% 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/Hoare/prob_285__3252300_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:28.275

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    set_se152467259iple_a : $tType).
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (35)
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__a, type,
    hoare_129598474rivs_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001tf__a, type,
    hoare_1775499016lids_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
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__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1356158809iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_set_nat : set_nat > set_nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1827677583iple_a : set_se152467259iple_a > set_se152467259iple_a > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J, type,
    ord_less_eq_o_nat : ($o > nat) > ($o > nat) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1048771374iple_a : ($o > set_Ho137910533iple_a) > ($o > set_Ho137910533iple_a) > $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__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1221261669iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1677170587iple_a : set_se152467259iple_a > set_se152467259iple_a > $o).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat, type,
    order_Greatest_nat : (nat > $o) > nat).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_929906668iple_a : (set_Ho137910533iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_1631207636at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_1673421321iple_a : (nat > set_Ho137910533iple_a) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_001t__Nat__Onat, type,
    order_194881289_a_nat : (set_Ho137910533iple_a > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_1710851741iple_a : (set_Ho137910533iple_a > set_Ho137910533iple_a) > $o).
thf(sy_c_Set_OBall_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    ball_H465710501iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > $o) > $o).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat, type,
    set_or601162459st_nat : nat > nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_or1668187408iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_se152467259iple_a).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat, type,
    set_ord_atLeast_nat : nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_or1109361786iple_a : set_Ho137910533iple_a > set_se152467259iple_a).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat, type,
    set_ord_atMost_nat : nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_or1587679614iple_a : set_Ho137910533iple_a > set_se152467259iple_a).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat, type,
    set_or1965240170an_nat : nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat, type,
    set_ord_lessThan_nat : nat > set_nat).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_c_member_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    member521824924iple_a : set_Ho137910533iple_a > set_se152467259iple_a > $o).
thf(sy_v_Ga, type,
    ga : set_Ho137910533iple_a).
thf(sy_v_tsa, type,
    tsa : set_Ho137910533iple_a).

% Relevant facts (189)
thf(fact_0_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A) => (member1332298086iple_a @ X @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_1_subset__antisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_2_order__refl, axiom,
    ((![X2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X2 @ X2)))). % order_refl
thf(fact_3_order__refl, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ X2 @ X2)))). % order_refl
thf(fact_4_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G : set_Ho137910533iple_a]: (^[Ts : set_Ho137910533iple_a]: (![N : nat]: (((![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ G)) => ((hoare_1926814542alid_a @ N @ X3))))) => ((![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ Ts)) => ((hoare_1926814542alid_a @ N @ X3)))))))))))). % hoare_valids_def
thf(fact_5_Ball__Collect, axiom,
    ((ball_H465710501iple_a = (^[A2 : set_Ho137910533iple_a]: (^[P : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ A2 @ (collec1600235172iple_a @ P))))))). % Ball_Collect
thf(fact_6_Ball__def, axiom,
    ((ball_H465710501iple_a = (^[A2 : set_Ho137910533iple_a]: (^[P : hoare_1678595023iple_a > $o]: (![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ A2)) => ((P @ X3))))))))). % Ball_def
thf(fact_7_ball__reg, axiom,
    ((![R : set_Ho137910533iple_a, P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ R) => ((P2 @ X) => (Q @ X)))) => ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ R) => (P2 @ X))) => (![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ R) => (Q @ X4)))))))). % ball_reg
thf(fact_8_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X2 @ A) => (member1332298086iple_a @ X2 @ B)))))). % in_mono
thf(fact_9_subsetD, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ C @ A) => (member1332298086iple_a @ C @ B)))))). % subsetD
