% 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_213__5369636_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:17.006

% 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 (31)
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_Osgn__class_Osgn_001t__Complex__Ocomplex, type,
    sgn_sgn_complex : complex > complex).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal, type,
    sgn_sgn_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_NthRoot_Oroot, type,
    root : nat > 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_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex, type,
    real_V306493662omplex : real > complex).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal, type,
    real_V1205483228l_real : real > real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_v_b, type,
    b : complex).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).
thf(sy_v_v____, type,
    v : complex).

% Relevant facts (235)
thf(fact_0__092_060open_062root_An_A_Icmod_Ab_J_A_094_An_A_061_Acmod_Ab_092_060close_062, axiom,
    (((power_power_real @ (root @ na @ (real_V638595069omplex @ b)) @ na) = (real_V638595069omplex @ b)))). % \<open>root n (cmod b) ^ n = cmod b\<close>
thf(fact_1_b, axiom,
    ((~ ((b = zero_zero_complex))))). % b
thf(fact_2_n, axiom,
    ((~ ((na = zero_zero_nat))))). % n
thf(fact_3_assms_I2_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(2)
thf(fact_4_of__real__power, axiom,
    ((![X : real, N : nat]: ((real_V1205483228l_real @ (power_power_real @ X @ N)) = (power_power_real @ (real_V1205483228l_real @ X) @ N))))). % of_real_power
thf(fact_5_of__real__power, axiom,
    ((![X : real, N : nat]: ((real_V306493662omplex @ (power_power_real @ X @ N)) = (power_power_complex @ (real_V306493662omplex @ X) @ N))))). % of_real_power
thf(fact_6_of__real__divide, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (divide_divide_real @ X @ Y)) = (divide_divide_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_divide
thf(fact_7_of__real__divide, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (divide_divide_real @ X @ Y)) = (divide1210191872omplex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_divide
thf(fact_8_th0, axiom,
    (((real_V638595069omplex @ (divide1210191872omplex @ (real_V306493662omplex @ (real_V638595069omplex @ b)) @ b)) = one_one_real))). % th0
thf(fact_9_of__real__eq__iff, axiom,
    ((![X : real, Y : real]: (((real_V1205483228l_real @ X) = (real_V1205483228l_real @ Y)) = (X = Y))))). % of_real_eq_iff
thf(fact_10_of__real__eq__iff, axiom,
    ((![X : real, Y : real]: (((real_V306493662omplex @ X) = (real_V306493662omplex @ Y)) = (X = Y))))). % of_real_eq_iff
thf(fact_11_v, axiom,
    ((ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ (divide1210191872omplex @ (real_V306493662omplex @ (real_V638595069omplex @ b)) @ b) @ (power_power_complex @ v @ na))) @ one_one_real))). % v
thf(fact_12_norm__power, axiom,
    ((![X : real, N : nat]: ((real_V646646907m_real @ (power_power_real @ X @ N)) = (power_power_real @ (real_V646646907m_real @ X) @ N))))). % norm_power
thf(fact_13_norm__power, axiom,
    ((![X : complex, N : nat]: ((real_V638595069omplex @ (power_power_complex @ X @ N)) = (power_power_real @ (real_V638595069omplex @ X) @ N))))). % norm_power
thf(fact_14_norm__divide, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (divide1210191872omplex @ A @ B)) = (divide_divide_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)))))). % norm_divide
thf(fact_15_norm__divide, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)))))). % norm_divide
thf(fact_16_power__divide, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_17_power__divide, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (divide_divide_real @ A @ B) @ N) = (divide_divide_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_divide
thf(fact_18_real__root__commute, axiom,
    ((![M : nat, N : nat, X : real]: ((root @ M @ (root @ N @ X)) = (root @ N @ (root @ M @ X)))))). % real_root_commute
thf(fact_19__092_060open_062_092_060exists_062v_O_Acmod_A_Icomplex__of__real_A_Icmod_Ab_J_A_P_Ab_A_L_Av_A_094_An_J_A_060_A1_092_060close_062, axiom,
    ((?[V : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ (divide1210191872omplex @ (real_V306493662omplex @ (real_V638595069omplex @ b)) @ b) @ (power_power_complex @ V @ na))) @ one_one_real)))). % \<open>\<exists>v. cmod (complex_of_real (cmod b) / b + v ^ n) < 1\<close>
