% 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_282__3252282_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:25.494

% Could-be-implicit typings (4)
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_I_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J_J, type,
    set_Ho1594419232le_a_o : $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).

% Explicit typings (19)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    comple1314025773le_a_o : set_Ho1594419232le_a_o > hoare_1678595023iple_a > $o).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    comple1228397400iple_a : set_se152467259iple_a > set_Ho137910533iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001tf__a, type,
    hoare_1775499016lids_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
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_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_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_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Obind_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    bind_H209762450iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > set_Ho137910533iple_a) > set_Ho137910533iple_a).
thf(sy_c_Set_Odisjnt_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    disjnt925686856iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Set_Oimage_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    image_122147803iple_a : ((hoare_1678595023iple_a > $o) > set_Ho137910533iple_a) > set_Ho1594419232le_a_o > set_se152467259iple_a).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_1577193952iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a) > set_Ho137910533iple_a > 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_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    pairwi531237284iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a > $o) > set_Ho137910533iple_a > $o).
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__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).

% Relevant facts (72)
thf(fact_0_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_1_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_2_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_3_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_4_emptyE, axiom,
    ((![A2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A2 @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_5_equals0D, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((A = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A2 @ A))))))). % equals0D
thf(fact_6_equals0I, axiom,
    ((![A : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A)))) => (A = bot_bo1298296729iple_a))))). % equals0I
thf(fact_7_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_8_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_9_Sup__empty, axiom,
    (((comple1228397400iple_a @ bot_bo922500559iple_a) = bot_bo1298296729iple_a))). % Sup_empty
thf(fact_10_Sup__bot__conv_I1_J, axiom,
    ((![A : set_se152467259iple_a]: (((comple1228397400iple_a @ A) = bot_bo1298296729iple_a) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A)) => ((X = bot_bo1298296729iple_a)))))))). % Sup_bot_conv(1)
thf(fact_11_Sup__bot__conv_I2_J, axiom,
    ((![A : set_se152467259iple_a]: ((bot_bo1298296729iple_a = (comple1228397400iple_a @ A)) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A)) => ((X = bot_bo1298296729iple_a)))))))). % Sup_bot_conv(2)
thf(fact_12_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_13_empty__Union__conv, axiom,
    ((![A : set_se152467259iple_a]: ((bot_bo1298296729iple_a = (comple1228397400iple_a @ A)) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A)) => ((X = bot_bo1298296729iple_a)))))))). % empty_Union_conv
thf(fact_14_Union__empty__conv, axiom,
    ((![A : set_se152467259iple_a]: (((comple1228397400iple_a @ A) = bot_bo1298296729iple_a) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A)) => ((X = bot_bo1298296729iple_a)))))))). % Union_empty_conv
thf(fact_15_Union__empty, axiom,
    (((comple1228397400iple_a @ bot_bo922500559iple_a) = bot_bo1298296729iple_a))). % Union_empty
thf(fact_16_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_17_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_18_is__singletonI_H, axiom,
    ((![A : set_Ho137910533iple_a]: ((~ ((A = bot_bo1298296729iple_a))) => ((![X2 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ A) => ((member1332298086iple_a @ Y @ A) => (X2 = Y)))) => (is_sin1784037339iple_a @ A)))))). % is_singletonI'
thf(fact_19_empty__bind, axiom,
    ((![F : hoare_1678595023iple_a > set_Ho137910533iple_a]: ((bind_H209762450iple_a @ bot_bo1298296729iple_a @ F) = bot_bo1298296729iple_a)))). % empty_bind
thf(fact_20_disjnt__self__iff__empty, axiom,
    ((![S : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ S @ S) = (S = bot_bo1298296729iple_a))))). % disjnt_self_iff_empty