thf(fact_10_equalityE, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (~ (((ord_le1221261669iple_a @ A @ B) => (~ ((ord_le1221261669iple_a @ B @ A)))))))))). % equalityE
thf(fact_11_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ A2)) => ((member1332298086iple_a @ X3 @ B2))))))))). % subset_eq
thf(fact_12_equalityD1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ A @ B))))). % equalityD1
thf(fact_13_dual__order_Oantisym, axiom,
    ((![B3 : set_Ho137910533iple_a, A3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B3 @ A3) => ((ord_le1221261669iple_a @ A3 @ B3) => (A3 = B3)))))). % dual_order.antisym
thf(fact_14_dual__order_Oantisym, axiom,
    ((![B3 : nat, A3 : nat]: ((ord_less_eq_nat @ B3 @ A3) => ((ord_less_eq_nat @ A3 @ B3) => (A3 = B3)))))). % dual_order.antisym
thf(fact_15_dual__order_Oeq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B4 @ A4)) & ((ord_le1221261669iple_a @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_16_dual__order_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ B4 @ A4)) & ((ord_less_eq_nat @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_17_dual__order_Otrans, axiom,
    ((![B3 : set_Ho137910533iple_a, A3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B3 @ A3) => ((ord_le1221261669iple_a @ C @ B3) => (ord_le1221261669iple_a @ C @ A3)))))). % dual_order.trans
thf(fact_18_dual__order_Otrans, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: ((ord_less_eq_nat @ B3 @ A3) => ((ord_less_eq_nat @ C @ B3) => (ord_less_eq_nat @ C @ A3)))))). % dual_order.trans
thf(fact_19_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A3 : nat, B3 : nat]: ((![A5 : nat, B5 : nat]: ((ord_less_eq_nat @ A5 @ B5) => (P2 @ A5 @ B5))) => ((![A5 : nat, B5 : nat]: ((P2 @ B5 @ A5) => (P2 @ A5 @ B5))) => (P2 @ A3 @ B3)))))). % linorder_wlog
thf(fact_20_dual__order_Orefl, axiom,
    ((![A3 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A3 @ A3)))). % dual_order.refl
thf(fact_21_dual__order_Orefl, axiom,
    ((![A3 : nat]: (ord_less_eq_nat @ A3 @ A3)))). % dual_order.refl
thf(fact_22_order__trans, axiom,
    ((![X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a, Z2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y2) => ((ord_le1221261669iple_a @ Y2 @ Z2) => (ord_le1221261669iple_a @ X2 @ Z2)))))). % order_trans
thf(fact_23_order__trans, axiom,
    ((![X2 : nat, Y2 : nat, Z2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => ((ord_less_eq_nat @ Y2 @ Z2) => (ord_less_eq_nat @ X2 @ Z2)))))). % order_trans
thf(fact_24_order__class_Oorder_Oantisym, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ B3) => ((ord_le1221261669iple_a @ B3 @ A3) => (A3 = B3)))))). % order_class.order.antisym
thf(fact_25_order__class_Oorder_Oantisym, axiom,
    ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ B3 @ A3) => (A3 = B3)))))). % order_class.order.antisym
thf(fact_26_ord__le__eq__trans, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ B3) => ((B3 = C) => (ord_le1221261669iple_a @ A3 @ C)))))). % ord_le_eq_trans
thf(fact_27_ord__le__eq__trans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((B3 = C) => (ord_less_eq_nat @ A3 @ C)))))). % ord_le_eq_trans
thf(fact_28_ord__eq__le__trans, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A3 = B3) => ((ord_le1221261669iple_a @ B3 @ C) => (ord_le1221261669iple_a @ A3 @ C)))))). % ord_eq_le_trans
thf(fact_29_ord__eq__le__trans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((A3 = B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A3 @ C)))))). % ord_eq_le_trans
thf(fact_30_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A4 @ B4)) & ((ord_le1221261669iple_a @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_31_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ A4 @ B4)) & ((ord_less_eq_nat @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_32_antisym__conv, axiom,
    ((![Y2 : set_Ho137910533iple_a, X2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y2 @ X2) => ((ord_le1221261669iple_a @ X2 @ Y2) = (X2 = Y2)))))). % antisym_conv
thf(fact_33_antisym__conv, axiom,
    ((![Y2 : nat, X2 : nat]: ((ord_less_eq_nat @ Y2 @ X2) => ((ord_less_eq_nat @ X2 @ Y2) = (X2 = Y2)))))). % antisym_conv
thf(fact_34_le__cases3, axiom,
    ((![X2 : nat, Y2 : nat, Z2 : nat]: (((ord_less_eq_nat @ X2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z2)))) => (((ord_less_eq_nat @ Y2 @ X2) => (~ ((ord_less_eq_nat @ X2 @ Z2)))) => (((ord_less_eq_nat @ X2 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y2)))) => (((ord_less_eq_nat @ Z2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X2)))) => (((ord_less_eq_nat @ Y2 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X2)))) => (~ (((ord_less_eq_nat @ Z2 @ X2) => (~ ((ord_less_eq_nat @ X2 @ Y2)))))))))))))). % le_cases3
thf(fact_35_order_Otrans, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ B3) => ((ord_le1221261669iple_a @ B3 @ C) => (ord_le1221261669iple_a @ A3 @ C)))))). % order.trans
thf(fact_36_order_Otrans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A3 @ C)))))). % order.trans
