% 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_378__5371226_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:40.421

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
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 (17)
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_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_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_r, type,
    r : real).
thf(sy_v_x____, type,
    x : real).
thf(sy_v_z____, type,
    z : complex).

% Relevant facts (248)
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_that_I3_J, axiom,
    ((~ ((ord_less_real @ x @ one_one_real))))). % that(3)
thf(fact_2_True, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % True
thf(fact_3_that_I1_J, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ z) @ r))). % that(1)
thf(fact_4_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_5_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_6_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_7_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_8_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_9_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_10_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_11_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_12_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_13_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_14_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_15_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_16_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_17_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_18_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_19_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_20_less__numeral__extra_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % less_numeral_extra(1)
thf(fact_21_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_22_zero__less__one, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % zero_less_one
thf(fact_23_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_24_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_25_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_26_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_27_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_28_one__reorient, axiom,
    ((![X3 : real]: ((one_one_real = X3) = (X3 = one_one_real))))). % one_reorient
thf(fact_29_one__reorient, axiom,
    ((![X3 : nat]: ((one_one_nat = X3) = (X3 = one_one_nat))))). % one_reorient
thf(fact_30_one__reorient, axiom,
    ((![X3 : complex]: ((one_one_complex = X3) = (X3 = one_one_complex))))). % one_reorient
thf(fact_31_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_32_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_33_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_34_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_35_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_36_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_37_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_38_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_39_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_40_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_41_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_42_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_43_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_real @ one_one_real @ one_one_real))). % le_numeral_extra(4)
thf(fact_44_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_45_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_real @ one_one_real @ one_one_real))))). % less_numeral_extra(4)
thf(fact_46_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_47_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ zero_zero_real))))). % not_one_le_zero
thf(fact_48_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_49_zero__le__one, axiom,
    ((ord_less_eq_real @ zero_zero_real @ one_one_real))). % zero_le_one
thf(fact_50_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_51_not__one__less__zero, axiom,
    ((~ ((ord_less_real @ one_one_real @ zero_zero_real))))). % not_one_less_zero
thf(fact_52_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_53_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_54_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_55_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % less_eq_real_def
thf(fact_56_complete__interval, axiom,
    ((![A : real, B : real, P : real > $o]: ((ord_less_real @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C : real]: ((ord_less_eq_real @ A @ C) & ((ord_less_eq_real @ C @ B) & ((![X4 : real]: (((ord_less_eq_real @ A @ X4) & (ord_less_real @ X4 @ C)) => (P @ X4))) & (![D : real]: ((![X : real]: (((ord_less_eq_real @ A @ X) & (ord_less_real @ X @ D)) => (P @ X))) => (ord_less_eq_real @ D @ C))))))))))))). % complete_interval
thf(fact_57_complete__interval, axiom,
    ((![A : nat, B : nat, P : nat > $o]: ((ord_less_nat @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C : nat]: ((ord_less_eq_nat @ A @ C) & ((ord_less_eq_nat @ C @ B) & ((![X4 : nat]: (((ord_less_eq_nat @ A @ X4) & (ord_less_nat @ X4 @ C)) => (P @ X4))) & (![D : nat]: ((![X : nat]: (((ord_less_eq_nat @ A @ X) & (ord_less_nat @ X @ D)) => (P @ X))) => (ord_less_eq_nat @ D @ C))))))))))))). % complete_interval
thf(fact_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_67_Collect__mem__eq, axiom,
    ((![A3 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A3))) = A3)))). % Collect_mem_eq
thf(fact_68_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C2)))))))). % order_subst1
thf(fact_69_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C2)))))))). % order_subst1
thf(fact_70_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C2)))))))). % order_subst1
thf(fact_71_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C2)))))))). % order_subst1
thf(fact_72_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C2))))))). % order_subst2
thf(fact_73_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C2))))))). % order_subst2
thf(fact_74_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C2))))))). % order_subst2
thf(fact_75_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C2))))))). % order_subst2
thf(fact_76_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_77_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_78_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_79_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_80_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_81_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_82_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_83_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_84_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_85_eq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_86_antisym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_87_antisym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((ord_less_eq_nat @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_88_linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_eq_real @ Y2 @ X3))))). % linear
thf(fact_89_linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) | (ord_less_eq_nat @ Y2 @ X3))))). % linear
thf(fact_90_eq__refl, axiom,
    ((![X3 : real, Y2 : real]: ((X3 = Y2) => (ord_less_eq_real @ X3 @ Y2))))). % eq_refl
thf(fact_91_eq__refl, axiom,
    ((![X3 : nat, Y2 : nat]: ((X3 = Y2) => (ord_less_eq_nat @ X3 @ Y2))))). % eq_refl
thf(fact_92_le__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % le_cases
