% 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_177__3251164_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:31.531

% Could-be-implicit typings (7)
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__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (28)
thf(sy_c_Com_Ocom_OBODY, type,
    body : pname > com).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Com__Opname_M_Eo_J, type,
    bot_bot_pname_o : pname > $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__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
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_It__Com__Opname_M_Eo_J, type,
    ord_less_eq_pname_o : (pname > $o) > (pname > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    ord_le1593108832le_a_o : (hoare_1678595023iple_a > $o) > (hoare_1678595023iple_a > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Com__Opname_J, type,
    ord_le865024672_pname : set_pname > set_pname > $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_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
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_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_2025567968iple_a : (pname > hoare_1678595023iple_a) > set_pname > set_Ho137910533iple_a).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Com__Opname, type,
    image_243573216_pname : (hoare_1678595023iple_a > pname) > set_Ho137910533iple_a > set_pname).
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__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
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__singleton_001t__Com__Opname, type,
    is_singleton_pname : set_pname > $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__Com__Opname, type,
    the_elem_pname : set_pname > pname).
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__Com__Opname, type,
    member_pname : pname > set_pname > $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_v_P, type,
    p : pname > a > state > $o).
thf(sy_v_Procs, type,
    procs : set_pname).
thf(sy_v_Q, type,
    q : pname > a > state > $o).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (249)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : a > state > $o, X2 : com, X3 : a > state > $o, Y1 : a > state > $o, Y2 : com, Y3 : a > state > $o]: (((hoare_719046530iple_a @ X1 @ X2 @ X3) = (hoare_719046530iple_a @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_triple_Oinduct, axiom,
    ((![P : hoare_1678595023iple_a > $o, Triple : hoare_1678595023iple_a]: ((![X1a : a > state > $o, X2a : com, X3a : a > state > $o]: (P @ (hoare_719046530iple_a @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_2_triple_Oexhaust, axiom,
    ((![Y : hoare_1678595023iple_a]: (~ ((![X12 : a > state > $o, X22 : com, X32 : a > state > $o]: (~ ((Y = (hoare_719046530iple_a @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_3_singleton__insert__inj__eq, axiom,
    ((![B : pname, A : pname, A2 : set_pname]: (((insert_pname @ B @ bot_bot_set_pname) = (insert_pname @ A @ A2)) = (((A = B)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq
thf(fact_4_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_5_singleton__insert__inj__eq_H, axiom,
    ((![A : pname, A2 : set_pname, B : pname]: (((insert_pname @ A @ A2) = (insert_pname @ B @ bot_bot_set_pname)) = (((A = B)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq'
thf(fact_6_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_7_singleton__conv, axiom,
    ((![A : pname]: ((collect_pname @ (^[X : pname]: (X = A))) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv
thf(fact_8_singleton__conv, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (X = A))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv
thf(fact_9_singleton__conv2, axiom,
    ((![A : pname]: ((collect_pname @ ((^[Y4 : pname]: (^[Z : pname]: (Y4 = Z))) @ A)) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_10_singleton__conv2, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ ((^[Y4 : hoare_1678595023iple_a]: (^[Z : hoare_1678595023iple_a]: (Y4 = Z))) @ A)) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv2
thf(fact_11_insert__subset, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ (insert_pname @ X4 @ A2) @ B2) = (((member_pname @ X4 @ B2)) & ((ord_le865024672_pname @ A2 @ B2))))))). % insert_subset
thf(fact_12_insert__subset, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B2) = (((member1332298086iple_a @ X4 @ B2)) & ((ord_le1221261669iple_a @ A2 @ B2))))))). % insert_subset
thf(fact_13_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_14_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_15_subset__empty, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % subset_empty
thf(fact_16_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_17_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_18_empty__subsetI, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A2)))). % empty_subsetI
thf(fact_19_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (insert1477804543iple_a @ A @ B2)) = (insert1477804543iple_a @ (F @ A) @ (image_1577193952iple_a @ F @ B2)))))). % image_insert
thf(fact_20_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > pname, A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((image_243573216_pname @ F @ (insert1477804543iple_a @ A @ B2)) = (insert_pname @ (F @ A) @ (image_243573216_pname @ F @ B2)))))). % image_insert
