot_set_nat))))))). % sup_bot.eq_neutr_iff
thf(fact_35_sup__eq__bot__iff, axiom,
    ((![X2 : set_nat, Y2 : set_nat]: (((sup_sup_set_nat @ X2 @ Y2) = bot_bot_set_nat) = (((X2 = bot_bot_set_nat)) & ((Y2 = bot_bot_set_nat))))))). % sup_eq_bot_iff
thf(fact_36_bot__eq__sup__iff, axiom,
    ((![X2 : set_nat, Y2 : set_nat]: ((bot_bot_set_nat = (sup_sup_set_nat @ X2 @ Y2)) = (((X2 = bot_bot_set_nat)) & ((Y2 = bot_bot_set_nat))))))). % bot_eq_sup_iff
thf(fact_37_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_38_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_39_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_40_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_41_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_42_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_43_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_44_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_45_mem__Collect__eq, axiom,
    ((![A : nat, P : nat > $o]: ((member_nat @ A @ (collect_nat @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_46_Collect__mem__eq, axiom,
    ((![A2 : set_nat]: ((collect_nat @ (^[X : nat]: (member_nat @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_47_Collect__cong, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((![X3 : nat]: ((P @ X3) = (Q @ X3))) => ((collect_nat @ P) = (collect_nat @ Q)))))). % Collect_cong
thf(fact_48_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_49_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_50_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_51_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_neqE_nat
thf(fact_52_bot__set__def, axiom,
    ((bot_bot_set_nat = (collect_nat @ bot_bot_nat_o)))). % bot_set_def
thf(fact_53_bot__nat__def, axiom,
    ((bot_bot_nat = zero_zero_nat))). % bot_nat_def
thf(fact_54_sup__set__def, axiom,
    ((sup_sup_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (collect_nat @ (sup_sup_nat_o @ (^[X : nat]: (member_nat @ X @ A3)) @ (^[X : nat]: (member_nat @ X @ B3))))))))). % sup_set_def
thf(fact_55_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_56_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_57_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_58_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_59_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_60_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_61_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_62_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_63_sup_Ostrict__coboundedI2, axiom,
    ((![C2 : set_nat, B2 : set_nat, A : set_nat]: ((ord_less_set_nat @ C2 @ B2) => (ord_less_set_nat @ C2 @ (sup_sup_set_nat @ A @ B2)))))). % sup.strict_coboundedI2
thf(fact_64_sup_Ostrict__coboundedI2, axiom,
    ((![C2 : nat, B2 : nat, A : nat]: ((ord_less_nat @ C2 @ B2) => (ord_less_nat @ C2 @ (sup_sup_nat @ A @ B2)))))). % sup.strict_coboundedI2
thf(fact_65_sup_Ostrict__coboundedI1, axiom,
    ((![C2 : set_nat, A : set_nat, B2 : set_nat]: ((ord_less_set_nat @ C2 @ A) => (ord_less_set_nat @ C2 @ (sup_sup_set_nat @ A @ B2)))))). % sup.strict_coboundedI1
thf(fact_66_sup_Ostrict__coboundedI1, axiom,
    ((![C2 : nat, A : nat, B2 : nat]: ((ord_less_nat @ C2 @ A) => (ord_less_nat @ C2 @ (sup_sup_nat @ A @ B2)))))). % sup.strict_coboundedI1
thf(fact_67_sup_Ostrict__order__iff, axiom,
