% 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_498__3255334_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:22.914

% Could-be-implicit typings (2)
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).

% Explicit typings (14)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    minus_1852999390iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    uminus922456654iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a).
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 > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1048771374iple_a : ($o > set_Ho137910533iple_a) > ($o > set_Ho137910533iple_a) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_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_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > 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_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
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_t, type,
    t : hoare_1678595023iple_a).

% Relevant facts (111)
thf(fact_0_singleton__insert__inj__eq, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A @ A2)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_1_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_2_insert__subset, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ A2) @ B2) = (((member1332298086iple_a @ X @ B2)) & ((ord_le1221261669iple_a @ A2 @ B2))))))). % insert_subset
thf(fact_3_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_4_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_5_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_6_subset__singletonD, axiom,
    ((![A2 : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) => ((A2 = bot_bo1298296729iple_a) | (A2 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_7_subset__singleton__iff, axiom,
    ((![X2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (((X2 = bot_bo1298296729iple_a)) | ((X2 = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_8_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B2))) => (A = B)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertCI
thf(fact_9_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) = (((A = B)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_10_insert__absorb2, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ X @ A2)) = (insert1477804543iple_a @ X @ A2))))). % insert_absorb2
thf(fact_11_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A2) => (member1332298086iple_a @ X3 @ B2))) => (ord_le1221261669iple_a @ A2 @ B2))))). % subsetI
thf(fact_12_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X4 : hoare_1678595023iple_a]: (~ ((P @ X4)))))))). % empty_Collect_eq
thf(fact_13_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X4 : hoare_1678595023iple_a]: (~ ((P @ X4)))))))). % Collect_empty_eq
thf(fact_14_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X4 @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_15_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_16_subset__antisym, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_17_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X4 : hoare_1678595023iple_a]: (member1332298086iple_a @ X4 @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_18_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_19_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_20_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_21_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X4 : hoare_1678595023iple_a]: (((P @ X4)) => ((Q @ X4)))))))). % Collect_mono_iff
thf(fact_22_set__eq__subset, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_23_subset__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ C2) => (ord_le1221261669iple_a @ A2 @ C2)))))). % subset_trans
thf(fact_24_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a]: ((P @ X3) => (Q @ X3))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_25_subset__refl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % subset_refl
thf(fact_26_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![T : hoare_1678595023iple_a]: (((member1332298086iple_a @ T @ A3)) => ((member1332298086iple_a @ T @ B3))))))))). % subset_iff
thf(fact_27_equalityD2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ B2 @ A2))))). % equalityD2
thf(fact_28_equalityD1, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ A2 @ B2))))). % equalityD1
thf(fact_29_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X4 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X4 @ A3)) => ((member1332298086iple_a @ X4 @ B3))))))))). % subset_eq
thf(fact_30_equalityE, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (~ (((ord_le1221261669iple_a @ A2 @ B2) => (~ ((ord_le1221261669iple_a @ B2 @ A2)))))))))). % equalityE
thf(fact_31_subsetD, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ C @ A2) => (member1332298086iple_a @ C @ B2)))))). % subsetD
thf(fact_32_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ X @ A2) => (member1332298086iple_a @ X @ B2)))))). % in_mono
thf(fact_33_mk__disjoint__insert, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => (?[B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ A @ B4)) & (~ ((member1332298086iple_a @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_34_insert__commute, axiom,
    ((![X : hoare_1678595023iple_a, Y3 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ Y3 @ A2)) = (insert1477804543iple_a @ Y3 @ (insert1477804543iple_a @ X @ A2)))))). % insert_commute
thf(fact_35_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B @ B2))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B @ C3))) & ((((~ ((member1332298086iple_a @ B @ C3)))) & ((((B2 = (insert1477804543iple_a @ A @ C3))) & ((~ ((member1332298086iple_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_36_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_37_insert__ident, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((~ ((member1332298086iple_a @ X @ B2))) => (((insert1477804543iple_a @ X @ A2) = (insert1477804543iple_a @ X @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_38_Set_Oset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A2) => (~ ((![B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X @ B4)) => (member1332298086iple_a @ X @ B4))))))))). % Set.set_insert
