% 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_240__5369838_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:25.485

% 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 (18)
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_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_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (231)
thf(fact_0_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_1_r, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N)) @ r)))). % r
thf(fact_2_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_3_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_4_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_5_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_6_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_7_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_8_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_9_diff__ge__0__iff__ge, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_eq_real @ B @ A))))). % diff_ge_0_iff_ge
thf(fact_10_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_11_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_12_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_13_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_14_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_15_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_16_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_17_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_18_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_19_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_20_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_21_not__gr__zero, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr_zero
thf(fact_22_le__zero__eq, axiom,
    ((![N2 : nat]: ((ord_less_eq_nat @ N2 @ zero_zero_nat) = (N2 = zero_zero_nat))))). % le_zero_eq
thf(fact_23_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_24_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_25_norm__triangle__ineq2, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq2
thf(fact_26_norm__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq2
thf(fact_27_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_28_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_29_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_30_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_31_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_32_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ C) @ B) = (minus_minus_complex @ (minus_minus_complex @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_33_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (minus_minus_real @ A @ C) @ B) = (minus_minus_real @ (minus_minus_real @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_34_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_35_diff__eq__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: (((minus_minus_complex @ A @ B) = (minus_minus_complex @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_36_diff__eq__diff__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_37_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_38_zero__less__iff__neq__zero, axiom,
    ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) = (~ ((N2 = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_39_gr__implies__not__zero, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_40_not__less__zero, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less_zero
thf(fact_41_gr__zeroI, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr_zeroI
thf(fact_42_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z2 : complex]: (Y2 = Z2))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_43_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: ((minus_minus_real @ A2 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_44_diff__eq__diff__less__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_eq_real @ A @ B) = (ord_less_eq_real @ C @ D)))))). % diff_eq_diff_less_eq
thf(fact_45_diff__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_right_mono
thf(fact_46_diff__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_left_mono
thf(fact_47_diff__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ D @ C) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_mono
thf(fact_48_diff__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_strict_right_mono
thf(fact_49_diff__strict__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => (ord_less_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_strict_left_mono
thf(fact_50_diff__eq__diff__less, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_real @ A @ B) = (ord_less_real @ C @ D)))))). % diff_eq_diff_less
thf(fact_51_diff__strict__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ D @ C) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_strict_mono
thf(fact_52_norm__minus__commute, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (minus_minus_complex @ A @ B)) = (real_V638595069omplex @ (minus_minus_complex @ B @ A)))))). % norm_minus_commute
thf(fact_53_norm__minus__commute, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (minus_minus_real @ A @ B)) = (real_V646646907m_real @ (minus_minus_real @ B @ A)))))). % norm_minus_commute
thf(fact_54_le__iff__diff__le__0, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (ord_less_eq_real @ (minus_minus_real @ A2 @ B2) @ zero_zero_real)))))). % le_iff_diff_le_0
thf(fact_55_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_56_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_57_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_58_Bolzano, axiom,
    ((![A : real, B : real, P : real > real > $o]: ((ord_less_eq_real @ A @ B) => ((![A3 : real, B3 : real, C2 : real]: ((P @ A3 @ B3) => ((P @ B3 @ C2) => ((ord_less_eq_real @ A3 @ B3) => ((ord_less_eq_real @ B3 @ C2) => (P @ A3 @ C2)))))) => ((![X : real]: ((ord_less_eq_real @ A @ X) => ((ord_less_eq_real @ X @ B) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![A3 : real, B3 : real]: (((ord_less_eq_real @ A3 @ X) & ((ord_less_eq_real @ X @ B3) & (ord_less_real @ (minus_minus_real @ B3 @ A3) @ D2))) => (P @ A3 @ B3)))))))) => (P @ A @ B))))))). % Bolzano
thf(fact_59_diff__is__0__eq, axiom,
    ((![M : nat, N2 : nat]: (((minus_minus_nat @ M @ N2) = zero_zero_nat) = (ord_less_eq_nat @ M @ N2))))). % diff_is_0_eq
thf(fact_60_diff__is__0__eq_H, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((minus_minus_nat @ M @ N2) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_61_diff__diff__cancel, axiom,
    ((![I : nat, N2 : nat]: ((ord_less_eq_nat @ I @ N2) => ((minus_minus_nat @ N2 @ (minus_minus_nat @ N2 @ I)) = I))))). % diff_diff_cancel
thf(fact_62_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_63_le0, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % le0
thf(fact_64_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_65_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_66_strict__mono__imp__increasing, axiom,
    ((![F : nat > nat, N2 : nat]: ((order_769474267at_nat @ F) => (ord_less_eq_nat @ N2 @ (F @ N2)))))). % strict_mono_imp_increasing
thf(fact_67_strict__mono__less, axiom,
    ((![F : real > real, X3 : real, Y3 : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y3)) = (ord_less_real @ X3 @ Y3)))))). % strict_mono_less
thf(fact_68_strict__mono__less, axiom,
    ((![F : nat > real, X3 : nat, Y3 : nat]: ((order_952716343t_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y3)) = (ord_less_nat @ X3 @ Y3)))))). % strict_mono_less