    ((ord_less_set_nat = (^[B4 : set_nat]: (^[A4 : set_nat]: (((A4 = (sup_sup_set_nat @ A4 @ B4))) & ((~ ((A4 = B4)))))))))). % sup.strict_order_iff
thf(fact_68_sup_Ostrict__order__iff, axiom,
    ((ord_less_nat = (^[B4 : nat]: (^[A4 : nat]: (((A4 = (sup_sup_nat @ A4 @ B4))) & ((~ ((A4 = B4)))))))))). % sup.strict_order_iff
thf(fact_69_sup_Ostrict__boundedE, axiom,
    ((![B2 : set_nat, C2 : set_nat, A : set_nat]: ((ord_less_set_nat @ (sup_sup_set_nat @ B2 @ C2) @ A) => (~ (((ord_less_set_nat @ B2 @ A) => (~ ((ord_less_set_nat @ C2 @ A)))))))))). % sup.strict_boundedE
thf(fact_70_sup_Ostrict__boundedE, axiom,
    ((![B2 : nat, C2 : nat, A : nat]: ((ord_less_nat @ (sup_sup_nat @ B2 @ C2) @ A) => (~ (((ord_less_nat @ B2 @ A) => (~ ((ord_less_nat @ C2 @ A)))))))))). % sup.strict_boundedE
thf(fact_71_less__supI2, axiom,
    ((![X2 : set_nat, B2 : set_nat, A : set_nat]: ((ord_less_set_nat @ X2 @ B2) => (ord_less_set_nat @ X2 @ (sup_sup_set_nat @ A @ B2)))))). % less_supI2
thf(fact_72_less__supI2, axiom,
    ((![X2 : nat, B2 : nat, A : nat]: ((ord_less_nat @ X2 @ B2) => (ord_less_nat @ X2 @ (sup_sup_nat @ A @ B2)))))). % less_supI2
thf(fact_73_less__supI1, axiom,
    ((![X2 : set_nat, A : set_nat, B2 : set_nat]: ((ord_less_set_nat @ X2 @ A) => (ord_less_set_nat @ X2 @ (sup_sup_set_nat @ A @ B2)))))). % less_supI1
thf(fact_74_less__supI1, axiom,
    ((![X2 : nat, A : nat, B2 : nat]: ((ord_less_nat @ X2 @ A) => (ord_less_nat @ X2 @ (sup_sup_nat @ A @ B2)))))). % less_supI1
thf(fact_75_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_76_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_77_ex__in__conv, axiom,
    ((![A2 : set_nat]: ((?[X : nat]: (member_nat @ X @ A2)) = (~ ((A2 = bot_bot_set_nat))))))). % ex_in_conv
thf(fact_78_equals0I, axiom,
    ((![A2 : set_nat]: ((![Y3 : nat]: (~ ((member_nat @ Y3 @ A2)))) => (A2 = bot_bot_set_nat))))). % equals0I
thf(fact_79_equals0D, axiom,
    ((![A2 : set_nat, A : nat]: ((A2 = bot_bot_set_nat) => (~ ((member_nat @ A @ A2))))))). % equals0D
thf(fact_80_emptyE, axiom,
    ((![A : nat]: (~ ((member_nat @ A @ bot_bot_set_nat)))))). % emptyE
thf(fact_81_mk__disjoint__insert, axiom,
    ((![A : nat, A2 : set_nat]: ((member_nat @ A @ A2) => (?[B5 : set_nat]: ((A2 = (insert_nat @ A @ B5)) & (~ ((member_nat @ A @ B5))))))))). % mk_disjoint_insert
thf(fact_82_insert__commute, axiom,
    ((![X2 : nat, Y2 : nat, A2 : set_nat]: ((insert_nat @ X2 @ (insert_nat @ Y2 @ A2)) = (insert_nat @ Y2 @ (insert_nat @ X2 @ A2)))))). % insert_commute
thf(fact_83_insert__eq__iff, axiom,
    ((![A : nat, A2 : set_nat, B2 : nat, B : set_nat]: ((~ ((member_nat @ A @ A2))) => ((~ ((member_nat @ B2 @ B))) => (((insert_nat @ A @ A2) = (insert_nat @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_nat]: (((A2 = (insert_nat @ B2 @ C3))) & ((((~ ((member_nat @ B2 @ C3)))) & ((((B = (insert_nat @ A @ C3))) & ((~ ((member_nat @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_84_insert__absorb, axiom,
    ((![A : nat, A2 : set_nat]: ((member_nat @ A @ A2) => ((insert_nat @ A @ A2) = A2))))). % insert_absorb
thf(fact_85_insert__ident, axiom,
    ((![X2 : nat, A2 : set_nat, B : set_nat]: ((~ ((member_nat @ X2 @ A2))) => ((~ ((member_nat @ X2 @ B))) => (((insert_nat @ X2 @ A2) = (insert_nat @ X2 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_86_Set_Oset__insert, axiom,
    ((![X2 : nat, A2 : set_nat]: ((member_nat @ X2 @ A2) => (~ ((![B5 : set_nat]: ((A2 = (insert_nat @ X2 @ B5)) => (member_nat @ X2 @ B5))))))))). % Set.set_insert