thf(fact_37_le__cases, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X2 @ Y2))) => (ord_less_eq_nat @ Y2 @ X2))))). % le_cases
thf(fact_38_eq__refl, axiom,
    ((![X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((X2 = Y2) => (ord_le1221261669iple_a @ X2 @ Y2))))). % eq_refl
thf(fact_39_eq__refl, axiom,
    ((![X2 : nat, Y2 : nat]: ((X2 = Y2) => (ord_less_eq_nat @ X2 @ Y2))))). % eq_refl
thf(fact_40_linear, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) | (ord_less_eq_nat @ Y2 @ X2))))). % linear
thf(fact_41_antisym, axiom,
    ((![X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y2) => ((ord_le1221261669iple_a @ Y2 @ X2) => (X2 = Y2)))))). % antisym
thf(fact_42_antisym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => ((ord_less_eq_nat @ Y2 @ X2) => (X2 = Y2)))))). % antisym
thf(fact_43_eq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[X3 : set_Ho137910533iple_a]: (^[Y3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X3 @ Y3)) & ((ord_le1221261669iple_a @ Y3 @ X3)))))))). % eq_iff
thf(fact_44_eq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[X3 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X3 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X3)))))))). % eq_iff
thf(fact_45_ord__le__eq__subst, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ B3) => (((F @ B3) = C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A3) @ C))))))). % ord_le_eq_subst
thf(fact_46_ord__le__eq__subst, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > nat, C : nat]: ((ord_le1221261669iple_a @ A3 @ B3) => (((F @ B3) = C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % ord_le_eq_subst
thf(fact_47_ord__le__eq__subst, axiom,
    ((![A3 : nat, B3 : nat, F : nat > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A3 @ B3) => (((F @ B3) = C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A3) @ C))))))). % ord_le_eq_subst
thf(fact_48_ord__le__eq__subst, axiom,
    ((![A3 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => (((F @ B3) = C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % ord_le_eq_subst
thf(fact_49_ord__eq__le__subst, axiom,
    ((![A3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A3 = (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A3 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_50_ord__eq__le__subst, axiom,
    ((![A3 : nat, F : set_Ho137910533iple_a > nat, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A3 = (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_51_ord__eq__le__subst, axiom,
    ((![A3 : set_Ho137910533iple_a, F : nat > set_Ho137910533iple_a, B3 : nat, C : nat]: ((A3 = (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A3 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_52_ord__eq__le__subst, axiom,
    ((![A3 : nat, F : nat > nat, B3 : nat, C : nat]: ((A3 = (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_53_order__subst2, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ B3) => ((ord_le1221261669iple_a @ (F @ B3) @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A3) @ C))))))). % order_subst2
thf(fact_54_order__subst2, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > nat, C : nat]: ((ord_le1221261669iple_a @ A3 @ B3) => ((ord_less_eq_nat @ (F @ B3) @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % order_subst2
thf(fact_55_order__subst2, axiom,
    ((![A3 : nat, B3 : nat, F : nat > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_le1221261669iple_a @ (F @ B3) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A3) @ C))))))). % order_subst2
thf(fact_56_order__subst2, axiom,
    ((![A3 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ (F @ B3) @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % order_subst2
thf(fact_57_order__subst1, axiom,
    ((![A3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A3 @ (F @ C)))))))). % order_subst1
thf(fact_58_order__subst1, axiom,
    ((![A3 : set_Ho137910533iple_a, F : nat > set_Ho137910533iple_a, B3 : nat, C : nat]: ((ord_le1221261669iple_a @ A3 @ (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A3 @ (F @ C)))))))). % order_subst1
thf(fact_59_order__subst1, axiom,
    ((![A3 : nat, F : set_Ho137910533iple_a > nat, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A3 @ (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % order_subst1
thf(fact_60_order__subst1, axiom,
    ((![A3 : nat, F : nat > nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % order_subst1
thf(fact_61_Collect__mono__iff, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P2) @ (collec1600235172iple_a @ Q)) = (![X3 : hoare_1678595023iple_a]: (((P2 @ X3)) => ((Q @ X3)))))))). % Collect_mono_iff
thf(fact_62_set__eq__subset, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A2 @ B2)) & ((ord_le1221261669iple_a @ B2 @ A2)))))))). % set_eq_subset
thf(fact_63_subset__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C2) => (ord_le1221261669iple_a @ A @ C2)))))). % subset_trans
