% 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_286__3252306_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:29.313

% Could-be-implicit typings (6)
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__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (33)
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_Ostate__not__singleton, type,
    hoare_405891322gleton : $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    inf_in1336607127iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    bot_bo431311916le_a_o : hoare_1678595023iple_a > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    bot_bo922500559iple_a : set_se152467259iple_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J, type,
    ord_less_eq_o_nat : ($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_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_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_emp901906557iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat, type,
    set_or1544565540an_nat : nat > nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_or769492057iple_a : set_Ho137910533iple_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_v_G_H, type,
    g : set_Ho137910533iple_a).
thf(sy_v_Ga, type,
    ga : set_Ho137910533iple_a).
thf(sy_v_tsa, type,
    tsa : set_Ho137910533iple_a).

% Relevant facts (162)
thf(fact_0_cut, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => ((hoare_129598474rivs_a @ G2 @ G) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % cut
thf(fact_1_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G3 : set_Ho137910533iple_a]: (^[Ts2 : set_Ho137910533iple_a]: (![N : nat]: (((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ G3)) => ((hoare_1926814542alid_a @ N @ X))))) => ((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ Ts2)) => ((hoare_1926814542alid_a @ N @ X)))))))))))). % hoare_valids_def
thf(fact_2_triples__valid__Suc, axiom,
    ((![Ts : set_Ho137910533iple_a, N2 : nat]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ Ts) => (hoare_1926814542alid_a @ (suc @ N2) @ X2))) => (![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ Ts) => (hoare_1926814542alid_a @ N2 @ X3))))))). % triples_valid_Suc
thf(fact_3_triple__valid__Suc, axiom,
    ((![N2 : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N2) @ T) => (hoare_1926814542alid_a @ N2 @ T))))). % triple_valid_Suc
thf(fact_4_empty, axiom,
    ((![G2 : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G2 @ bot_bo1298296729iple_a)))). % empty
thf(fact_5_thin, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => ((ord_le1221261669iple_a @ G @ G2) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % thin
thf(fact_6_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G2) => (hoare_129598474rivs_a @ G2 @ Ts))))). % asm
thf(fact_7_weaken, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts3 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts3) => ((ord_le1221261669iple_a @ Ts @ Ts3) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % weaken
thf(fact_8_single__stateE, axiom,
    ((hoare_405891322gleton => (![T2 : state]: (~ ((![S : state]: (S = T2)))))))). % single_stateE
thf(fact_9_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T3 : state]: (~ ((S2 = T3)))))))). % state_not_singleton_def
thf(fact_10_ball__empty, axiom,
    ((![P : hoare_1678595023iple_a > $o, X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ bot_bo1298296729iple_a) => (P @ X3))))). % ball_empty
thf(fact_11_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_12_all__not__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X @ A)))) = (A = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_13_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_14_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X : hoare_1678595023iple_a]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_15_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ A) => (member1332298086iple_a @ X2 @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_16_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_17_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_18_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_19_emptyE, axiom,
