% 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_776__5375446_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:02.430

% 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 (19)
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_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex, type,
    neg_nu484426047omplex : complex > complex).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal, type,
    neg_nu1973887165c_real : 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_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal, type,
    arcosh_real : real > real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_k____, type,
    k : nat).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_t____, type,
    t : real).

% Relevant facts (228)
thf(fact_0_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_1_t_I2_J, axiom,
    ((ord_less_real @ t @ one_one_real))). % t(2)
thf(fact_2_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_3__092_060open_062_092_060And_062d2_O_A_I0_058_058_063_Ha_J_A_060_Ad2_A_092_060Longrightarrow_062_A_092_060exists_062e_0620_058_058_063_Ha_O_Ae_A_060_A_I1_058_058_063_Ha_J_A_092_060and_062_Ae_A_060_Ad2_092_060close_062, axiom,
    ((![D2 : real]: ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ one_one_real) & (ord_less_real @ E @ D2)))))))). % \<open>\<And>d2. (0::?'a) < d2 \<Longrightarrow> \<exists>e>0::?'a. e < (1::?'a) \<and> e < d2\<close>
thf(fact_4_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_5_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_6_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_7_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_real @ one_one_real @ one_one_real))). % le_numeral_extra(4)
thf(fact_8_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_9_zero__le__one, axiom,
    ((ord_less_eq_real @ zero_zero_real @ one_one_real))). % zero_le_one
thf(fact_10_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_11_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ zero_zero_real))))). % not_one_le_zero
thf(fact_12_complete__real, axiom,
    ((![S : set_real]: ((?[X2 : real]: (member_real @ X2 @ S)) => ((?[Z : real]: (![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z)))) => (?[Y : real]: ((![X2 : real]: ((member_real @ X2 @ S) => (ord_less_eq_real @ X2 @ Y))) & (![Z : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z))) => (ord_less_eq_real @ Y @ Z)))))))))). % complete_real
thf(fact_13_one__reorient, axiom,
    ((![X : real]: ((one_one_real = X) = (X = one_one_real))))). % one_reorient
thf(fact_14_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_15_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) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_16_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) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_17_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) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_18_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) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_19_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_20_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_21_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_22_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_23_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_24_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_25_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_26_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_27_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_28_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_29_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_30_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_31_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_32_lt__ex, axiom,
    ((![X : real]: (?[Y : real]: (ord_less_real @ Y @ X))))). % lt_ex
thf(fact_33_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_34_neqE, axiom,
    ((![X : real, Y2 : real]: ((~ ((X = Y2))) => ((~ ((ord_less_real @ X @ Y2))) => (ord_less_real @ Y2 @ X)))))). % neqE
thf(fact_35_neq__iff, axiom,
    ((![X : real, Y2 : real]: ((~ ((X = Y2))) = (((ord_less_real @ X @ Y2)) | ((ord_less_real @ Y2 @ X))))))). % neq_iff
thf(fact_36_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_37_dense, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (?[Z2 : real]: ((ord_less_real @ X @ Z2) & (ord_less_real @ Z2 @ Y2))))))). % dense
thf(fact_38_less__imp__neq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_neq
thf(fact_39_less__asym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_asym
thf(fact_40_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_41_less__trans, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_real @ X @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X @ Z3)))))). % less_trans
thf(fact_42_less__linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) | ((X = Y2) | (ord_less_real @ Y2 @ X)))))). % less_linear
thf(fact_43_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_44_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_45_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_46_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_47_less__imp__not__eq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_not_eq
thf(fact_48_less__not__sym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_not_sym