thf(fact_64_Collect__mono, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: ((P2 @ X) => (Q @ X))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P2) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_65_subset__refl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % subset_refl
thf(fact_66_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![T : hoare_1678595023iple_a]: (((member1332298086iple_a @ T @ A2)) => ((member1332298086iple_a @ T @ B2))))))))). % subset_iff
thf(fact_67_equalityD2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ B @ A))))). % equalityD2
thf(fact_68_mem__Collect__eq, axiom,
    ((![A3 : hoare_1678595023iple_a, P2 : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A3 @ (collec1600235172iple_a @ P2)) = (P2 @ A3))))). % mem_Collect_eq
thf(fact_69_Collect__mem__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X3 : hoare_1678595023iple_a]: (member1332298086iple_a @ X3 @ A))) = A)))). % Collect_mem_eq
thf(fact_70_Collect__cong, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: ((P2 @ X) = (Q @ X))) => ((collec1600235172iple_a @ P2) = (collec1600235172iple_a @ Q)))))). % Collect_cong
thf(fact_71_Greatest__equality, axiom,
    ((![P2 : set_Ho137910533iple_a > $o, X2 : set_Ho137910533iple_a]: ((P2 @ X2) => ((![Y4 : set_Ho137910533iple_a]: ((P2 @ Y4) => (ord_le1221261669iple_a @ Y4 @ X2))) => ((order_929906668iple_a @ P2) = X2)))))). % Greatest_equality
thf(fact_72_Greatest__equality, axiom,
    ((![P2 : nat > $o, X2 : nat]: ((P2 @ X2) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ X2))) => ((order_Greatest_nat @ P2) = X2)))))). % Greatest_equality
thf(fact_73_GreatestI2__order, axiom,
    ((![P2 : set_Ho137910533iple_a > $o, X2 : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P2 @ X2) => ((![Y4 : set_Ho137910533iple_a]: ((P2 @ Y4) => (ord_le1221261669iple_a @ Y4 @ X2))) => ((![X : set_Ho137910533iple_a]: ((P2 @ X) => ((![Y5 : set_Ho137910533iple_a]: ((P2 @ Y5) => (ord_le1221261669iple_a @ Y5 @ X))) => (Q @ X)))) => (Q @ (order_929906668iple_a @ P2)))))))). % GreatestI2_order
thf(fact_74_GreatestI2__order, axiom,
    ((![P2 : nat > $o, X2 : nat, Q : nat > $o]: ((P2 @ X2) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ X2))) => ((![X : nat]: ((P2 @ X) => ((![Y5 : nat]: ((P2 @ Y5) => (ord_less_eq_nat @ Y5 @ X))) => (Q @ X)))) => (Q @ (order_Greatest_nat @ P2)))))))). % GreatestI2_order
thf(fact_75_triples__valid__Suc, axiom,
    ((![Ts2 : set_Ho137910533iple_a, N2 : nat]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ Ts2) => (hoare_1926814542alid_a @ (suc @ N2) @ X))) => (![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ Ts2) => (hoare_1926814542alid_a @ N2 @ X4))))))). % triples_valid_Suc
thf(fact_76_le__rel__bool__arg__iff, axiom,
    ((ord_le1048771374iple_a = (^[X5 : $o > set_Ho137910533iple_a]: (^[Y6 : $o > set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_le1221261669iple_a @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_77_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_nat = (^[X5 : $o > nat]: (^[Y6 : $o > nat]: (((ord_less_eq_nat @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_nat @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_78_verit__la__disequality, axiom,
    ((![A3 : nat, B3 : nat]: ((A3 = B3) | ((~ ((ord_less_eq_nat @ A3 @ B3))) | (~ ((ord_less_eq_nat @ B3 @ A3)))))))). % verit_la_disequality
thf(fact_79_antimono__def, axiom,
    ((order_1710851741iple_a = (^[F2 : set_Ho137910533iple_a > set_Ho137910533iple_a]: (![X3 : set_Ho137910533iple_a]: (![Y3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X3 @ Y3)) => ((ord_le1221261669iple_a @ (F2 @ Y3) @ (F2 @ X3)))))))))). % antimono_def
thf(fact_80_antimono__def, axiom,
    ((order_194881289_a_nat = (^[F2 : set_Ho137910533iple_a > nat]: (![X3 : set_Ho137910533iple_a]: (![Y3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X3 @ Y3)) => ((ord_less_eq_nat @ (F2 @ Y3) @ (F2 @ X3)))))))))). % antimono_def
thf(fact_81_antimono__def, axiom,
    ((order_1673421321iple_a = (^[F2 : nat > set_Ho137910533iple_a]: (![X3 : nat]: (![Y3 : nat]: (((ord_less_eq_nat @ X3 @ Y3)) => ((ord_le1221261669iple_a @ (F2 @ Y3) @ (F2 @ X3)))))))))). % antimono_def
thf(fact_82_antimono__def, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![X3 : nat]: (![Y3 : nat]: (((ord_less_eq_nat @ X3 @ Y3)) => ((ord_less_eq_nat @ (F2 @ Y3) @ (F2 @ X3)))))))))). % antimono_def
thf(fact_83_antimonoI, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a]: ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X)))) => (order_1710851741iple_a @ F))))). % antimonoI
thf(fact_84_antimonoI, axiom,
    ((![F : set_Ho137910533iple_a > nat]: ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X)))) => (order_194881289_a_nat @ F))))). % antimonoI
thf(fact_85_antimonoI, axiom,
    ((![F : nat > set_Ho137910533iple_a]: ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X)))) => (order_1673421321iple_a @ F))))). % antimonoI
thf(fact_86_antimonoI, axiom,
    ((![F : nat > nat]: ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X)))) => (order_1631207636at_nat @ F))))). % antimonoI