thf(fact_21_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_22_image__eqI, axiom,
    ((![B : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((B = (F @ X3)) => ((member1332298086iple_a @ X3 @ A) => (member1332298086iple_a @ B @ (image_1577193952iple_a @ F @ A))))))). % image_eqI
thf(fact_23_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_24_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_25_insertCI, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A2 @ B2))) => (A2 = B)) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B @ B2)))))). % insertCI
thf(fact_26_insert__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B @ A)) = (((A2 = B)) | ((member1332298086iple_a @ A2 @ A))))))). % insert_iff
thf(fact_27_insert__subset, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X3 @ A) @ B2) = (((member1332298086iple_a @ X3 @ B2)) & ((ord_le1221261669iple_a @ A @ B2))))))). % insert_subset
thf(fact_28_UnionI, axiom,
    ((![X4 : set_Ho137910533iple_a, C2 : set_se152467259iple_a, A : hoare_1678595023iple_a]: ((member521824924iple_a @ X4 @ C2) => ((member1332298086iple_a @ A @ X4) => (member1332298086iple_a @ A @ (comple1228397400iple_a @ C2))))))). % UnionI
thf(fact_29_Union__iff, axiom,
    ((![A : hoare_1678595023iple_a, C2 : set_se152467259iple_a]: ((member1332298086iple_a @ A @ (comple1228397400iple_a @ C2)) = (?[X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ C2)) & ((member1332298086iple_a @ A @ X)))))))). % Union_iff
thf(fact_30_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((image_1577193952iple_a @ F @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_31_mem__Collect__eq, axiom,
    ((![A2 : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A2 @ (collec1600235172iple_a @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_32_Collect__mem__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_33_Collect__cong, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X2 : hoare_1678595023iple_a]: ((P @ X2) = (Q @ X2))) => ((collec1600235172iple_a @ P) = (collec1600235172iple_a @ Q)))))). % Collect_cong
thf(fact_34_empty__is__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (image_1577193952iple_a @ F @ A)) = (A = bot_bo1298296729iple_a))))). % empty_is_image
thf(fact_35_image__is__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (((image_1577193952iple_a @ F @ A) = bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % image_is_empty
thf(fact_36_singleton__insert__inj__eq_H, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (((A2 = B)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_37_singleton__insert__inj__eq, axiom,
    ((![B : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A2 @ A)) = (((A2 = B)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_38_singletonI, axiom,
    ((![A2 : hoare_1678595023iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_39_disjnt__insert1, axiom,
    ((![A2 : hoare_1678595023iple_a, X4 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ (insert1477804543iple_a @ A2 @ X4) @ Y2) = (((~ ((member1332298086iple_a @ A2 @ Y2)))) & ((disjnt925686856iple_a @ X4 @ Y2))))))). % disjnt_insert1
thf(fact_40_disjnt__insert2, axiom,
    ((![Y2 : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a, X4 : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ Y2 @ (insert1477804543iple_a @ A2 @ X4)) = (((~ ((member1332298086iple_a @ A2 @ Y2)))) & ((disjnt925686856iple_a @ Y2 @ X4))))))). % disjnt_insert2
thf(fact_41_is__singletonI, axiom,
    ((![X3 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_42_imageI, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A) => (member1332298086iple_a @ (F @ X3) @ (image_1577193952iple_a @ F @ A)))))). % imageI
thf(fact_43_insertE, axiom,
    ((![A2 : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B @ A)) => ((~ ((A2 = B))) => (member1332298086iple_a @ A2 @ A)))))). % insertE
thf(fact_44_insertI1, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ B2))))). % insertI1
thf(fact_45_insertI2, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((member1332298086iple_a @ A2 @ B2) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B @ B2)))))). % insertI2
thf(fact_46_pairwiseD, axiom,
    ((![R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, S : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a, Y3 : hoare_1678595023iple_a]: ((pairwi531237284iple_a @ R @ S) => ((member1332298086iple_a @ X3 @ S) => ((member1332298086iple_a @ Y3 @ S) => ((~ ((X3 = Y3))) => (R @ X3 @ Y3)))))))). % pairwiseD
thf(fact_47_pairwiseI, axiom,
    ((![S : set_Ho137910533iple_a, R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: ((![X2 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ S) => ((member1332298086iple_a @ Y @ S) => ((~ ((X2 = Y))) => (R @ X2 @ Y))))) => (pairwi531237284iple_a @ R @ S))))). % pairwiseI
thf(fact_48_disjnt__iff, axiom,
