% 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_443__5371908_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:10.168

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (36)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal, type,
    minus_minus_real : real > 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__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_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_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal, type,
    semiri2110766477t_real : nat > real).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > 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__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
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_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_v_N1____, type,
    n1 : nat).
thf(sy_v_N2____, type,
    n2 : nat).
thf(sy_v_d____, type,
    d : real).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_s____, type,
    s : real).
thf(sy_v_z____, type,
    z : complex).

% Relevant facts (225)
thf(fact_0_N2, axiom,
    ((ord_less_real @ (divide_divide_real @ (numeral_numeral_real @ (bit0 @ one)) @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))) @ (semiri2110766477t_real @ n2)))). % N2
thf(fact_1__092_060open_062_092_060exists_062n_O_A2_A_P_A_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_060_Areal_An_092_060close_062, axiom,
    ((?[N : nat]: (ord_less_real @ (divide_divide_real @ (numeral_numeral_real @ (bit0 @ one)) @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))) @ (semiri2110766477t_real @ N))))). % \<open>\<exists>n. 2 / \<bar>cmod (poly p z) - - s\<bar> < real n\<close>
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062N2_O_A2_A_P_A_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_060_Areal_AN2_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![N2 : nat]: (~ ((ord_less_real @ (divide_divide_real @ (numeral_numeral_real @ (bit0 @ one)) @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))) @ (semiri2110766477t_real @ N2))))))))). % \<open>\<And>thesis. (\<And>N2. 2 / \<bar>cmod (poly p z) - - s\<bar> < real N2 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_e, axiom,
    ((ord_less_real @ zero_zero_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))))). % e
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_e2, axiom,
    ((ord_less_real @ zero_zero_real @ (divide_divide_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s))) @ (numeral_numeral_real @ (bit0 @ one)))))). % e2
thf(fact_6_norm__divide__numeral, axiom,
    ((![A : complex, W : num]: ((real_V638595069omplex @ (divide1210191872omplex @ A @ (numera632737353omplex @ W))) = (divide_divide_real @ (real_V638595069omplex @ A) @ (numeral_numeral_real @ W)))))). % norm_divide_numeral
thf(fact_7_norm__divide__numeral, axiom,
    ((![A : real, W : num]: ((real_V646646907m_real @ (divide_divide_real @ A @ (numeral_numeral_real @ W))) = (divide_divide_real @ (real_V646646907m_real @ A) @ (numeral_numeral_real @ W)))))). % norm_divide_numeral
thf(fact_8_norm__neg__numeral, axiom,
    ((![W : num]: ((real_V638595069omplex @ (uminus1204672759omplex @ (numera632737353omplex @ W))) = (numeral_numeral_real @ W))))). % norm_neg_numeral
thf(fact_9_norm__neg__numeral, axiom,
    ((![W : num]: ((real_V646646907m_real @ (uminus_uminus_real @ (numeral_numeral_real @ W))) = (numeral_numeral_real @ W))))). % norm_neg_numeral
thf(fact_10_norm__of__nat, axiom,
    ((![N3 : nat]: ((real_V638595069omplex @ (semiri356525583omplex @ N3)) = (semiri2110766477t_real @ N3))))). % norm_of_nat
thf(fact_11_norm__of__nat, axiom,
    ((![N3 : nat]: ((real_V646646907m_real @ (semiri2110766477t_real @ N3)) = (semiri2110766477t_real @ N3))))). % norm_of_nat
thf(fact_12_th000_I3_J, axiom,
    ((ord_less_real @ zero_zero_real @ (semiri2110766477t_real @ (suc @ (plus_plus_nat @ n1 @ n2)))))). % th000(3)
thf(fact_13_norm__numeral, axiom,
    ((![W : num]: ((real_V638595069omplex @ (numera632737353omplex @ W)) = (numeral_numeral_real @ W))))). % norm_numeral
thf(fact_14_norm__numeral, axiom,
    ((![W : num]: ((real_V646646907m_real @ (numeral_numeral_real @ W)) = (numeral_numeral_real @ W))))). % norm_numeral
thf(fact_15_of__nat__add, axiom,
    ((![M : nat, N3 : nat]: ((semiri1382578993at_nat @ (plus_plus_nat @ M @ N3)) = (plus_plus_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N3)))))). % of_nat_add
