% 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_515__5372926_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:47.481

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (18)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
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_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001tf__a, type,
    real_V1022479215norm_a : a > 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_c____, type,
    c : a).
thf(sy_v_cs____, type,
    cs : poly_a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (248)
thf(fact_0_c0, axiom,
    ((~ ((c = zero_zero_a))))). % c0
thf(fact_1_ex, axiom,
    ((~ ((p = zero_zero_poly_a))))). % ex
thf(fact_2_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_3_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_4_zero__less__norm__iff, axiom,
    ((![X3 : a]: ((ord_less_real @ zero_zero_real @ (real_V1022479215norm_a @ X3)) = (~ ((X3 = zero_zero_a))))))). % zero_less_norm_iff
thf(fact_5_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_6_norm__zero, axiom,
    (((real_V1022479215norm_a @ zero_zero_a) = zero_zero_real))). % norm_zero
thf(fact_7_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_8_norm__eq__zero, axiom,
    ((![X3 : a]: (((real_V1022479215norm_a @ X3) = zero_zero_real) = (X3 = zero_zero_a))))). % norm_eq_zero
thf(fact_9_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_10_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_11_norm__not__less__zero, axiom,
    ((![X3 : a]: (~ ((ord_less_real @ (real_V1022479215norm_a @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_12_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_13_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_14_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_15_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_16_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_17_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_18_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_19_True, axiom,
    ((cs = zero_zero_poly_a))). % True
thf(fact_20_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_21_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X3 : a]: ((zero_zero_a = X3) = (X3 = zero_zero_a))))). % zero_reorient
thf(fact_23_zero__reorient, axiom,
    ((![X3 : poly_a]: ((zero_zero_poly_a = X3) = (X3 = zero_zero_poly_a))))). % zero_reorient
thf(fact_24_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_25_pCons_Oprems, axiom,
    ((~ (((pCons_a @ c @ cs) = zero_zero_poly_a))))). % pCons.prems
thf(fact_26_psize__eq__0__iff, axiom,
    ((![P2 : poly_a]: (((fundam247907092size_a @ P2) = zero_zero_nat) = (P2 = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_27_norm__le__zero__iff, axiom,
    ((![X3 : a]: ((ord_less_eq_real @ (real_V1022479215norm_a @ X3) @ zero_zero_real) = (X3 = zero_zero_a))))). % norm_le_zero_iff
thf(fact_28_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_29_ex__gt__or__lt, axiom,
    ((![A : real]: (?[B : real]: ((ord_less_real @ A @ B) | (ord_less_real @ B @ A)))))). % ex_gt_or_lt
thf(fact_30_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_31_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (~ ((A = B2))))))). % dual_order.strict_implies_not_eq
thf(fact_32_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_nat @ B2 @ A) => (~ ((A = B2))))))). % dual_order.strict_implies_not_eq
thf(fact_33_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_34_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y3 : real]: (ord_less_real @ Y3 @ X4))))). % linordered_field_no_lb
thf(fact_35_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((A = B2))))))). % order.strict_implies_not_eq
thf(fact_36_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((A = B2))))))). % order.strict_implies_not_eq
thf(fact_37_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_38_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_39_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_40_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)))) => (?[Y3 : real]: ((![X4 : real]: ((member_real @ X4 @ S2) => (ord_less_eq_real @ X4 @ Y3))) & (![Z : real]: ((![X : real]: ((member_real @ X @ S2) => (ord_less_eq_real @ X @ Z))) => (ord_less_eq_real @ Y3 @ Z)))))))))). % complete_real
thf(fact_41_dual__order_Oantisym, axiom,
    ((![B2 : real, A : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_eq_real @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_42_dual__order_Oantisym, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_eq_nat @ B2 @ A) => ((ord_less_eq_nat @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_43_dual__order_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z2 : real]: (Y4 = Z2))) = (^[A2 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A2)) & ((ord_less_eq_real @ A2 @ B3)))))))). % dual_order.eq_iff
thf(fact_44_dual__order_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[A2 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A2)) & ((ord_less_eq_nat @ A2 @ B3)))))))). % dual_order.eq_iff
thf(fact_45_dual__order_Otrans, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_eq_real @ C @ B2) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_46_dual__order_Otrans, axiom,
    ((![B2 : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B2 @ A) => ((ord_less_eq_nat @ C @ B2) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_47_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B2 : real]: ((![A3 : real, B : real]: ((ord_less_eq_real @ A3 @ B) => (P @ A3 @ B))) => ((![A3 : real, B : real]: ((P @ B @ A3) => (P @ A3 @ B))) => (P @ A @ B2)))))). % linorder_wlog
thf(fact_48_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B2 : nat]: ((![A3 : nat, B : nat]: ((ord_less_eq_nat @ A3 @ B) => (P @ A3 @ B))) => ((![A3 : nat, B : nat]: ((P @ B @ A3) => (P @ A3 @ B))) => (P @ A @ B2)))))). % linorder_wlog