thf(fact_93_le__cases, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y2))) => (ord_less_eq_nat @ Y2 @ X3))))). % le_cases
thf(fact_94_order_Otrans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C2) => (ord_less_eq_real @ A @ C2)))))). % order.trans
thf(fact_95_order_Otrans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C2) => (ord_less_eq_nat @ A @ C2)))))). % order.trans
thf(fact_96_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_97_le__cases3, axiom,
    ((![X3 : nat, Y2 : nat, Z3 : nat]: (((ord_less_eq_nat @ X3 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z3)))) => (((ord_less_eq_nat @ Y2 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Z3)))) => (((ord_less_eq_nat @ X3 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y2)))) => (((ord_less_eq_nat @ Z3 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X3)))) => (((ord_less_eq_nat @ Y2 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X3)))) => (~ (((ord_less_eq_nat @ Z3 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Y2)))))))))))))). % le_cases3
thf(fact_98_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_99_antisym__conv, axiom,
    ((![Y2 : nat, X3 : nat]: ((ord_less_eq_nat @ Y2 @ X3) => ((ord_less_eq_nat @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv
thf(fact_100_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_101_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_102_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((A = B) => ((ord_less_eq_real @ B @ C2) => (ord_less_eq_real @ A @ C2)))))). % ord_eq_le_trans
thf(fact_103_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C2) => (ord_less_eq_nat @ A @ C2)))))). % ord_eq_le_trans
thf(fact_104_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((B = C2) => (ord_less_eq_real @ A @ C2)))))). % ord_le_eq_trans
thf(fact_105_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C2) => (ord_less_eq_nat @ A @ C2)))))). % ord_le_eq_trans
thf(fact_106_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_107_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_108_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_109_order__trans, axiom,
    ((![X3 : nat, Y2 : nat, Z3 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((ord_less_eq_nat @ Y2 @ Z3) => (ord_less_eq_nat @ X3 @ Z3)))))). % order_trans
thf(fact_110_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_111_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_112_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A4 : real, B3 : real]: ((ord_less_eq_real @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : real, B3 : real]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_113_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B3 : nat]: ((ord_less_eq_nat @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : nat, B3 : nat]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_114_dual__order_Otrans, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C2 @ B) => (ord_less_eq_real @ C2 @ A)))))). % dual_order.trans
thf(fact_115_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C2 @ B) => (ord_less_eq_nat @ C2 @ A)))))). % dual_order.trans
thf(fact_116_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_117_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_118_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_119_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_120_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_121_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_122_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_123_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_124_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_real @ A @ B) => (((F @ B) = C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_125_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => (((F @ B) = C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_126_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => (((F @ B) = C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_127_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_128_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_less_subst1
thf(fact_129_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_less_subst1
thf(fact_130_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_less_subst1
thf(fact_131_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_less_subst1
thf(fact_132_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_less_subst2
thf(fact_133_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_less_subst2
thf(fact_134_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_less_subst2
thf(fact_135_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_less_subst2
thf(fact_136_lt__ex, axiom,
    ((![X3 : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X3))))). % lt_ex
thf(fact_137_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_138_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_139_neqE, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % neqE
thf(fact_140_neqE, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) => ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_nat @ Y2 @ X3)))))). % neqE
thf(fact_141_neq__iff, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) = (((ord_less_real @ X3 @ Y2)) | ((ord_less_real @ Y2 @ X3))))))). % neq_iff
thf(fact_142_neq__iff, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) = (((ord_less_nat @ X3 @ Y2)) | ((ord_less_nat @ Y2 @ X3))))))). % neq_iff
thf(fact_143_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_144_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_145_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_146_less__imp__neq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_147_less__imp__neq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_148_less__asym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_asym
thf(fact_149_less__asym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_asym
thf(fact_150_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_151_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_152_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_153_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_154_less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) | ((X3 = Y2) | (ord_less_real @ Y2 @ X3)))))). % less_linear
thf(fact_155_less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) | ((X3 = Y2) | (ord_less_nat @ Y2 @ X3)))))). % less_linear
thf(fact_156_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_157_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_158_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((A = B) => ((ord_less_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % ord_eq_less_trans
thf(fact_159_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((A = B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % ord_eq_less_trans
thf(fact_160_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_real @ A @ B) => ((B = C2) => (ord_less_real @ A @ C2)))))). % ord_less_eq_trans
thf(fact_161_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((B = C2) => (ord_less_nat @ A @ C2)))))). % ord_less_eq_trans
thf(fact_162_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_163_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_164_less__imp__not__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_165_less__imp__not__eq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_166_less__not__sym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_not_sym
thf(fact_167_less__not__sym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_not_sym