thf(fact_16_of__nat__add, axiom,
    ((![M : nat, N3 : nat]: ((semiri2110766477t_real @ (plus_plus_nat @ M @ N3)) = (plus_plus_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N3)))))). % of_nat_add
thf(fact_17_of__nat__add, axiom,
    ((![M : nat, N3 : nat]: ((semiri356525583omplex @ (plus_plus_nat @ M @ N3)) = (plus_plus_complex @ (semiri356525583omplex @ M) @ (semiri356525583omplex @ N3)))))). % of_nat_add
thf(fact_18_abs__neg__numeral, axiom,
    ((![N3 : num]: ((abs_abs_real @ (uminus_uminus_real @ (numeral_numeral_real @ N3))) = (numeral_numeral_real @ N3))))). % abs_neg_numeral
thf(fact_19_diff__minus__eq__add, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ A @ (uminus_uminus_real @ B)) = (plus_plus_real @ A @ B))))). % diff_minus_eq_add
thf(fact_20_diff__minus__eq__add, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ A @ (uminus1204672759omplex @ B)) = (plus_plus_complex @ A @ B))))). % diff_minus_eq_add
thf(fact_21_uminus__add__conv__diff, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ B) = (minus_minus_real @ B @ A))))). % uminus_add_conv_diff
thf(fact_22_uminus__add__conv__diff, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ B) = (minus_minus_complex @ B @ A))))). % uminus_add_conv_diff
thf(fact_23_semiring__norm_I168_J, axiom,
    ((![V : num, W : num, Y2 : real]: ((plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ V)) @ (plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ W)) @ Y2)) = (plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W))) @ Y2))))). % semiring_norm(168)
thf(fact_24_semiring__norm_I168_J, axiom,
    ((![V : num, W : num, Y2 : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ V)) @ (plus_plus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ W)) @ Y2)) = (plus_plus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ (plus_plus_num @ V @ W))) @ Y2))))). % semiring_norm(168)
thf(fact_25_add__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N3 : num]: ((plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ (numeral_numeral_real @ N3))) = (uminus_uminus_real @ (plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N3))))))). % add_neg_numeral_simps(3)
thf(fact_26_add__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N3 : num]: ((plus_plus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (uminus1204672759omplex @ (numera632737353omplex @ N3))) = (uminus1204672759omplex @ (plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N3))))))). % add_neg_numeral_simps(3)
thf(fact_27_d_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ d))). % d(1)
thf(fact_28_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_29_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_30_numeral__eq__iff, axiom,
    ((![M : num, N3 : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N3)) = (M = N3))))). % numeral_eq_iff
thf(fact_31_numeral__eq__iff, axiom,
    ((![M : num, N3 : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N3)) = (M = N3))))). % numeral_eq_iff
thf(fact_32_numeral__eq__iff, axiom,
    ((![M : num, N3 : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N3)) = (M = N3))))). % numeral_eq_iff
thf(fact_33_semiring__norm_I87_J, axiom,
    ((![M : num, N3 : num]: (((bit0 @ M) = (bit0 @ N3)) = (M = N3))))). % semiring_norm(87)
thf(fact_34_neg__equal__iff__equal, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = (uminus_uminus_real @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_35_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_36_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_37_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_38_Suc__diff__diff, axiom,
    ((![M : nat, N3 : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ (suc @ M) @ N3) @ (suc @ K)) = (minus_minus_nat @ (minus_minus_nat @ M @ N3) @ K))))). % Suc_diff_diff
thf(fact_39_diff__Suc__Suc, axiom,
    ((![M : nat, N3 : nat]: ((minus_minus_nat @ (suc @ M) @ (suc @ N3)) = (minus_minus_nat @ M @ N3))))). % diff_Suc_Suc
thf(fact_40_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_41_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_42_of__nat__eq__iff, axiom,
    ((![M : nat, N3 : nat]: (((semiri2110766477t_real @ M) = (semiri2110766477t_real @ N3)) = (M = N3))))). % of_nat_eq_iff
thf(fact_43_of__nat__eq__iff, axiom,
    ((![M : nat, N3 : nat]: (((semiri356525583omplex @ M) = (semiri356525583omplex @ N3)) = (M = N3))))). % of_nat_eq_iff