thf(fact_20__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_Acmod_A_Icomplex__of__real_A_Icmod_Ab_J_A_P_Ab_A_L_Av_A_094_An_J_A_060_A1_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![V : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ (divide1210191872omplex @ (real_V306493662omplex @ (real_V638595069omplex @ b)) @ b) @ (power_power_complex @ V @ na))) @ one_one_real)))))))). % \<open>\<And>thesis. (\<And>v. cmod (complex_of_real (cmod b) / b + v ^ n) < 1 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_21_sgn__eq, axiom,
    ((sgn_sgn_complex = (^[Z : complex]: (divide1210191872omplex @ Z @ (real_V306493662omplex @ (real_V638595069omplex @ Z))))))). % sgn_eq
thf(fact_22_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_23_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_24_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_25_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_26_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_27_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_28_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_29_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_30_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_31_of__real__eq__0__iff, axiom,
    ((![X : real]: (((real_V306493662omplex @ X) = zero_zero_complex) = (X = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_32_of__real__eq__0__iff, axiom,
    ((![X : real]: (((real_V1205483228l_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_33_sgn__zero, axiom,
    (((sgn_sgn_complex @ zero_zero_complex) = zero_zero_complex))). % sgn_zero
thf(fact_34_sgn__zero, axiom,
    (((sgn_sgn_real @ zero_zero_real) = zero_zero_real))). % sgn_zero
thf(fact_35_of__real__add, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (plus_plus_real @ X @ Y)) = (plus_plus_complex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_add
thf(fact_36_of__real__add, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (plus_plus_real @ X @ Y)) = (plus_plus_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_add
thf(fact_37_sgn__one, axiom,
    (((sgn_sgn_complex @ one_one_complex) = one_one_complex))). % sgn_one
thf(fact_38_sgn__one, axiom,
    (((sgn_sgn_real @ one_one_real) = one_one_real))). % sgn_one
thf(fact_39_power__sgn, axiom,
    ((![A : real, N : nat]: ((sgn_sgn_real @ (power_power_real @ A @ N)) = (power_power_real @ (sgn_sgn_real @ A) @ N))))). % power_sgn
thf(fact_40_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_41_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_42_power__strict__increasing__iff, axiom,
    ((![B : real, X : nat, Y : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X) @ (power_power_real @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_43_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_44_zero__less__norm__iff, axiom,
    ((![X : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X)) = (~ ((X = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_45_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_46_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_47_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_48_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_49_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_50_of__real__eq__1__iff, axiom,
    ((![X : real]: (((real_V306493662omplex @ X) = one_one_complex) = (X = one_one_real))))). % of_real_eq_1_iff
thf(fact_51_of__real__eq__1__iff, axiom,
    ((![X : real]: (((real_V1205483228l_real @ X) = one_one_real) = (X = one_one_real))))). % of_real_eq_1_iff
thf(fact_52_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_53_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_54_norm__sgn, axiom,
    ((![X : complex]: (((X = zero_zero_complex) => ((real_V638595069omplex @ (sgn_sgn_complex @ X)) = zero_zero_real)) & ((~ ((X = zero_zero_complex))) => ((real_V638595069omplex @ (sgn_sgn_complex @ X)) = one_one_real)))))). % norm_sgn
thf(fact_55_norm__sgn, axiom,
    ((![X : real]: (((X = zero_zero_real) => ((real_V646646907m_real @ (sgn_sgn_real @ X)) = zero_zero_real)) & ((~ ((X = zero_zero_real))) => ((real_V646646907m_real @ (sgn_sgn_real @ X)) = one_one_real)))))). % norm_sgn
thf(fact_56_sgn__of__real, axiom,
    ((![R : real]: ((sgn_sgn_complex @ (real_V306493662omplex @ R)) = (real_V306493662omplex @ (sgn_sgn_real @ R)))))). % sgn_of_real
thf(fact_57_sgn__of__real, axiom,
    ((![R : real]: ((sgn_sgn_real @ (real_V1205483228l_real @ R)) = (real_V1205483228l_real @ (sgn_sgn_real @ R)))))). % sgn_of_real
