% 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_753__5375260_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:49.513

% 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 (16)
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_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_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_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
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_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).
thf(sy_v_w____, type,
    w : complex).

% Relevant facts (166)
thf(fact_0_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_1_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_2_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_3__092_060open_0621_A_L_Aw_A_094_Ak_A_K_Aa_A_N_A1_A_061_A0_A_N_A1_092_060close_062, axiom,
    (((minus_minus_complex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) @ one_one_complex) = (minus_minus_complex @ zero_zero_complex @ one_one_complex)))). % \<open>1 + w ^ k * a - 1 = 0 - 1\<close>
thf(fact_4_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062w_O_A1_A_L_Aw_A_094_Ak_A_K_Aa_A_061_A0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![W : complex]: (~ (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ W @ k) @ a)) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>w. 1 + w ^ k * a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N)) = one_one_complex)))). % minus_one_mult_self
thf(fact_7_left__minus__one__mult__self, axiom,
    ((![N : nat, A : complex]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_8_mult__minus1, axiom,
    ((![Z : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z) = (uminus1204672759omplex @ Z))))). % mult_minus1
thf(fact_9_mult__minus1__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z))))). % mult_minus1_right
thf(fact_10_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_11_mult__minus__left, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_left
thf(fact_12_minus__mult__minus, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (times_times_complex @ A @ B))))). % minus_mult_minus
thf(fact_13_mult__minus__right, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ A @ (uminus1204672759omplex @ B)) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_right
thf(fact_14_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_15_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_16_power__minus, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ N) = (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ A @ N)))))). % power_minus
thf(fact_17_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_18_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_19_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_20_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_21_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_22_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_23_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_24_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_25_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_26_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_27_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_28_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_29_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_30_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_31_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_32_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_33_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_34_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_35_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_36_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_37_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_38_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_39_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_40_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_41_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_42_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_43_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_44_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_45_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_46_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_47_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_48_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_49_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_50_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_51_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_52_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_53_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_54_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_55_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_56_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_57_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_58_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_59_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_60_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_61_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_62_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_63_minus__add__distrib, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)))))). % minus_add_distrib
thf(fact_64_minus__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ (plus_plus_complex @ A @ B)) = B)))). % minus_add_cancel
thf(fact_65_add__minus__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ A @ (plus_plus_complex @ (uminus1204672759omplex @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_66_minus__diff__eq, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (minus_minus_complex @ A @ B)) = (minus_minus_complex @ B @ A))))). % minus_diff_eq