thf(fact_44_diff__diff__left, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ I @ (plus_plus_nat @ J @ K)))))). % diff_diff_left
thf(fact_45_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_57_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_58_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_59_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_60_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_61_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_62_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_63_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_64_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_65_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_66_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_67_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_68_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_69_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_70_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_71_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_72_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_73_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_74_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_75_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_76_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_77_numeral__less__iff, axiom,
    ((![M : num, N3 : num]: ((ord_less_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N3)) = (ord_less_num @ M @ N3))))). % numeral_less_iff
thf(fact_78_numeral__less__iff, axiom,
    ((![M : num, N3 : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N3)) = (ord_less_num @ M @ N3))))). % numeral_less_iff
thf(fact_79_neg__equal__zero, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = A) = (A = zero_zero_real))))). % neg_equal_zero
thf(fact_80_equal__neg__zero, axiom,
    ((![A : real]: ((A = (uminus_uminus_real @ A)) = (A = zero_zero_real))))). % equal_neg_zero
thf(fact_81_neg__equal__0__iff__equal, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % neg_equal_0_iff_equal
thf(fact_82_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_83_neg__0__equal__iff__equal, axiom,
    ((![A : real]: ((zero_zero_real = (uminus_uminus_real @ A)) = (zero_zero_real = A))))). % neg_0_equal_iff_equal
thf(fact_84_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_85_add_Oinverse__neutral, axiom,
    (((uminus_uminus_real @ zero_zero_real) = zero_zero_real))). % add.inverse_neutral
thf(fact_86_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_87_add__diff__cancel__right_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_88_add__diff__cancel__right_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_89_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_90_add__diff__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_right
thf(fact_91_add__diff__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_92_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_93_add__diff__cancel__left_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_94_add__diff__cancel__left_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_95_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_96_add__diff__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_left
thf(fact_97_add__diff__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_98_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_99_diff__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_100_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_101_add__diff__cancel, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_102_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_103_neg__less__iff__less, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ B))))). % neg_less_iff_less
thf(fact_104_minus__add__distrib, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)))))). % minus_add_distrib
thf(fact_105_minus__add__distrib, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)))))). % minus_add_distrib
thf(fact_106_minus__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ (plus_plus_real @ A @ B)) = B)))). % minus_add_cancel
thf(fact_107_minus__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ (plus_plus_complex @ A @ B)) = B)))). % minus_add_cancel