thf(fact_39_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B2) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertI2
thf(fact_40_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B2))))). % insertI1
thf(fact_41_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) => ((~ ((A = B))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_42_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) => (A = B))))). % singleton_inject
thf(fact_43_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_44_doubleton__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_45_mem__Collect__eq, axiom,
    ((![A : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A @ (collec1600235172iple_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_46_Collect__mem__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X4 : hoare_1678595023iple_a]: (member1332298086iple_a @ X4 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_47_singleton__iff, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B = A))))). % singleton_iff
thf(fact_48_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B = A))))). % singletonD
thf(fact_49_subset__insertI2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ B2)))))). % subset_insertI2
thf(fact_50_subset__insertI, axiom,
    ((![B2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B2 @ (insert1477804543iple_a @ A @ B2))))). % subset_insertI
thf(fact_51_subset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X @ B2)) = (ord_le1221261669iple_a @ A2 @ B2)))))). % subset_insert
thf(fact_52_insert__mono, axiom,
    ((![C2 : set_Ho137910533iple_a, D2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C2 @ D2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A @ C2) @ (insert1477804543iple_a @ A @ D2)))))). % insert_mono
thf(fact_53_the__elem__eq, axiom,
    ((![X : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) = X)))). % the_elem_eq
thf(fact_54_order__refl, axiom,
    ((![X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ X)))). % order_refl
thf(fact_55_is__singletonI, axiom,
    ((![X : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_56_insert__subsetI, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, X2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A2) => ((ord_le1221261669iple_a @ X2 @ A2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ X2) @ A2)))))). % insert_subsetI
thf(fact_57_subset__emptyI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X3 @ A2)))) => (ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_58_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_59_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_60_bot_Oextremum, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % bot.extremum
thf(fact_61_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_62_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_63_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = (insert1477804543iple_a @ (the_el434698138iple_a @ A3) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_64_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X3 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A2) => ((member1332298086iple_a @ Y @ A2) => (X3 = Y)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_65_order__subst1, axiom,
    ((![A : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X3 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y) => (ord_le1221261669iple_a @ (F @ X3) @ (F @ Y)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % order_subst1
thf(fact_66_order__subst2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ (F @ B) @ C) => ((![X3 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y) => (ord_le1221261669iple_a @ (F @ X3) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % order_subst2
thf(fact_67_ord__eq__le__subst, axiom,
    ((![A : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A = (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X3 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y) => (ord_le1221261669iple_a @ (F @ X3) @ (F @ Y)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_68_ord__le__eq__subst, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => (((F @ B) = C) => ((![X3 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y) => (ord_le1221261669iple_a @ (F @ X3) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_69_eq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[X4 : set_Ho137910533iple_a]: (^[Y4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X4 @ Y4)) & ((ord_le1221261669iple_a @ Y4 @ X4)))))))). % eq_iff
thf(fact_70_antisym, axiom,
    ((![X : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y3) => ((ord_le1221261669iple_a @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_71_eq__refl, axiom,
    ((![X : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((X = Y3) => (ord_le1221261669iple_a @ X @ Y3))))). % eq_refl
thf(fact_72_order_Otrans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ A @ C)))))). % order.trans
thf(fact_73_antisym__conv, axiom,
    ((![Y3 : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y3 @ X) => ((ord_le1221261669iple_a @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_74_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A4 @ B5)) & ((ord_le1221261669iple_a @ B5 @ A4)))))))). % order_class.order.eq_iff
thf(fact_75_ord__eq__le__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A = B) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ A @ C)))))). % ord_eq_le_trans
thf(fact_76_ord__le__eq__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((B = C) => (ord_le1221261669iple_a @ A @ C)))))). % ord_le_eq_trans
thf(fact_77_order__class_Oorder_Oantisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_78_order__trans, axiom,