thf(fact_49_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_50_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_51_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_52_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_53_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B2 : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_54_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_55_ord__le__eq__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((B2 = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_56_ord__le__eq__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((B2 = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_57_ord__eq__le__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((A = B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_58_ord__eq__le__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((A = B2) => ((ord_less_eq_nat @ B2 @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_59_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z2 : real]: (Y4 = Z2))) = (^[A2 : real]: (^[B3 : real]: (((ord_less_eq_real @ A2 @ B3)) & ((ord_less_eq_real @ B3 @ A2)))))))). % order_class.order.eq_iff
thf(fact_60_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[A2 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A2 @ B3)) & ((ord_less_eq_nat @ B3 @ A2)))))))). % order_class.order.eq_iff
thf(fact_61_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_62_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_63_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_64_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_65_order_Otrans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_66_order_Otrans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ B2 @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_67_le__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % le_cases
thf(fact_68_le__cases, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y2))) => (ord_less_eq_nat @ Y2 @ X3))))). % le_cases
thf(fact_69_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_70_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_71_eq__refl, axiom,
    ((![X3 : real, Y2 : real]: ((X3 = Y2) => (ord_less_eq_real @ X3 @ Y2))))). % eq_refl
thf(fact_72_eq__refl, axiom,
    ((![X3 : nat, Y2 : nat]: ((X3 = Y2) => (ord_less_eq_nat @ X3 @ Y2))))). % eq_refl
thf(fact_73_linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_eq_real @ Y2 @ X3))))). % linear
thf(fact_74_linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) | (ord_less_eq_nat @ Y2 @ X3))))). % linear
thf(fact_75_antisym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_76_antisym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((ord_less_eq_nat @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_77_eq__iff, axiom,
    (((^[Y4 : real]: (^[Z2 : real]: (Y4 = Z2))) = (^[X2 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X2 @ Y5)) & ((ord_less_eq_real @ Y5 @ X2)))))))). % eq_iff
thf(fact_78_eq__iff, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[X2 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X2 @ Y5)) & ((ord_less_eq_nat @ Y5 @ X2)))))))). % eq_iff
thf(fact_79_ord__le__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => (((F @ B2) = C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_80_ord__le__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B2) => (((F @ B2) = C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_81_ord__le__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B2) => (((F @ B2) = C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_82_ord__le__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => (((F @ B2) = C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_83_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_84_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_85_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((A = (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_86_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((A = (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_87_order__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_88_order__subst2, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B2) => ((ord_less_eq_nat @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_89_order__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_90_order__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_91_order__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_92_order__subst1, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_93_order__subst1, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_94_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_95_leD, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_real @ X3 @ Y2))))))). % leD
thf(fact_96_leD, axiom,
    ((![Y2 : nat, X3 : nat]: ((ord_less_eq_nat @ Y2 @ X3) => (~ ((ord_less_nat @ X3 @ Y2))))))). % leD
thf(fact_97_leI, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % leI
thf(fact_98_leI, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_eq_nat @ Y2 @ X3))))). % leI
thf(fact_99_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y5 : real]: (((ord_less_real @ X2 @ Y5)) | ((X2 = Y5)))))))). % le_less
thf(fact_100_le__less, axiom,
    ((ord_less_eq_nat = (^[X2 : nat]: (^[Y5 : nat]: (((ord_less_nat @ X2 @ Y5)) | ((X2 = Y5)))))))). % le_less
thf(fact_101_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X2 @ Y5)) & ((~ ((X2 = Y5)))))))))). % less_le
thf(fact_102_less__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X2 @ Y5)) & ((~ ((X2 = Y5)))))))))). % less_le