thf(fact_87_insertI2, axiom,
    ((![A : nat, B : set_nat, B2 : nat]: ((member_nat @ A @ B) => (member_nat @ A @ (insert_nat @ B2 @ B)))))). % insertI2
thf(fact_88_insertI1, axiom,
    ((![A : nat, B : set_nat]: (member_nat @ A @ (insert_nat @ A @ B))))). % insertI1
thf(fact_89_insertE, axiom,
    ((![A : nat, B2 : nat, A2 : set_nat]: ((member_nat @ A @ (insert_nat @ B2 @ A2)) => ((~ ((A = B2))) => (member_nat @ A @ A2)))))). % insertE
thf(fact_90_sum_Oreindex__bij__witness, axiom,
    ((![S2 : set_nat, I : nat > nat, J : nat > nat, T2 : set_nat, H : nat > complex, G : nat > complex]: ((![A5 : nat]: ((member_nat @ A5 @ S2) => ((I @ (J @ A5)) = A5))) => ((![A5 : nat]: ((member_nat @ A5 @ S2) => (member_nat @ (J @ A5) @ T2))) => ((![B6 : nat]: ((member_nat @ B6 @ T2) => ((J @ (I @ B6)) = B6))) => ((![B6 : nat]: ((member_nat @ B6 @ T2) => (member_nat @ (I @ B6) @ S2))) => ((![A5 : nat]: ((member_nat @ A5 @ S2) => ((H @ (J @ A5)) = (G @ A5)))) => ((groups59700922omplex @ G @ S2) = (groups59700922omplex @ H @ T2)))))))))). % sum.reindex_bij_witness
thf(fact_91_sum_Oreindex__bij__witness, axiom,
    ((![S2 : set_nat, I : nat > nat, J : nat > nat, T2 : set_nat, H : nat > nat, G : nat > nat]: ((![A5 : nat]: ((member_nat @ A5 @ S2) => ((I @ (J @ A5)) = A5))) => ((![A5 : nat]: ((member_nat @ A5 @ S2) => (member_nat @ (J @ A5) @ T2))) => ((![B6 : nat]: ((member_nat @ B6 @ T2) => ((J @ (I @ B6)) = B6))) => ((![B6 : nat]: ((member_nat @ B6 @ T2) => (member_nat @ (I @ B6) @ S2))) => ((![A5 : nat]: ((member_nat @ A5 @ S2) => ((H @ (J @ A5)) = (G @ A5)))) => ((groups1842438620at_nat @ G @ S2) = (groups1842438620at_nat @ H @ T2)))))))))). % sum.reindex_bij_witness
thf(fact_92_sum_Oeq__general__inverses, axiom,
    ((![B : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B) => ((member_nat @ (K @ Y3) @ A2) & ((H @ (K @ Y3)) = Y3)))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H @ X3) @ B) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => ((groups59700922omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_93_sum_Oeq__general__inverses, axiom,
    ((![B : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y3 : nat]: ((member_nat @ Y3 @ B) => ((member_nat @ (K @ Y3) @ A2) & ((H @ (K @ Y3)) = Y3)))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H @ X3) @ B) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => ((groups1842438620at_nat @ Phi @ A2) = (groups1842438620at_nat @ Gamma @ B))))))). % sum.eq_general_inverses
thf(fact_94_sum_Oeq__general, axiom,
    ((![B : set_nat, A2 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B) => (?[X4 : nat]: (((member_nat @ X4 @ A2) & ((H @ X4) = Y3)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y3)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H @ X3) @ B) & ((Gamma @ (H @ X3)) = (Phi @ X3))))) => ((groups59700922omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B))))))). % sum.eq_general
thf(fact_95_sum_Oeq__general, axiom,
    ((![B : set_nat, A2 : set_nat, H : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y3 : nat]: ((member_nat @ Y3 @ B) => (?[X4 : nat]: (((member_nat @ X4 @ A2) & ((H @ X4) = Y3)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y3)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H @ X3) @ B) & ((Gamma @ (H @ X3)) = (Phi @ X3))))) => ((groups1842438620at_nat @ Phi @ A2) = (groups1842438620at_nat @ Gamma @ B))))))). % sum.eq_general
