% 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_758__5375302_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:52.479

% Could-be-implicit typings (3)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (26)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex, type,
    invers502456322omplex : complex > complex).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal, type,
    inverse_inverse_real : real > real).
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_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
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_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
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_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_k____, type,
    k : nat).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_thesis____, type,
    thesis : $o).
thf(sy_v_w____, type,
    w : complex).

% Relevant facts (246)
thf(fact_0__092_060open_062_092_060And_062d2_O_A_I0_058_058_063_Ha_J_A_060_Ad2_A_092_060Longrightarrow_062_A_092_060exists_062e_0620_058_058_063_Ha_O_Ae_A_060_A_I1_058_058_063_Ha_J_A_092_060and_062_Ae_A_060_Ad2_092_060close_062, axiom,
    ((![D2 : real]: ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ one_one_real) & (ord_less_real @ E @ D2)))))))). % \<open>\<And>d2. (0::?'a) < d2 \<Longrightarrow> \<exists>e>0::?'a. e < (1::?'a) \<and> e < d2\<close>
thf(fact_1_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_2_inv0, axiom,
    ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m))))). % inv0
thf(fact_3_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_4_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_5_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_6_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_7_left__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % left_inverse
thf(fact_8_left__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ (inverse_inverse_real @ A) @ A) = one_one_real))))). % left_inverse
thf(fact_9_right__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ A @ (invers502456322omplex @ A)) = one_one_complex))))). % right_inverse
thf(fact_10_right__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ A @ (inverse_inverse_real @ A)) = one_one_real))))). % right_inverse
thf(fact_11_power__strict__decreasing__iff, axiom,
    ((![B : real, M : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_12_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_13_not__real__square__gt__zero, axiom,
    ((![X3 : real]: ((~ ((ord_less_real @ zero_zero_real @ (times_times_real @ X3 @ X3)))) = (X3 = zero_zero_real))))). % not_real_square_gt_zero
thf(fact_14_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_15_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_16_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_17_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_18_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_19_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_20_inverse__less__iff__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less
thf(fact_21_inverse__less__iff__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less_neg
thf(fact_22_inverse__negative__iff__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % inverse_negative_iff_negative
thf(fact_23_inverse__positive__iff__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % inverse_positive_iff_positive
thf(fact_24_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_25_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_26_inverse__inverse__eq, axiom,
    ((![A : real]: ((inverse_inverse_real @ (inverse_inverse_real @ A)) = A)))). % inverse_inverse_eq
thf(fact_27_inverse__inverse__eq, axiom,
    ((![A : complex]: ((invers502456322omplex @ (invers502456322omplex @ A)) = A)))). % inverse_inverse_eq
thf(fact_28_inverse__eq__iff__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_29_inverse__eq__iff__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_30_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_31_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_32_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_33_inverse__nonzero__iff__nonzero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % inverse_nonzero_iff_nonzero
thf(fact_34_inverse__nonzero__iff__nonzero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % inverse_nonzero_iff_nonzero
thf(fact_35_inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % inverse_zero
thf(fact_36_inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % inverse_zero
thf(fact_37_nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X3 @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_38_inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)))))). % inverse_mult_distrib
thf(fact_39_inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)))))). % inverse_mult_distrib
thf(fact_40_inverse__1, axiom,
    (((inverse_inverse_real @ one_one_real) = one_one_real))). % inverse_1
thf(fact_41_inverse__1, axiom,
    (((invers502456322omplex @ one_one_complex) = one_one_complex))). % inverse_1
thf(fact_42_inverse__eq__1__iff, axiom,
    ((![X3 : real]: (((inverse_inverse_real @ X3) = one_one_real) = (X3 = one_one_real))))). % inverse_eq_1_iff
thf(fact_43_inverse__eq__1__iff, axiom,
    ((![X3 : complex]: (((invers502456322omplex @ X3) = one_one_complex) = (X3 = one_one_complex))))). % inverse_eq_1_iff
thf(fact_44_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_45_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_46_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_47_sum__squares__eq__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: (((plus_plus_real @ (times_times_real @ X3 @ X3) @ (times_times_real @ Y2 @ Y2)) = zero_zero_real) = (((X3 = zero_zero_real)) & ((Y2 = zero_zero_real))))))). % sum_squares_eq_zero_iff