thf(fact_87_antimonoE, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a, X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((order_1710851741iple_a @ F) => ((ord_le1221261669iple_a @ X2 @ Y2) => (ord_le1221261669iple_a @ (F @ Y2) @ (F @ X2))))))). % antimonoE
thf(fact_88_antimonoE, axiom,
    ((![F : set_Ho137910533iple_a > nat, X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((order_194881289_a_nat @ F) => ((ord_le1221261669iple_a @ X2 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X2))))))). % antimonoE
thf(fact_89_antimonoE, axiom,
    ((![F : nat > set_Ho137910533iple_a, X2 : nat, Y2 : nat]: ((order_1673421321iple_a @ F) => ((ord_less_eq_nat @ X2 @ Y2) => (ord_le1221261669iple_a @ (F @ Y2) @ (F @ X2))))))). % antimonoE
thf(fact_90_antimonoE, axiom,
    ((![F : nat > nat, X2 : nat, Y2 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X2))))))). % antimonoE
thf(fact_91_antimonoD, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a, X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((order_1710851741iple_a @ F) => ((ord_le1221261669iple_a @ X2 @ Y2) => (ord_le1221261669iple_a @ (F @ Y2) @ (F @ X2))))))). % antimonoD
thf(fact_92_antimonoD, axiom,
    ((![F : set_Ho137910533iple_a > nat, X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((order_194881289_a_nat @ F) => ((ord_le1221261669iple_a @ X2 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X2))))))). % antimonoD
thf(fact_93_antimonoD, axiom,
    ((![F : nat > set_Ho137910533iple_a, X2 : nat, Y2 : nat]: ((order_1673421321iple_a @ F) => ((ord_less_eq_nat @ X2 @ Y2) => (ord_le1221261669iple_a @ (F @ Y2) @ (F @ X2))))))). % antimonoD
thf(fact_94_antimonoD, axiom,
    ((![F : nat > nat, X2 : nat, Y2 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X2))))))). % antimonoD
thf(fact_95_weaken, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts3 : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts3) => ((ord_le1221261669iple_a @ Ts2 @ Ts3) => (hoare_129598474rivs_a @ G2 @ Ts2)))))). % weaken
thf(fact_96_triple__valid__Suc, axiom,
    ((![N2 : nat, T2 : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N2) @ T2) => (hoare_1926814542alid_a @ N2 @ T2))))). % triple_valid_Suc
thf(fact_97_thin, axiom,
    ((![G3 : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G3 @ Ts2) => ((ord_le1221261669iple_a @ G3 @ G2) => (hoare_129598474rivs_a @ G2 @ Ts2)))))). % thin
thf(fact_98_asm, axiom,
    ((![Ts2 : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts2 @ G2) => (hoare_129598474rivs_a @ G2 @ Ts2))))). % asm
thf(fact_99_Suc__le__mono, axiom,
    ((![N2 : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N2) @ (suc @ M)) = (ord_less_eq_nat @ N2 @ M))))). % Suc_le_mono
thf(fact_100_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_101_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_102_antimono__iff__le__Suc, axiom,
    ((order_1673421321iple_a = (^[F2 : nat > set_Ho137910533iple_a]: (![N : nat]: (ord_le1221261669iple_a @ (F2 @ (suc @ N)) @ (F2 @ N))))))). % antimono_iff_le_Suc
thf(fact_103_antimono__iff__le__Suc, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![N : nat]: (ord_less_eq_nat @ (F2 @ (suc @ N)) @ (F2 @ N))))))). % antimono_iff_le_Suc
thf(fact_104_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_105_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_106_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_107_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_108_GreatestI__nat, axiom,
    ((![P2 : nat > $o, K : nat, B3 : nat]: ((P2 @ K) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ B3))) => (P2 @ (order_Greatest_nat @ P2))))))). % GreatestI_nat
thf(fact_109_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_110_Greatest__le__nat, axiom,
    ((![P2 : nat > $o, K : nat, B3 : nat]: ((P2 @ K) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ B3))) => (ord_less_eq_nat @ K @ (order_Greatest_nat @ P2))))))). % Greatest_le_nat
thf(fact_111_GreatestI__ex__nat, axiom,
    ((![P2 : nat > $o, B3 : nat]: ((?[X_1 : nat]: (P2 @ X_1)) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ B3))) => (P2 @ (order_Greatest_nat @ P2))))))). % GreatestI_ex_nat
thf(fact_112_Nat_Oex__has__greatest__nat, axiom,
    ((![P2 : nat > $o, K : nat, B3 : nat]: ((P2 @ K) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ B3))) => (?[X : nat]: ((P2 @ X) & (![Y5 : nat]: ((P2 @ Y5) => (ord_less_eq_nat @ Y5 @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_113_Suc__inject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))). % Suc_inject
thf(fact_114_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_115_transitive__stepwise__le, axiom,
    ((![M : nat, N2 : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((![X : nat]: (R @ X @ X)) => ((![X : nat, Y4 : nat, Z3 : nat]: ((R @ X @ Y4) => ((R @ Y4 @ Z3) => (R @ X @ Z3)))) => ((![N3 : nat]: (R @ N3 @ (suc @ N3))) => (R @ M @ N2)))))))). % transitive_stepwise_le
thf(fact_116_nat__induct__at__least, axiom,
    ((![M : nat, N2 : nat, P2 : nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((P2 @ M) => ((![N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((P2 @ N3) => (P2 @ (suc @ N3))))) => (P2 @ N2))))))). % nat_induct_at_least