thf(fact_103_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_104_order__le__less__subst1, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_105_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_106_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_107_order__le__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_108_order__le__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B2) => ((ord_less_nat @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_109_order__le__less__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_110_order__le__less__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_nat @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_111_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_112_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((ord_less_nat @ A @ (F @ B2)) => ((ord_less_eq_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_113_order__less__le__subst1, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_114_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_nat @ A @ (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => (ord_less_eq_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_115_order__less__le__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_116_order__less__le__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B2) => ((ord_less_eq_real @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_117_order__less__le__subst2, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B2) => ((ord_less_eq_nat @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_118_order__less__le__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B2) => ((ord_less_eq_nat @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_119_not__le, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) = (ord_less_real @ Y2 @ X3))))). % not_le
thf(fact_120_not__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y2))) = (ord_less_nat @ Y2 @ X3))))). % not_le
thf(fact_121_not__less, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (ord_less_eq_real @ Y2 @ X3))))). % not_less
thf(fact_122_not__less, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) = (ord_less_eq_nat @ Y2 @ X3))))). % not_less
thf(fact_123_le__neq__trans, axiom,
    ((![A : real, B2 : real]: ((ord_less_eq_real @ A @ B2) => ((~ ((A = B2))) => (ord_less_real @ A @ B2)))))). % le_neq_trans
thf(fact_124_le__neq__trans, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_eq_nat @ A @ B2) => ((~ ((A = B2))) => (ord_less_nat @ A @ B2)))))). % le_neq_trans
thf(fact_125_antisym__conv1, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_126_antisym__conv1, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((ord_less_eq_nat @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_127_antisym__conv2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_128_antisym__conv2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_129_less__imp__le, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Y2))))). % less_imp_le
thf(fact_130_less__imp__le, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_eq_nat @ X3 @ Y2))))). % less_imp_le
thf(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_le__less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_real @ Y2 @ X3))))). % le_less_linear
thf(fact_138_le__less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_eq_nat @ X3 @ Y2) | (ord_less_nat @ Y2 @ X3))))). % le_less_linear
thf(fact_139_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_140_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_141_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X2 @ Y5)) & ((~ ((ord_less_eq_real @ Y5 @ X2)))))))))). % less_le_not_le
thf(fact_142_less__le__not__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X2 @ Y5)) & ((~ ((ord_less_eq_nat @ Y5 @ X2)))))))))). % less_le_not_le
thf(fact_143_not__le__imp__less, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_eq_real @ Y2 @ X3))) => (ord_less_real @ X3 @ Y2))))). % not_le_imp_less
thf(fact_144_not__le__imp__less, axiom,
    ((![Y2 : nat, X3 : nat]: ((~ ((ord_less_eq_nat @ Y2 @ X3))) => (ord_less_nat @ X3 @ Y2))))). % not_le_imp_less
thf(fact_145_order_Ostrict__trans1, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_eq_real @ A @ B2) => ((ord_less_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_146_order_Ostrict__trans1, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_nat @ B2 @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_147_order_Ostrict__trans2, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_eq_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_148_order_Ostrict__trans2, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_nat @ A @ B2) => ((ord_less_eq_nat @ B2 @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_149_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B3 : real]: (((ord_less_real @ A2 @ B3)) | ((A2 = B3)))))))). % order.order_iff_strict
thf(fact_150_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B3 : nat]: (((ord_less_nat @ A2 @ B3)) | ((A2 = B3)))))))). % order.order_iff_strict
thf(fact_151_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B3 : real]: (((ord_less_eq_real @ A2 @ B3)) & ((~ ((A2 = B3)))))))))). % order.strict_iff_order
thf(fact_152_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A2 @ B3)) & ((~ ((A2 = B3)))))))))). % order.strict_iff_order
thf(fact_153_dual__order_Ostrict__trans1, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_eq_real @ B2 @ A) => ((ord_less_real @ C @ B2) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_154_dual__order_Ostrict__trans1, axiom,
    ((![B2 : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B2 @ A) => ((ord_less_nat @ C @ B2) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_155_dual__order_Ostrict__trans2, axiom,
    ((![B2 : real, A : real, C : real]: ((ord_less_real @ B2 @ A) => ((ord_less_eq_real @ C @ B2) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_156_dual__order_Ostrict__trans2, axiom,
    ((![B2 : nat, A : nat, C : nat]: ((ord_less_nat @ B2 @ A) => ((ord_less_eq_nat @ C @ B2) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_157_dense__ge__bounded, axiom,