thf(fact_48_power__inject__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M) = (power_power_real @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_49_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_50_power__strict__increasing__iff, axiom,
    ((![B : real, X3 : nat, Y2 : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X3) @ (power_power_real @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_51_power__strict__increasing__iff, axiom,
    ((![B : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X3) @ (power_power_nat @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_52_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_53_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_54_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_55_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_56_power__mult, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M @ N)) = (power_power_real @ (power_power_real @ A @ M) @ N))))). % power_mult
thf(fact_57_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_58_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_59_power__eq__imp__eq__norm, axiom,
    ((![W : complex, N : nat, Z2 : complex]: (((power_power_complex @ W @ N) = (power_power_complex @ Z2 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V638595069omplex @ W) = (real_V638595069omplex @ Z2))))))). % power_eq_imp_eq_norm
thf(fact_60_power__eq__imp__eq__norm, axiom,
    ((![W : real, N : nat, Z2 : real]: (((power_power_real @ W @ N) = (power_power_real @ Z2 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V646646907m_real @ W) = (real_V646646907m_real @ Z2))))))). % power_eq_imp_eq_norm
thf(fact_61_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X : real]: (((ord_less_real @ zero_zero_real @ X) & ((power_power_real @ X @ N) = A)) & (![Y : real]: (((ord_less_real @ zero_zero_real @ Y) & ((power_power_real @ Y @ N) = A)) => (Y = X)))))))))). % realpow_pos_nth_unique
thf(fact_62_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R : real]: ((ord_less_real @ zero_zero_real @ R) & ((power_power_real @ R @ N) = A)))))))). % realpow_pos_nth
thf(fact_63_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_64_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_65_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_66_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_67_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_68_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_69_one__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ one_one_real @ (power_power_real @ A @ N))))))). % one_less_power
thf(fact_70_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_71_power__eq__1__iff, axiom,
    ((![W : complex, N : nat]: (((power_power_complex @ W @ N) = one_one_complex) => (((real_V638595069omplex @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_72_power__eq__1__iff, axiom,
    ((![W : real, N : nat]: (((power_power_real @ W @ N) = one_one_real) => (((real_V646646907m_real @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_73_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_74_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y3 : real]: (ord_less_real @ Y3 @ X4))))). % linordered_field_no_lb
thf(fact_75_inverse__eq__imp__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_76_inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_77_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_78_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_79_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_80_power__commuting__commutes, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = (times_times_real @ Y2 @ X3)) => ((times_times_real @ (power_power_real @ X3 @ N) @ Y2) = (times_times_real @ Y2 @ (power_power_real @ X3 @ N))))))). % power_commuting_commutes
thf(fact_81_power__commuting__commutes, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = (times_times_complex @ Y2 @ X3)) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ Y2) = (times_times_complex @ Y2 @ (power_power_complex @ X3 @ N))))))). % power_commuting_commutes
thf(fact_82_power__commuting__commutes, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = (times_times_nat @ Y2 @ X3)) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ Y2) = (times_times_nat @ Y2 @ (power_power_nat @ X3 @ N))))))). % power_commuting_commutes
thf(fact_83_power__mult__distrib, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (times_times_real @ A @ B) @ N) = (times_times_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_mult_distrib
thf(fact_84_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_85_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_86_power__commutes, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ A @ N) @ A) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_commutes
thf(fact_87_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_88_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_89_nonzero__imp__inverse__nonzero, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => (~ (((inverse_inverse_real @ A) = zero_zero_real))))))). % nonzero_imp_inverse_nonzero
thf(fact_90_nonzero__imp__inverse__nonzero, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (~ (((invers502456322omplex @ A) = zero_zero_complex))))))). % nonzero_imp_inverse_nonzero
thf(fact_91_nonzero__inverse__inverse__eq, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((inverse_inverse_real @ (inverse_inverse_real @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_92_nonzero__inverse__inverse__eq, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((invers502456322omplex @ (invers502456322omplex @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_93_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) => ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_94_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_95_inverse__zero__imp__zero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) => (A = zero_zero_real))))). % inverse_zero_imp_zero