thf(fact_58_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_59_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_60_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_61_sgn__zero__iff, axiom,
    ((![X : complex]: (((sgn_sgn_complex @ X) = zero_zero_complex) = (X = zero_zero_complex))))). % sgn_zero_iff
thf(fact_62_sgn__zero__iff, axiom,
    ((![X : real]: (((sgn_sgn_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % sgn_zero_iff
thf(fact_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_norm__add__less, axiom,
    ((![X : complex, R : real, Y : complex, S : real]: ((ord_less_real @ (real_V638595069omplex @ X) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y) @ S) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X @ Y)) @ (plus_plus_real @ R @ S))))))). % norm_add_less
thf(fact_77_norm__add__less, axiom,
    ((![X : real, R : real, Y : real, S : real]: ((ord_less_real @ (real_V646646907m_real @ X) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y) @ S) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X @ Y)) @ (plus_plus_real @ R @ S))))))). % norm_add_less
thf(fact_78_norm__triangle__lt, axiom,
    ((![X : complex, Y : complex, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V638595069omplex @ X) @ (real_V638595069omplex @ Y)) @ E) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X @ Y)) @ E))))). % norm_triangle_lt
thf(fact_79_norm__triangle__lt, axiom,
    ((![X : real, Y : real, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V646646907m_real @ X) @ (real_V646646907m_real @ Y)) @ E) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X @ Y)) @ E))))). % norm_triangle_lt
thf(fact_80_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z2 : real]: (![X2 : real]: ((P @ X2) => (ord_less_real @ X2 @ Z2)))) => (?[S2 : real]: (![Y2 : real]: ((?[X3 : real]: (((P @ X3)) & ((ord_less_real @ Y2 @ X3)))) = (ord_less_real @ Y2 @ S2))))))))). % real_sup_exists
thf(fact_81_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_82_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_83_norm__less__p1, axiom,
    ((![X : complex]: (ord_less_real @ (real_V638595069omplex @ X) @ (real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ (real_V638595069omplex @ X)) @ one_one_complex)))))). % norm_less_p1
thf(fact_84_norm__less__p1, axiom,
    ((![X : real]: (ord_less_real @ (real_V646646907m_real @ X) @ (real_V646646907m_real @ (plus_plus_real @ (real_V1205483228l_real @ (real_V646646907m_real @ X)) @ one_one_real)))))). % norm_less_p1
thf(fact_85_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_86_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_87_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_88_real__root__divide, axiom,
    ((![N : nat, X : real, Y : real]: ((root @ N @ (divide_divide_real @ X @ Y)) = (divide_divide_real @ (root @ N @ X) @ (root @ N @ Y)))))). % real_root_divide
thf(fact_89_power__one__over, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ one_one_complex @ A) @ N) = (divide1210191872omplex @ one_one_complex @ (power_power_complex @ A @ N)))))). % power_one_over
thf(fact_90_power__one__over, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (divide_divide_real @ one_one_real @ A) @ N) = (divide_divide_real @ one_one_real @ (power_power_real @ A @ N)))))). % power_one_over
thf(fact_91_nonzero__norm__divide, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((real_V638595069omplex @ (divide1210191872omplex @ A @ B)) = (divide_divide_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B))))))). % nonzero_norm_divide
thf(fact_92_nonzero__norm__divide, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => ((real_V646646907m_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B))))))). % nonzero_norm_divide
thf(fact_93_sgn__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((sgn_sgn_real @ A) = one_one_real))))). % sgn_pos
thf(fact_94_divide__less__0__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ (divide_divide_real @ one_one_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % divide_less_0_1_iff
thf(fact_95_divide__less__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ A @ B)))))). % divide_less_eq_1_neg
thf(fact_96_divide__less__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ B @ A)))))). % divide_less_eq_1_pos
thf(fact_97_less__divide__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ B @ A)))))). % less_divide_eq_1_neg
thf(fact_98_less__divide__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ A @ B)))))). % less_divide_eq_1_pos
thf(fact_99_zero__less__divide__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_real @ one_one_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_divide_1_iff