thf(fact_108_add__minus__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ (uminus_uminus_real @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_109_add__minus__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ A @ (plus_plus_complex @ (uminus1204672759omplex @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_110_of__nat__0, axiom,
    (((semiri2110766477t_real @ zero_zero_nat) = zero_zero_real))). % of_nat_0
thf(fact_111_of__nat__0, axiom,
    (((semiri356525583omplex @ zero_zero_nat) = zero_zero_complex))). % of_nat_0
thf(fact_112_of__nat__0__eq__iff, axiom,
    ((![N3 : nat]: ((zero_zero_real = (semiri2110766477t_real @ N3)) = (zero_zero_nat = N3))))). % of_nat_0_eq_iff
thf(fact_113_of__nat__0__eq__iff, axiom,
    ((![N3 : nat]: ((zero_zero_complex = (semiri356525583omplex @ N3)) = (zero_zero_nat = N3))))). % of_nat_0_eq_iff
thf(fact_114_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2110766477t_real @ M) = zero_zero_real) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_115_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri356525583omplex @ M) = zero_zero_complex) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_116_neg__numeral__eq__iff, axiom,
    ((![M : num, N3 : num]: (((uminus_uminus_real @ (numeral_numeral_real @ M)) = (uminus_uminus_real @ (numeral_numeral_real @ N3))) = (M = N3))))). % neg_numeral_eq_iff
thf(fact_117_neg__numeral__eq__iff, axiom,
    ((![M : num, N3 : num]: (((uminus1204672759omplex @ (numera632737353omplex @ M)) = (uminus1204672759omplex @ (numera632737353omplex @ N3))) = (M = N3))))). % neg_numeral_eq_iff
thf(fact_118_minus__diff__eq, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (minus_minus_real @ A @ B)) = (minus_minus_real @ B @ A))))). % minus_diff_eq
thf(fact_119_minus__diff__eq, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (minus_minus_complex @ A @ B)) = (minus_minus_complex @ B @ A))))). % minus_diff_eq
thf(fact_120_of__nat__less__iff, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N3)) = (ord_less_nat @ M @ N3))))). % of_nat_less_iff
thf(fact_121_semiring__norm_I83_J, axiom,
    ((![N3 : num]: (~ ((one = (bit0 @ N3))))))). % semiring_norm(83)
thf(fact_122_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_123_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_124_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_125_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_126_of__nat__numeral, axiom,
    ((![N3 : num]: ((semiri1382578993at_nat @ (numeral_numeral_nat @ N3)) = (numeral_numeral_nat @ N3))))). % of_nat_numeral
thf(fact_127_of__nat__numeral, axiom,
    ((![N3 : num]: ((semiri2110766477t_real @ (numeral_numeral_nat @ N3)) = (numeral_numeral_real @ N3))))). % of_nat_numeral
thf(fact_128_of__nat__numeral, axiom,
    ((![N3 : num]: ((semiri356525583omplex @ (numeral_numeral_nat @ N3)) = (numera632737353omplex @ N3))))). % of_nat_numeral
thf(fact_129_abs__add__abs, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) = (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_add_abs
thf(fact_130_abs__numeral, axiom,
    ((![N3 : num]: ((abs_abs_real @ (numeral_numeral_real @ N3)) = (numeral_numeral_real @ N3))))). % abs_numeral
thf(fact_131_abs__minus__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus_cancel
thf(fact_132_add__Suc__right, axiom,
    ((![M : nat, N3 : nat]: ((plus_plus_nat @ M @ (suc @ N3)) = (suc @ (plus_plus_nat @ M @ N3)))))). % add_Suc_right
thf(fact_133_norm__minus__cancel, axiom,
    ((![X3 : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X3)) = (real_V638595069omplex @ X3))))). % norm_minus_cancel
thf(fact_134_norm__minus__cancel, axiom,
    ((![X3 : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X3)) = (real_V646646907m_real @ X3))))). % norm_minus_cancel
thf(fact_135_abs__of__nat, axiom,
    ((![N3 : nat]: ((abs_abs_real @ (semiri2110766477t_real @ N3)) = (semiri2110766477t_real @ N3))))). % abs_of_nat
thf(fact_136_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_137_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_138_semiring__norm_I6_J, axiom,
    ((![M : num, N3 : num]: ((plus_plus_num @ (bit0 @ M) @ (bit0 @ N3)) = (bit0 @ (plus_plus_num @ M @ N3)))))). % semiring_norm(6)
thf(fact_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_add__less__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel1
thf(fact_149_diff__gt__0__iff__gt, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_real @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_150_diff__add__zero, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ A @ (plus_plus_nat @ A @ B)) = zero_zero_nat)))). % diff_add_zero