thf(fact_96_inverse__zero__imp__zero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) => (A = zero_zero_complex))))). % inverse_zero_imp_zero
thf(fact_97_field__class_Ofield__inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % field_class.field_inverse_zero
thf(fact_98_field__class_Ofield__inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % field_class.field_inverse_zero
thf(fact_99_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y2 : real, X3 : real]: (((times_times_real @ Y2 @ X3) = (times_times_real @ X3 @ Y2)) => ((times_times_real @ (inverse_inverse_real @ Y2) @ X3) = (times_times_real @ X3 @ (inverse_inverse_real @ Y2))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_100_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y2 : complex, X3 : complex]: (((times_times_complex @ Y2 @ X3) = (times_times_complex @ X3 @ Y2)) => ((times_times_complex @ (invers502456322omplex @ Y2) @ X3) = (times_times_complex @ X3 @ (invers502456322omplex @ Y2))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_101_power__inverse, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (inverse_inverse_real @ A) @ N) = (inverse_inverse_real @ (power_power_real @ A @ N)))))). % power_inverse
thf(fact_102_power__inverse, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (invers502456322omplex @ A) @ N) = (invers502456322omplex @ (power_power_complex @ A @ N)))))). % power_inverse
thf(fact_103_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_104_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_105_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_real @ zero_zero_real @ N) = one_one_real)) & ((~ ((N = zero_zero_nat))) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real)))))). % power_0_left
thf(fact_106_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_107_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_108_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_strict_increasing
thf(fact_109_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_110_power__less__imp__less__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_111_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_112_left__right__inverse__power, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = one_one_real) => ((times_times_real @ (power_power_real @ X3 @ N) @ (power_power_real @ Y2 @ N)) = one_one_real))))). % left_right_inverse_power
thf(fact_113_left__right__inverse__power, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ (power_power_complex @ Y2 @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_114_left__right__inverse__power, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ (power_power_nat @ Y2 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_115_positive__imp__inverse__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)))))). % positive_imp_inverse_positive
thf(fact_116_negative__imp__inverse__negative, axiom,
    ((![A : real]: ((ord_less_real @ A @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real))))). % negative_imp_inverse_negative
thf(fact_117_inverse__positive__imp__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) => ((~ ((A = zero_zero_real))) => (ord_less_real @ zero_zero_real @ A)))))). % inverse_positive_imp_positive
thf(fact_118_inverse__negative__imp__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) => ((~ ((A = zero_zero_real))) => (ord_less_real @ A @ zero_zero_real)))))). % inverse_negative_imp_negative
thf(fact_119_less__imp__inverse__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less_neg
thf(fact_120_inverse__less__imp__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less_neg
thf(fact_121_less__imp__inverse__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less
thf(fact_122_inverse__less__imp__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less
thf(fact_123_nonzero__inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_124_nonzero__inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ B) @ (invers502456322omplex @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_125_power__add, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (plus_plus_nat @ M @ N)) = (times_times_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)))))). % power_add
thf(fact_126_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_127_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_128_inverse__unique, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ B) = one_one_real) => ((inverse_inverse_real @ A) = B))))). % inverse_unique
thf(fact_129_inverse__unique, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = one_one_complex) => ((invers502456322omplex @ A) = B))))). % inverse_unique
thf(fact_130_norm__triangle__lt, axiom,
    ((![X3 : complex, Y2 : complex, E2 : real]: ((ord_less_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)) @ E2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ E2))))). % norm_triangle_lt
thf(fact_131_norm__triangle__lt, axiom,
    ((![X3 : real, Y2 : real, E2 : real]: ((ord_less_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)) @ E2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ E2))))). % norm_triangle_lt
thf(fact_132_norm__add__less, axiom,
    ((![X3 : complex, R2 : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R2) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ (plus_plus_real @ R2 @ S2))))))). % norm_add_less
thf(fact_133_norm__add__less, axiom,
    ((![X3 : real, R2 : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R2) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ (plus_plus_real @ R2 @ S2))))))). % norm_add_less
thf(fact_134_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_135_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_136_norm__mult, axiom,