thf(fact_49_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_50_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_51_antisym__conv3, axiom,
    ((![Y2 : real, X : real]: ((~ ((ord_less_real @ Y2 @ X))) => ((~ ((ord_less_real @ X @ Y2))) = (X = Y2)))))). % antisym_conv3
thf(fact_52_less__imp__not__eq2, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((Y2 = X))))))). % less_imp_not_eq2
thf(fact_53_less__imp__triv, axiom,
    ((![X : real, Y2 : real, P : $o]: ((ord_less_real @ X @ Y2) => ((ord_less_real @ Y2 @ X) => P))))). % less_imp_triv
thf(fact_54_linorder__cases, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => ((~ ((X = Y2))) => (ord_less_real @ Y2 @ X)))))). % linorder_cases
thf(fact_55_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_56_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_57_less__imp__not__less, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_imp_not_less
thf(fact_58_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B2 : real]: ((ord_less_real @ A3 @ B2) => (P @ A3 @ B2))) => ((![A3 : real]: (P @ A3 @ A3)) => ((![A3 : real, B2 : real]: ((P @ B2 @ A3) => (P @ A3 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_59_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_60_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) = (((ord_less_real @ Y2 @ X)) | ((X = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_61_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_62_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_63_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_64_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_65_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y2 : real]: ((~ ((X = Y2))) => ((~ ((ord_less_real @ X @ Y2))) => (ord_less_real @ Y2 @ X)))))). % linorder_neqE_linordered_idom
thf(fact_66_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_67_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_68_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_69_not__one__less__zero, axiom,
    ((~ ((ord_less_real @ one_one_real @ zero_zero_real))))). % not_one_less_zero
thf(fact_70_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_71_zero__less__one, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % zero_less_one
thf(fact_72_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_73_less__numeral__extra_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % less_numeral_extra(1)
thf(fact_74_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_12 : real]: (P @ X_12)) => ((?[Z : real]: (![X3 : real]: ((P @ X3) => (ord_less_real @ X3 @ Z)))) => (?[S2 : real]: (![Y3 : real]: ((?[X4 : real]: (((P @ X4)) & ((ord_less_real @ Y3 @ X4)))) = (ord_less_real @ Y3 @ S2))))))))). % real_sup_exists
thf(fact_75_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_76_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_77_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_78_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_79_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B3 : real]: (^[A4 : real]: (((ord_less_eq_real @ B3 @ A4)) & ((~ ((A4 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_80_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B3 : nat]: (^[A4 : nat]: (((ord_less_eq_nat @ B3 @ A4)) & ((~ ((A4 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_81_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B3 : real]: (^[A4 : real]: (((ord_less_real @ B3 @ A4)) | ((A4 = B3)))))))). % dual_order.order_iff_strict
thf(fact_82_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B3 : nat]: (^[A4 : nat]: (((ord_less_nat @ B3 @ A4)) | ((A4 = B3)))))))). % dual_order.order_iff_strict
thf(fact_83_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_84_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_85_dense__le__bounded, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_real @ X @ Y2) => ((![W : real]: ((ord_less_real @ X @ W) => ((ord_less_real @ W @ Y2) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_le_bounded
thf(fact_86_dense__ge__bounded, axiom,
    ((![Z3 : real, X : real, Y2 : real]: ((ord_less_real @ Z3 @ X) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X) => (ord_less_eq_real @ Y2 @ W)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_ge_bounded
thf(fact_87_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_88_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_89_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_90_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_91_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ A4 @ B3)) & ((~ ((A4 = B3)))))))))). % order.strict_iff_order
thf(fact_92_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A4 @ B3)) & ((~ ((A4 = B3)))))))))). % order.strict_iff_order
thf(fact_93_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A4 : real]: (^[B3 : real]: (((ord_less_real @ A4 @ B3)) | ((A4 = B3)))))))). % order.order_iff_strict
thf(fact_94_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A4 : nat]: (^[B3 : nat]: (((ord_less_nat @ A4 @ B3)) | ((A4 = B3)))))))). % order.order_iff_strict
thf(fact_95_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_96_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_97_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_98_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_99_not__le__imp__less, axiom,
