% 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_273__5370262_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:50.251

% 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 (20)
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_952716343t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Nat__Onat, type,
    order_297469111al_nat : (real > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_1818878995l_real : (real > real) > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (244)
thf(fact_0_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_1_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_2_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_3_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_4_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_5_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_6_abs__le__self__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % abs_le_self_iff
thf(fact_7_abs__le__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ zero_zero_real) = (A = zero_zero_real))))). % abs_le_zero_iff
thf(fact_8_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_9_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_10_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_11_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_12_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_13_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_14_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_15_dense__eq0__I, axiom,
    ((![X3 : real]: ((![E : real]: ((ord_less_real @ zero_zero_real @ E) => (ord_less_eq_real @ (abs_abs_real @ X3) @ E))) => (X3 = zero_zero_real))))). % dense_eq0_I
thf(fact_16_abs__of__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_pos
thf(fact_17_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_18_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_19_abs__abs, axiom,
    ((![A : complex]: ((abs_abs_complex @ (abs_abs_complex @ A)) = (abs_abs_complex @ A))))). % abs_abs
thf(fact_20_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_21_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_23_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_24_linorder__neqE__linordered__idom, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_neqE_linordered_idom
thf(fact_25_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_26_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_27_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_28_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_29_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_30_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_31_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_32_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_33_abs__le__D1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) => (ord_less_eq_real @ A @ B))))). % abs_le_D1
thf(fact_34_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_35_abs__not__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (abs_abs_real @ A) @ zero_zero_real)))))). % abs_not_less_zero
thf(fact_36__092_060open_062cmod_A_Is_A0_J_A_092_060le_062_Ar_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (s @ zero_zero_nat)) @ r))). % \<open>cmod (s 0) \<le> r\<close>
thf(fact_37_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_38_strict__mono__less, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y2)) = (ord_less_real @ X3 @ Y2)))))). % strict_mono_less
thf(fact_39_strict__mono__less, axiom,
    ((![F : nat > real, X3 : nat, Y2 : nat]: ((order_952716343t_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % strict_mono_less
thf(fact_40_strict__mono__less, axiom,
    ((![F : real > nat, X3 : real, Y2 : real]: ((order_297469111al_nat @ F) => ((ord_less_nat @ (F @ X3) @ (F @ Y2)) = (ord_less_real @ X3 @ Y2)))))). % strict_mono_less
thf(fact_41_strict__mono__less, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ (F @ X3) @ (F @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % strict_mono_less
thf(fact_42_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F2 : real > real]: (![X2 : real]: (![Y3 : real]: (((ord_less_real @ X2 @ Y3)) => ((ord_less_real @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_43_strict__mono__def, axiom,
    ((order_297469111al_nat = (^[F2 : real > nat]: (![X2 : real]: (![Y3 : real]: (((ord_less_real @ X2 @ Y3)) => ((ord_less_nat @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_44_strict__mono__def, axiom,
    ((order_952716343t_real = (^[F2 : nat > real]: (![X2 : nat]: (![Y3 : nat]: (((ord_less_nat @ X2 @ Y3)) => ((ord_less_real @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_45_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F2 : nat > nat]: (![X2 : nat]: (![Y3 : nat]: (((ord_less_nat @ X2 @ Y3)) => ((ord_less_nat @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_46_strict__monoI, axiom,
    ((![F : real > real]: ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (order_1818878995l_real @ F))))). % strict_monoI
thf(fact_47_strict__monoI, axiom,
    ((![F : real > nat]: ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (order_297469111al_nat @ F))))). % strict_monoI
thf(fact_48_strict__monoI, axiom,
    ((![F : nat > real]: ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (order_952716343t_real @ F))))). % strict_monoI
thf(fact_49_strict__monoI, axiom,
    ((![F : nat > nat]: ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (order_769474267at_nat @ F))))). % strict_monoI
thf(fact_50_strict__monoD, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ X3 @ Y2) => (ord_less_real @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_51_strict__monoD, axiom,
    ((![F : real > nat, X3 : real, Y2 : real]: ((order_297469111al_nat @ F) => ((ord_less_real @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_52_strict__monoD, axiom,
    ((![F : nat > real, X3 : nat, Y2 : nat]: ((order_952716343t_real @ F) => ((ord_less_nat @ X3 @ Y2) => (ord_less_real @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_53_strict__monoD, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_54_strict__mono__less__eq, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_eq_real @ (F @ X3) @ (F @ Y2)) = (ord_less_eq_real @ X3 @ Y2)))))). % strict_mono_less_eq
thf(fact_55_strict__mono__less__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_eq_nat @ (F @ X3) @ (F @ Y2)) = (ord_less_eq_nat @ X3 @ Y2)))))). % strict_mono_less_eq
thf(fact_56_strict__mono__leD, axiom,
    ((![R : real > real, M : real, N : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M @ N) => (ord_less_eq_real @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_57_strict__mono__leD, axiom,
    ((![R : nat > nat, M : nat, N : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_58_zero__complex_Osimps_I2_J, axiom,
    (((im @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(2)
thf(fact_59_abs__Im__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (abs_abs_real @ (im @ X3)) @ (real_V638595069omplex @ X3))))). % abs_Im_le_cmod
thf(fact_60_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_61_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_62_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_63_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_64_eq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((ord_less_eq_real @ Y3 @ X2)))))))). % eq_iff
thf(fact_65_antisym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_66_linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_eq_real @ Y2 @ X3))))). % linear
thf(fact_67_eq__refl, axiom,
    ((![X3 : real, Y2 : real]: ((X3 = Y2) => (ord_less_eq_real @ X3 @ Y2))))). % eq_refl
thf(fact_68_le__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % le_cases