thf(fact_96_sum_Ocong, axiom,
    ((![A2 : set_nat, B : set_nat, G : nat > complex, H : nat > complex]: ((A2 = B) => ((![X3 : nat]: ((member_nat @ X3 @ B) => ((G @ X3) = (H @ X3)))) => ((groups59700922omplex @ G @ A2) = (groups59700922omplex @ H @ B))))))). % sum.cong
thf(fact_97_sum_Ocong, axiom,
    ((![A2 : set_nat, B : set_nat, G : nat > nat, H : nat > nat]: ((A2 = B) => ((![X3 : nat]: ((member_nat @ X3 @ B) => ((G @ X3) = (H @ X3)))) => ((groups1842438620at_nat @ G @ A2) = (groups1842438620at_nat @ H @ B))))))). % sum.cong
thf(fact_98_sup__left__commute, axiom,
    ((![X2 : set_nat, Y2 : set_nat, Z2 : set_nat]: ((sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ Y2 @ Z2)) = (sup_sup_set_nat @ Y2 @ (sup_sup_set_nat @ X2 @ Z2)))))). % sup_left_commute
thf(fact_99_sup_Oleft__commute, axiom,
    ((![B2 : set_nat, A : set_nat, C2 : set_nat]: ((sup_sup_set_nat @ B2 @ (sup_sup_set_nat @ A @ C2)) = (sup_sup_set_nat @ A @ (sup_sup_set_nat @ B2 @ C2)))))). % sup.left_commute
thf(fact_100_sup__commute, axiom,
    ((sup_sup_set_nat = (^[X : set_nat]: (^[Y4 : set_nat]: (sup_sup_set_nat @ Y4 @ X)))))). % sup_commute
thf(fact_101_sup_Ocommute, axiom,
    ((sup_sup_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (sup_sup_set_nat @ B4 @ A4)))))). % sup.commute
thf(fact_102_sup__assoc, axiom,
    ((![X2 : set_nat, Y2 : set_nat, Z2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ X2 @ Y2) @ Z2) = (sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ Y2 @ Z2)))))). % sup_assoc
thf(fact_103_sup_Oassoc, axiom,
    ((![A : set_nat, B2 : set_nat, C2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A @ B2) @ C2) = (sup_sup_set_nat @ A @ (sup_sup_set_nat @ B2 @ C2)))))). % sup.assoc
thf(fact_104_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_nat, K : set_nat, B2 : set_nat, A : set_nat]: ((B = (sup_sup_set_nat @ K @ B2)) => ((sup_sup_set_nat @ A @ B) = (sup_sup_set_nat @ K @ (sup_sup_set_nat @ A @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_105_boolean__algebra__cancel_Osup1, axiom,
    ((![A2 : set_nat, K : set_nat, A : set_nat, B2 : set_nat]: ((A2 = (sup_sup_set_nat @ K @ A)) => ((sup_sup_set_nat @ A2 @ B2) = (sup_sup_set_nat @ K @ (sup_sup_set_nat @ A @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_106_inf__sup__aci_I5_J, axiom,
    ((sup_sup_set_nat = (^[X : set_nat]: (^[Y4 : set_nat]: (sup_sup_set_nat @ Y4 @ X)))))). % inf_sup_aci(5)
thf(fact_107_inf__sup__aci_I6_J, axiom,
    ((![X2 : set_nat, Y2 : set_nat, Z2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ X2 @ Y2) @ Z2) = (sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ Y2 @ Z2)))))). % inf_sup_aci(6)
thf(fact_108_inf__sup__aci_I7_J, axiom,
    ((![X2 : set_nat, Y2 : set_nat, Z2 : set_nat]: ((sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ Y2 @ Z2)) = (sup_sup_set_nat @ Y2 @ (sup_sup_set_nat @ X2 @ Z2)))))). % inf_sup_aci(7)
thf(fact_109_inf__sup__aci_I8_J, axiom,
    ((![X2 : set_nat, Y2 : set_nat]: ((sup_sup_set_nat @ X2 @ (sup_sup_set_nat @ X2 @ Y2)) = (sup_sup_set_nat @ X2 @ Y2))))). % inf_sup_aci(8)
thf(fact_110_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_111_Un__left__commute, axiom,
    ((![A2 : set_nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ B @ C)) = (sup_sup_set_nat @ B @ (sup_sup_set_nat @ A2 @ C)))))). % Un_left_commute