thf(fact_21_image__insert, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : pname, B2 : set_pname]: ((image_2025567968iple_a @ F @ (insert_pname @ A @ B2)) = (insert1477804543iple_a @ (F @ A) @ (image_2025567968iple_a @ F @ B2)))))). % image_insert
thf(fact_22_image__insert, axiom,
    ((![F : pname > pname, A : pname, B2 : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A @ B2)) = (insert_pname @ (F @ A) @ (image_pname_pname @ F @ B2)))))). % image_insert
thf(fact_23_insert__image, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((insert1477804543iple_a @ (F @ X4) @ (image_2025567968iple_a @ F @ A2)) = (image_2025567968iple_a @ F @ A2)))))). % insert_image
thf(fact_24_insert__image, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X4 @ A2) => ((insert_pname @ (F @ X4) @ (image_pname_pname @ F @ A2)) = (image_pname_pname @ F @ A2)))))). % insert_image
thf(fact_25_insert__image, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => ((insert1477804543iple_a @ (F @ X4) @ (image_1577193952iple_a @ F @ A2)) = (image_1577193952iple_a @ F @ A2)))))). % insert_image
thf(fact_26_insert__image, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > pname]: ((member1332298086iple_a @ X4 @ A2) => ((insert_pname @ (F @ X4) @ (image_243573216_pname @ F @ A2)) = (image_243573216_pname @ F @ A2)))))). % insert_image
thf(fact_27_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((image_1577193952iple_a @ F @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_28_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > pname]: ((image_243573216_pname @ F @ bot_bo1298296729iple_a) = bot_bot_set_pname)))). % image_empty
thf(fact_29_image__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ F @ bot_bot_set_pname) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_30_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_31_empty__is__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (image_1577193952iple_a @ F @ A2)) = (A2 = bot_bo1298296729iple_a))))). % empty_is_image
thf(fact_32_empty__is__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((bot_bo1298296729iple_a = (image_2025567968iple_a @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_33_empty__is__image, axiom,
    ((![F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a]: ((bot_bot_set_pname = (image_243573216_pname @ F @ A2)) = (A2 = bot_bo1298296729iple_a))))). % empty_is_image
thf(fact_34_empty__is__image, axiom,
    ((![F : pname > pname, A2 : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_35_image__eqI, axiom,
    ((![B : pname, F : pname > pname, X4 : pname, A2 : set_pname]: ((B = (F @ X4)) => ((member_pname @ X4 @ A2) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_36_image__eqI, axiom,
    ((![B : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X4 : pname, A2 : set_pname]: ((B = (F @ X4)) => ((member_pname @ X4 @ A2) => (member1332298086iple_a @ B @ (image_2025567968iple_a @ F @ A2))))))). % image_eqI
thf(fact_37_image__eqI, axiom,
    ((![B : pname, F : hoare_1678595023iple_a > pname, X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((B = (F @ X4)) => ((member1332298086iple_a @ X4 @ A2) => (member_pname @ B @ (image_243573216_pname @ F @ A2))))))). % image_eqI
thf(fact_38_image__eqI, axiom,
    ((![B : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((B = (F @ X4)) => ((member1332298086iple_a @ X4 @ A2) => (member1332298086iple_a @ B @ (image_1577193952iple_a @ F @ A2))))))). % image_eqI
thf(fact_39_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_40_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X : pname]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_41_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_42_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X : pname]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_43_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_44_all__not__in__conv, axiom,
    ((![A2 : set_pname]: ((![X : pname]: (~ ((member_pname @ X @ A2)))) = (A2 = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_45_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_46_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_47_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_48_subset__antisym, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ A2 @ B2) => ((ord_le865024672_pname @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_49_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => (member1332298086iple_a @ X5 @ B2))) => (ord_le1221261669iple_a @ A2 @ B2))))). % subsetI
thf(fact_50_subsetI, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((![X5 : pname]: ((member_pname @ X5 @ A2) => (member_pname @ X5 @ B2))) => (ord_le865024672_pname @ A2 @ B2))))). % subsetI
thf(fact_51_insert__absorb2, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ X4 @ A2)) = (insert1477804543iple_a @ X4 @ A2))))). % insert_absorb2
thf(fact_52_insert__absorb2, axiom,
    ((![X4 : pname, A2 : set_pname]: ((insert_pname @ X4 @ (insert_pname @ X4 @ A2)) = (insert_pname @ X4 @ A2))))). % insert_absorb2
thf(fact_53_insert__iff, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) = (((A = B)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_54_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_55_insertCI, axiom,
    ((![A : pname, B2 : set_pname, B : pname]: (((~ ((member_pname @ A @ B2))) => (A = B)) => (member_pname @ A @ (insert_pname @ B @ B2)))))). % insertCI