thf(fact_100_sgn__less, axiom,
    ((![A : real]: ((ord_less_real @ (sgn_sgn_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % sgn_less
thf(fact_101_sgn__greater, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (sgn_sgn_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % sgn_greater
thf(fact_102_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_103_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_104_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_105_sgn__sgn, axiom,
    ((![A : complex]: ((sgn_sgn_complex @ (sgn_sgn_complex @ A)) = (sgn_sgn_complex @ A))))). % sgn_sgn
thf(fact_106_sgn__sgn, axiom,
    ((![A : real]: ((sgn_sgn_real @ (sgn_sgn_real @ A)) = (sgn_sgn_real @ A))))). % sgn_sgn
thf(fact_107_div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % div_0
thf(fact_108_div__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ zero_zero_complex @ A) = zero_zero_complex)))). % div_0
thf(fact_109_div__0, axiom,
    ((![A : real]: ((divide_divide_real @ zero_zero_real @ A) = zero_zero_real)))). % div_0
thf(fact_110_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_111_divide__eq__0__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = zero_zero_real) = (((A = zero_zero_real)) | ((B = zero_zero_real))))))). % divide_eq_0_iff
thf(fact_112_div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_113_div__by__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % div_by_0
thf(fact_114_div__by__0, axiom,
    ((![A : real]: ((divide_divide_real @ A @ zero_zero_real) = zero_zero_real)))). % div_by_0
thf(fact_115_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_116_divide__cancel__left, axiom,
    ((![C : real, A : real, B : real]: (((divide_divide_real @ C @ A) = (divide_divide_real @ C @ B)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_left
thf(fact_117_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_118_divide__cancel__right, axiom,
    ((![A : real, C : real, B : real]: (((divide_divide_real @ A @ C) = (divide_divide_real @ B @ C)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_right
thf(fact_119_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_120_division__ring__divide__zero, axiom,
    ((![A : real]: ((divide_divide_real @ A @ zero_zero_real) = zero_zero_real)))). % division_ring_divide_zero
thf(fact_121_div__by__1, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ one_one_complex) = A)))). % div_by_1
thf(fact_122_div__by__1, axiom,
    ((![A : real]: ((divide_divide_real @ A @ one_one_real) = A)))). % div_by_1
thf(fact_123_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_124_sgn__0, axiom,
    (((sgn_sgn_complex @ zero_zero_complex) = zero_zero_complex))). % sgn_0
thf(fact_125_sgn__0, axiom,
    (((sgn_sgn_real @ zero_zero_real) = zero_zero_real))). % sgn_0
thf(fact_126_sgn__1, axiom,
    (((sgn_sgn_complex @ one_one_complex) = one_one_complex))). % sgn_1
thf(fact_127_sgn__1, axiom,
    (((sgn_sgn_real @ one_one_real) = one_one_real))). % sgn_1
thf(fact_128_sgn__divide, axiom,
    ((![A : complex, B : complex]: ((sgn_sgn_complex @ (divide1210191872omplex @ A @ B)) = (divide1210191872omplex @ (sgn_sgn_complex @ A) @ (sgn_sgn_complex @ B)))))). % sgn_divide
thf(fact_129_sgn__divide, axiom,
    ((![A : real, B : real]: ((sgn_sgn_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (sgn_sgn_real @ A) @ (sgn_sgn_real @ B)))))). % sgn_divide
thf(fact_130_real__root__zero, axiom,
    ((![N : nat]: ((root @ N @ zero_zero_real) = zero_zero_real)))). % real_root_zero
thf(fact_131_zero__eq__1__divide__iff, axiom,
    ((![A : real]: ((zero_zero_real = (divide_divide_real @ one_one_real @ A)) = (A = zero_zero_real))))). % zero_eq_1_divide_iff
thf(fact_132_one__divide__eq__0__iff, axiom,
    ((![A : real]: (((divide_divide_real @ one_one_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % one_divide_eq_0_iff
thf(fact_133_eq__divide__eq__1, axiom,
    ((![B : real, A : real]: ((one_one_real = (divide_divide_real @ B @ A)) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % eq_divide_eq_1
thf(fact_134_divide__eq__eq__1, axiom,
    ((![B : real, A : real]: (((divide_divide_real @ B @ A) = one_one_real) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % divide_eq_eq_1
thf(fact_135_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_136_divide__self__if, axiom,
    ((![A : real]: (((A = zero_zero_real) => ((divide_divide_real @ A @ A) = zero_zero_real)) & ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real)))))). % divide_self_if