thf(fact_151_neg__less__0__iff__less, axiom,
    ((![A : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ zero_zero_real) = (ord_less_real @ zero_zero_real @ A))))). % neg_less_0_iff_less
thf(fact_152_neg__0__less__iff__less, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ zero_zero_real))))). % neg_0_less_iff_less
thf(fact_153_neg__less__pos, axiom,
    ((![A : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ A) = (ord_less_real @ zero_zero_real @ A))))). % neg_less_pos
thf(fact_154_less__neg__neg, axiom,
    ((![A : real]: ((ord_less_real @ A @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ zero_zero_real))))). % less_neg_neg
thf(fact_155_add_Oright__inverse, axiom,
    ((![A : real]: ((plus_plus_real @ A @ (uminus_uminus_real @ A)) = zero_zero_real)))). % add.right_inverse
thf(fact_156_add_Oright__inverse, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ (uminus1204672759omplex @ A)) = zero_zero_complex)))). % add.right_inverse
thf(fact_157_add_Oleft__inverse, axiom,
    ((![A : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ A) = zero_zero_real)))). % add.left_inverse
thf(fact_158_add_Oleft__inverse, axiom,
    ((![A : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ A) = zero_zero_complex)))). % add.left_inverse
thf(fact_159_diff__0, axiom,
    ((![A : real]: ((minus_minus_real @ zero_zero_real @ A) = (uminus_uminus_real @ A))))). % diff_0
thf(fact_160_diff__0, axiom,
    ((![A : complex]: ((minus_minus_complex @ zero_zero_complex @ A) = (uminus1204672759omplex @ A))))). % diff_0
thf(fact_161_of__nat__0__less__iff, axiom,
    ((![N3 : nat]: ((ord_less_real @ zero_zero_real @ (semiri2110766477t_real @ N3)) = (ord_less_nat @ zero_zero_nat @ N3))))). % of_nat_0_less_iff
thf(fact_162_neg__numeral__less__iff, axiom,
    ((![M : num, N3 : num]: ((ord_less_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ (numeral_numeral_real @ N3))) = (ord_less_num @ N3 @ M))))). % neg_numeral_less_iff
thf(fact_163_zero__less__abs__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (abs_abs_real @ A)) = (~ ((A = zero_zero_real))))))). % zero_less_abs_iff
thf(fact_164_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_165_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_166_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_167_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_168_numeral__plus__numeral, axiom,
    ((![M : num, N3 : num]: ((plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N3)) = (numeral_numeral_real @ (plus_plus_num @ M @ N3)))))). % numeral_plus_numeral
thf(fact_169_numeral__plus__numeral, axiom,
    ((![M : num, N3 : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N3)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N3)))))). % numeral_plus_numeral
thf(fact_170_numeral__plus__numeral, axiom,
    ((![M : num, N3 : num]: ((plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N3)) = (numera632737353omplex @ (plus_plus_num @ M @ N3)))))). % numeral_plus_numeral
thf(fact_171_add__numeral__left, axiom,
    ((![V : num, W : num, Z2 : real]: ((plus_plus_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ (numeral_numeral_real @ W) @ Z2)) = (plus_plus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W)) @ Z2))))). % add_numeral_left
thf(fact_172_add__numeral__left, axiom,
    ((![V : num, W : num, Z2 : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z2)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z2))))). % add_numeral_left
thf(fact_173_add__numeral__left, axiom,
    ((![V : num, W : num, Z2 : complex]: ((plus_plus_complex @ (numera632737353omplex @ V) @ (plus_plus_complex @ (numera632737353omplex @ W) @ Z2)) = (plus_plus_complex @ (numera632737353omplex @ (plus_plus_num @ V @ W)) @ Z2))))). % add_numeral_left
thf(fact_174_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_175_Suc__numeral, axiom,
    ((![N3 : num]: ((suc @ (numeral_numeral_nat @ N3)) = (numeral_numeral_nat @ (plus_plus_num @ N3 @ one)))))). % Suc_numeral
thf(fact_176_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_177_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_178_add__2__eq__Suc_H, axiom,
    ((![N3 : nat]: ((plus_plus_nat @ N3 @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (suc @ N3)))))). % add_2_eq_Suc'
thf(fact_179_add__2__eq__Suc, axiom,