thf(fact_56_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_57_image__is__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((image_1577193952iple_a @ F @ A2) = bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % image_is_empty
thf(fact_58_image__is__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname]: (((image_2025567968iple_a @ F @ A2) = bot_bo1298296729iple_a) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_59_image__is__empty, axiom,
    ((![F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a]: (((image_243573216_pname @ F @ A2) = bot_bot_set_pname) = (A2 = bot_bo1298296729iple_a))))). % image_is_empty
thf(fact_60_image__is__empty, axiom,
    ((![F : pname > pname, A2 : set_pname]: (((image_pname_pname @ F @ A2) = bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_61_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B : pname, F : pname > pname]: ((member_pname @ X4 @ A2) => ((B = (F @ X4)) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_62_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((B = (F @ X4)) => (member1332298086iple_a @ B @ (image_2025567968iple_a @ F @ A2))))))). % rev_image_eqI
thf(fact_63_rev__image__eqI, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : pname, F : hoare_1678595023iple_a > pname]: ((member1332298086iple_a @ X4 @ A2) => ((B = (F @ X4)) => (member_pname @ B @ (image_243573216_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_64_rev__image__eqI, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => ((B = (F @ X4)) => (member1332298086iple_a @ B @ (image_1577193952iple_a @ F @ A2))))))). % rev_image_eqI
thf(fact_65_ball__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ (image_2025567968iple_a @ F @ A2)) => (P @ X5))) => (![X6 : pname]: ((member_pname @ X6 @ A2) => (P @ (F @ X6)))))))). % ball_imageD
thf(fact_66_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_1678595023iple_a, G : pname > hoare_1678595023iple_a]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F @ X5) = (G @ X5)))) => ((image_2025567968iple_a @ F @ M) = (image_2025567968iple_a @ G @ N))))))). % image_cong
thf(fact_67_bex__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P : hoare_1678595023iple_a > $o]: ((?[X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ (image_2025567968iple_a @ F @ A2)) & (P @ X6))) => (?[X5 : pname]: ((member_pname @ X5 @ A2) & (P @ (F @ X5)))))))). % bex_imageD
thf(fact_68_image__iff, axiom,
    ((![Z2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ Z2 @ (image_2025567968iple_a @ F @ A2)) = (?[X : pname]: (((member_pname @ X @ A2)) & ((Z2 = (F @ X))))))))). % image_iff
thf(fact_69_imageI, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X4 @ A2) => (member_pname @ (F @ X4) @ (image_pname_pname @ F @ A2)))))). % imageI
thf(fact_70_imageI, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => (member1332298086iple_a @ (F @ X4) @ (image_2025567968iple_a @ F @ A2)))))). % imageI
thf(fact_71_imageI, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > pname]: ((member1332298086iple_a @ X4 @ A2) => (member_pname @ (F @ X4) @ (image_243573216_pname @ F @ A2)))))). % imageI
thf(fact_72_imageI, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => (member1332298086iple_a @ (F @ X4) @ (image_1577193952iple_a @ F @ A2)))))). % imageI
thf(fact_73_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_74_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X : pname]: (member_pname @ X @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_75_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_76_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y5 : pname]: (~ ((member_pname @ Y5 @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_77_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_78_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_79_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_80_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_81_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_82_Collect__mono__iff, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)) = (![X : pname]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_83_set__eq__subset, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_84_set__eq__subset, axiom,
    (((^[Y4 : set_pname]: (^[Z : set_pname]: (Y4 = Z))) = (^[A3 : set_pname]: (^[B3 : set_pname]: (((ord_le865024672_pname @ A3 @ B3)) & ((ord_le865024672_pname @ B3 @ A3)))))))). % set_eq_subset
thf(fact_85_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_86_subset__trans, axiom,
    ((![A2 : set_pname, B2 : set_pname, C2 : set_pname]: ((ord_le865024672_pname @ A2 @ B2) => ((ord_le865024672_pname @ B2 @ C2) => (ord_le865024672_pname @ A2 @ C2)))))). % subset_trans
thf(fact_87_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((P @ X5) => (Q @ X5))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_88_Collect__mono, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X5 : pname]: ((P @ X5) => (Q @ X5))) => (ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_mono
thf(fact_89_subset__refl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % subset_refl
thf(fact_90_subset__refl, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ A2 @ A2)))). % subset_refl
thf(fact_91_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_92_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (![T : pname]: (((member_pname @ T @ A3)) => ((member_pname @ T @ B3))))))))). % subset_iff