    ((![Y2 : real, X : real]: ((~ ((ord_less_eq_real @ Y2 @ X))) => (ord_less_real @ X @ Y2))))). % not_le_imp_less
thf(fact_100_not__le__imp__less, axiom,
    ((![Y2 : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y2 @ X))) => (ord_less_nat @ X @ Y2))))). % not_le_imp_less
thf(fact_101_less__le__not__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((ord_less_eq_real @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_102_less__le__not__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((~ ((ord_less_eq_nat @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_103_le__imp__less__or__eq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_real @ X @ Y2) | (X = Y2)))))). % le_imp_less_or_eq
thf(fact_104_le__imp__less__or__eq, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_nat @ X @ Y2) | (X = Y2)))))). % le_imp_less_or_eq
thf(fact_105_le__less__linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) | (ord_less_real @ Y2 @ X))))). % le_less_linear
thf(fact_106_le__less__linear, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) | (ord_less_nat @ Y2 @ X))))). % le_less_linear
thf(fact_107_dense__le, axiom,
    ((![Y2 : real, Z3 : real]: ((![X3 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Z3))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_le
thf(fact_108_dense__ge, axiom,
    ((![Z3 : real, Y2 : real]: ((![X3 : real]: ((ord_less_real @ Z3 @ X3) => (ord_less_eq_real @ Y2 @ X3))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_ge
thf(fact_109_less__le__trans, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_real @ X @ Z3)))))). % less_le_trans
thf(fact_110_less__le__trans, axiom,
    ((![X : nat, Y2 : nat, Z3 : nat]: ((ord_less_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ Z3) => (ord_less_nat @ X @ Z3)))))). % less_le_trans
thf(fact_111_le__less__trans, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X @ Z3)))))). % le_less_trans
thf(fact_112_le__less__trans, axiom,
    ((![X : nat, Y2 : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_nat @ Y2 @ Z3) => (ord_less_nat @ X @ Z3)))))). % le_less_trans
thf(fact_113_less__imp__le, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_eq_real @ X @ Y2))))). % less_imp_le
thf(fact_114_less__imp__le, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_eq_nat @ X @ Y2))))). % less_imp_le
thf(fact_115_antisym__conv2, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((~ ((ord_less_real @ X @ Y2))) = (X = Y2)))))). % antisym_conv2
thf(fact_116_antisym__conv2, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((~ ((ord_less_nat @ X @ Y2))) = (X = Y2)))))). % antisym_conv2
thf(fact_117_antisym__conv1, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => ((ord_less_eq_real @ X @ Y2) = (X = Y2)))))). % antisym_conv1
thf(fact_118_antisym__conv1, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_nat @ X @ Y2))) => ((ord_less_eq_nat @ X @ Y2) = (X = Y2)))))). % antisym_conv1
thf(fact_119_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_120_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_121_not__less, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) = (ord_less_eq_real @ Y2 @ X))))). % not_less
thf(fact_122_not__less, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_nat @ X @ Y2))) = (ord_less_eq_nat @ Y2 @ X))))). % not_less
thf(fact_123_not__le, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_eq_real @ X @ Y2))) = (ord_less_real @ Y2 @ X))))). % not_le
thf(fact_124_not__le, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X @ Y2))) = (ord_less_nat @ Y2 @ X))))). % not_le
thf(fact_125_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_126_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_127_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_128_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_129_order__less__le__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_130_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_131_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_132_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_133_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_134_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_135_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_136_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_137_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_138_less__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_139_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_140_le__less, axiom,
    ((ord_less_eq_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_nat @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_141_leI, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => (ord_less_eq_real @ Y2 @ X))))). % leI
thf(fact_142_leI, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_nat @ X @ Y2))) => (ord_less_eq_nat @ Y2 @ X))))). % leI
thf(fact_143_leD, axiom,
    ((![Y2 : real, X : real]: ((ord_less_eq_real @ Y2 @ X) => (~ ((ord_less_real @ X @ Y2))))))). % leD
thf(fact_144_leD, axiom,
    ((![Y2 : nat, X : nat]: ((ord_less_eq_nat @ Y2 @ X) => (~ ((ord_less_nat @ X @ Y2))))))). % leD
thf(fact_145_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_146_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_real @ one_one_real @ one_one_real))))). % less_numeral_extra(4)