thf(fact_69_strict__mono__less, axiom,
    ((![F : real > nat, X3 : real, Y3 : real]: ((order_297469111al_nat @ F) => ((ord_less_nat @ (F @ X3) @ (F @ Y3)) = (ord_less_real @ X3 @ Y3)))))). % strict_mono_less
thf(fact_70_strict__mono__less, axiom,
    ((![F : nat > nat, X3 : nat, Y3 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ (F @ X3) @ (F @ Y3)) = (ord_less_nat @ X3 @ Y3)))))). % strict_mono_less
thf(fact_71_neq0__conv, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N2))))). % neq0_conv
thf(fact_72_less__nat__zero__code, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_73_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_74_diff__0__eq__0, axiom,
    ((![N2 : nat]: ((minus_minus_nat @ zero_zero_nat @ N2) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_75_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_76_zero__less__diff, axiom,
    ((![N2 : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N2 @ M)) = (ord_less_nat @ M @ N2))))). % zero_less_diff
thf(fact_77_gr0I, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr0I
thf(fact_78_not__gr0, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr0
thf(fact_79_diff__less, axiom,
    ((![N2 : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N2) @ M)))))). % diff_less
thf(fact_80_not__less0, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less0
thf(fact_81_less__zeroE, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_zeroE
thf(fact_82_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_83_diff__less__mono2, axiom,
    ((![M : nat, N2 : nat, L : nat]: ((ord_less_nat @ M @ N2) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N2) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_84_gr__implies__not0, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not0
thf(fact_85_diffs0__imp__equal, axiom,
    ((![M : nat, N2 : nat]: (((minus_minus_nat @ M @ N2) = zero_zero_nat) => (((minus_minus_nat @ N2 @ M) = zero_zero_nat) => (M = N2)))))). % diffs0_imp_equal
thf(fact_86_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_87_infinite__descent0, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2)))))))) => (P @ N2)))))). % infinite_descent0
thf(fact_88_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N2 : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N2) @ K))))). % less_imp_diff_less
thf(fact_89_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_90_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_91_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_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_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_92_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_93_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_94_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_95_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_96_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_97_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_98_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_99_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_100_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, 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_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_101_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_102_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_103_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_104_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_105_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_106_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[X2 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X2 @ Y5)) & ((ord_less_eq_nat @ Y5 @ X2)))))))). % eq_iff
thf(fact_107_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[X2 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X2 @ Y5)) & ((ord_less_eq_real @ Y5 @ X2)))))))). % eq_iff
thf(fact_108_antisym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_109_antisym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_110_linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) | (ord_less_eq_nat @ Y3 @ X3))))). % linear
thf(fact_111_linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) | (ord_less_eq_real @ Y3 @ X3))))). % linear
thf(fact_112_eq__refl, axiom,
    ((![X3 : nat, Y3 : nat]: ((X3 = Y3) => (ord_less_eq_nat @ X3 @ Y3))))). % eq_refl
thf(fact_113_eq__refl, axiom,
    ((![X3 : real, Y3 : real]: ((X3 = Y3) => (ord_less_eq_real @ X3 @ Y3))))). % eq_refl
thf(fact_114_le__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y3))) => (ord_less_eq_nat @ Y3 @ X3))))). % le_cases
thf(fact_115_le__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X3 @ Y3))) => (ord_less_eq_real @ Y3 @ X3))))). % le_cases
thf(fact_116_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_117_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_118_le__cases3, axiom,
    ((![X3 : nat, Y3 : nat, Z3 : nat]: (((ord_less_eq_nat @ X3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z3)))) => (((ord_less_eq_nat @ Y3 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Z3)))) => (((ord_less_eq_nat @ X3 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y3)))) => (((ord_less_eq_nat @ Z3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X3)))) => (((ord_less_eq_nat @ Y3 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X3)))) => (~ (((ord_less_eq_nat @ Z3 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_119_le__cases3, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: (((ord_less_eq_real @ X3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z3)))) => (((ord_less_eq_real @ Y3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z3)))) => (((ord_less_eq_real @ X3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y3)))) => (((ord_less_eq_real @ Z3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X3)))) => (((ord_less_eq_real @ Y3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X3)))) => (~ (((ord_less_eq_real @ Z3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_120_antisym__conv, axiom,
    ((![Y3 : nat, X3 : nat]: ((ord_less_eq_nat @ Y3 @ X3) => ((ord_less_eq_nat @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_121_antisym__conv, axiom,
    ((![Y3 : real, X3 : real]: ((ord_less_eq_real @ Y3 @ X3) => ((ord_less_eq_real @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_122_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_123_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_124_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_125_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_126_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_127_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_128_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_129_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_130_order__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ Z3) => (ord_less_eq_nat @ X3 @ Z3)))))). % order_trans
thf(fact_131_order__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_eq_real @ X3 @ Z3)))))). % order_trans
thf(fact_132_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_133_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_134_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_135_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_136_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_137_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_138_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_139_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_140_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_lt__ex, axiom,