    ((![A2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A2 @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_20_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X4 @ A) => (member1332298086iple_a @ X4 @ B)))))). % in_mono
thf(fact_21_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_22_equals0D, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((A = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A2 @ A))))))). % equals0D
thf(fact_23_equals0I, axiom,
    ((![A : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A)))) => (A = bot_bo1298296729iple_a))))). % equals0I
thf(fact_24_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_25_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A3)) => ((member1332298086iple_a @ X @ B2))))))))). % subset_eq
thf(fact_26_equalityD1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ A @ B))))). % equalityD1
thf(fact_27_equalityD2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ B @ A))))). % equalityD2
thf(fact_28_ex__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((?[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A)) = (~ ((A = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_29_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![T3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ T3 @ A3)) => ((member1332298086iple_a @ T3 @ B2))))))))). % subset_iff
thf(fact_30_subset__refl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % subset_refl
thf(fact_31_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X2 : hoare_1678595023iple_a]: ((P @ X2) => (Q @ X2))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_32_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_33_set__eq__subset, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B2)) & ((ord_le1221261669iple_a @ B2 @ A3)))))))). % set_eq_subset
thf(fact_34_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X : hoare_1678595023iple_a]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_35_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_36_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_37_order__refl, axiom,
    ((![X4 : nat]: (ord_less_eq_nat @ X4 @ X4)))). % order_refl
thf(fact_38_order__refl, axiom,
    ((![X4 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X4 @ X4)))). % order_refl
thf(fact_39_Ball__Collect, axiom,
    ((ball_H465710501iple_a = (^[A3 : set_Ho137910533iple_a]: (^[P2 : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ A3 @ (collec1600235172iple_a @ P2))))))). % Ball_Collect
thf(fact_40_subset__emptyI, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X2 @ A)))) => (ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_41_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ N4) @ (F @ (suc @ N4)))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N2) @ (F @ N3))))))). % lift_Suc_mono_le
thf(fact_42_lift__Suc__mono__le, axiom,
    ((![F : nat > set_Ho137910533iple_a, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_le1221261669iple_a @ (F @ N4) @ (F @ (suc @ N4)))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_le1221261669iple_a @ (F @ N2) @ (F @ N3))))))). % lift_Suc_mono_le
thf(fact_43_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ (suc @ N4)) @ (F @ N4))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N3) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_44_lift__Suc__antimono__le, axiom,
    ((![F : nat > set_Ho137910533iple_a, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_le1221261669iple_a @ (F @ (suc @ N4)) @ (F @ N4))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_le1221261669iple_a @ (F @ N3) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_45_bot_Oextremum, axiom,
    ((![A2 : nat]: (ord_less_eq_nat @ bot_bot_nat @ A2)))). % bot.extremum
thf(fact_46_bot_Oextremum, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % bot.extremum
thf(fact_47_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_48_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_49_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_50_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_51_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_52_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_53_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (?[X2 : nat]: ((P @ X2) & (![Y3 : nat]: ((P @ Y3) => (ord_less_eq_nat @ Y3 @ X2)))))))))). % Nat.ex_has_greatest_nat