thf(fact_147_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_148_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_149_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_150_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_151_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_152_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_153_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % less_eq_real_def
thf(fact_154_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_155_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_156_dual__order_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z4 : real]: (Y5 = Z4))) = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A4)) & ((ord_less_eq_real @ A4 @ B3)))))))). % dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[A4 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A4)) & ((ord_less_eq_nat @ A4 @ B3)))))))). % dual_order.eq_iff
thf(fact_158_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_159_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_160_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B2 : real]: ((ord_less_eq_real @ A3 @ B2) => (P @ A3 @ B2))) => ((![A3 : real, B2 : real]: ((P @ B2 @ A3) => (P @ A3 @ B2))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_161_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B2 : nat]: ((ord_less_eq_nat @ A3 @ B2) => (P @ A3 @ B2))) => ((![A3 : nat, B2 : nat]: ((P @ B2 @ A3) => (P @ A3 @ B2))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_162_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_163_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_164_order__trans, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_eq_real @ X @ Z3)))))). % order_trans
thf(fact_165_order__trans, axiom,
    ((![X : nat, Y2 : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ Z3) => (ord_less_eq_nat @ X @ Z3)))))). % order_trans
thf(fact_166_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_167_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_168_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_169_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_170_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_171_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_172_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z4 : real]: (Y5 = Z4))) = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ A4 @ B3)) & ((ord_less_eq_real @ B3 @ A4)))))))). % order_class.order.eq_iff
thf(fact_173_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[A4 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A4 @ B3)) & ((ord_less_eq_nat @ B3 @ A4)))))))). % order_class.order.eq_iff
thf(fact_174_antisym__conv, axiom,
    ((![Y2 : real, X : real]: ((ord_less_eq_real @ Y2 @ X) => ((ord_less_eq_real @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_175_antisym__conv, axiom,
    ((![Y2 : nat, X : nat]: ((ord_less_eq_nat @ Y2 @ X) => ((ord_less_eq_nat @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_176_le__cases3, axiom,
    ((![X : real, Y2 : real, Z3 : real]: (((ord_less_eq_real @ X @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z3)))) => (((ord_less_eq_real @ Y2 @ X) => (~ ((ord_less_eq_real @ X @ Z3)))) => (((ord_less_eq_real @ X @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y2)))) => (((ord_less_eq_real @ Z3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X)))) => (((ord_less_eq_real @ Y2 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X)))) => (~ (((ord_less_eq_real @ Z3 @ X) => (~ ((ord_less_eq_real @ X @ Y2)))))))))))))). % le_cases3
thf(fact_177_le__cases3, axiom,
    ((![X : nat, Y2 : nat, Z3 : nat]: (((ord_less_eq_nat @ X @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z3)))) => (((ord_less_eq_nat @ Y2 @ X) => (~ ((ord_less_eq_nat @ X @ Z3)))) => (((ord_less_eq_nat @ X @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y2)))) => (((ord_less_eq_nat @ Z3 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X)))) => (((ord_less_eq_nat @ Y2 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X)))) => (~ (((ord_less_eq_nat @ Z3 @ X) => (~ ((ord_less_eq_nat @ X @ Y2)))))))))))))). % le_cases3
thf(fact_178_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_179_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_180_le__cases, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_eq_real @ X @ Y2))) => (ord_less_eq_real @ Y2 @ X))))). % le_cases
thf(fact_181_le__cases, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X @ Y2))) => (ord_less_eq_nat @ Y2 @ X))))). % le_cases
thf(fact_182_eq__refl, axiom,
    ((![X : real, Y2 : real]: ((X = Y2) => (ord_less_eq_real @ X @ Y2))))). % eq_refl
thf(fact_183_eq__refl, axiom,
    ((![X : nat, Y2 : nat]: ((X = Y2) => (ord_less_eq_nat @ X @ Y2))))). % eq_refl
thf(fact_184_linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) | (ord_less_eq_real @ Y2 @ X))))). % linear
thf(fact_185_linear, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) | (ord_less_eq_nat @ Y2 @ X))))). % linear
thf(fact_186_antisym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_187_antisym, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_188_eq__iff, axiom,
    (((^[Y5 : real]: (^[Z4 : real]: (Y5 = Z4))) = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((ord_less_eq_real @ Y4 @ X4)))))))). % eq_iff
thf(fact_189_eq__iff, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X4)))))))). % eq_iff
