% 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_751__5375246_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:48.130

% 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_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_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_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 (210)
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_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_3_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_4__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_5_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_6_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_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_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_12_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_13_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_14_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_15_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_16_add__diff__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_17_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_18_add__diff__cancel__left_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ A) = B)))). % add_diff_cancel_left'
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_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_21_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_22_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_23_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_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__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_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__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_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__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_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__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
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_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_45_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_46_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_47_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_48_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_49_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_50_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_51_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_52_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_53_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_54_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_55_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_56_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_57_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_58_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_59_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_60_add__diff__cancel__right_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ B) = A)))). % add_diff_cancel_right'
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, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_63_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_64_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_65_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_66_power__add, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_67_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_68_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_69_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_70_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_71_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_72_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_73_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_74_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_75_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_76_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_77_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_78_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_79_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_80_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_81_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_82_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_83_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_84_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_85_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_86_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A2)))))). % add.commute
thf(fact_87_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_88_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_89_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_90_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_91_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_92_group__cancel_Oadd2, axiom,
    ((![B3 : complex, K : complex, B : complex, A : complex]: ((B3 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B3) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_93_group__cancel_Oadd2, axiom,
    ((![B3 : nat, K : nat, B : nat, A : nat]: ((B3 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_94_group__cancel_Oadd1, axiom,
    ((![A3 : complex, K : complex, A : complex, B : complex]: ((A3 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A3 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_95_group__cancel_Oadd1, axiom,
    ((![A3 : nat, K : nat, A : nat, B : nat]: ((A3 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A3 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_96_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_97_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_98_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_99_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_100_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_101_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_add_Ogroup__left__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.group_left_neutral
thf(fact_116_add_Ocomm__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.comm_neutral
thf(fact_117_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_118_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_119_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_120_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_121_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_122_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z : complex]: (Y2 = Z))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_123_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_124_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_125_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_126_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_127_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_128_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_129_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_130_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_131_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_132_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_133_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_134_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_add__implies__diff, axiom,
    ((![C : nat, B : nat, A : nat]: (((plus_plus_nat @ C @ B) = A) => (C = (minus_minus_nat @ A @ B)))))). % add_implies_diff
thf(fact_147_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_148_diff__diff__add, axiom,
    ((![A : nat, B : nat, C : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ B) @ C) = (minus_minus_nat @ A @ (plus_plus_nat @ B @ C)))))). % diff_diff_add
thf(fact_149_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_150_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_151_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_152_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_153_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_154_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_155_group__cancel_Osub1, axiom,
    ((![A3 : complex, K : complex, A : complex, B : complex]: ((A3 = (plus_plus_complex @ K @ A)) => ((minus_minus_complex @ A3 @ B) = (plus_plus_complex @ K @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub1
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_mult__diff__mult, axiom,
    ((![X : complex, Y : complex, A : complex, B : complex]: ((minus_minus_complex @ (times_times_complex @ X @ Y) @ (times_times_complex @ A @ B)) = (plus_plus_complex @ (times_times_complex @ X @ (minus_minus_complex @ Y @ B)) @ (times_times_complex @ (minus_minus_complex @ X @ A) @ B)))))). % mult_diff_mult
thf(fact_171_add__scale__eq__noteq, axiom,
    ((![R : complex, A : complex, B : complex, C : complex, D : complex]: ((~ ((R = zero_zero_complex))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_complex @ A @ (times_times_complex @ R @ C)) = (plus_plus_complex @ B @ (times_times_complex @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_172_add__scale__eq__noteq, axiom,
    ((![R : nat, A : nat, B : nat, C : nat, D : nat]: ((~ ((R = zero_zero_nat))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_nat @ A @ (times_times_nat @ R @ C)) = (plus_plus_nat @ B @ (times_times_nat @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_173_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_174_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_175_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_176_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_177_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_178_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_179_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_180_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_181_diff__diff__left, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ I @ (plus_plus_nat @ J @ K)))))). % diff_diff_left
thf(fact_182_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_183_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_184_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_185_Nat_Odiff__cancel, axiom,
    ((![K : nat, M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (minus_minus_nat @ M @ N))))). % Nat.diff_cancel
thf(fact_186_diff__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ K) @ (plus_plus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))). % diff_cancel2
thf(fact_187_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_188_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_189_diff__add__inverse, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ (plus_plus_nat @ N @ M) @ N) = M)))). % diff_add_inverse
thf(fact_190_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_191_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_192_diff__add__inverse2, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ N) @ N) = M)))). % diff_add_inverse2
thf(fact_193_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_194_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_195_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_196_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_197_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_198_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_199_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_200_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_201_mult__eq__if, axiom,
    ((times_times_nat = (^[M2 : nat]: (^[N2 : nat]: (if_nat @ (M2 = zero_zero_nat) @ zero_zero_nat @ (plus_plus_nat @ N2 @ (times_times_nat @ (minus_minus_nat @ M2 @ one_one_nat) @ N2)))))))). % mult_eq_if
thf(fact_202_diff__add__0, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ N @ (plus_plus_nat @ N @ M)) = zero_zero_nat)))). % diff_add_0
thf(fact_203_add__0__iff, axiom,
    ((![B : complex, A : complex]: ((B = (plus_plus_complex @ B @ A)) = (A = zero_zero_complex))))). % add_0_iff
thf(fact_204_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_205_crossproduct__eq, axiom,
    ((![W2 : complex, Y : complex, X : complex, Z2 : complex]: (((plus_plus_complex @ (times_times_complex @ W2 @ Y) @ (times_times_complex @ X @ Z2)) = (plus_plus_complex @ (times_times_complex @ W2 @ Z2) @ (times_times_complex @ X @ Y))) = (((W2 = X)) | ((Y = Z2))))))). % crossproduct_eq
thf(fact_206_crossproduct__eq, axiom,
    ((![W2 : nat, Y : nat, X : nat, Z2 : nat]: (((plus_plus_nat @ (times_times_nat @ W2 @ Y) @ (times_times_nat @ X @ Z2)) = (plus_plus_nat @ (times_times_nat @ W2 @ Z2) @ (times_times_nat @ X @ Y))) = (((W2 = X)) | ((Y = Z2))))))). % crossproduct_eq
thf(fact_207_crossproduct__noteq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ D)) = (plus_plus_complex @ (times_times_complex @ A @ D) @ (times_times_complex @ B @ C))))))))). % crossproduct_noteq
thf(fact_208_crossproduct__noteq, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D)) = (plus_plus_nat @ (times_times_nat @ A @ D) @ (times_times_nat @ B @ C))))))))). % crossproduct_noteq
thf(fact_209_add__diff__add, axiom,
    ((![A : complex, C : complex, B : complex, D : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ D)) = (plus_plus_complex @ (minus_minus_complex @ A @ B) @ (minus_minus_complex @ C @ D)))))). % add_diff_add

% 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 (1)
thf(conj_0, conjecture,
    (((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)))).