thf(fact_67_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_68_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_69_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_70_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_71_diff__add__zero, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ A @ (plus_plus_nat @ A @ B)) = zero_zero_nat)))). % diff_add_zero
thf(fact_72_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_73_add_Oright__inverse, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ (uminus1204672759omplex @ A)) = zero_zero_complex)))). % add.right_inverse
thf(fact_74_add_Oleft__inverse, axiom,
    ((![A : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ A) = zero_zero_complex)))). % add.left_inverse
thf(fact_75_diff__0, axiom,
    ((![A : complex]: ((minus_minus_complex @ zero_zero_complex @ A) = (uminus1204672759omplex @ A))))). % diff_0
thf(fact_76_uminus__add__conv__diff, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ B) = (minus_minus_complex @ B @ A))))). % uminus_add_conv_diff
thf(fact_77_diff__minus__eq__add, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ A @ (uminus1204672759omplex @ B)) = (plus_plus_complex @ A @ B))))). % diff_minus_eq_add
thf(fact_78_add__neg__numeral__special_I7_J, axiom,
    (((plus_plus_complex @ one_one_complex @ (uminus1204672759omplex @ one_one_complex)) = zero_zero_complex))). % add_neg_numeral_special(7)
thf(fact_79_add__neg__numeral__special_I8_J, axiom,
    (((plus_plus_complex @ (uminus1204672759omplex @ one_one_complex) @ one_one_complex) = zero_zero_complex))). % add_neg_numeral_special(8)
thf(fact_80_diff__numeral__special_I12_J, axiom,
    (((minus_minus_complex @ (uminus1204672759omplex @ one_one_complex) @ (uminus1204672759omplex @ one_one_complex)) = zero_zero_complex))). % diff_numeral_special(12)
thf(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_82_is__num__normalize_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % is_num_normalize(1)
thf(fact_83_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_84_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_85_group__cancel_Oadd1, axiom,
    ((![A2 : complex, K : complex, A : complex, B : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A2 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_86_group__cancel_Oadd2, axiom,
    ((![B2 : complex, K : complex, B : complex, A : complex]: ((B2 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B2) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_87_group__cancel_Osub1, axiom,
    ((![A2 : complex, K : complex, A : complex, B : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((minus_minus_complex @ A2 @ B) = (plus_plus_complex @ K @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub1
thf(fact_88_diff__eq__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((minus_minus_complex @ A @ B) = C) = (A = (plus_plus_complex @ C @ B)))))). % diff_eq_eq
thf(fact_89_eq__diff__eq, axiom,
    ((![A : complex, C : complex, B : complex]: ((A = (minus_minus_complex @ C @ B)) = ((plus_plus_complex @ A @ B) = C))))). % eq_diff_eq
thf(fact_90_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_91_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_92_add__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ B) @ C))))). % add_diff_eq
thf(fact_93_diff__diff__eq2, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_diff_eq2
thf(fact_94_add_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.assoc
thf(fact_95_diff__add__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_add_eq
thf(fact_96_add_Oleft__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_97_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_98_add_Oright__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_99_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z2 : complex]: (Y2 = Z2))) = (^[A3 : complex]: (^[B3 : complex]: ((minus_minus_complex @ A3 @ B3) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_100_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A3 : complex]: (^[B3 : complex]: (plus_plus_complex @ B3 @ A3)))))). % add.commute
thf(fact_101_add_Ocomm__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.comm_neutral
thf(fact_102_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_103_add_Ogroup__left__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.group_left_neutral
thf(fact_104_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.left_commute
thf(fact_105_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_106_diff__add__eq__diff__diff__swap, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ (minus_minus_complex @ A @ C) @ B))))). % diff_add_eq_diff_diff_swap
thf(fact_107_diff__diff__add, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ A @ (plus_plus_complex @ B @ C)))))). % diff_diff_add
thf(fact_108_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_109_add__implies__diff, axiom,
    ((![C : complex, B : complex, A : complex]: (((plus_plus_complex @ C @ B) = A) => (C = (minus_minus_complex @ A @ B)))))). % add_implies_diff
thf(fact_110_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_111_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_112_square__diff__square__factored, axiom,
    ((![X : complex, Y : complex]: ((minus_minus_complex @ (times_times_complex @ X @ X) @ (times_times_complex @ Y @ Y)) = (times_times_complex @ (plus_plus_complex @ X @ Y) @ (minus_minus_complex @ X @ Y)))))). % square_diff_square_factored
thf(fact_113_eq__add__iff2, axiom,
    ((![A : complex, E : complex, C : complex, B : complex, D : complex]: (((plus_plus_complex @ (times_times_complex @ A @ E) @ C) = (plus_plus_complex @ (times_times_complex @ B @ E) @ D)) = (C = (plus_plus_complex @ (times_times_complex @ (minus_minus_complex @ B @ A) @ E) @ D)))))). % eq_add_iff2