thf(fact_54_transitive__stepwise__le, axiom,
    ((![M : nat, N2 : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((![X2 : nat]: (R @ X2 @ X2)) => ((![X2 : nat, Y : nat, Z2 : nat]: ((R @ X2 @ Y) => ((R @ Y @ Z2) => (R @ X2 @ Z2)))) => ((![N4 : nat]: (R @ N4 @ (suc @ N4))) => (R @ M @ N2)))))))). % transitive_stepwise_le
thf(fact_55_nat__induct__at__least, axiom,
    ((![M : nat, N2 : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((P @ M) => ((![N4 : nat]: ((ord_less_eq_nat @ M @ N4) => ((P @ N4) => (P @ (suc @ N4))))) => (P @ N2))))))). % nat_induct_at_least
thf(fact_56_full__nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N4 : nat]: ((![M2 : nat]: ((ord_less_eq_nat @ (suc @ M2) @ N4) => (P @ M2))) => (P @ N4))) => (P @ N2))))). % full_nat_induct
thf(fact_57_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_58_Suc__n__not__le__n, axiom,
    ((![N2 : nat]: (~ ((ord_less_eq_nat @ (suc @ N2) @ N2)))))). % Suc_n_not_le_n
thf(fact_59_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_60_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_61_le__SucI, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => (ord_less_eq_nat @ M @ (suc @ N2)))))). % le_SucI
thf(fact_62_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_63_Suc__leD, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ (suc @ M) @ N2) => (ord_less_eq_nat @ M @ N2))))). % Suc_leD
thf(fact_64_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_65_dual__order_Oantisym, axiom,
    ((![B3 : nat, A2 : nat]: ((ord_less_eq_nat @ B3 @ A2) => ((ord_less_eq_nat @ A2 @ B3) => (A2 = B3)))))). % dual_order.antisym
thf(fact_66_dual__order_Oantisym, axiom,
    ((![B3 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B3 @ A2) => ((ord_le1221261669iple_a @ A2 @ B3) => (A2 = B3)))))). % dual_order.antisym
thf(fact_67_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ B4 @ A4)) & ((ord_less_eq_nat @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_68_dual__order_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = 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_69_dual__order_Otrans, axiom,
    ((![B3 : nat, A2 : nat, C : nat]: ((ord_less_eq_nat @ B3 @ A2) => ((ord_less_eq_nat @ C @ B3) => (ord_less_eq_nat @ C @ A2)))))). % dual_order.trans
thf(fact_70_dual__order_Otrans, axiom,
    ((![B3 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B3 @ A2) => ((ord_le1221261669iple_a @ C @ B3) => (ord_le1221261669iple_a @ C @ A2)))))). % dual_order.trans
thf(fact_71_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A2 : nat, B3 : nat]: ((![A5 : nat, B5 : nat]: ((ord_less_eq_nat @ A5 @ B5) => (P @ A5 @ B5))) => ((![A5 : nat, B5 : nat]: ((P @ B5 @ A5) => (P @ A5 @ B5))) => (P @ A2 @ B3)))))). % linorder_wlog
thf(fact_72_dual__order_Orefl, axiom,
    ((![A2 : nat]: (ord_less_eq_nat @ A2 @ A2)))). % dual_order.refl
thf(fact_73_dual__order_Orefl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % dual_order.refl
thf(fact_74_order__trans, axiom,
    ((![X4 : nat, Y4 : nat, Z3 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => ((ord_less_eq_nat @ Y4 @ Z3) => (ord_less_eq_nat @ X4 @ Z3)))))). % order_trans
thf(fact_75_order__trans, axiom,
    ((![X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a, Z3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y4) => ((ord_le1221261669iple_a @ Y4 @ Z3) => (ord_le1221261669iple_a @ X4 @ Z3)))))). % order_trans
thf(fact_76_order__class_Oorder_Oantisym, axiom,
    ((![A2 : nat, B3 : nat]: ((ord_less_eq_nat @ A2 @ B3) => ((ord_less_eq_nat @ B3 @ A2) => (A2 = B3)))))). % order_class.order.antisym
thf(fact_77_order__class_Oorder_Oantisym, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B3) => ((ord_le1221261669iple_a @ B3 @ A2) => (A2 = B3)))))). % order_class.order.antisym
thf(fact_78_ord__le__eq__trans, axiom,
    ((![A2 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A2 @ B3) => ((B3 = C) => (ord_less_eq_nat @ A2 @ C)))))). % ord_le_eq_trans