thf(fact_69_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_70_le__cases3, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: (((ord_less_eq_real @ X3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z3)))) => (((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z3)))) => (((ord_less_eq_real @ X3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y2)))) => (((ord_less_eq_real @ Z3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X3)))) => (((ord_less_eq_real @ Y2 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X3)))) => (~ (((ord_less_eq_real @ Z3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y2)))))))))))))). % le_cases3
thf(fact_71_antisym__conv, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv
thf(fact_72_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_73_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_74_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_75_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_76_order__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_eq_real @ X3 @ Z3)))))). % order_trans
thf(fact_77_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_78_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_79_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_80_dual__order_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_81_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_82_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_83_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_84_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_85_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_86_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_87_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_88_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_89_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_90_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_91_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_92_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_93_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_94_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_95_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_96_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_97_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_98_lt__ex, axiom,
    ((![X3 : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X3))))). % lt_ex
thf(fact_99_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_100_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_101_neqE, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % neqE
thf(fact_102_neqE, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) => ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_nat @ Y2 @ X3)))))). % neqE
thf(fact_103_neq__iff, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) = (((ord_less_real @ X3 @ Y2)) | ((ord_less_real @ Y2 @ X3))))))). % neq_iff
thf(fact_104_neq__iff, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) = (((ord_less_nat @ X3 @ Y2)) | ((ord_less_nat @ Y2 @ X3))))))). % neq_iff
thf(fact_105_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_106_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_107_dense, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (?[Z4 : real]: ((ord_less_real @ X3 @ Z4) & (ord_less_real @ Z4 @ Y2))))))). % dense
thf(fact_108_less__imp__neq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_109_less__imp__neq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_110_less__asym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_asym
thf(fact_111_less__asym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_asym
thf(fact_112_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_113_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_114_less__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_trans
thf(fact_115_less__trans, axiom,
    ((![X3 : nat, Y2 : nat, Z3 : nat]: ((ord_less_nat @ X3 @ Y2) => ((ord_less_nat @ Y2 @ Z3) => (ord_less_nat @ X3 @ Z3)))))). % less_trans
thf(fact_116_less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) | ((X3 = Y2) | (ord_less_real @ Y2 @ X3)))))). % less_linear
thf(fact_117_less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) | ((X3 = Y2) | (ord_less_nat @ Y2 @ X3)))))). % less_linear
thf(fact_118_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_119_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_120_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_121_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_122_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_123_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_124_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_125_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_126_less__imp__not__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_127_less__imp__not__eq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_128_less__not__sym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_not_sym
thf(fact_129_less__not__sym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_not_sym
thf(fact_130_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X : nat]: ((![Y : nat]: ((ord_less_nat @ Y @ X) => (P @ Y))) => (P @ X))) => (P @ A))))). % less_induct
thf(fact_131_antisym__conv3, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_real @ Y2 @ X3))) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_132_antisym__conv3, axiom,
    ((![Y2 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y2 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_133_less__imp__not__eq2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_134_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_135_less__imp__triv, axiom,
    ((![X3 : real, Y2 : real, P : $o]: ((ord_less_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ X3) => P))))). % less_imp_triv
thf(fact_136_less__imp__triv, axiom,
    ((![X3 : nat, Y2 : nat, P : $o]: ((ord_less_nat @ X3 @ Y2) => ((ord_less_nat @ Y2 @ X3) => P))))). % less_imp_triv
thf(fact_137_linorder__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_cases
thf(fact_138_linorder__cases, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_nat @ Y2 @ X3)))))). % linorder_cases
thf(fact_139_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_140_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_141_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_142_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_143_less__imp__not__less, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_imp_not_less
thf(fact_144_less__imp__not__less, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_imp_not_less
thf(fact_145_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N3)) => ((~ ((P3 @ M2))))))))))))). % exists_least_iff
thf(fact_146_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real]: (P @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_147_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat]: (P @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_148_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_149_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_150_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (((ord_less_real @ Y2 @ X3)) | ((X3 = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_151_not__less__iff__gr__or__eq, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) = (((ord_less_nat @ Y2 @ X3)) | ((X3 = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_152_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_153_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_154_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_155_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_156_strict__mono__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => (((F @ X3) = (F @ Y2)) = (X3 = Y2)))))). % strict_mono_eq
thf(fact_157_leD, axiom,
    ((![Y2 : nat, X3 : nat]: ((ord_less_eq_nat @ Y2 @ X3) => (~ ((ord_less_nat @ X3 @ Y2))))))). % leD
thf(fact_158_leD, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_real @ X3 @ Y2))))))). % leD
thf(fact_159_leI, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_eq_nat @ Y2 @ X3))))). % leI