    ((![X3 : complex, Y2 : complex]: ((real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) = (times_times_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)))))). % norm_mult
thf(fact_137_norm__mult, axiom,
    ((![X3 : real, Y2 : real]: ((real_V646646907m_real @ (times_times_real @ X3 @ Y2)) = (times_times_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)))))). % norm_mult
thf(fact_138_norm__power, axiom,
    ((![X3 : complex, N : nat]: ((real_V638595069omplex @ (power_power_complex @ X3 @ N)) = (power_power_real @ (real_V638595069omplex @ X3) @ N))))). % norm_power
thf(fact_139_norm__power, axiom,
    ((![X3 : real, N : nat]: ((real_V646646907m_real @ (power_power_real @ X3 @ N)) = (power_power_real @ (real_V646646907m_real @ X3) @ N))))). % norm_power
thf(fact_140_norm__inverse, axiom,
    ((![A : complex]: ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A)))))). % norm_inverse
thf(fact_141_norm__inverse, axiom,
    ((![A : real]: ((real_V646646907m_real @ (inverse_inverse_real @ A)) = (inverse_inverse_real @ (real_V646646907m_real @ A)))))). % norm_inverse
thf(fact_142_sum__squares__gt__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ (times_times_real @ X3 @ X3) @ (times_times_real @ Y2 @ Y2))) = (((~ ((X3 = zero_zero_real)))) | ((~ ((Y2 = zero_zero_real))))))))). % sum_squares_gt_zero_iff
thf(fact_143_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_strict_decreasing
thf(fact_144_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_145_power__less__power__Suc, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (times_times_real @ A @ (power_power_real @ A @ N))))))). % power_less_power_Suc
thf(fact_146_power__less__power__Suc, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_less_power_Suc
thf(fact_147_power__gt1__lemma, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => (ord_less_real @ one_one_real @ (times_times_real @ A @ (power_power_real @ A @ N))))))). % power_gt1_lemma
thf(fact_148_power__gt1__lemma, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ one_one_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_gt1_lemma
thf(fact_149_one__less__inverse__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ one_one_real @ (inverse_inverse_real @ X3)) = (((ord_less_real @ zero_zero_real @ X3)) & ((ord_less_real @ X3 @ one_one_real))))))). % one_less_inverse_iff
thf(fact_150_one__less__inverse, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ one_one_real @ (inverse_inverse_real @ A))))))). % one_less_inverse
thf(fact_151_division__ring__inverse__add, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => ((plus_plus_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (times_times_real @ (times_times_real @ (inverse_inverse_real @ A) @ (plus_plus_real @ A @ B)) @ (inverse_inverse_real @ B)))))))). % division_ring_inverse_add
thf(fact_152_division__ring__inverse__add, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((plus_plus_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)) = (times_times_complex @ (times_times_complex @ (invers502456322omplex @ A) @ (plus_plus_complex @ A @ B)) @ (invers502456322omplex @ B)))))))). % division_ring_inverse_add
thf(fact_153_inverse__add, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => ((plus_plus_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (times_times_real @ (times_times_real @ (plus_plus_real @ A @ B) @ (inverse_inverse_real @ A)) @ (inverse_inverse_real @ B)))))))). % inverse_add
thf(fact_154_inverse__add, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((plus_plus_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)) = (times_times_complex @ (times_times_complex @ (plus_plus_complex @ A @ B) @ (invers502456322omplex @ A)) @ (invers502456322omplex @ B)))))))). % inverse_add
thf(fact_155_field__class_Ofield__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ (inverse_inverse_real @ A) @ A) = one_one_real))))). % field_class.field_inverse
thf(fact_156_field__class_Ofield__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % field_class.field_inverse
thf(fact_157_norm__mult__less, axiom,
    ((![X3 : complex, R2 : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R2) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) @ (times_times_real @ R2 @ S2))))))). % norm_mult_less
thf(fact_158_norm__mult__less, axiom,
    ((![X3 : real, R2 : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R2) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (times_times_real @ X3 @ Y2)) @ (times_times_real @ R2 @ S2))))))). % norm_mult_less
thf(fact_159_nonzero__norm__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A))))))). % nonzero_norm_inverse
thf(fact_160_nonzero__norm__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((real_V646646907m_real @ (inverse_inverse_real @ A)) = (inverse_inverse_real @ (real_V646646907m_real @ A))))))). % nonzero_norm_inverse
thf(fact_161_power__Suc__less, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (times_times_real @ A @ (power_power_real @ A @ N)) @ (power_power_real @ A @ N))))))). % power_Suc_less
thf(fact_162_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_163_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_164_add__gr__0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_165_mult__cancel__right2, axiom,
    ((![A : real, C : real]: (((times_times_real @ A @ C) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_right2
thf(fact_166_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_167_mult__cancel__right1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_right1
thf(fact_168_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_169_mult__cancel__left2, axiom,
    ((![C : real, A : real]: (((times_times_real @ C @ A) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_left2
thf(fact_170_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_171_mult__cancel__left1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_left1