thf(fact_79_ord__le__eq__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B3) => ((B3 = C) => (ord_le1221261669iple_a @ A2 @ C)))))). % ord_le_eq_trans
thf(fact_80_ord__eq__le__trans, axiom,
    ((![A2 : nat, B3 : nat, C : nat]: ((A2 = B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A2 @ C)))))). % ord_eq_le_trans
thf(fact_81_ord__eq__le__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A2 = B3) => ((ord_le1221261669iple_a @ B3 @ C) => (ord_le1221261669iple_a @ A2 @ C)))))). % ord_eq_le_trans
thf(fact_82_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ A4 @ B4)) & ((ord_less_eq_nat @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_83_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = 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_84_antisym__conv, axiom,
    ((![Y4 : nat, X4 : nat]: ((ord_less_eq_nat @ Y4 @ X4) => ((ord_less_eq_nat @ X4 @ Y4) = (X4 = Y4)))))). % antisym_conv
thf(fact_85_antisym__conv, axiom,
    ((![Y4 : set_Ho137910533iple_a, X4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y4 @ X4) => ((ord_le1221261669iple_a @ X4 @ Y4) = (X4 = Y4)))))). % antisym_conv
thf(fact_86_le__cases3, axiom,
    ((![X4 : nat, Y4 : nat, Z3 : nat]: (((ord_less_eq_nat @ X4 @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ Z3)))) => (((ord_less_eq_nat @ Y4 @ X4) => (~ ((ord_less_eq_nat @ X4 @ Z3)))) => (((ord_less_eq_nat @ X4 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y4)))) => (((ord_less_eq_nat @ Z3 @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ X4)))) => (((ord_less_eq_nat @ Y4 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X4)))) => (~ (((ord_less_eq_nat @ Z3 @ X4) => (~ ((ord_less_eq_nat @ X4 @ Y4)))))))))))))). % le_cases3
thf(fact_87_order_Otrans, axiom,
    ((![A2 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A2 @ B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A2 @ C)))))). % order.trans
thf(fact_88_order_Otrans, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B3) => ((ord_le1221261669iple_a @ B3 @ C) => (ord_le1221261669iple_a @ A2 @ C)))))). % order.trans
thf(fact_89_le__cases, axiom,
    ((![X4 : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X4 @ Y4))) => (ord_less_eq_nat @ Y4 @ X4))))). % le_cases
thf(fact_90_eq__refl, axiom,
    ((![X4 : nat, Y4 : nat]: ((X4 = Y4) => (ord_less_eq_nat @ X4 @ Y4))))). % eq_refl
thf(fact_91_eq__refl, axiom,
    ((![X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((X4 = Y4) => (ord_le1221261669iple_a @ X4 @ Y4))))). % eq_refl
thf(fact_92_linear, axiom,
    ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) | (ord_less_eq_nat @ Y4 @ X4))))). % linear
thf(fact_93_antisym, axiom,
    ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => ((ord_less_eq_nat @ Y4 @ X4) => (X4 = Y4)))))). % antisym
thf(fact_94_antisym, axiom,
    ((![X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y4) => ((ord_le1221261669iple_a @ Y4 @ X4) => (X4 = Y4)))))). % antisym
thf(fact_95_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[X : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X @ Y5)) & ((ord_less_eq_nat @ Y5 @ X)))))))). % eq_iff
thf(fact_96_eq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[X : set_Ho137910533iple_a]: (^[Y5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X @ Y5)) & ((ord_le1221261669iple_a @ Y5 @ X)))))))). % eq_iff
thf(fact_97_ord__le__eq__subst, axiom,
    ((![A2 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A2 @ B3) => (((F @ B3) = C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_98_ord__le__eq__subst, axiom,
    ((![A2 : nat, B3 : nat, F : nat > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A2 @ B3) => (((F @ B3) = C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_99_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > nat, C : nat]: ((ord_le1221261669iple_a @ A2 @ B3) => (((F @ B3) = C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_100_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B3) => (((F @ B3) = C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_101_ord__eq__le__subst, axiom,