thf(fact_137_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_138_divide__self, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real))))). % divide_self
thf(fact_139_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_140_one__eq__divide__iff, axiom,
    ((![A : real, B : real]: ((one_one_real = (divide_divide_real @ A @ B)) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_141_div__self, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) => ((divide_divide_nat @ A @ A) = one_one_nat))))). % div_self
thf(fact_142_div__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % div_self
thf(fact_143_div__self, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real))))). % div_self
thf(fact_144_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_145_divide__eq__1__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = one_one_real) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_146_root__0, axiom,
    ((![X : real]: ((root @ zero_zero_nat @ X) = zero_zero_real)))). % root_0
thf(fact_147_real__root__eq__iff, axiom,
    ((![N : nat, X : real, Y : real]: ((ord_less_nat @ zero_zero_nat @ N) => (((root @ N @ X) = (root @ N @ Y)) = (X = Y)))))). % real_root_eq_iff
thf(fact_148_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_149_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_150_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_151_real__root__eq__0__iff, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => (((root @ N @ X) = zero_zero_real) = (X = zero_zero_real)))))). % real_root_eq_0_iff
thf(fact_152_real__root__less__iff, axiom,
    ((![N : nat, X : real, Y : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ (root @ N @ X) @ (root @ N @ Y)) = (ord_less_real @ X @ Y)))))). % real_root_less_iff
thf(fact_153_real__root__eq__1__iff, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => (((root @ N @ X) = one_one_real) = (X = one_one_real)))))). % real_root_eq_1_iff
thf(fact_154_real__root__one, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((root @ N @ one_one_real) = one_one_real))))). % real_root_one
thf(fact_155_real__root__lt__0__iff, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ (root @ N @ X) @ zero_zero_real) = (ord_less_real @ X @ zero_zero_real)))))). % real_root_lt_0_iff
thf(fact_156_real__root__gt__0__iff, axiom,
    ((![N : nat, Y : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ (root @ N @ Y)) = (ord_less_real @ zero_zero_real @ Y)))))). % real_root_gt_0_iff
thf(fact_157_real__root__lt__1__iff, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ (root @ N @ X) @ one_one_real) = (ord_less_real @ X @ one_one_real)))))). % real_root_lt_1_iff
thf(fact_158_real__root__gt__1__iff, axiom,
    ((![N : nat, Y : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ one_one_real @ (root @ N @ Y)) = (ord_less_real @ one_one_real @ Y)))))). % real_root_gt_1_iff
thf(fact_159_IH, axiom,
    ((![M2 : nat]: ((ord_less_nat @ M2 @ na) => ((~ ((M2 = zero_zero_nat))) => (?[Z3 : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ b @ (power_power_complex @ Z3 @ M2)))) @ one_one_real))))))). % IH
thf(fact_160_sgn__root, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((sgn_sgn_real @ (root @ N @ X)) = (sgn_sgn_real @ X)))))). % sgn_root
thf(fact_161_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_162_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X2 : real]: (((ord_less_real @ zero_zero_real @ X2) & ((power_power_real @ X2 @ N) = A)) & (![Y2 : real]: (((ord_less_real @ zero_zero_real @ Y2) & ((power_power_real @ Y2 @ N) = A)) => (Y2 = X2)))))))))). % realpow_pos_nth_unique
thf(fact_163_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R2 : real]: ((ord_less_real @ zero_zero_real @ R2) & ((power_power_real @ R2 @ N) = A)))))))). % realpow_pos_nth
thf(fact_164_real__root__gt__zero, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ X) => (ord_less_real @ zero_zero_real @ (root @ N @ X))))))). % real_root_gt_zero
thf(fact_165_real__root__strict__increasing, axiom,
    ((![N : nat, N2 : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ N @ N2) => ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ X @ one_one_real) => (ord_less_real @ (root @ N @ X) @ (root @ N2 @ X))))))))). % real_root_strict_increasing
thf(fact_166_real__root__pow__pos, axiom,
    ((![N : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ X) => ((power_power_real @ (root @ N @ X) @ N) = X)))))). % real_root_pow_pos
thf(fact_167_norm__not__less__zero, axiom,