    ((![X3 : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X3))))). % lt_ex
thf(fact_159_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_160_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_161_neqE, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) => ((~ ((ord_less_real @ X3 @ Y3))) => (ord_less_real @ Y3 @ X3)))))). % neqE
thf(fact_162_neqE, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) => ((~ ((ord_less_nat @ X3 @ Y3))) => (ord_less_nat @ Y3 @ X3)))))). % neqE
thf(fact_163_neq__iff, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) = (((ord_less_real @ X3 @ Y3)) | ((ord_less_real @ Y3 @ X3))))))). % neq_iff
thf(fact_164_neq__iff, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) = (((ord_less_nat @ X3 @ Y3)) | ((ord_less_nat @ Y3 @ X3))))))). % neq_iff
thf(fact_165_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_166_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_167_dense, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (?[Z4 : real]: ((ord_less_real @ X3 @ Z4) & (ord_less_real @ Z4 @ Y3))))))). % dense
thf(fact_168_less__imp__neq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_169_less__imp__neq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_170_less__asym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_asym
thf(fact_171_less__asym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_asym
thf(fact_172_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_173_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_174_less__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_trans
thf(fact_175_less__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z3 : nat]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ Z3) => (ord_less_nat @ X3 @ Z3)))))). % less_trans
thf(fact_176_less__linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) | ((X3 = Y3) | (ord_less_real @ Y3 @ X3)))))). % less_linear
thf(fact_177_less__linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) | ((X3 = Y3) | (ord_less_nat @ Y3 @ X3)))))). % less_linear
thf(fact_178_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_179_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_180_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_181_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_182_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_183_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_184_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_185_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_186_less__imp__not__eq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_187_less__imp__not__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_188_less__not__sym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_not_sym
thf(fact_189_less__not__sym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_not_sym
thf(fact_190_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_191_antisym__conv3, axiom,
    ((![Y3 : real, X3 : real]: ((~ ((ord_less_real @ Y3 @ X3))) => ((~ ((ord_less_real @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_192_antisym__conv3, axiom,
    ((![Y3 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y3 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_193_less__imp__not__eq2, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_194_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_195_less__imp__triv, axiom,
    ((![X3 : real, Y3 : real, P : $o]: ((ord_less_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ X3) => P))))). % less_imp_triv
thf(fact_196_less__imp__triv, axiom,
    ((![X3 : nat, Y3 : nat, P : $o]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ X3) => P))))). % less_imp_triv
thf(fact_197_linorder__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_real @ Y3 @ X3)))))). % linorder_cases
thf(fact_198_linorder__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_nat @ Y3 @ X3)))))). % linorder_cases
thf(fact_199_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_200_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_201_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_202_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_203_less__imp__not__less, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_imp_not_less
thf(fact_204_less__imp__not__less, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_imp_not_less
thf(fact_205_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N4 : nat]: (((P3 @ N4)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N4)) => ((~ ((P3 @ M3))))))))))))). % exists_least_iff
thf(fact_206_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_207_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_208_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_209_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_210_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) = (((ord_less_real @ Y3 @ X3)) | ((X3 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_211_not__less__iff__gr__or__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) = (((ord_less_nat @ Y3 @ X3)) | ((X3 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_212_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_213_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_214_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_215_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_216_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_217_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_218_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_219_le__antisym, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((ord_less_eq_nat @ N2 @ M) => (M = N2)))))). % le_antisym
thf(fact_220_nat__le__linear, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) | (ord_less_eq_nat @ N2 @ M))))). % nat_le_linear
thf(fact_221_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B))) => (?[X : nat]: ((P @ X) & (![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_222_less__eq__nat_Osimps_I1_J, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % less_eq_nat.simps(1)
thf(fact_223_le__0__eq, axiom,
    ((![N2 : nat]: ((ord_less_eq_nat @ N2 @ zero_zero_nat) = (N2 = zero_zero_nat))))). % le_0_eq
thf(fact_224_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_225_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_226_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N4 : nat]: (((ord_less_eq_nat @ M3 @ N4)) & ((~ ((M3 = N4)))))))))). % nat_less_le
thf(fact_227_less__imp__le__nat, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (ord_less_eq_nat @ M @ N2))))). % less_imp_le_nat
thf(fact_228_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N4 : nat]: (((ord_less_nat @ M3 @ N4)) | ((M3 = N4)))))))). % le_eq_less_or_eq
thf(fact_229_less__or__eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: (((ord_less_nat @ M @ N2) | (M = N2)) => (ord_less_eq_nat @ M @ N2))))). % less_or_eq_imp_le
thf(fact_230_le__neq__implies__less, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((~ ((M = N2))) => (ord_less_nat @ M @ N2)))))). % le_neq_implies_less

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[F2 : nat > nat, Z : complex]: ((order_769474267at_nat @ F2) & (![E : real]: ((~ ((ord_less_real @ zero_zero_real @ E))) | (?[N5 : nat]: (![N3 : nat]: ((~ ((ord_less_eq_nat @ N5 @ N3))) | (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (s @ (F2 @ N3)) @ Z)) @ E)))))))))).