    ((![N3 : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N3) = (suc @ (suc @ N3)))))). % add_2_eq_Suc
thf(fact_180_diff__numeral__simps_I2_J, axiom,
    ((![M : num, N3 : num]: ((minus_minus_real @ (numeral_numeral_real @ M) @ (uminus_uminus_real @ (numeral_numeral_real @ N3))) = (numeral_numeral_real @ (plus_plus_num @ M @ N3)))))). % diff_numeral_simps(2)
thf(fact_181_diff__numeral__simps_I2_J, axiom,
    ((![M : num, N3 : num]: ((minus_minus_complex @ (numera632737353omplex @ M) @ (uminus1204672759omplex @ (numera632737353omplex @ N3))) = (numera632737353omplex @ (plus_plus_num @ M @ N3)))))). % diff_numeral_simps(2)
thf(fact_182_diff__numeral__simps_I3_J, axiom,
    ((![M : num, N3 : num]: ((minus_minus_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (numeral_numeral_real @ N3)) = (uminus_uminus_real @ (numeral_numeral_real @ (plus_plus_num @ M @ N3))))))). % diff_numeral_simps(3)
thf(fact_183_diff__numeral__simps_I3_J, axiom,
    ((![M : num, N3 : num]: ((minus_minus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (numera632737353omplex @ N3)) = (uminus1204672759omplex @ (numera632737353omplex @ (plus_plus_num @ M @ N3))))))). % diff_numeral_simps(3)
thf(fact_184__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060forall_062w_O_A0_A_060_Acmod_A_Iw_A_N_Az_J_A_092_060and_062_Acmod_A_Iw_A_N_Az_J_A_060_Ad_A_092_060longrightarrow_062_Acmod_A_Ipoly_Ap_Aw_A_N_Apoly_Ap_Az_J_A_060_A_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_P_A2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![D : real]: ((ord_less_real @ zero_zero_real @ D) => (~ ((![W2 : complex]: (((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z))) & (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z)) @ D)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ p @ W2) @ (poly_complex2 @ p @ z))) @ (divide_divide_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s))) @ (numeral_numeral_real @ (bit0 @ one)))))))))))))). % \<open>\<And>thesis. (\<And>d. \<lbrakk>0 < d; \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < \<bar>cmod (poly p z) - - s\<bar> / 2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_185__092_060open_062_092_060exists_062d_0620_O_A_092_060forall_062w_O_A0_A_060_Acmod_A_Iw_A_N_Az_J_A_092_060and_062_Acmod_A_Iw_A_N_Az_J_A_060_Ad_A_092_060longrightarrow_062_Acmod_A_Ipoly_Ap_Aw_A_N_Apoly_Ap_Az_J_A_060_A_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_P_A2_092_060close_062, axiom,
    ((?[D : real]: ((ord_less_real @ zero_zero_real @ D) & (![W2 : complex]: (((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z))) & (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z)) @ D)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ p @ W2) @ (poly_complex2 @ p @ z))) @ (divide_divide_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s))) @ (numeral_numeral_real @ (bit0 @ one)))))))))). % \<open>\<exists>d>0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < \<bar>cmod (poly p z) - - s\<bar> / 2\<close>
thf(fact_186_d_I2_J, axiom,
    ((![W2 : complex]: (((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z))) & (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ W2 @ z)) @ d)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ p @ W2) @ (poly_complex2 @ p @ z))) @ (divide_divide_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s))) @ (numeral_numeral_real @ (bit0 @ one)))))))). % d(2)
thf(fact_187_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_188_add__One__commute, axiom,
    ((![N3 : num]: ((plus_plus_num @ one @ N3) = (plus_plus_num @ N3 @ one))))). % add_One_commute
thf(fact_189_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N : nat]: ((P @ (suc @ N)) => (P @ N))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_190_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_191_diff__add__inverse2, axiom,
    ((![M : nat, N3 : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ N3) @ N3) = M)))). % diff_add_inverse2
thf(fact_192_diff__add__inverse, axiom,
    ((![N3 : nat, M : nat]: ((minus_minus_nat @ (plus_plus_nat @ N3 @ M) @ N3) = M)))). % diff_add_inverse
thf(fact_193_diff__cancel2, axiom,
    ((![M : nat, K : nat, N3 : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ K) @ (plus_plus_nat @ N3 @ K)) = (minus_minus_nat @ M @ N3))))). % diff_cancel2
thf(fact_194_Nat_Odiff__cancel, axiom,