thf(fact_112_Un__left__absorb, axiom,
    ((![A2 : set_nat, B : set_nat]: ((sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ A2 @ B)) = (sup_sup_set_nat @ A2 @ B))))). % Un_left_absorb
thf(fact_113_Un__commute, axiom,
    ((sup_sup_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (sup_sup_set_nat @ B3 @ A3)))))). % Un_commute
thf(fact_114_Un__absorb, axiom,
    ((![A2 : set_nat]: ((sup_sup_set_nat @ A2 @ A2) = A2)))). % Un_absorb
thf(fact_115_Un__assoc, axiom,
    ((![A2 : set_nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A2 @ B) @ C) = (sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ B @ C)))))). % Un_assoc
thf(fact_116_ball__Un, axiom,
    ((![A2 : set_nat, B : set_nat, P : nat > $o]: ((![X : nat]: (((member_nat @ X @ (sup_sup_set_nat @ A2 @ B))) => ((P @ X)))) = (((![X : nat]: (((member_nat @ X @ A2)) => ((P @ X))))) & ((![X : nat]: (((member_nat @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_117_bex__Un, axiom,
    ((![A2 : set_nat, B : set_nat, P : nat > $o]: ((?[X : nat]: (((member_nat @ X @ (sup_sup_set_nat @ A2 @ B))) & ((P @ X)))) = (((?[X : nat]: (((member_nat @ X @ A2)) & ((P @ X))))) | ((?[X : nat]: (((member_nat @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_118_UnI2, axiom,
    ((![C2 : nat, B : set_nat, A2 : set_nat]: ((member_nat @ C2 @ B) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnI2
thf(fact_119_UnI1, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ A2) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnI1
thf(fact_120_UnE, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)) => ((~ ((member_nat @ C2 @ A2))) => (member_nat @ C2 @ B)))))). % UnE
thf(fact_121_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_122_empty__def, axiom,
    ((bot_bot_set_nat = (collect_nat @ (^[X : nat]: $false))))). % empty_def
thf(fact_123_insert__Collect, axiom,
    ((![A : nat, P : nat > $o]: ((insert_nat @ A @ (collect_nat @ P)) = (collect_nat @ (^[U : nat]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_124_insert__compr, axiom,
    ((insert_nat = (^[A4 : nat]: (^[B3 : set_nat]: (collect_nat @ (^[X : nat]: (((X = A4)) | ((member_nat @ X @ B3)))))))))). % insert_compr
thf(fact_125_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B : set_nat, A2 : set_nat]: ((groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (G @ I2) @ B)) @ A2) = (groups59700922omplex @ (^[J2 : nat]: (groups59700922omplex @ (^[I2 : nat]: (G @ I2 @ J2)) @ A2)) @ B))))). % sum.swap
thf(fact_126_sum_Oswap, axiom,
    ((![G : nat > nat > nat, B : set_nat, A2 : set_nat]: ((groups1842438620at_nat @ (^[I2 : nat]: (groups1842438620at_nat @ (G @ I2) @ B)) @ A2) = (groups1842438620at_nat @ (^[J2 : nat]: (groups1842438620at_nat @ (^[I2 : nat]: (G @ I2 @ J2)) @ A2)) @ B))))). % sum.swap
thf(fact_127_Collect__disj__eq, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((collect_nat @ (^[X : nat]: (((P @ X)) | ((Q @ X))))) = (sup_sup_set_nat @ (collect_nat @ P) @ (collect_nat @ Q)))))). % Collect_disj_eq
thf(fact_128_Un__def, axiom,
    ((sup_sup_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (collect_nat @ (^[X : nat]: (((member_nat @ X @ A3)) | ((member_nat @ X @ B3)))))))))). % Un_def
thf(fact_129_root__summation, axiom,
    ((![K : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ K @ N) => ((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)) @ (set_or562006527an_nat @ zero_zero_nat @ N)) = zero_zero_complex)))))). % root_summation