    ((![Z3 : real, X3 : real, Y2 : real]: ((ord_less_real @ Z3 @ X3) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X3) => (ord_less_eq_real @ Y2 @ W)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_ge_bounded
thf(fact_158_dense__le__bounded, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((![W : real]: ((ord_less_real @ X3 @ W) => ((ord_less_real @ W @ Y2) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_le_bounded
thf(fact_159_order_Ostrict__implies__order, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (ord_less_eq_real @ A @ B2))))). % order.strict_implies_order
thf(fact_160_order_Ostrict__implies__order, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (ord_less_eq_nat @ A @ B2))))). % order.strict_implies_order
thf(fact_161_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B3 : real]: (^[A2 : real]: (((ord_less_real @ B3 @ A2)) | ((A2 = B3)))))))). % dual_order.order_iff_strict
thf(fact_162_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B3 : nat]: (^[A2 : nat]: (((ord_less_nat @ B3 @ A2)) | ((A2 = B3)))))))). % dual_order.order_iff_strict
thf(fact_163_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B3 : real]: (^[A2 : real]: (((ord_less_eq_real @ B3 @ A2)) & ((~ ((A2 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_164_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B3 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B3 @ A2)) & ((~ ((A2 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_165_dual__order_Ostrict__implies__order, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (ord_less_eq_real @ B2 @ A))))). % dual_order.strict_implies_order
thf(fact_166_dual__order_Ostrict__implies__order, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_nat @ B2 @ A) => (ord_less_eq_nat @ B2 @ A))))). % dual_order.strict_implies_order
thf(fact_167_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B2 : real]: ((~ ((A = B2))) => ((ord_less_eq_real @ A @ B2) => (ord_less_real @ A @ B2)))))). % order.not_eq_order_implies_strict
thf(fact_168_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B2 : nat]: ((~ ((A = B2))) => ((ord_less_eq_nat @ A @ B2) => (ord_less_nat @ A @ B2)))))). % order.not_eq_order_implies_strict
thf(fact_169_complete__interval, axiom,
    ((![A : real, B2 : real, P : real > $o]: ((ord_less_real @ A @ B2) => ((P @ A) => ((~ ((P @ B2))) => (?[C2 : real]: ((ord_less_eq_real @ A @ C2) & ((ord_less_eq_real @ C2 @ B2) & ((![X4 : real]: (((ord_less_eq_real @ A @ X4) & (ord_less_real @ X4 @ C2)) => (P @ X4))) & (![D : real]: ((![X : real]: (((ord_less_eq_real @ A @ X) & (ord_less_real @ X @ D)) => (P @ X))) => (ord_less_eq_real @ D @ C2))))))))))))). % complete_interval
thf(fact_170_complete__interval, axiom,
    ((![A : nat, B2 : nat, P : nat > $o]: ((ord_less_nat @ A @ B2) => ((P @ A) => ((~ ((P @ B2))) => (?[C2 : nat]: ((ord_less_eq_nat @ A @ C2) & ((ord_less_eq_nat @ C2 @ B2) & ((![X4 : nat]: (((ord_less_eq_nat @ A @ X4) & (ord_less_nat @ X4 @ C2)) => (P @ X4))) & (![D : nat]: ((![X : nat]: (((ord_less_eq_nat @ A @ X) & (ord_less_nat @ X @ D)) => (P @ X))) => (ord_less_eq_nat @ D @ C2))))))))))))). % complete_interval
thf(fact_171_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_172_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_173_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_174_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y5 : real]: (((ord_less_real @ X2 @ Y5)) | ((X2 = Y5)))))))). % less_eq_real_def
thf(fact_175_norm__ge__zero, axiom,
    ((![X3 : a]: (ord_less_eq_real @ zero_zero_real @ (real_V1022479215norm_a @ X3))))). % norm_ge_zero
thf(fact_176_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_177_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_178_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((A = (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_179_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((A = (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_180_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((A = (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_181_ord__less__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => (((F @ B2) = C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_182_ord__less__eq__subst, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B2) => (((F @ B2) = C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_183_ord__less__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B2) => (((F @ B2) = C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_184_ord__less__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B2) => (((F @ B2) = C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_185_order__less__subst1, axiom,
    ((![A : real, F : real > real, B2 : real, C : real]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_186_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B2 : nat, C : nat]: ((ord_less_real @ A @ (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_187_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B2 : real, C : real]: ((ord_less_nat @ A @ (F @ B2)) => ((ord_less_real @ B2 @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_188_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_nat @ A @ (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_189_order__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > real, C : real]: ((ord_less_real @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_190_order__less__subst2, axiom,
    ((![A : real, B2 : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B2) => ((ord_less_nat @ (F @ B2) @ C) => ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_191_order__less__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B2) => ((ord_less_real @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_real @ (F @ X) @ (F @ Y3)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_192_order__less__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B2) => ((ord_less_nat @ (F @ B2) @ C) => ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_nat @ (F @ X) @ (F @ Y3)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_193_lt__ex, axiom,