    ((![A2 : nat, F : nat > nat, B3 : nat, C : nat]: ((A2 = (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_102_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, F : nat > set_Ho137910533iple_a, B3 : nat, C : nat]: ((A2 = (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_103_ord__eq__le__subst, axiom,
    ((![A2 : nat, F : set_Ho137910533iple_a > nat, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A2 = (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_104_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A2 = (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_105_order__subst2, axiom,
    ((![A2 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A2 @ B3) => ((ord_less_eq_nat @ (F @ B3) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A2) @ C))))))). % order_subst2
thf(fact_106_order__subst2, axiom,
    ((![A2 : nat, B3 : nat, F : nat > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A2 @ B3) => ((ord_le1221261669iple_a @ (F @ B3) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % order_subst2
thf(fact_107_order__subst2, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > nat, C : nat]: ((ord_le1221261669iple_a @ A2 @ B3) => ((ord_less_eq_nat @ (F @ B3) @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A2) @ C))))))). % order_subst2
thf(fact_108_order__subst2, axiom,
    ((![A2 : set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B3) => ((ord_le1221261669iple_a @ (F @ B3) @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % order_subst2
thf(fact_109_order__subst1, axiom,
    ((![A2 : nat, F : nat > nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A2 @ (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_110_order__subst1, axiom,
    ((![A2 : nat, F : set_Ho137910533iple_a > nat, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_less_eq_nat @ A2 @ (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_111_order__subst1, axiom,
    ((![A2 : set_Ho137910533iple_a, F : nat > set_Ho137910533iple_a, B3 : nat, C : nat]: ((ord_le1221261669iple_a @ A2 @ (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_112_order__subst1, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B3 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (F @ B3)) => ((ord_le1221261669iple_a @ B3 @ C) => ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ X2) @ (F @ Y)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_113_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_114_Suc__inject, axiom,
    ((![X4 : nat, Y4 : nat]: (((suc @ X4) = (suc @ Y4)) => (X4 = Y4))))). % Suc_inject
thf(fact_115_bot_Oextremum__uniqueI, axiom,
    ((![A2 : nat]: ((ord_less_eq_nat @ A2 @ bot_bot_nat) => (A2 = bot_bot_nat))))). % bot.extremum_uniqueI
thf(fact_116_bot_Oextremum__uniqueI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) => (A2 = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_117_bot_Oextremum__unique, axiom,
    ((![A2 : nat]: ((ord_less_eq_nat @ A2 @ bot_bot_nat) = (A2 = bot_bot_nat))))). % bot.extremum_unique
thf(fact_118_bot_Oextremum__unique, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_119_Suc__le__D__lemma, axiom,
    ((![N2 : nat, M3 : nat, P : nat > $o]: ((ord_less_eq_nat @ (suc @ N2) @ M3) => ((![M4 : nat]: ((ord_less_eq_nat @ N2 @ M4) => (P @ (suc @ M4)))) => (P @ M3)))))). % Suc_le_D_lemma
thf(fact_120_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_121_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_122_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_123_Greatest__equality, axiom,
    ((![P : set_Ho137910533iple_a > $o, X4 : set_Ho137910533iple_a]: ((P @ X4) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X4))) => ((order_929906668iple_a @ P) = X4)))))). % Greatest_equality
thf(fact_124_Greatest__equality, axiom,
    ((![P : nat > $o, X4 : nat]: ((P @ X4) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X4))) => ((order_Greatest_nat @ P) = X4)))))). % Greatest_equality
thf(fact_125_GreatestI2__order, axiom,
    ((![P : set_Ho137910533iple_a > $o, X4 : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P @ X4) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X4))) => ((![X2 : set_Ho137910533iple_a]: ((P @ X2) => ((![Y3 : set_Ho137910533iple_a]: ((P @ Y3) => (ord_le1221261669iple_a @ Y3 @ X2))) => (Q @ X2)))) => (Q @ (order_929906668iple_a @ P)))))))). % GreatestI2_order
