% 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_249__5369922_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:31.483

% 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 (15)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex, type,
    real_V1560324349omplex : real > complex > complex).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal, type,
    real_V453051771R_real : real > real > real).
thf(sy_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (216)
thf(fact_0__092_060open_0620_A_092_060le_062_Acmod_A_Is_A0_J_092_060close_062, axiom,
    ((ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ (s @ zero_zero_nat))))). % \<open>0 \<le> cmod (s 0)\<close>
thf(fact_1__092_060open_062cmod_A_Is_A0_J_A_092_060le_062_Ar_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (s @ zero_zero_nat)) @ r))). % \<open>cmod (s 0) \<le> r\<close>
thf(fact_2_r, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N)) @ r)))). % r
thf(fact_3_le__zero__eq, axiom,
    ((![N2 : nat]: ((ord_less_eq_nat @ N2 @ zero_zero_nat) = (N2 = zero_zero_nat))))). % le_zero_eq
thf(fact_4_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_5_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_6_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_7_norm__le__zero__iff, axiom,
    ((![X : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X) @ zero_zero_real) = (X = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_8_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_9_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_10_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_11_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_12_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_13_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_14_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_15_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero
thf(fact_16_norm__ge__zero, axiom,
    ((![X : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X))))). % norm_ge_zero
thf(fact_17_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_18_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_19_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_20_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_21_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_22_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_23_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_24_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_25_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_26_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_27_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_28_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_29_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_30_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_31_order__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_eq_real @ X @ Z3)))))). % order_trans
thf(fact_32_order__trans, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z3) => (ord_less_eq_nat @ X @ Z3)))))). % order_trans
thf(fact_33_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_34_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_35_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_36_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_37_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_38_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_39_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_40_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_41_antisym__conv, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_42_antisym__conv, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_43_le__cases3, axiom,
    ((![X : real, Y3 : real, Z3 : real]: (((ord_less_eq_real @ X @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z3)))) => (((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_eq_real @ X @ Z3)))) => (((ord_less_eq_real @ X @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y3)))) => (((ord_less_eq_real @ Z3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X)))) => (((ord_less_eq_real @ Y3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X)))) => (~ (((ord_less_eq_real @ Z3 @ X) => (~ ((ord_less_eq_real @ X @ Y3)))))))))))))). % le_cases3
thf(fact_44_le__cases3, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: (((ord_less_eq_nat @ X @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z3)))) => (((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_eq_nat @ X @ Z3)))) => (((ord_less_eq_nat @ X @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y3)))) => (((ord_less_eq_nat @ Z3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X)))) => (((ord_less_eq_nat @ Y3 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X)))) => (~ (((ord_less_eq_nat @ Z3 @ X) => (~ ((ord_less_eq_nat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_45_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_46_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_47_le__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % le_cases
thf(fact_48_le__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % le_cases
thf(fact_49_eq__refl, axiom,
    ((![X : real, Y3 : real]: ((X = Y3) => (ord_less_eq_real @ X @ Y3))))). % eq_refl
thf(fact_50_eq__refl, axiom,
    ((![X : nat, Y3 : nat]: ((X = Y3) => (ord_less_eq_nat @ X @ Y3))))). % eq_refl
thf(fact_51_linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_eq_real @ Y3 @ X))))). % linear
thf(fact_52_linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_eq_nat @ Y3 @ X))))). % linear
thf(fact_53_antisym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_54_antisym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_55_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((ord_less_eq_real @ Y4 @ X4)))))))). % eq_iff
thf(fact_56_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X4)))))))). % eq_iff
thf(fact_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_le0, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % le0
thf(fact_74_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_75_zero__less__norm__iff, axiom,
    ((![X : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X)) = (~ ((X = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_76_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_77_scaleR__mono_H, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (real_V453051771R_real @ A @ C) @ (real_V453051771R_real @ B @ D))))))))). % scaleR_mono'
thf(fact_78_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_79_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_80_scaleR__mono, axiom,
    ((![A : real, B : real, X : real, Y3 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_eq_real @ zero_zero_real @ X) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X) @ (real_V453051771R_real @ B @ Y3))))))))). % scaleR_mono