thf(fact_114_eq__add__iff1, axiom,
    ((![A : complex, E : complex, C : complex, B : complex, D : complex]: (((plus_plus_complex @ (times_times_complex @ A @ E) @ C) = (plus_plus_complex @ (times_times_complex @ B @ E) @ D)) = ((plus_plus_complex @ (times_times_complex @ (minus_minus_complex @ A @ B) @ E) @ C) = D))))). % eq_add_iff1
thf(fact_115_ab__group__add__class_Oab__diff__conv__add__uminus, axiom,
    ((minus_minus_complex = (^[A3 : complex]: (^[B3 : complex]: (plus_plus_complex @ A3 @ (uminus1204672759omplex @ B3))))))). % ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_116_diff__conv__add__uminus, axiom,
    ((minus_minus_complex = (^[A3 : complex]: (^[B3 : complex]: (plus_plus_complex @ A3 @ (uminus1204672759omplex @ B3))))))). % diff_conv_add_uminus
thf(fact_117_group__cancel_Osub2, axiom,
    ((![B2 : complex, K : complex, B : complex, A : complex]: ((B2 = (plus_plus_complex @ K @ B)) => ((minus_minus_complex @ A @ B2) = (plus_plus_complex @ (uminus1204672759omplex @ K) @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub2
thf(fact_118_neg__eq__iff__add__eq__0, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = B) = ((plus_plus_complex @ A @ B) = zero_zero_complex))))). % neg_eq_iff_add_eq_0
thf(fact_119_eq__neg__iff__add__eq__0, axiom,
    ((![A : complex, B : complex]: ((A = (uminus1204672759omplex @ B)) = ((plus_plus_complex @ A @ B) = zero_zero_complex))))). % eq_neg_iff_add_eq_0
thf(fact_120_add_Oinverse__unique, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = zero_zero_complex) => ((uminus1204672759omplex @ A) = B))))). % add.inverse_unique
thf(fact_121_ab__group__add__class_Oab__left__minus, axiom,
    ((![A : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ A) = zero_zero_complex)))). % ab_group_add_class.ab_left_minus
thf(fact_122_add__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = zero_zero_complex) = (B = (uminus1204672759omplex @ A)))))). % add_eq_0_iff
thf(fact_123_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_124_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_125_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_126_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_127_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_128_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_129_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_130_combine__common__factor, axiom,
    ((![A : complex, E : complex, B : complex, C : complex]: ((plus_plus_complex @ (times_times_complex @ A @ E) @ (plus_plus_complex @ (times_times_complex @ B @ E) @ C)) = (plus_plus_complex @ (times_times_complex @ (plus_plus_complex @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_131_distrib__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % distrib_right
thf(fact_132_distrib__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % distrib_left
thf(fact_133_comm__semiring__class_Odistrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_134_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_135_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_136_minus__diff__commute, axiom,
    ((![B : complex, A : complex]: ((minus_minus_complex @ (uminus1204672759omplex @ B) @ A) = (minus_minus_complex @ (uminus1204672759omplex @ A) @ B))))). % minus_diff_commute
thf(fact_137_add_Oinverse__distrib__swap, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ B) @ (uminus1204672759omplex @ A)))))). % add.inverse_distrib_swap
thf(fact_138_group__cancel_Oneg1, axiom,
    ((![A2 : complex, K : complex, A : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((uminus1204672759omplex @ A2) = (plus_plus_complex @ (uminus1204672759omplex @ K) @ (uminus1204672759omplex @ A))))))). % group_cancel.neg1
thf(fact_139_is__num__normalize_I8_J, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ B) @ (uminus1204672759omplex @ A)))))). % is_num_normalize(8)
thf(fact_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_151_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_152_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_153_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_154_square__diff__one__factored, axiom,
    ((![X : complex]: ((minus_minus_complex @ (times_times_complex @ X @ X) @ one_one_complex) = (times_times_complex @ (plus_plus_complex @ X @ one_one_complex) @ (minus_minus_complex @ X @ one_one_complex)))))). % square_diff_one_factored
thf(fact_155_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_156_mult_Ocommute, axiom,
    ((times_times_complex = (^[A3 : complex]: (^[B3 : complex]: (times_times_complex @ B3 @ A3)))))). % mult.commute
thf(fact_157_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_158_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_159_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_160_minus__equation__iff, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = B) = ((uminus1204672759omplex @ B) = A))))). % minus_equation_iff
thf(fact_161_equation__minus__iff, axiom,
    ((![A : complex, B : complex]: ((A = (uminus1204672759omplex @ B)) = (B = (uminus1204672759omplex @ A)))))). % equation_minus_iff
thf(fact_162_zero__neq__neg__one, axiom,
    ((~ ((zero_zero_complex = (uminus1204672759omplex @ one_one_complex)))))). % zero_neq_neg_one
thf(fact_163_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_164_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_165_minus__mult__commute, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (times_times_complex @ A @ (uminus1204672759omplex @ B)))))). % minus_mult_commute

% Conjectures (1)
thf(conj_0, conjecture,
    (((times_times_complex @ (power_power_complex @ w @ k) @ a) = (uminus1204672759omplex @ one_one_complex)))).