thf(fact_126_GreatestI2__order, axiom,
    ((![P : nat > $o, X4 : nat, Q : nat > $o]: ((P @ X4) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X4))) => ((![X2 : nat]: ((P @ X2) => ((![Y3 : nat]: ((P @ Y3) => (ord_less_eq_nat @ Y3 @ X2))) => (Q @ X2)))) => (Q @ (order_Greatest_nat @ P)))))))). % GreatestI2_order
thf(fact_127_bounded__Max__nat, axiom,
    ((![P : nat > $o, X4 : nat, M5 : nat]: ((P @ X4) => ((![X2 : nat]: ((P @ X2) => (ord_less_eq_nat @ X2 @ M5))) => (~ ((![M4 : nat]: ((P @ M4) => (~ ((![X3 : nat]: ((P @ X3) => (ord_less_eq_nat @ X3 @ M4)))))))))))))). % bounded_Max_nat
thf(fact_128_antimonoD, axiom,
    ((![F : nat > nat, X4 : nat, Y4 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X4))))))). % antimonoD
thf(fact_129_antimonoD, axiom,
    ((![F : nat > set_Ho137910533iple_a, X4 : nat, Y4 : nat]: ((order_1673421321iple_a @ F) => ((ord_less_eq_nat @ X4 @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X4))))))). % antimonoD
thf(fact_130_antimonoD, axiom,
    ((![F : set_Ho137910533iple_a > nat, X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((order_194881289_a_nat @ F) => ((ord_le1221261669iple_a @ X4 @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X4))))))). % antimonoD
thf(fact_131_antimonoD, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a, X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((order_1710851741iple_a @ F) => ((ord_le1221261669iple_a @ X4 @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X4))))))). % antimonoD
thf(fact_132_antimonoE, axiom,
    ((![F : nat > nat, X4 : nat, Y4 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X4))))))). % antimonoE
thf(fact_133_antimonoE, axiom,
    ((![F : nat > set_Ho137910533iple_a, X4 : nat, Y4 : nat]: ((order_1673421321iple_a @ F) => ((ord_less_eq_nat @ X4 @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X4))))))). % antimonoE
thf(fact_134_antimonoE, axiom,
    ((![F : set_Ho137910533iple_a > nat, X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((order_194881289_a_nat @ F) => ((ord_le1221261669iple_a @ X4 @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X4))))))). % antimonoE
thf(fact_135_antimonoE, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a, X4 : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((order_1710851741iple_a @ F) => ((ord_le1221261669iple_a @ X4 @ Y4) => (ord_le1221261669iple_a @ (F @ Y4) @ (F @ X4))))))). % antimonoE
thf(fact_136_antimonoI, axiom,
    ((![F : nat > nat]: ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ Y) @ (F @ X2)))) => (order_1631207636at_nat @ F))))). % antimonoI
thf(fact_137_antimonoI, axiom,
    ((![F : nat > set_Ho137910533iple_a]: ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ Y) @ (F @ X2)))) => (order_1673421321iple_a @ F))))). % antimonoI
thf(fact_138_antimonoI, axiom,
    ((![F : set_Ho137910533iple_a > nat]: ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_less_eq_nat @ (F @ Y) @ (F @ X2)))) => (order_194881289_a_nat @ F))))). % antimonoI
thf(fact_139_antimonoI, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a]: ((![X2 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y) => (ord_le1221261669iple_a @ (F @ Y) @ (F @ X2)))) => (order_1710851741iple_a @ F))))). % antimonoI
thf(fact_140_antimono__def, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![X : nat]: (![Y5 : nat]: (((ord_less_eq_nat @ X @ Y5)) => ((ord_less_eq_nat @ (F2 @ Y5) @ (F2 @ X)))))))))). % antimono_def
thf(fact_141_antimono__def, axiom,
    ((order_1673421321iple_a = (^[F2 : nat > set_Ho137910533iple_a]: (![X : nat]: (![Y5 : nat]: (((ord_less_eq_nat @ X @ Y5)) => ((ord_le1221261669iple_a @ (F2 @ Y5) @ (F2 @ X)))))))))). % antimono_def
thf(fact_142_antimono__def, axiom,
    ((order_194881289_a_nat = (^[F2 : set_Ho137910533iple_a > nat]: (![X : set_Ho137910533iple_a]: (![Y5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X @ Y5)) => ((ord_less_eq_nat @ (F2 @ Y5) @ (F2 @ X)))))))))). % antimono_def
thf(fact_143_antimono__def, axiom,