thf(fact_81_scaleR__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ zero_zero_real @ (real_V453051771R_real @ A @ B))))))). % scaleR_nonpos_nonpos
thf(fact_82_scaleR__nonpos__nonneg, axiom,
    ((![A : real, X : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ zero_zero_real @ X) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X) @ zero_zero_real)))))). % scaleR_nonpos_nonneg
thf(fact_83_scaleR__nonneg__nonpos, axiom,
    ((![A : real, X : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ X @ zero_zero_real) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X) @ zero_zero_real)))))). % scaleR_nonneg_nonpos
thf(fact_84_scaleR__nonneg__nonneg, axiom,
    ((![A : real, X : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ X) => (ord_less_eq_real @ zero_zero_real @ (real_V453051771R_real @ A @ X))))))). % scaleR_nonneg_nonneg
thf(fact_85_not__gr__zero, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr_zero
thf(fact_86_scale__zero__right, axiom,
    ((![A : real]: ((real_V453051771R_real @ A @ zero_zero_real) = zero_zero_real)))). % scale_zero_right
thf(fact_87_scale__zero__right, axiom,
    ((![A : real]: ((real_V1560324349omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % scale_zero_right
thf(fact_88_scale__cancel__right, axiom,
    ((![A : real, X : real, B : real]: (((real_V453051771R_real @ A @ X) = (real_V453051771R_real @ B @ X)) = (((A = B)) | ((X = zero_zero_real))))))). % scale_cancel_right
thf(fact_89_scale__cancel__right, axiom,
    ((![A : real, X : complex, B : real]: (((real_V1560324349omplex @ A @ X) = (real_V1560324349omplex @ B @ X)) = (((A = B)) | ((X = zero_zero_complex))))))). % scale_cancel_right
thf(fact_90_scale__eq__0__iff, axiom,
    ((![A : real, X : real]: (((real_V453051771R_real @ A @ X) = zero_zero_real) = (((A = zero_zero_real)) | ((X = zero_zero_real))))))). % scale_eq_0_iff
thf(fact_91_scale__eq__0__iff, axiom,
    ((![A : real, X : complex]: (((real_V1560324349omplex @ A @ X) = zero_zero_complex) = (((A = zero_zero_real)) | ((X = zero_zero_complex))))))). % scale_eq_0_iff
thf(fact_92_scale__zero__left, axiom,
    ((![X : real]: ((real_V453051771R_real @ zero_zero_real @ X) = zero_zero_real)))). % scale_zero_left
thf(fact_93_scale__zero__left, axiom,
    ((![X : complex]: ((real_V1560324349omplex @ zero_zero_real @ X) = zero_zero_complex)))). % scale_zero_left
thf(fact_94_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_95_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_96_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_97_le__antisym, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((ord_less_eq_nat @ N2 @ M) => (M = N2)))))). % le_antisym
thf(fact_98_nat__le__linear, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) | (ord_less_eq_nat @ N2 @ M))))). % nat_le_linear
thf(fact_99_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X3 : nat]: ((P @ X3) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_100_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_101_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_102_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_103_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_104_lt__ex, axiom,
    ((![X : real]: (?[Y : real]: (ord_less_real @ Y @ X))))). % lt_ex
thf(fact_105_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_106_neqE, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) => ((~ ((ord_less_real @ X @ Y3))) => (ord_less_real @ Y3 @ X)))))). % neqE
thf(fact_107_neq__iff, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) = (((ord_less_real @ X @ Y3)) | ((ord_less_real @ Y3 @ X))))))). % neq_iff
thf(fact_108_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_109_dense, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (?[Z4 : real]: ((ord_less_real @ X @ Z4) & (ord_less_real @ Z4 @ Y3))))))). % dense
thf(fact_110_less__imp__neq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_111_less__asym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_asym
thf(fact_112_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_113_less__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_real @ X @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X @ Z3)))))). % less_trans
thf(fact_114_less__linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) | ((X = Y3) | (ord_less_real @ Y3 @ X)))))). % less_linear
thf(fact_115_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_116_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_117_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_118_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_119_less__imp__not__eq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_120_less__not__sym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_not_sym
thf(fact_121_antisym__conv3, axiom,
    ((![Y3 : real, X : real]: ((~ ((ord_less_real @ Y3 @ X))) => ((~ ((ord_less_real @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_122_less__imp__not__eq2, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_123_less__imp__triv, axiom,
    ((![X : real, Y3 : real, P : $o]: ((ord_less_real @ X @ Y3) => ((ord_less_real @ Y3 @ X) => P))))). % less_imp_triv
thf(fact_124_linorder__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_real @ Y3 @ X)))))). % linorder_cases
thf(fact_125_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_126_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_127_less__imp__not__less, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_imp_not_less
thf(fact_128_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real]: (P @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_129_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_130_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) = (((ord_less_real @ Y3 @ X)) | ((X = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_131_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_132_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_133_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]: (![Y5 : real]: ((?[X4 : real]: (((P @ X4)) & ((ord_less_real @ Y5 @ X4)))) = (ord_less_real @ Y5 @ S2))))))))). % real_sup_exists
thf(fact_134_scaleR__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (real_V453051771R_real @ C @ A) @ (real_V453051771R_real @ C @ B)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ A @ B)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ B @ A))))))))). % scaleR_le_cancel_left
thf(fact_135_scaleR__le__cancel__left__neg, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ C @ zero_zero_real) => ((ord_less_eq_real @ (real_V453051771R_real @ C @ A) @ (real_V453051771R_real @ C @ B)) = (ord_less_eq_real @ B @ A)))))). % scaleR_le_cancel_left_neg
thf(fact_136_scaleR__le__cancel__left__pos, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ zero_zero_real @ C) => ((ord_less_eq_real @ (real_V453051771R_real @ C @ A) @ (real_V453051771R_real @ C @ B)) = (ord_less_eq_real @ A @ B)))))). % scaleR_le_cancel_left_pos