    ((![X : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_168_norm__not__less__zero, axiom,
    ((![X : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_169_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_170_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_171_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_172_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_173_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_174_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y3 : real]: (ord_less_real @ Y3 @ X4))))). % linordered_field_no_lb
thf(fact_175_power__eq__imp__eq__norm, axiom,
    ((![W : complex, N : nat, Z4 : complex]: (((power_power_complex @ W @ N) = (power_power_complex @ Z4 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V638595069omplex @ W) = (real_V638595069omplex @ Z4))))))). % power_eq_imp_eq_norm
thf(fact_176_power__eq__imp__eq__norm, axiom,
    ((![W : real, N : nat, Z4 : real]: (((power_power_real @ W @ N) = (power_power_real @ Z4 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V646646907m_real @ W) = (real_V646646907m_real @ Z4))))))). % power_eq_imp_eq_norm
thf(fact_177_real__root__less__mono, axiom,
    ((![N : nat, X : real, Y : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ X @ Y) => (ord_less_real @ (root @ N @ X) @ (root @ N @ Y))))))). % real_root_less_mono
thf(fact_178_real__root__power, axiom,
    ((![N : nat, X : real, K : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((root @ N @ (power_power_real @ X @ K)) = (power_power_real @ (root @ N @ X) @ K)))))). % real_root_power
thf(fact_179_nonzero__of__real__divide, axiom,
    ((![Y : real, X : real]: ((~ ((Y = zero_zero_real))) => ((real_V306493662omplex @ (divide_divide_real @ X @ Y)) = (divide1210191872omplex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y))))))). % nonzero_of_real_divide
thf(fact_180_nonzero__of__real__divide, axiom,
    ((![Y : real, X : real]: ((~ ((Y = zero_zero_real))) => ((real_V1205483228l_real @ (divide_divide_real @ X @ Y)) = (divide_divide_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y))))))). % nonzero_of_real_divide
thf(fact_181_real__root__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, X : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ N @ N2) => ((ord_less_real @ one_one_real @ X) => (ord_less_real @ (root @ N2 @ X) @ (root @ N @ X)))))))). % real_root_strict_decreasing
thf(fact_182_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_183_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_184_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_185_add__divide__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (divide1210191872omplex @ A @ C) @ (divide1210191872omplex @ B @ C)))))). % add_divide_distrib
thf(fact_186_add__divide__distrib, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C)))))). % add_divide_distrib
thf(fact_187_sgn__eq__0__iff, axiom,
    ((![A : complex]: (((sgn_sgn_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % sgn_eq_0_iff
thf(fact_188_sgn__eq__0__iff, axiom,
    ((![A : real]: (((sgn_sgn_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % sgn_eq_0_iff
thf(fact_189_sgn__0__0, axiom,
    ((![A : real]: (((sgn_sgn_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % sgn_0_0
thf(fact_190_same__sgn__sgn__add, axiom,
    ((![B : real, A : real]: (((sgn_sgn_real @ B) = (sgn_sgn_real @ A)) => ((sgn_sgn_real @ (plus_plus_real @ A @ B)) = (sgn_sgn_real @ A)))))). % same_sgn_sgn_add
thf(fact_191_add__less__zeroD, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ (plus_plus_real @ X @ Y) @ zero_zero_real) => ((ord_less_real @ X @ zero_zero_real) | (ord_less_real @ Y @ zero_zero_real)))))). % add_less_zeroD
thf(fact_192_not__one__less__zero, axiom,
    ((~ ((ord_less_real @ one_one_real @ zero_zero_real))))). % not_one_less_zero
thf(fact_193_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_194_zero__less__one, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % zero_less_one
thf(fact_195_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_196_divide__strict__right__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ zero_zero_real) => (ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono_neg
thf(fact_197_divide__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono
thf(fact_198_zero__less__divide__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_real @ A @ B)) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ zero_zero_real @ B)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ B @ zero_zero_real))))))))). % zero_less_divide_iff