    ((order_1710851741iple_a = (^[F2 : set_Ho137910533iple_a > set_Ho137910533iple_a]: (![X : set_Ho137910533iple_a]: (![Y5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X @ Y5)) => ((ord_le1221261669iple_a @ (F2 @ Y5) @ (F2 @ X)))))))))). % antimono_def
thf(fact_144_GreatestI__ex__nat, axiom,
    ((![P : nat > $o, B3 : nat]: ((?[X_1 : nat]: (P @ X_1)) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_ex_nat
thf(fact_145_Greatest__le__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (ord_less_eq_nat @ K @ (order_Greatest_nat @ P))))))). % Greatest_le_nat
thf(fact_146_GreatestI__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_nat
thf(fact_147_Collect__empty__eq__bot, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (P = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_148_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_149_greaterThan__subset__iff, axiom,
    ((![X4 : nat, Y4 : nat]: ((ord_less_eq_set_nat @ (set_or1965240170an_nat @ X4) @ (set_or1965240170an_nat @ Y4)) = (ord_less_eq_nat @ Y4 @ X4))))). % greaterThan_subset_iff
thf(fact_150_verit__la__disequality, axiom,
    ((![A2 : nat, B3 : nat]: ((A2 = B3) | ((~ ((ord_less_eq_nat @ A2 @ B3))) | (~ ((ord_less_eq_nat @ B3 @ A2)))))))). % verit_la_disequality
thf(fact_151_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_152_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_153_lessThan__subset__iff, axiom,
    ((![X4 : nat, Y4 : nat]: ((ord_less_eq_set_nat @ (set_ord_lessThan_nat @ X4) @ (set_ord_lessThan_nat @ Y4)) = (ord_less_eq_nat @ X4 @ Y4))))). % lessThan_subset_iff
thf(fact_154_greaterThanLessThan__empty, axiom,
    ((![L : nat, K : nat]: ((ord_less_eq_nat @ L @ K) => ((set_or1544565540an_nat @ K @ L) = bot_bot_set_nat))))). % greaterThanLessThan_empty
thf(fact_155_greaterThanLessThan__empty, axiom,
    ((![L : set_Ho137910533iple_a, K : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ L @ K) => ((set_or769492057iple_a @ K @ L) = bot_bo922500559iple_a))))). % greaterThanLessThan_empty
thf(fact_156_Int__subset__iff, axiom,
    ((![C2 : set_Ho137910533iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C2 @ (inf_in1336607127iple_a @ A @ B)) = (((ord_le1221261669iple_a @ C2 @ A)) & ((ord_le1221261669iple_a @ C2 @ B))))))). % Int_subset_iff
thf(fact_157_Int__emptyI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ A) => (~ ((member1332298086iple_a @ X2 @ B))))) => ((inf_in1336607127iple_a @ A @ B) = bot_bo1298296729iple_a))))). % Int_emptyI
thf(fact_158_disjoint__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((inf_in1336607127iple_a @ A @ B) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A)) => ((~ ((member1332298086iple_a @ X @ B)))))))))). % disjoint_iff
thf(fact_159_Int__empty__left, axiom,
    ((![B : set_Ho137910533iple_a]: ((inf_in1336607127iple_a @ bot_bo1298296729iple_a @ B) = bot_bo1298296729iple_a)))). % Int_empty_left
thf(fact_160_Int__empty__right, axiom,
    ((![A : set_Ho137910533iple_a]: ((inf_in1336607127iple_a @ A @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % Int_empty_right
thf(fact_161_disjoint__iff__not__equal, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((inf_in1336607127iple_a @ A @ B) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A)) => ((![Y5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y5 @ B)) => ((~ ((X = Y5))))))))))))). % disjoint_iff_not_equal

% Conjectures (5)
thf(conj_0, hypothesis,
    ((hoare_129598474rivs_a @ g @ tsa))).
thf(conj_1, hypothesis,
    ((![N5 : nat]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ g) => (hoare_1926814542alid_a @ N5 @ X2))) => (![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ tsa) => (hoare_1926814542alid_a @ N5 @ X3))))))).
thf(conj_2, hypothesis,
    ((hoare_129598474rivs_a @ ga @ g))).
thf(conj_3, hypothesis,
    ((![N5 : nat]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ ga) => (hoare_1926814542alid_a @ N5 @ X2))) => (![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ g) => (hoare_1926814542alid_a @ N5 @ X3))))))).
thf(conj_4, conjecture,
    ((![N4 : nat]: ((?[X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ ga) & (~ ((hoare_1926814542alid_a @ N4 @ X3))))) | (![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ tsa) => (hoare_1926814542alid_a @ N4 @ X2))))))).