thf(fact_130_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > complex, A2 : set_nat]: ((~ (((groups59700922omplex @ G @ A2) = zero_zero_complex))) => (~ ((![A5 : nat]: ((member_nat @ A5 @ A2) => ((G @ A5) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_131_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > nat, A2 : set_nat]: ((~ (((groups1842438620at_nat @ G @ A2) = zero_zero_nat))) => (~ ((![A5 : nat]: ((member_nat @ A5 @ A2) => ((G @ A5) = zero_zero_nat))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_132_sum_Oneutral, axiom,
    ((![A2 : set_nat, G : nat > complex]: ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((G @ X3) = zero_zero_complex))) => ((groups59700922omplex @ G @ A2) = zero_zero_complex))))). % sum.neutral
thf(fact_133_sum_Oneutral, axiom,
    ((![A2 : set_nat, G : nat > nat]: ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((G @ X3) = zero_zero_nat))) => ((groups1842438620at_nat @ G @ A2) = zero_zero_nat))))). % sum.neutral
thf(fact_134_singleton__inject, axiom,
    ((![A : nat, B2 : nat]: (((insert_nat @ A @ bot_bot_set_nat) = (insert_nat @ B2 @ bot_bot_set_nat)) => (A = B2))))). % singleton_inject
thf(fact_135_insert__not__empty, axiom,
    ((![A : nat, A2 : set_nat]: (~ (((insert_nat @ A @ A2) = bot_bot_set_nat)))))). % insert_not_empty
thf(fact_136_doubleton__eq__iff, axiom,
    ((![A : nat, B2 : nat, C2 : nat, D : nat]: (((insert_nat @ A @ (insert_nat @ B2 @ bot_bot_set_nat)) = (insert_nat @ C2 @ (insert_nat @ D @ bot_bot_set_nat))) = (((((A = C2)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C2))))))))). % doubleton_eq_iff
thf(fact_137_singleton__iff, axiom,
    ((![B2 : nat, A : nat]: ((member_nat @ B2 @ (insert_nat @ A @ bot_bot_set_nat)) = (B2 = A))))). % singleton_iff
thf(fact_138_singletonD, axiom,
    ((![B2 : nat, A : nat]: ((member_nat @ B2 @ (insert_nat @ A @ bot_bot_set_nat)) => (B2 = A))))). % singletonD
thf(fact_139_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_140_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_141_Un__empty__right, axiom,
    ((![A2 : set_nat]: ((sup_sup_set_nat @ A2 @ bot_bot_set_nat) = A2)))). % Un_empty_right
thf(fact_142_Un__empty__left, axiom,
    ((![B : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ B) = B)))). % Un_empty_left
thf(fact_143_sum__distrib__right, axiom,
    ((![F : nat > complex, A2 : set_nat, R : complex]: ((times_times_complex @ (groups59700922omplex @ F @ A2) @ R) = (groups59700922omplex @ (^[N3 : nat]: (times_times_complex @ (F @ N3) @ R)) @ A2))))). % sum_distrib_right
thf(fact_144_sum__distrib__right, axiom,
    ((![F : nat > nat, A2 : set_nat, R : nat]: ((times_times_nat @ (groups1842438620at_nat @ F @ A2) @ R) = (groups1842438620at_nat @ (^[N3 : nat]: (times_times_nat @ (F @ N3) @ R)) @ A2))))). % sum_distrib_right
thf(fact_145_sum__distrib__left, axiom,
    ((![R : complex, F : nat > complex, A2 : set_nat]: ((times_times_complex @ R @ (groups59700922omplex @ F @ A2)) = (groups59700922omplex @ (^[N3 : nat]: (times_times_complex @ R @ (F @ N3))) @ A2))))). % sum_distrib_left
thf(fact_146_sum__distrib__left, axiom,
    ((![R : nat, F : nat > nat, A2 : set_nat]: ((times_times_nat @ R @ (groups1842438620at_nat @ F @ A2)) = (groups1842438620at_nat @ (^[N3 : nat]: (times_times_nat @ R @ (F @ N3))) @ A2))))). % sum_distrib_left
