% 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/Fundamental_Theorem_Algebra/prob_738__5375086_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:40.919

% Could-be-implicit typings (2)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (17)
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_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_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_If_001t__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex, type,
    neg_nu484426047omplex : complex > complex).
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__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_k____, type,
    k : nat).

% Relevant facts (175)
thf(fact_0_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_1_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_2_power__Suc0__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_3_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_4_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N)) = zero_zero_complex)))). % power_0_Suc
thf(fact_5_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_6_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_7_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_8_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_9_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_10_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_11_diff__Suc__Suc, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (suc @ M) @ (suc @ N)) = (minus_minus_nat @ M @ N))))). % diff_Suc_Suc
thf(fact_12_Suc__diff__diff, axiom,
    ((![M : nat, N : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ (suc @ M) @ N) @ (suc @ K)) = (minus_minus_nat @ (minus_minus_nat @ M @ N) @ K))))). % Suc_diff_diff
thf(fact_13_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_14_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_15_power__eq__if, axiom,
    ((power_power_complex = (^[P : complex]: (^[M2 : nat]: (if_complex @ (M2 = zero_zero_nat) @ one_one_complex @ (times_times_complex @ P @ (power_power_complex @ P @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_16_power__eq__if, axiom,
    ((power_power_nat = (^[P : nat]: (^[M2 : nat]: (if_nat @ (M2 = zero_zero_nat) @ one_one_nat @ (times_times_nat @ P @ (power_power_nat @ P @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_17_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_18_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_19_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_20_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_21_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_22_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_23_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_24_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_25_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_26_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_27_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_28_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_29_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_30_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_31_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_32_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_33_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_34_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_35_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_36_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_37_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_38_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_39_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_40_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_41_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_42_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_43_diff__Suc__1, axiom,
    ((![N : nat]: ((minus_minus_nat @ (suc @ N) @ one_one_nat) = N)))). % diff_Suc_1
thf(fact_44_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_45_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_46_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_47_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_48_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_49_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_50_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_51_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_52_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_53_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_54_diff__Suc__eq__diff__pred, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ M @ (suc @ N)) = (minus_minus_nat @ (minus_minus_nat @ M @ one_one_nat) @ N))))). % diff_Suc_eq_diff_pred
thf(fact_55_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_56_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_69_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M3 : nat]: (N = (suc @ M3))))))). % not0_implies_Suc
thf(fact_85_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_86_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_87_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_88_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_89_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_90_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_91_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P2 @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P2 @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P2 @ X3 @ Y3) => (P2 @ (suc @ X3) @ (suc @ Y3)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_92_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((P2 @ N2) => (P2 @ (suc @ N2)))) => (P2 @ N)))))). % nat_induct
thf(fact_93_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_94_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_95_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_96_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_97_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_98_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_99_zero__induct__lemma, axiom,
    ((![P2 : nat > $o, K : nat, I : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_100_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_101_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_102_power__Suc2, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (suc @ N)) = (times_times_complex @ (power_power_complex @ A @ N) @ A))))). % power_Suc2
thf(fact_103_power__Suc2, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ (power_power_nat @ A @ N) @ A))))). % power_Suc2
thf(fact_104_power__Suc, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (suc @ N)) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_Suc
thf(fact_105_power__Suc, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_Suc
thf(fact_106_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_107_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_108_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_109_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_110_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_111_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_112_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_113_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_114_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_115_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_116_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_117_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_118_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_119_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_120_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_121_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_122_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_123_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_124_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_125_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_126_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_127_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_128_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_129_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_130_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_131_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_132_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_133_diff__eq__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: (((minus_minus_complex @ A @ B) = (minus_minus_complex @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_134_cancel__ab__semigroup__add__class_Odiff__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))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_135_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ C) @ B) = (minus_minus_complex @ (minus_minus_complex @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_136_eq__iff__diff__eq__0, axiom,
    (((^[Y4 : complex]: (^[Z : complex]: (Y4 = Z))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_137_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_138_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_139_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_140_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_141_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_142_exists__least__lemma, axiom,
    ((![P2 : nat > $o]: ((~ ((P2 @ zero_zero_nat))) => ((?[X_1 : nat]: (P2 @ X_1)) => (?[N2 : nat]: ((~ ((P2 @ N2))) & (P2 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_143_inf__period_I1_J, axiom,
    ((![P2 : complex > $o, D2 : complex, Q : complex > $o]: ((![X3 : complex, K2 : complex]: ((P2 @ X3) = (P2 @ (minus_minus_complex @ X3 @ (times_times_complex @ K2 @ D2))))) => ((![X3 : complex, K2 : complex]: ((Q @ X3) = (Q @ (minus_minus_complex @ X3 @ (times_times_complex @ K2 @ D2))))) => (![X4 : complex, K3 : complex]: ((((P2 @ X4)) & ((Q @ X4))) = (((P2 @ (minus_minus_complex @ X4 @ (times_times_complex @ K3 @ D2)))) & ((Q @ (minus_minus_complex @ X4 @ (times_times_complex @ K3 @ D2)))))))))))). % inf_period(1)
thf(fact_144_inf__period_I2_J, axiom,
    ((![P2 : complex > $o, D2 : complex, Q : complex > $o]: ((![X3 : complex, K2 : complex]: ((P2 @ X3) = (P2 @ (minus_minus_complex @ X3 @ (times_times_complex @ K2 @ D2))))) => ((![X3 : complex, K2 : complex]: ((Q @ X3) = (Q @ (minus_minus_complex @ X3 @ (times_times_complex @ K2 @ D2))))) => (![X4 : complex, K3 : complex]: ((((P2 @ X4)) | ((Q @ X4))) = (((P2 @ (minus_minus_complex @ X4 @ (times_times_complex @ K3 @ D2)))) | ((Q @ (minus_minus_complex @ X4 @ (times_times_complex @ K3 @ D2)))))))))))). % inf_period(2)
thf(fact_145_dbl__inc__simps_I2_J, axiom,
    (((neg_nu484426047omplex @ zero_zero_complex) = one_one_complex))). % dbl_inc_simps(2)
thf(fact_146_power__minus__mult, axiom,
    ((![N : nat, A : complex]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_times_complex @ (power_power_complex @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_power_complex @ A @ N)))))). % power_minus_mult
thf(fact_147_power__minus__mult, axiom,
    ((![N : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_times_nat @ (power_power_nat @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_power_nat @ A @ N)))))). % power_minus_mult
thf(fact_148_Suc__diff__1, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((suc @ (minus_minus_nat @ N @ one_one_nat)) = N))))). % Suc_diff_1
thf(fact_149_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_150_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_151_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_152_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_153_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_154_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_155_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_156_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_157_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_158_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_159_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_160_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_161_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_162_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_163_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_164_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_165_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_166_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_167_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_168_Suc__pred, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((suc @ (minus_minus_nat @ N @ (suc @ zero_zero_nat))) = N))))). % Suc_pred
thf(fact_169_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_170_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_171_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_172_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_173_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_174_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

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $true @ X @ Y) = X)))).

% Conjectures (2)
thf(conj_0, hypothesis,
    ($true)).
thf(conj_1, conjecture,
    ((~ (((times_times_complex @ zero_zero_complex @ (power_power_complex @ zero_zero_complex @ (minus_minus_nat @ k @ (suc @ zero_zero_nat)))) = (times_times_complex @ one_one_complex @ (power_power_complex @ one_one_complex @ (minus_minus_nat @ k @ (suc @ zero_zero_nat))))))))).