    ((disjnt925686856iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (~ ((((member1332298086iple_a @ X @ A3)) & ((member1332298086iple_a @ X @ B3))))))))))). % disjnt_iff
thf(fact_49_Set_Oset__insert, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X3 @ A) => (~ ((![B4 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ X3 @ B4)) => (member1332298086iple_a @ X3 @ B4))))))))). % Set.set_insert
thf(fact_50_insert__ident, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((~ ((member1332298086iple_a @ X3 @ B2))) => (((insert1477804543iple_a @ X3 @ A) = (insert1477804543iple_a @ X3 @ B2)) = (A = B2))))))). % insert_ident
thf(fact_51_image__subsetI, axiom,
    ((![A : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X2 @ A) => (member1332298086iple_a @ (F @ X2) @ B2))) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ A) @ B2))))). % image_subsetI
thf(fact_52_insert__absorb, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => ((insert1477804543iple_a @ A2 @ A) = A))))). % insert_absorb
thf(fact_53_insert__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A2 @ A))) => ((~ ((member1332298086iple_a @ B @ B2))) => (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B @ B2)) = (((((A2 = B)) => ((A = B2)))) & ((((~ ((A2 = B)))) => ((?[C3 : set_Ho137910533iple_a]: (((A = (insert1477804543iple_a @ B @ C3))) & ((((~ ((member1332298086iple_a @ B @ C3)))) & ((((B2 = (insert1477804543iple_a @ A2 @ C3))) & ((~ ((member1332298086iple_a @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_54_rev__image__eqI, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A) => ((B = (F @ X3)) => (member1332298086iple_a @ B @ (image_1577193952iple_a @ F @ A))))))). % rev_image_eqI
thf(fact_55_subset__insert, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ B2)) = (ord_le1221261669iple_a @ A @ B2)))))). % subset_insert
thf(fact_56_Sup__SUP__eq, axiom,
    ((comple1314025773le_a_o = (^[S2 : set_Ho1594419232le_a_o]: (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ (comple1228397400iple_a @ (image_122147803iple_a @ collec1600235172iple_a @ S2)))))))). % Sup_SUP_eq
thf(fact_57_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X3 : hoare_1678595023iple_a, S3 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X3 @ S3)) = (((![Y4 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y4 @ S3)) & ((~ ((Y4 = X3)))))) => ((((R2 @ X3 @ Y4)) & ((R2 @ Y4 @ X3))))))) & ((pairwi531237284iple_a @ R2 @ S3))))))). % pairwise_insert
thf(fact_58_subset__singletonD, axiom,
    ((![A : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) => ((A = bot_bo1298296729iple_a) | (A = (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_59_mk__disjoint__insert, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => (?[B4 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ A2 @ B4)) & (~ ((member1332298086iple_a @ A2 @ B4))))))))). % mk_disjoint_insert
thf(fact_60_UnionE, axiom,
    ((![A : hoare_1678595023iple_a, C2 : set_se152467259iple_a]: ((member1332298086iple_a @ A @ (comple1228397400iple_a @ C2)) => (~ ((![X5 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ X5) => (~ ((member521824924iple_a @ X5 @ C2))))))))))). % UnionE
thf(fact_61_subset__singleton__iff, axiom,
    ((![X4 : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X4 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (((X4 = bot_bo1298296729iple_a)) | ((X4 = (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_62_pairwise__singleton, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (pairwi531237284iple_a @ P @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % pairwise_singleton
thf(fact_63_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_64_is__singletonE, axiom,
    ((![A : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A) => (~ ((![X2 : hoare_1678595023iple_a]: (~ ((A = (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_65_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) => (B = A2))))). % singletonD
thf(fact_66_singleton__iff, axiom,
    ((![B : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (B = A2))))). % singleton_iff
thf(fact_67_doubleton__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A2 = C)) & ((B = D)))) | ((((A2 = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_68_insert__not__empty, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A2 @ A) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_69_singleton__inject, axiom,
    ((![A2 : hoare_1678595023iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) => (A2 = B))))). % singleton_inject
thf(fact_70_bot_Oextremum, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % bot.extremum
thf(fact_71_bot_Oextremum__unique, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % bot.extremum_unique

% Conjectures (1)
thf(conj_0, conjecture,
    ((hoare_1775499016lids_a @ ga @ bot_bo1298296729iple_a))).