thf(fact_147_sum__product, axiom,
    ((![F : nat > complex, A2 : set_nat, G : nat > complex, B : set_nat]: ((times_times_complex @ (groups59700922omplex @ F @ A2) @ (groups59700922omplex @ G @ B)) = (groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (F @ I2) @ (G @ J2))) @ B)) @ A2))))). % sum_product
thf(fact_148_sum__product, axiom,
    ((![F : nat > nat, A2 : set_nat, G : nat > nat, B : set_nat]: ((times_times_nat @ (groups1842438620at_nat @ F @ A2) @ (groups1842438620at_nat @ G @ B)) = (groups1842438620at_nat @ (^[I2 : nat]: (groups1842438620at_nat @ (^[J2 : nat]: (times_times_nat @ (F @ I2) @ (G @ J2))) @ B)) @ A2))))). % sum_product
thf(fact_149_sum__subtractf, axiom,
    ((![F : nat > complex, G : nat > complex, A2 : set_nat]: ((groups59700922omplex @ (^[X : nat]: (minus_minus_complex @ (F @ X) @ (G @ X))) @ A2) = (minus_minus_complex @ (groups59700922omplex @ F @ A2) @ (groups59700922omplex @ G @ A2)))))). % sum_subtractf
thf(fact_150_Collect__conv__if2, axiom,
    ((![P : nat > $o, A : nat]: (((P @ A) => ((collect_nat @ (^[X : nat]: (((A = X)) & ((P @ X))))) = (insert_nat @ A @ bot_bot_set_nat))) & ((~ ((P @ A))) => ((collect_nat @ (^[X : nat]: (((A = X)) & ((P @ X))))) = bot_bot_set_nat)))))). % Collect_conv_if2
thf(fact_151_Collect__conv__if, axiom,
    ((![P : nat > $o, A : nat]: (((P @ A) => ((collect_nat @ (^[X : nat]: (((X = A)) & ((P @ X))))) = (insert_nat @ A @ bot_bot_set_nat))) & ((~ ((P @ A))) => ((collect_nat @ (^[X : nat]: (((X = A)) & ((P @ X))))) = bot_bot_set_nat)))))). % Collect_conv_if
thf(fact_152_insert__def, axiom,
    ((insert_nat = (^[A4 : nat]: (sup_sup_set_nat @ (collect_nat @ (^[X : nat]: (X = A4)))))))). % insert_def
thf(fact_153_singleton__Un__iff, axiom,
    ((![X2 : nat, A2 : set_nat, B : set_nat]: (((insert_nat @ X2 @ bot_bot_set_nat) = (sup_sup_set_nat @ A2 @ B)) = (((((A2 = bot_bot_set_nat)) & ((B = (insert_nat @ X2 @ bot_bot_set_nat))))) | ((((((A2 = (insert_nat @ X2 @ bot_bot_set_nat))) & ((B = bot_bot_set_nat)))) | ((((A2 = (insert_nat @ X2 @ bot_bot_set_nat))) & ((B = (insert_nat @ X2 @ bot_bot_set_nat)))))))))))). % singleton_Un_iff
thf(fact_154_Un__singleton__iff, axiom,
    ((![A2 : set_nat, B : set_nat, X2 : nat]: (((sup_sup_set_nat @ A2 @ B) = (insert_nat @ X2 @ bot_bot_set_nat)) = (((((A2 = bot_bot_set_nat)) & ((B = (insert_nat @ X2 @ bot_bot_set_nat))))) | ((((((A2 = (insert_nat @ X2 @ bot_bot_set_nat))) & ((B = bot_bot_set_nat)))) | ((((A2 = (insert_nat @ X2 @ bot_bot_set_nat))) & ((B = (insert_nat @ X2 @ bot_bot_set_nat)))))))))))). % Un_singleton_iff
thf(fact_155_insert__is__Un, axiom,
    ((insert_nat = (^[A4 : nat]: (sup_sup_set_nat @ (insert_nat @ A4 @ bot_bot_set_nat)))))). % insert_is_Un
thf(fact_156_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_157_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_158_atLeastLessThan__empty__iff2, axiom,
    ((![A : nat, B2 : nat]: ((bot_bot_set_nat = (set_or562006527an_nat @ A @ B2)) = (~ ((ord_less_nat @ A @ B2))))))). % atLeastLessThan_empty_iff2

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J2 @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J2))) @ (set_or562006527an_nat @ i @ n)) = (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J2 @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J2))) @ (sup_sup_set_nat @ (insert_nat @ i @ bot_bot_set_nat) @ (set_or1544565540an_nat @ i @ n)))))).