thf(fact_117_full__nat__induct, axiom,
    ((![P2 : nat > $o, N2 : nat]: ((![N3 : nat]: ((![M2 : nat]: ((ord_less_eq_nat @ (suc @ M2) @ N3) => (P2 @ M2))) => (P2 @ N3))) => (P2 @ N2))))). % full_nat_induct
thf(fact_118_not__less__eq__eq, axiom,
    ((![M : nat, N2 : nat]: ((~ ((ord_less_eq_nat @ M @ N2))) = (ord_less_eq_nat @ (suc @ N2) @ M))))). % not_less_eq_eq
thf(fact_119_Suc__n__not__le__n, axiom,
    ((![N2 : nat]: (~ ((ord_less_eq_nat @ (suc @ N2) @ N2)))))). % Suc_n_not_le_n
thf(fact_120_le__Suc__eq, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ (suc @ N2)) = (((ord_less_eq_nat @ M @ N2)) | ((M = (suc @ N2)))))))). % le_Suc_eq
thf(fact_121_Suc__le__D, axiom,
    ((![N2 : nat, M3 : nat]: ((ord_less_eq_nat @ (suc @ N2) @ M3) => (?[M4 : nat]: (M3 = (suc @ M4))))))). % Suc_le_D
thf(fact_122_le__SucI, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => (ord_less_eq_nat @ M @ (suc @ N2)))))). % le_SucI
thf(fact_123_le__SucE, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ (suc @ N2)) => ((~ ((ord_less_eq_nat @ M @ N2))) => (M = (suc @ N2))))))). % le_SucE
thf(fact_124_Suc__leD, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ (suc @ M) @ N2) => (ord_less_eq_nat @ M @ N2))))). % Suc_leD
thf(fact_125_lift__Suc__antimono__le, axiom,
    ((![F : nat > set_Ho137910533iple_a, N2 : nat, N4 : nat]: ((![N3 : nat]: (ord_le1221261669iple_a @ (F @ (suc @ N3)) @ (F @ N3))) => ((ord_less_eq_nat @ N2 @ N4) => (ord_le1221261669iple_a @ (F @ N4) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_126_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N2 : nat, N4 : nat]: ((![N3 : nat]: (ord_less_eq_nat @ (F @ (suc @ N3)) @ (F @ N3))) => ((ord_less_eq_nat @ N2 @ N4) => (ord_less_eq_nat @ (F @ N4) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_127_lift__Suc__mono__le, axiom,
    ((![F : nat > set_Ho137910533iple_a, N2 : nat, N4 : nat]: ((![N3 : nat]: (ord_le1221261669iple_a @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_eq_nat @ N2 @ N4) => (ord_le1221261669iple_a @ (F @ N2) @ (F @ N4))))))). % lift_Suc_mono_le
thf(fact_128_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N2 : nat, N4 : nat]: ((![N3 : nat]: (ord_less_eq_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_eq_nat @ N2 @ N4) => (ord_less_eq_nat @ (F @ N2) @ (F @ N4))))))). % lift_Suc_mono_le
thf(fact_129_Suc__le__D__lemma, axiom,
    ((![N2 : nat, M3 : nat, P2 : nat > $o]: ((ord_less_eq_nat @ (suc @ N2) @ M3) => ((![M4 : nat]: ((ord_less_eq_nat @ N2 @ M4) => (P2 @ (suc @ M4)))) => (P2 @ M3)))))). % Suc_le_D_lemma
thf(fact_130_bounded__Max__nat, axiom,
    ((![P2 : nat > $o, X2 : nat, M5 : nat]: ((P2 @ X2) => ((![X : nat]: ((P2 @ X) => (ord_less_eq_nat @ X @ M5))) => (~ ((![M4 : nat]: ((P2 @ M4) => (~ ((![X4 : nat]: ((P2 @ X4) => (ord_less_eq_nat @ X4 @ M4)))))))))))))). % bounded_Max_nat
thf(fact_131_greaterThan__subset__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_set_nat @ (set_or1965240170an_nat @ X2) @ (set_or1965240170an_nat @ Y2)) = (ord_less_eq_nat @ Y2 @ X2))))). % greaterThan_subset_iff
thf(fact_132_greaterThan__eq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: (((set_or1965240170an_nat @ X2) = (set_or1965240170an_nat @ Y2)) = (X2 = Y2))))). % greaterThan_eq_iff
thf(fact_133_lessThan__subset__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_set_nat @ (set_ord_lessThan_nat @ X2) @ (set_ord_lessThan_nat @ Y2)) = (ord_less_eq_nat @ X2 @ Y2))))). % lessThan_subset_iff
thf(fact_134_atLeast__subset__iff, axiom,
    ((![X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((ord_le1677170587iple_a @ (set_or1109361786iple_a @ X2) @ (set_or1109361786iple_a @ Y2)) = (ord_le1221261669iple_a @ Y2 @ X2))))). % atLeast_subset_iff
thf(fact_135_atLeast__subset__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_set_nat @ (set_ord_atLeast_nat @ X2) @ (set_ord_atLeast_nat @ Y2)) = (ord_less_eq_nat @ Y2 @ X2))))). % atLeast_subset_iff
thf(fact_136_atMost__subset__iff, axiom,
    ((![X2 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((ord_le1677170587iple_a @ (set_or1587679614iple_a @ X2) @ (set_or1587679614iple_a @ Y2)) = (ord_le1221261669iple_a @ X2 @ Y2))))). % atMost_subset_iff
thf(fact_137_atMost__subset__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_set_nat @ (set_ord_atMost_nat @ X2) @ (set_ord_atMost_nat @ Y2)) = (ord_less_eq_nat @ X2 @ Y2))))). % atMost_subset_iff
thf(fact_138_atMost__eq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: (((set_ord_atMost_nat @ X2) = (set_ord_atMost_nat @ Y2)) = (X2 = Y2))))). % atMost_eq_iff
thf(fact_139_lessThan__eq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: (((set_ord_lessThan_nat @ X2) = (set_ord_lessThan_nat @ Y2)) = (X2 = Y2))))). % lessThan_eq_iff
thf(fact_140_atLeast__eq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: (((set_ord_atLeast_nat @ X2) = (set_ord_atLeast_nat @ Y2)) = (X2 = Y2))))). % atLeast_eq_iff
thf(fact_141_atMost__iff, axiom,