thf(fact_190_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_191_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_192_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_193_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_194_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_195_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_196_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_197_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_198_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) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_199_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) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_200_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) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_201_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) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_202_complete__interval, axiom,
    ((![A : real, B : real, P : real > $o]: ((ord_less_real @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C2 : real]: ((ord_less_eq_real @ A @ C2) & ((ord_less_eq_real @ C2 @ B) & ((![X2 : real]: (((ord_less_eq_real @ A @ X2) & (ord_less_real @ X2 @ C2)) => (P @ X2))) & (![D : real]: ((![X3 : real]: (((ord_less_eq_real @ A @ X3) & (ord_less_real @ X3 @ D)) => (P @ X3))) => (ord_less_eq_real @ D @ C2))))))))))))). % complete_interval
thf(fact_203_complete__interval, axiom,
    ((![A : nat, B : nat, P : nat > $o]: ((ord_less_nat @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C2 : nat]: ((ord_less_eq_nat @ A @ C2) & ((ord_less_eq_nat @ C2 @ B) & ((![X2 : nat]: (((ord_less_eq_nat @ A @ X2) & (ord_less_nat @ X2 @ C2)) => (P @ X2))) & (![D : nat]: ((![X3 : nat]: (((ord_less_eq_nat @ A @ X3) & (ord_less_nat @ X3 @ D)) => (P @ X3))) => (ord_less_eq_nat @ D @ C2))))))))))))). % complete_interval
thf(fact_204_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : real, A5 : real]: ((~ ((ord_less_eq_real @ B4 @ A5))) = (ord_less_real @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_205_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : nat, A5 : nat]: ((~ ((ord_less_eq_nat @ B4 @ A5))) = (ord_less_nat @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_206_pinf_I6_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (~ ((ord_less_eq_real @ X2 @ T))))))))). % pinf(6)
thf(fact_207_pinf_I6_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X2 : nat]: ((ord_less_nat @ Z2 @ X2) => (~ ((ord_less_eq_nat @ X2 @ T))))))))). % pinf(6)
thf(fact_208_pinf_I8_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (ord_less_eq_real @ T @ X2))))))). % pinf(8)
thf(fact_209_pinf_I8_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X2 : nat]: ((ord_less_nat @ Z2 @ X2) => (ord_less_eq_nat @ T @ X2))))))). % pinf(8)
thf(fact_210_minf_I6_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => (ord_less_eq_real @ X2 @ T))))))). % minf(6)
thf(fact_211_minf_I6_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z2) => (ord_less_eq_nat @ X2 @ T))))))). % minf(6)
thf(fact_212_minf_I8_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ X2 @ Z2) => (~ ((ord_less_eq_real @ T @ X2))))))))). % minf(8)
thf(fact_213_minf_I8_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z2) => (~ ((ord_less_eq_nat @ T @ X2))))))))). % minf(8)
thf(fact_214_dbl__inc__simps_I2_J, axiom,
    (((neg_nu1973887165c_real @ zero_zero_real) = one_one_real))). % dbl_inc_simps(2)
thf(fact_215_dbl__inc__simps_I2_J, axiom,
    (((neg_nu484426047omplex @ zero_zero_complex) = one_one_complex))). % dbl_inc_simps(2)
thf(fact_216_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_217_arcosh__1, axiom,
    (((arcosh_real @ one_one_real) = zero_zero_real))). % arcosh_1
thf(fact_218_verit__la__disequality, axiom,
    ((![A : real, B : real]: ((A = B) | ((~ ((ord_less_eq_real @ A @ B))) | (~ ((ord_less_eq_real @ B @ A)))))))). % verit_la_disequality
thf(fact_219_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_220_ex__gt__or__lt, axiom,
    ((![A : real]: (?[B2 : real]: ((ord_less_real @ A @ B2) | (ord_less_real @ B2 @ A)))))). % ex_gt_or_lt
thf(fact_221_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_222_pinf_I1_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z : real]: (![X3 : real]: ((ord_less_real @ Z @ X3) => ((P @ X3) = (P2 @ X3))))) => ((?[Z : real]: (![X3 : real]: ((ord_less_real @ Z @ X3) => ((Q @ X3) = (Q2 @ X3))))) => (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((((P @ X2)) & ((Q @ X2))) = (((P2 @ X2)) & ((Q2 @ X2)))))))))))). % pinf(1)
thf(fact_223_pinf_I2_J, axiom,
    ((![P : real > $o, P2 : real > $o, Q : real > $o, Q2 : real > $o]: ((?[Z : real]: (![X3 : real]: ((ord_less_real @ Z @ X3) => ((P @ X3) = (P2 @ X3))))) => ((?[Z : real]: (![X3 : real]: ((ord_less_real @ Z @ X3) => ((Q @ X3) = (Q2 @ X3))))) => (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => ((((P @ X2)) | ((Q @ X2))) = (((P2 @ X2)) | ((Q2 @ X2)))))))))))). % pinf(2)
thf(fact_224_pinf_I3_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (~ ((X2 = T))))))))). % pinf(3)
thf(fact_225_pinf_I4_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (~ ((X2 = T))))))))). % pinf(4)
thf(fact_226_pinf_I5_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (~ ((ord_less_real @ X2 @ T))))))))). % pinf(5)
thf(fact_227_pinf_I7_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X2 : real]: ((ord_less_real @ Z2 @ X2) => (ord_less_real @ T @ X2))))))). % pinf(7)

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ t @ one_one_real))).