    ((![X3 : real]: (?[Y3 : real]: (ord_less_real @ Y3 @ X3))))). % lt_ex
thf(fact_194_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_195_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_196_neqE, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % neqE
thf(fact_197_neqE, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) => ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_nat @ Y2 @ X3)))))). % neqE
thf(fact_198_neq__iff, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) = (((ord_less_real @ X3 @ Y2)) | ((ord_less_real @ Y2 @ X3))))))). % neq_iff
thf(fact_199_neq__iff, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) = (((ord_less_nat @ X3 @ Y2)) | ((ord_less_nat @ Y2 @ X3))))))). % neq_iff
thf(fact_200_order_Oasym, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((ord_less_real @ B2 @ A))))))). % order.asym
thf(fact_201_order_Oasym, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((ord_less_nat @ B2 @ A))))))). % order.asym
thf(fact_202_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_203_less__imp__neq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_204_less__imp__neq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_205_less__asym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_asym
thf(fact_206_less__asym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_asym
thf(fact_207_less__asym_H, axiom,
    ((![A : real, B2 : real]: ((ord_less_real @ A @ B2) => (~ ((ord_less_real @ B2 @ A))))))). % less_asym'
thf(fact_208_less__asym_H, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((ord_less_nat @ B2 @ A))))))). % less_asym'
thf(fact_209_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_210_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_211_less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) | ((X3 = Y2) | (ord_less_real @ Y2 @ X3)))))). % less_linear
thf(fact_212_less__linear, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) | ((X3 = Y2) | (ord_less_nat @ Y2 @ X3)))))). % less_linear
thf(fact_213_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_214_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_215_ord__eq__less__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((A = B2) => ((ord_less_real @ B2 @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_216_ord__eq__less__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((A = B2) => ((ord_less_nat @ B2 @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_217_ord__less__eq__trans, axiom,
    ((![A : real, B2 : real, C : real]: ((ord_less_real @ A @ B2) => ((B2 = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_218_ord__less__eq__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_nat @ A @ B2) => ((B2 = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_219_dual__order_Oasym, axiom,
    ((![B2 : real, A : real]: ((ord_less_real @ B2 @ A) => (~ ((ord_less_real @ A @ B2))))))). % dual_order.asym
thf(fact_220_dual__order_Oasym, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_nat @ B2 @ A) => (~ ((ord_less_nat @ A @ B2))))))). % dual_order.asym
thf(fact_221_less__imp__not__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_222_less__imp__not__eq, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_223_less__not__sym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_not_sym
thf(fact_224_less__not__sym, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((ord_less_nat @ Y2 @ X3))))))). % less_not_sym
thf(fact_225_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_226_antisym__conv3, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_real @ Y2 @ X3))) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_227_antisym__conv3, axiom,
    ((![Y2 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y2 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_228_less__imp__not__eq2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_229_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_230_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_231_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_232_linorder__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_cases
thf(fact_233_linorder__cases, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_nat @ Y2 @ X3)))))). % linorder_cases
thf(fact_234_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_235_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_236_pCons_Ohyps_I2_J, axiom,
    ((![D3 : real, A : a]: ((~ ((cs = zero_zero_poly_a))) => (?[R : real]: (![Z : a]: ((ord_less_eq_real @ R @ (real_V1022479215norm_a @ Z)) => (ord_less_eq_real @ D3 @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ A @ cs) @ Z)))))))))). % pCons.hyps(2)
thf(fact_237_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_238_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_239_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_240_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_241_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_242_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_243_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_244_nat__less__le, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M2 @ N2)) & ((~ ((M2 = N2)))))))))). % nat_less_le
thf(fact_245_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K : nat]: ((ord_less_eq_nat @ K @ N) & ((![I : nat]: ((ord_less_nat @ I @ K) => (~ ((P @ I))))) & (P @ K))))))))). % ex_least_nat_le
thf(fact_246_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_247_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_nat @ M2 @ N2)) | ((M2 = N2)))))))). % le_eq_less_or_eq

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_real @ zero_zero_real @ (real_V1022479215norm_a @ c)))).