    ((![I : set_Ho137910533iple_a, K : set_Ho137910533iple_a]: ((member521824924iple_a @ I @ (set_or1587679614iple_a @ K)) = (ord_le1221261669iple_a @ I @ K))))). % atMost_iff
thf(fact_142_atMost__iff, axiom,
    ((![I : nat, K : nat]: ((member_nat @ I @ (set_ord_atMost_nat @ K)) = (ord_less_eq_nat @ I @ K))))). % atMost_iff
thf(fact_143_atLeast__iff, axiom,
    ((![I : set_Ho137910533iple_a, K : set_Ho137910533iple_a]: ((member521824924iple_a @ I @ (set_or1109361786iple_a @ K)) = (ord_le1221261669iple_a @ K @ I))))). % atLeast_iff
thf(fact_144_atLeast__iff, axiom,
    ((![I : nat, K : nat]: ((member_nat @ I @ (set_ord_atLeast_nat @ K)) = (ord_less_eq_nat @ K @ I))))). % atLeast_iff
thf(fact_145_atLeast__Suc__greaterThan, axiom,
    ((![K : nat]: ((set_ord_atLeast_nat @ (suc @ K)) = (set_or1965240170an_nat @ K))))). % atLeast_Suc_greaterThan
thf(fact_146_lessThan__Suc__atMost, axiom,
    ((![K : nat]: ((set_ord_lessThan_nat @ (suc @ K)) = (set_ord_atMost_nat @ K))))). % lessThan_Suc_atMost
thf(fact_147_not__Iic__eq__Ici, axiom,
    ((![H : nat, L : nat]: (~ (((set_ord_atMost_nat @ H) = (set_ord_atLeast_nat @ L))))))). % not_Iic_eq_Ici
thf(fact_148_not__Ici__le__Iic, axiom,
    ((![L2 : nat, H2 : nat]: (~ ((ord_less_eq_set_nat @ (set_ord_atLeast_nat @ L2) @ (set_ord_atMost_nat @ H2))))))). % not_Ici_le_Iic
thf(fact_149_Ioi__le__Ico, axiom,
    ((![A3 : nat]: (ord_less_eq_set_nat @ (set_or1965240170an_nat @ A3) @ (set_ord_atLeast_nat @ A3))))). % Ioi_le_Ico
thf(fact_150_Ici__subset__Ioi__iff, axiom,
    ((![A3 : nat, B3 : nat]: ((ord_less_eq_set_nat @ (set_ord_atLeast_nat @ A3) @ (set_or1965240170an_nat @ B3)) = (ord_less_nat @ B3 @ A3))))). % Ici_subset_Ioi_iff
thf(fact_151_Icc__subset__Ici__iff, axiom,
    ((![L2 : set_Ho137910533iple_a, H : set_Ho137910533iple_a, L : set_Ho137910533iple_a]: ((ord_le1677170587iple_a @ (set_or1668187408iple_a @ L2 @ H) @ (set_or1109361786iple_a @ L)) = (((~ ((ord_le1221261669iple_a @ L2 @ H)))) | ((ord_le1221261669iple_a @ L @ L2))))))). % Icc_subset_Ici_iff
thf(fact_152_Icc__subset__Ici__iff, axiom,
    ((![L2 : nat, H : nat, L : nat]: ((ord_less_eq_set_nat @ (set_or601162459st_nat @ L2 @ H) @ (set_ord_atLeast_nat @ L)) = (((~ ((ord_less_eq_nat @ L2 @ H)))) | ((ord_less_eq_nat @ L @ L2))))))). % Icc_subset_Ici_iff
thf(fact_153_Icc__subset__Iic__iff, axiom,
    ((![L2 : set_Ho137910533iple_a, H : set_Ho137910533iple_a, H2 : set_Ho137910533iple_a]: ((ord_le1677170587iple_a @ (set_or1668187408iple_a @ L2 @ H) @ (set_or1587679614iple_a @ H2)) = (((~ ((ord_le1221261669iple_a @ L2 @ H)))) | ((ord_le1221261669iple_a @ H @ H2))))))). % Icc_subset_Iic_iff
thf(fact_154_Icc__subset__Iic__iff, axiom,
    ((![L2 : nat, H : nat, H2 : nat]: ((ord_less_eq_set_nat @ (set_or601162459st_nat @ L2 @ H) @ (set_ord_atMost_nat @ H2)) = (((~ ((ord_less_eq_nat @ L2 @ H)))) | ((ord_less_eq_nat @ H @ H2))))))). % Icc_subset_Iic_iff
thf(fact_155_lessI, axiom,
    ((![N2 : nat]: (ord_less_nat @ N2 @ (suc @ N2))))). % lessI
thf(fact_156_Suc__mono, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (ord_less_nat @ (suc @ M) @ (suc @ N2)))))). % Suc_mono
thf(fact_157_Suc__less__eq, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N2)) = (ord_less_nat @ M @ N2))))). % Suc_less_eq
thf(fact_158_psubsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((~ ((A = B))) => (ord_le1356158809iple_a @ A @ B)))))). % psubsetI
thf(fact_159_Icc__eq__Icc, axiom,
    ((![L2 : set_Ho137910533iple_a, H : set_Ho137910533iple_a, L : set_Ho137910533iple_a, H2 : set_Ho137910533iple_a]: (((set_or1668187408iple_a @ L2 @ H) = (set_or1668187408iple_a @ L @ H2)) = (((((L2 = L)) & ((H = H2)))) | ((((~ ((ord_le1221261669iple_a @ L2 @ H)))) & ((~ ((ord_le1221261669iple_a @ L @ H2))))))))))). % Icc_eq_Icc
thf(fact_160_Icc__eq__Icc, axiom,
    ((![L2 : nat, H : nat, L : nat, H2 : nat]: (((set_or601162459st_nat @ L2 @ H) = (set_or601162459st_nat @ L @ H2)) = (((((L2 = L)) & ((H = H2)))) | ((((~ ((ord_less_eq_nat @ L2 @ H)))) & ((~ ((ord_less_eq_nat @ L @ H2))))))))))). % Icc_eq_Icc
thf(fact_161_atLeastAtMost__iff, axiom,
    ((![I : set_Ho137910533iple_a, L2 : set_Ho137910533iple_a, U : set_Ho137910533iple_a]: ((member521824924iple_a @ I @ (set_or1668187408iple_a @ L2 @ U)) = (((ord_le1221261669iple_a @ L2 @ I)) & ((ord_le1221261669iple_a @ I @ U))))))). % atLeastAtMost_iff
thf(fact_162_atLeastAtMost__iff, axiom,
    ((![I : nat, L2 : nat, U : nat]: ((member_nat @ I @ (set_or601162459st_nat @ L2 @ U)) = (((ord_less_eq_nat @ L2 @ I)) & ((ord_less_eq_nat @ I @ U))))))). % atLeastAtMost_iff
thf(fact_163_lessThan__iff, axiom,
    ((![I : nat, K : nat]: ((member_nat @ I @ (set_ord_lessThan_nat @ K)) = (ord_less_nat @ I @ K))))). % lessThan_iff
thf(fact_164_greaterThan__iff, axiom,
    ((![I : nat, K : nat]: ((member_nat @ I @ (set_or1965240170an_nat @ K)) = (ord_less_nat @ K @ I))))). % greaterThan_iff
thf(fact_165_atLeastatMost__subset__iff, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a, D : set_Ho137910533iple_a]: ((ord_le1677170587iple_a @ (set_or1668187408iple_a @ A3 @ B3) @ (set_or1668187408iple_a @ C @ D)) = (((~ ((ord_le1221261669iple_a @ A3 @ B3)))) | ((((ord_le1221261669iple_a @ C @ A3)) & ((ord_le1221261669iple_a @ B3 @ D))))))))). % atLeastatMost_subset_iff