thf(fact_199_divide__less__cancel, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ B)))) & ((((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ A)))) & ((~ ((C = zero_zero_real))))))))))). % divide_less_cancel
thf(fact_200_divide__less__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (divide_divide_real @ A @ B) @ zero_zero_real) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ B @ zero_zero_real)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ zero_zero_real @ B))))))))). % divide_less_0_iff
thf(fact_201_divide__pos__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_pos_pos
thf(fact_202_divide__pos__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_pos_neg
thf(fact_203_divide__neg__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_neg_pos
thf(fact_204_divide__neg__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_neg_neg
thf(fact_205_add__mono1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ A @ one_one_real) @ (plus_plus_real @ B @ one_one_real)))))). % add_mono1
thf(fact_206_add__mono1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ one_one_nat) @ (plus_plus_nat @ B @ one_one_nat)))))). % add_mono1
thf(fact_207_less__add__one, axiom,
    ((![A : real]: (ord_less_real @ A @ (plus_plus_real @ A @ one_one_real))))). % less_add_one
thf(fact_208_less__add__one, axiom,
    ((![A : nat]: (ord_less_nat @ A @ (plus_plus_nat @ A @ one_one_nat))))). % less_add_one
thf(fact_209_right__inverse__eq, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => (((divide1210191872omplex @ A @ B) = one_one_complex) = (A = B)))))). % right_inverse_eq
thf(fact_210_right__inverse__eq, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => (((divide_divide_real @ A @ B) = one_one_real) = (A = B)))))). % right_inverse_eq
thf(fact_211_zero__less__two, axiom,
    ((ord_less_real @ zero_zero_real @ (plus_plus_real @ one_one_real @ one_one_real)))). % zero_less_two
thf(fact_212_zero__less__two, axiom,
    ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ one_one_nat @ one_one_nat)))). % zero_less_two
thf(fact_213_less__divide__eq__1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ A @ B)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ B @ A))))))))). % less_divide_eq_1
thf(fact_214_divide__less__eq__1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ B @ A)))) | ((((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ A @ B)))) | ((A = zero_zero_real))))))))). % divide_less_eq_1
thf(fact_215_less__half__sum, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_real @ A @ (divide_divide_real @ (plus_plus_real @ A @ B) @ (plus_plus_real @ one_one_real @ one_one_real))))))). % less_half_sum
thf(fact_216_gt__half__sum, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (divide_divide_real @ (plus_plus_real @ A @ B) @ (plus_plus_real @ one_one_real @ one_one_real)) @ B))))). % gt_half_sum
thf(fact_217_sgn__1__pos, axiom,
    ((![A : real]: (((sgn_sgn_real @ A) = one_one_real) = (ord_less_real @ zero_zero_real @ A))))). % sgn_1_pos
thf(fact_218_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_219_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_220_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_221_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_222_less__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel1
thf(fact_223_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_224_add__less__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel2
thf(fact_225_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_226_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_227_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_228_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_229_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_230_mult__zero__left, axiom,
    ((![A : real]: ((times_times_real @ zero_zero_real @ A) = zero_zero_real)))). % mult_zero_left
thf(fact_231_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_232_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_233_mult__zero__right, axiom,
    ((![A : real]: ((times_times_real @ A @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_234_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right

% Conjectures (1)
thf(conj_0, conjecture,
    (((power_power_complex @ (divide1210191872omplex @ v @ (real_V306493662omplex @ (root @ na @ (real_V638595069omplex @ b)))) @ na) = (divide1210191872omplex @ (power_power_complex @ v @ na) @ (real_V306493662omplex @ (real_V638595069omplex @ b)))))).