thf(fact_160_leI, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % leI
thf(fact_161_le__less, axiom,
    ((ord_less_eq_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_nat @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_162_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_163_less__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_164_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_165_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_166_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_167_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_168_order__le__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_169_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_170_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_171_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_172_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_173_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_174_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_175_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_176_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_177_not__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y2))) = (ord_less_nat @ Y2 @ X3))))). % not_le
thf(fact_178_not__le, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) = (ord_less_real @ Y2 @ X3))))). % not_le
thf(fact_179_not__less, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) = (ord_less_eq_nat @ Y2 @ X3))))). % not_less
thf(fact_180_not__less, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (ord_less_eq_real @ Y2 @ X3))))). % not_less
thf(fact_181_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_182_le__neq__trans, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((~ ((A = B))) => (ord_less_real @ A @ B)))))). % le_neq_trans
thf(fact_183_antisym__conv1, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((ord_less_eq_nat @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_184_antisym__conv1, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_185_antisym__conv2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_186_antisym__conv2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_187_less__imp__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_eq_nat @ X3 @ Y2))))). % less_imp_le
thf(fact_188_less__imp__le, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Y2))))). % less_imp_le
thf(fact_189_le__less__trans, axiom,
    ((![X3 : nat, Y2 : nat, Z3 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((ord_less_nat @ Y2 @ Z3) => (ord_less_nat @ X3 @ Z3)))))). % le_less_trans
thf(fact_190_le__less__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % le_less_trans
thf(fact_191_less__le__trans, axiom,
    ((![X3 : nat, Y2 : nat, Z3 : nat]: ((ord_less_nat @ X3 @ Y2) => ((ord_less_eq_nat @ Y2 @ Z3) => (ord_less_nat @ X3 @ Z3)))))). % less_le_trans
thf(fact_192_less__le__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_le_trans
thf(fact_193_dense__ge, axiom,
    ((![Z3 : real, Y2 : real]: ((![X : real]: ((ord_less_real @ Z3 @ X) => (ord_less_eq_real @ Y2 @ X))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_ge
thf(fact_194_dense__le, axiom,
    ((![Y2 : real, Z3 : real]: ((![X : real]: ((ord_less_real @ X @ Y2) => (ord_less_eq_real @ X @ Z3))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_le
thf(fact_195_le__less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) | (ord_less_nat @ Y2 @ X3))))). % le_less_linear
thf(fact_196_le__less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_real @ Y2 @ X3))))). % le_less_linear
thf(fact_197_le__imp__less__or__eq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((ord_less_nat @ X3 @ Y2) | (X3 = Y2)))))). % le_imp_less_or_eq
thf(fact_198_le__imp__less__or__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_real @ X3 @ Y2) | (X3 = Y2)))))). % le_imp_less_or_eq
thf(fact_199_less__le__not__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((~ ((ord_less_eq_nat @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_200_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((ord_less_eq_real @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_201_not__le__imp__less, axiom,
    ((![Y2 : nat, X3 : nat]: ((~ ((ord_less_eq_nat @ Y2 @ X3))) => (ord_less_nat @ X3 @ Y2))))). % not_le_imp_less
thf(fact_202_not__le__imp__less, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_eq_real @ Y2 @ X3))) => (ord_less_real @ X3 @ Y2))))). % not_le_imp_less
thf(fact_203_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_204_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_205_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_206_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_207_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_208_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_real @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_209_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_210_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_211_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_212_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_213_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_214_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_215_dense__ge__bounded, axiom,
    ((![Z3 : real, X3 : real, Y2 : real]: ((ord_less_real @ Z3 @ X3) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X3) => (ord_less_eq_real @ Y2 @ W)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_ge_bounded
thf(fact_216_dense__le__bounded, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((![W : real]: ((ord_less_real @ X3 @ W) => ((ord_less_real @ W @ Y2) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_le_bounded
thf(fact_217_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_218_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_219_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_nat @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_220_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_real @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_221_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_222_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_223_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_224_dual__order_Ostrict__implies__order, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (ord_less_eq_real @ B @ A))))). % dual_order.strict_implies_order
thf(fact_225_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_226_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B : real]: ((~ ((A = B))) => ((ord_less_eq_real @ A @ B) => (ord_less_real @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_227_norm__le__zero__iff, axiom,
    ((![X3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ zero_zero_real) = (X3 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_228_norm__le__zero__iff, axiom,
    ((![X3 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X3) @ zero_zero_real) = (X3 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_229_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_230_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_231_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_232_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_233_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_234_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_235_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_236_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_237_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_238_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_239_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_240_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_241_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_242_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_243_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[K : real]: ((ord_less_real @ zero_zero_real @ K) & (![N4 : nat]: (ord_less_eq_real @ (abs_abs_real @ (im @ (s @ (f @ (g @ N4))))) @ K)))))).