    ((![X : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a, Z2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y3) => ((ord_le1221261669iple_a @ Y3 @ Z2) => (ord_le1221261669iple_a @ X @ Z2)))))). % order_trans
thf(fact_79_dual__order_Orefl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % dual_order.refl
thf(fact_80_dual__order_Otrans, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ A) => ((ord_le1221261669iple_a @ C @ B) => (ord_le1221261669iple_a @ C @ A)))))). % dual_order.trans
thf(fact_81_dual__order_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B5 @ A4)) & ((ord_le1221261669iple_a @ A4 @ B5)))))))). % dual_order.eq_iff
thf(fact_82_dual__order_Oantisym, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ A) => ((ord_le1221261669iple_a @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_83_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X3 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_84_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X4 : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_85_subset__Compl__singleton, axiom,
    ((![A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (uminus922456654iple_a @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a))) = (~ ((member1332298086iple_a @ B @ A2))))))). % subset_Compl_singleton
thf(fact_86_GreatestI2__order, axiom,
    ((![P : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P @ X) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X))) => ((![X3 : set_Ho137910533iple_a]: ((P @ X3) => ((![Y5 : set_Ho137910533iple_a]: ((P @ Y5) => (ord_le1221261669iple_a @ Y5 @ X3))) => (Q @ X3)))) => (Q @ (order_929906668iple_a @ P)))))))). % GreatestI2_order
thf(fact_87_Greatest__equality, axiom,
    ((![P : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a]: ((P @ X) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X))) => ((order_929906668iple_a @ P) = X)))))). % Greatest_equality
thf(fact_88_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X4 : hoare_1678595023iple_a]: (member1332298086iple_a @ X4 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_89_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_90_Compl__iff, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (uminus922456654iple_a @ A2)) = (~ ((member1332298086iple_a @ C @ A2))))))). % Compl_iff
thf(fact_91_ComplI, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ C @ A2))) => (member1332298086iple_a @ C @ (uminus922456654iple_a @ A2)))))). % ComplI
thf(fact_92_Compl__subset__Compl__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ A2) @ (uminus922456654iple_a @ B2)) = (ord_le1221261669iple_a @ B2 @ A2))))). % Compl_subset_Compl_iff
thf(fact_93_Compl__anti__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ B2) @ (uminus922456654iple_a @ A2)))))). % Compl_anti_mono
thf(fact_94_ComplD, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (uminus922456654iple_a @ A2)) => (~ ((member1332298086iple_a @ C @ A2))))))). % ComplD
thf(fact_95_subset__Compl__self__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (uminus922456654iple_a @ A2)) = (A2 = bot_bo1298296729iple_a))))). % subset_Compl_self_eq
thf(fact_96_compl__le__compl__iff, axiom,
    ((![X : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ (uminus922456654iple_a @ Y3)) = (ord_le1221261669iple_a @ Y3 @ X))))). % compl_le_compl_iff
thf(fact_97_compl__le__swap2, axiom,
    ((![Y3 : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ Y3) @ X) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ Y3))))). % compl_le_swap2
thf(fact_98_compl__le__swap1, axiom,
    ((![Y3 : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y3 @ (uminus922456654iple_a @ X)) => (ord_le1221261669iple_a @ X @ (uminus922456654iple_a @ Y3)))))). % compl_le_swap1
thf(fact_99_compl__mono, axiom,
    ((![X : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y3) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ Y3) @ (uminus922456654iple_a @ X)))))). % compl_mono
thf(fact_100_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_101_Compl__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((uminus922456654iple_a @ (insert1477804543iple_a @ X @ A2)) = (minus_1852999390iple_a @ (uminus922456654iple_a @ A2) @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))))). % Compl_insert
thf(fact_102_Diff__iff, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A2 @ B2)) = (((member1332298086iple_a @ C @ A2)) & ((~ ((member1332298086iple_a @ C @ B2))))))))). % Diff_iff
thf(fact_103_DiffI, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ A2) => ((~ ((member1332298086iple_a @ C @ B2))) => (member1332298086iple_a @ C @ (minus_1852999390iple_a @ A2 @ B2))))))). % DiffI
thf(fact_104_Diff__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Diff_empty
thf(fact_105_empty__Diff, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A2) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_106_Diff__cancel, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ A2) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_107_Diff__insert0, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ X @ B2)) = (minus_1852999390iple_a @ A2 @ B2)))))). % Diff_insert0
thf(fact_108_insert__Diff1, axiom,
    ((![X : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ B2) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X @ A2) @ B2) = (minus_1852999390iple_a @ A2 @ B2)))))). % insert_Diff1
thf(fact_109_Diff__eq__empty__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: (((minus_1852999390iple_a @ A2 @ B2) = bot_bo1298296729iple_a) = (ord_le1221261669iple_a @ A2 @ B2))))). % Diff_eq_empty_iff
thf(fact_110_insert__Diff__single, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ A @ (minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) = (insert1477804543iple_a @ A @ A2))))). % insert_Diff_single

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_le1221261669iple_a @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a) @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a)))).