thf(fact_166_atLeastatMost__subset__iff, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: ((ord_less_eq_set_nat @ (set_or601162459st_nat @ A3 @ B3) @ (set_or601162459st_nat @ C @ D)) = (((~ ((ord_less_eq_nat @ A3 @ B3)))) | ((((ord_less_eq_nat @ C @ A3)) & ((ord_less_eq_nat @ B3 @ D))))))))). % atLeastatMost_subset_iff
thf(fact_167_lessThan__strict__subset__iff, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_set_nat @ (set_ord_lessThan_nat @ M) @ (set_ord_lessThan_nat @ N2)) = (ord_less_nat @ M @ N2))))). % lessThan_strict_subset_iff
thf(fact_168_not__Ici__eq__Icc, axiom,
    ((![L : nat, L2 : nat, H : nat]: (~ (((set_ord_atLeast_nat @ L) = (set_or601162459st_nat @ L2 @ H))))))). % not_Ici_eq_Icc
thf(fact_169_atLeastatMost__psubset__iff, axiom,
    ((![A3 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a, D : set_Ho137910533iple_a]: ((ord_le1827677583iple_a @ (set_or1668187408iple_a @ A3 @ B3) @ (set_or1668187408iple_a @ C @ D)) = (((((~ ((ord_le1221261669iple_a @ A3 @ B3)))) | ((((ord_le1221261669iple_a @ C @ A3)) & ((((ord_le1221261669iple_a @ B3 @ D)) & ((((ord_le1356158809iple_a @ C @ A3)) | ((ord_le1356158809iple_a @ B3 @ D)))))))))) & ((ord_le1221261669iple_a @ C @ D))))))). % atLeastatMost_psubset_iff
thf(fact_170_atLeastatMost__psubset__iff, axiom,
    ((![A3 : nat, B3 : nat, C : nat, D : nat]: ((ord_less_set_nat @ (set_or601162459st_nat @ A3 @ B3) @ (set_or601162459st_nat @ C @ D)) = (((((~ ((ord_less_eq_nat @ A3 @ B3)))) | ((((ord_less_eq_nat @ C @ A3)) & ((((ord_less_eq_nat @ B3 @ D)) & ((((ord_less_nat @ C @ A3)) | ((ord_less_nat @ B3 @ D)))))))))) & ((ord_less_eq_nat @ C @ D))))))). % atLeastatMost_psubset_iff
thf(fact_171_verit__comp__simplify1_I1_J, axiom,
    ((![A3 : nat]: (~ ((ord_less_nat @ A3 @ A3)))))). % verit_comp_simplify1(1)
thf(fact_172_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B3 : nat, A3 : nat]: ((ord_less_nat @ B3 @ A3) => (~ ((A3 = B3))))))). % dual_order.strict_implies_not_eq
thf(fact_173_order_Ostrict__implies__not__eq, axiom,
    ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (~ ((A3 = B3))))))). % order.strict_implies_not_eq
thf(fact_174_not__less__iff__gr__or__eq, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X2 @ Y2))) = (((ord_less_nat @ Y2 @ X2)) | ((X2 = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_175_dual__order_Ostrict__trans, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: ((ord_less_nat @ B3 @ A3) => ((ord_less_nat @ C @ B3) => (ord_less_nat @ C @ A3)))))). % dual_order.strict_trans
thf(fact_176_linorder__less__wlog, axiom,
    ((![P2 : nat > nat > $o, A3 : nat, B3 : nat]: ((![A5 : nat, B5 : nat]: ((ord_less_nat @ A5 @ B5) => (P2 @ A5 @ B5))) => ((![A5 : nat]: (P2 @ A5 @ A5)) => ((![A5 : nat, B5 : nat]: ((P2 @ B5 @ A5) => (P2 @ A5 @ B5))) => (P2 @ A3 @ B3))))))). % linorder_less_wlog
thf(fact_177_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X6 : nat]: (P3 @ X6))) = (^[P : nat > $o]: (?[N : nat]: (((P @ N)) & ((![M6 : nat]: (((ord_less_nat @ M6 @ N)) => ((~ ((P @ M6))))))))))))). % exists_least_iff
thf(fact_178_less__imp__not__less, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_imp_not_less
thf(fact_179_order_Ostrict__trans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((ord_less_nat @ A3 @ B3) => ((ord_less_nat @ B3 @ C) => (ord_less_nat @ A3 @ C)))))). % order.strict_trans
thf(fact_180_dual__order_Oirrefl, axiom,
    ((![A3 : nat]: (~ ((ord_less_nat @ A3 @ A3)))))). % dual_order.irrefl
thf(fact_181_linorder__cases, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X2 @ Y2))) => ((~ ((X2 = Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_cases
thf(fact_182_less__imp__triv, axiom,
    ((![X2 : nat, Y2 : nat, P2 : $o]: ((ord_less_nat @ X2 @ Y2) => ((ord_less_nat @ Y2 @ X2) => P2))))). % less_imp_triv
thf(fact_183_less__imp__not__eq2, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((Y2 = X2))))))). % less_imp_not_eq2
thf(fact_184_antisym__conv3, axiom,
    ((![Y2 : nat, X2 : nat]: ((~ ((ord_less_nat @ Y2 @ X2))) => ((~ ((ord_less_nat @ X2 @ Y2))) = (X2 = Y2)))))). % antisym_conv3
thf(fact_185_less__induct, axiom,
    ((![P2 : nat > $o, A3 : nat]: ((![X : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X) => (P2 @ Y5))) => (P2 @ X))) => (P2 @ A3))))). % less_induct
thf(fact_186_less__not__sym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_not_sym
thf(fact_187_less__imp__not__eq, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((X2 = Y2))))))). % less_imp_not_eq
thf(fact_188_dual__order_Oasym, axiom,
    ((![B3 : nat, A3 : nat]: ((ord_less_nat @ B3 @ A3) => (~ ((ord_less_nat @ A3 @ B3))))))). % dual_order.asym

% Conjectures (2)
thf(conj_0, hypothesis,
    ((ord_le1221261669iple_a @ tsa @ ga))).
thf(conj_1, conjecture,
    ((![N3 : nat]: ((?[X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ ga) & (~ ((hoare_1926814542alid_a @ N3 @ X4))))) | (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ tsa) => (hoare_1926814542alid_a @ N3 @ X))))))).