thf(fact_172_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_173_zero__less__double__add__iff__zero__less__single__add, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_double_add_iff_zero_less_single_add
thf(fact_174_double__add__less__zero__iff__single__add__less__zero, axiom,
    ((![A : real]: ((ord_less_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % double_add_less_zero_iff_single_add_less_zero
thf(fact_175_less__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel2
thf(fact_176_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_177_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_178_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_179_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_180_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_181_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_182__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,
    ((~ ((![W2 : complex]: (~ (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ W2 @ k) @ a)) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>w. 1 + w ^ k * a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_183_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_184_mult__zero__left, axiom,
    ((![A : real]: ((times_times_real @ zero_zero_real @ A) = zero_zero_real)))). % mult_zero_left
thf(fact_185_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_186_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_187_mult__zero__right, axiom,
    ((![A : real]: ((times_times_real @ A @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_188_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_189_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_190_mult__eq__0__iff, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ B) = zero_zero_real) = (((A = zero_zero_real)) | ((B = zero_zero_real))))))). % mult_eq_0_iff
thf(fact_191_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_192_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_193_mult__cancel__left, axiom,
    ((![C : real, A : real, B : real]: (((times_times_real @ C @ A) = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((A = B))))))). % mult_cancel_left
thf(fact_194_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_195_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_196_mult__cancel__right, axiom,
    ((![A : real, C : real, B : real]: (((times_times_real @ A @ C) = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((A = B))))))). % mult_cancel_right
thf(fact_197_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_198_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_199_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_200_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_201_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_202_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_203_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_204_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_205_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_206_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_207_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_208_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_209_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_210_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_211_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_212_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_213_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_214_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_215_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_216_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_217_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_218_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_219_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y2 : nat]: (((plus_plus_nat @ X3 @ Y2) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_220_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y2)) = (((X3 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_221_add__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_real @ A @ B))))). % add_less_cancel_left
thf(fact_222_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_223_add__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_real @ A @ B))))). % add_less_cancel_right
thf(fact_224_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_225_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_226_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_227_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_228_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_229_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_230_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_231_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_232_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_233_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_234_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_235_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_236_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_237_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_238_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_239_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_240_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_241_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_242_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_243_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_244_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_245_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

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![T : real]: ((ord_less_real @ zero_zero_real @ T) => ((ord_less_real @ T @ one_one_real) => ((ord_less_real @ T @ (inverse_inverse_real @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m))) => thesis)))))).
thf(conj_1, conjecture,
    (thesis)).