    ((![K : nat, M : nat, N3 : nat]: ((minus_minus_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N3)) = (minus_minus_nat @ M @ N3))))). % Nat.diff_cancel
thf(fact_195_add_Ogroup__left__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.group_left_neutral
thf(fact_196_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_197_add_Ocomm__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.comm_neutral
thf(fact_198_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_199_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_200_zero__neq__numeral, axiom,
    ((![N3 : num]: (~ ((zero_zero_real = (numeral_numeral_real @ N3))))))). % zero_neq_numeral
thf(fact_201_zero__neq__numeral, axiom,
    ((![N3 : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N3))))))). % zero_neq_numeral
thf(fact_202_zero__neq__numeral, axiom,
    ((![N3 : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N3))))))). % zero_neq_numeral
thf(fact_203_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[A2 : real]: (^[B2 : real]: ((minus_minus_real @ A2 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_204_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : complex]: (^[Z3 : complex]: (Y3 = Z3))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_205_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_206_Suc__nat__number__of__add, axiom,
    ((![V : num, N3 : nat]: ((suc @ (plus_plus_nat @ (numeral_numeral_nat @ V) @ N3)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ one)) @ N3))))). % Suc_nat_number_of_add
thf(fact_207_pos__add__strict, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % pos_add_strict
thf(fact_208_pos__add__strict, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ B @ C) => (ord_less_real @ B @ (plus_plus_real @ A @ C))))))). % pos_add_strict
thf(fact_209_canonically__ordered__monoid__add__class_OlessE, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((![C2 : nat]: ((B = (plus_plus_nat @ A @ C2)) => (C2 = zero_zero_nat))))))))). % canonically_ordered_monoid_add_class.lessE
thf(fact_210_add__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_pos
thf(fact_211_add__pos__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_pos_pos
thf(fact_212_add__neg__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_neg
thf(fact_213_add__neg__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_neg_neg
thf(fact_214_not__numeral__less__zero, axiom,
    ((![N3 : num]: (~ ((ord_less_real @ (numeral_numeral_real @ N3) @ zero_zero_real)))))). % not_numeral_less_zero
thf(fact_215_not__numeral__less__zero, axiom,
    ((![N3 : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N3) @ zero_zero_nat)))))). % not_numeral_less_zero
thf(fact_216_zero__less__numeral, axiom,
    ((![N3 : num]: (ord_less_real @ zero_zero_real @ (numeral_numeral_real @ N3))))). % zero_less_numeral
thf(fact_217_zero__less__numeral, axiom,
    ((![N3 : num]: (ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ N3))))). % zero_less_numeral
thf(fact_218_less__iff__diff__less__0, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (ord_less_real @ (minus_minus_real @ A2 @ B2) @ zero_zero_real)))))). % less_iff_diff_less_0
thf(fact_219_neg__eq__iff__add__eq__0, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = B) = ((plus_plus_real @ A @ B) = zero_zero_real))))). % neg_eq_iff_add_eq_0
thf(fact_220_neg__eq__iff__add__eq__0, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = B) = ((plus_plus_complex @ A @ B) = zero_zero_complex))))). % neg_eq_iff_add_eq_0
thf(fact_221_eq__neg__iff__add__eq__0, axiom,
    ((![A : real, B : real]: ((A = (uminus_uminus_real @ B)) = ((plus_plus_real @ A @ B) = zero_zero_real))))). % eq_neg_iff_add_eq_0
thf(fact_222_eq__neg__iff__add__eq__0, axiom,
    ((![A : complex, B : complex]: ((A = (uminus1204672759omplex @ B)) = ((plus_plus_complex @ A @ B) = zero_zero_complex))))). % eq_neg_iff_add_eq_0
thf(fact_223_add_Oinverse__unique, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = zero_zero_real) => ((uminus_uminus_real @ A) = B))))). % add.inverse_unique
thf(fact_224_add_Oinverse__unique, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = zero_zero_complex) => ((uminus1204672759omplex @ A) = B))))). % add.inverse_unique

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_real @ (divide_divide_real @ (numeral_numeral_real @ (bit0 @ one)) @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))) @ (semiri2110766477t_real @ (suc @ (plus_plus_nat @ n1 @ n2)))))).