thf(fact_168_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_169_antisym__conv3, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_real @ Y2 @ X3))) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_170_antisym__conv3, axiom,
    ((![Y2 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y2 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_171_less__imp__not__eq2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_172_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_173_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_174_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_175_linorder__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_cases
thf(fact_176_linorder__cases, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_nat @ Y2 @ X3)))))). % linorder_cases
thf(fact_177_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_178_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_179_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % order.strict_trans
thf(fact_180_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % order.strict_trans
thf(fact_181_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_182_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_183_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X5 : nat]: (P2 @ X5))) = (^[P3 : nat > $o]: (?[N2 : nat]: (((P3 @ N2)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N2)) => ((~ ((P3 @ M2))))))))))))). % exists_least_iff
thf(fact_184_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A4 : real, B3 : real]: ((ord_less_real @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : real]: (P @ A4 @ A4)) => ((![A4 : real, B3 : real]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_185_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B3 : nat]: ((ord_less_nat @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : nat]: (P @ A4 @ A4)) => ((![A4 : nat, B3 : nat]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_186_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C2 @ B) => (ord_less_real @ C2 @ A)))))). % dual_order.strict_trans
thf(fact_187_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C2 @ B) => (ord_less_nat @ C2 @ A)))))). % dual_order.strict_trans
thf(fact_188_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_189_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_190_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_191_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_192_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_193_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_194_ex__gt__or__lt, axiom,
    ((![A : real]: (?[B3 : real]: ((ord_less_real @ A @ B3) | (ord_less_real @ B3 @ A)))))). % ex_gt_or_lt
thf(fact_195_complete__real, axiom,
    ((![S2 : set_real]: ((?[X4 : real]: (member_real @ X4 @ S2)) => ((?[Z : real]: (![X : real]: ((member_real @ X @ S2) => (ord_less_eq_real @ X @ Z)))) => (?[Y4 : real]: ((![X4 : real]: ((member_real @ X4 @ S2) => (ord_less_eq_real @ X4 @ Y4))) & (![Z : real]: ((![X : real]: ((member_real @ X @ S2) => (ord_less_eq_real @ X @ Z))) => (ord_less_eq_real @ Y4 @ Z)))))))))). % complete_real
thf(fact_196_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_197_leD, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_real @ X3 @ Y2))))))). % leD
thf(fact_198_leD, axiom,
    ((![Y2 : nat, X3 : nat]: ((ord_less_eq_nat @ Y2 @ X3) => (~ ((ord_less_nat @ X3 @ Y2))))))). % leD
thf(fact_199_leI, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % leI
thf(fact_200_leI, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_eq_nat @ Y2 @ X3))))). % leI
thf(fact_201_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_202_le__less, axiom,
    ((ord_less_eq_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_nat @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_203_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_204_less__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_205_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_le_less_subst1
thf(fact_206_order__le__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_le_less_subst1
thf(fact_207_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_le_less_subst1
thf(fact_208_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_le_less_subst1
thf(fact_209_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_le_less_subst2
thf(fact_210_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_le_less_subst2
thf(fact_211_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_le_less_subst2
thf(fact_212_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_le_less_subst2
thf(fact_213_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C2 : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_less_le_subst1
thf(fact_214_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C2 : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_less_le_subst1
thf(fact_215_order__less__le__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C2 : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C2)))))))). % order_less_le_subst1
thf(fact_216_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_less_le_subst1
thf(fact_217_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_less_le_subst2
thf(fact_218_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C2))))))). % order_less_le_subst2
thf(fact_219_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C2) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_less_le_subst2
thf(fact_220_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C2) => ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_less_le_subst2
thf(fact_221_not__le, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) = (ord_less_real @ Y2 @ X3))))). % not_le
thf(fact_222_not__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y2))) = (ord_less_nat @ Y2 @ X3))))). % not_le
thf(fact_223_not__less, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (ord_less_eq_real @ Y2 @ X3))))). % not_less
thf(fact_224_not__less, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) = (ord_less_eq_nat @ Y2 @ X3))))). % not_less
thf(fact_225_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_226_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_227_antisym__conv1, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_228_antisym__conv1, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((ord_less_eq_nat @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_229_antisym__conv2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_230_antisym__conv2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_231_less__imp__le, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Y2))))). % less_imp_le
thf(fact_232_less__imp__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_eq_nat @ X3 @ Y2))))). % less_imp_le
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_le__less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_real @ Y2 @ X3))))). % le_less_linear
thf(fact_240_le__less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) | (ord_less_nat @ Y2 @ X3))))). % le_less_linear
thf(fact_241_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_242_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_243_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_244_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_245_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_246_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_247_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

% Conjectures (1)
thf(conj_0, conjecture,
    ($false)).