thf(fact_137_scale__right__imp__eq, axiom,
    ((![X : real, A : real, B : real]: ((~ ((X = zero_zero_real))) => (((real_V453051771R_real @ A @ X) = (real_V453051771R_real @ B @ X)) => (A = B)))))). % scale_right_imp_eq
thf(fact_138_scale__right__imp__eq, axiom,
    ((![X : complex, A : real, B : real]: ((~ ((X = zero_zero_complex))) => (((real_V1560324349omplex @ A @ X) = (real_V1560324349omplex @ B @ X)) => (A = B)))))). % scale_right_imp_eq
thf(fact_139_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_140_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_141_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_142_gr__zeroI, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr_zeroI
thf(fact_143_not__less__zero, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less_zero
thf(fact_144_gr__implies__not__zero, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_145_zero__less__iff__neq__zero, axiom,
    ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) = (~ ((N2 = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_146_leD, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_real @ X @ Y3))))))). % leD
thf(fact_147_leD, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_nat @ X @ Y3))))))). % leD
thf(fact_148_leI, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % leI
thf(fact_149_leI, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % leI
thf(fact_150_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_151_le__less, axiom,
    ((ord_less_eq_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_nat @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_152_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_153_less__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_not__le, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) = (ord_less_real @ Y3 @ X))))). % not_le
thf(fact_167_not__le, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) = (ord_less_nat @ Y3 @ X))))). % not_le
thf(fact_168_not__less, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) = (ord_less_eq_real @ Y3 @ X))))). % not_less
thf(fact_169_not__less, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) = (ord_less_eq_nat @ Y3 @ X))))). % not_less
thf(fact_170_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_171_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_172_antisym__conv1, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv1
thf(fact_173_antisym__conv1, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv1
thf(fact_174_antisym__conv2, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((~ ((ord_less_real @ X @ Y3))) = (X = Y3)))))). % antisym_conv2
thf(fact_175_antisym__conv2, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((~ ((ord_less_nat @ X @ Y3))) = (X = Y3)))))). % antisym_conv2
thf(fact_176_less__imp__le, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_eq_real @ X @ Y3))))). % less_imp_le
thf(fact_177_less__imp__le, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_eq_nat @ X @ Y3))))). % less_imp_le
thf(fact_178_le__less__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X @ Z3)))))). % le_less_trans
thf(fact_179_le__less__trans, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ Z3) => (ord_less_nat @ X @ Z3)))))). % le_less_trans
thf(fact_180_less__le__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_real @ X @ Z3)))))). % less_le_trans
thf(fact_181_less__le__trans, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: ((ord_less_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z3) => (ord_less_nat @ X @ Z3)))))). % less_le_trans
thf(fact_182_dense__ge, axiom,
    ((![Z3 : real, Y3 : real]: ((![X3 : real]: ((ord_less_real @ Z3 @ X3) => (ord_less_eq_real @ Y3 @ X3))) => (ord_less_eq_real @ Y3 @ Z3))))). % dense_ge
thf(fact_183_dense__le, axiom,
    ((![Y3 : real, Z3 : real]: ((![X3 : real]: ((ord_less_real @ X3 @ Y3) => (ord_less_eq_real @ X3 @ Z3))) => (ord_less_eq_real @ Y3 @ Z3))))). % dense_le
thf(fact_184_le__less__linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_real @ Y3 @ X))))). % le_less_linear
thf(fact_185_le__less__linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_nat @ Y3 @ X))))). % le_less_linear
thf(fact_186_le__imp__less__or__eq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_real @ X @ Y3) | (X = Y3)))))). % le_imp_less_or_eq
thf(fact_187_le__imp__less__or__eq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_nat @ X @ Y3) | (X = Y3)))))). % le_imp_less_or_eq
thf(fact_188_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_189_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_190_not__le__imp__less, axiom,
    ((![Y3 : real, X : real]: ((~ ((ord_less_eq_real @ Y3 @ X))) => (ord_less_real @ X @ Y3))))). % not_le_imp_less
thf(fact_191_not__le__imp__less, axiom,
    ((![Y3 : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y3 @ X))) => (ord_less_nat @ X @ Y3))))). % not_le_imp_less
thf(fact_192_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_193_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_194_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_195_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_196_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_real @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_197_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_198_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_199_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_200_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_201_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_202_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_203_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_204_dense__ge__bounded, axiom,
    ((![Z3 : real, X : real, Y3 : real]: ((ord_less_real @ Z3 @ X) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X) => (ord_less_eq_real @ Y3 @ W)))) => (ord_less_eq_real @ Y3 @ Z3)))))). % dense_ge_bounded
thf(fact_205_dense__le__bounded, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_real @ X @ Y3) => ((![W : real]: ((ord_less_real @ X @ W) => ((ord_less_real @ W @ Y3) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y3 @ Z3)))))). % dense_le_bounded
thf(fact_206_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_207_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_208_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_real @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_209_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_nat @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_210_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_211_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_212_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_213_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_214_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_215_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ zero_zero_real @ r))).