thf(fact_93_equalityD2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ B2 @ A2))))). % equalityD2
thf(fact_94_equalityD2, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (ord_le865024672_pname @ B2 @ A2))))). % equalityD2
thf(fact_95_equalityD1, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ A2 @ B2))))). % equalityD1
thf(fact_96_equalityD1, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (ord_le865024672_pname @ A2 @ B2))))). % equalityD1
thf(fact_97_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_98_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_99_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_100_Collect__mem__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_101_Collect__cong, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((P @ X5) = (Q @ X5))) => ((collec1600235172iple_a @ P) = (collec1600235172iple_a @ Q)))))). % Collect_cong
thf(fact_102_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A3)) => ((member1332298086iple_a @ X @ B3))))))))). % subset_eq
thf(fact_103_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (![X : pname]: (((member_pname @ X @ A3)) => ((member_pname @ X @ B3))))))))). % subset_eq
thf(fact_104_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_105_equalityE, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (~ (((ord_le865024672_pname @ A2 @ B2) => (~ ((ord_le865024672_pname @ B2 @ A2)))))))))). % equalityE
thf(fact_106_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_107_subsetD, axiom,
    ((![A2 : set_pname, B2 : set_pname, C : pname]: ((ord_le865024672_pname @ A2 @ B2) => ((member_pname @ C @ A2) => (member_pname @ C @ B2)))))). % subsetD
thf(fact_108_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ X4 @ A2) => (member1332298086iple_a @ X4 @ B2)))))). % in_mono
thf(fact_109_in__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, X4 : pname]: ((ord_le865024672_pname @ A2 @ B2) => ((member_pname @ X4 @ A2) => (member_pname @ X4 @ B2)))))). % in_mono
thf(fact_110_mk__disjoint__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => (?[B4 : set_pname]: ((A2 = (insert_pname @ A @ B4)) & (~ ((member_pname @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_111_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_112_insert__commute, axiom,
    ((![X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ Y @ A2)) = (insert1477804543iple_a @ Y @ (insert1477804543iple_a @ X4 @ A2)))))). % insert_commute
thf(fact_113_insert__commute, axiom,
    ((![X4 : pname, Y : pname, A2 : set_pname]: ((insert_pname @ X4 @ (insert_pname @ Y @ A2)) = (insert_pname @ Y @ (insert_pname @ X4 @ A2)))))). % insert_commute
thf(fact_114_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B : pname, B2 : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B @ B2))) => (((insert_pname @ A @ A2) = (insert_pname @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_pname]: (((A2 = (insert_pname @ B @ C3))) & ((((~ ((member_pname @ B @ C3)))) & ((((B2 = (insert_pname @ A @ C3))) & ((~ ((member_pname @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_115_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_116_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_117_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_118_insert__ident, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : set_pname]: ((~ ((member_pname @ X4 @ A2))) => ((~ ((member_pname @ X4 @ B2))) => (((insert_pname @ X4 @ A2) = (insert_pname @ X4 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_119_insert__ident, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((~ ((member1332298086iple_a @ X4 @ B2))) => (((insert1477804543iple_a @ X4 @ A2) = (insert1477804543iple_a @ X4 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_120_Set_Oset__insert, axiom,
    ((![X4 : pname, A2 : set_pname]: ((member_pname @ X4 @ A2) => (~ ((![B4 : set_pname]: ((A2 = (insert_pname @ X4 @ B4)) => (member_pname @ X4 @ B4))))))))). % Set.set_insert
thf(fact_121_Set_Oset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ A2) => (~ ((![B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X4 @ B4)) => (member1332298086iple_a @ X4 @ B4))))))))). % Set.set_insert
thf(fact_122_insertI2, axiom,
    ((![A : pname, B2 : set_pname, B : pname]: ((member_pname @ A @ B2) => (member_pname @ A @ (insert_pname @ B @ B2)))))). % insertI2
thf(fact_123_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_124_insertI1, axiom,
    ((![A : pname, B2 : set_pname]: (member_pname @ A @ (insert_pname @ A @ B2))))). % insertI1
thf(fact_125_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B2))))). % insertI1
thf(fact_126_insertE, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) => ((~ ((A = B))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_127_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_128_Compr__image__eq, axiom,
    ((![F : pname > pname, A2 : set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F @ A2))) & ((P @ X))))) = (image_pname_pname @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_129_Compr__image__eq, axiom,
    ((![F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_243573216_pname @ F @ A2))) & ((P @ X))))) = (image_243573216_pname @ F @ (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_130_Compr__image__eq, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ (image_2025567968iple_a @ F @ A2))) & ((P @ X))))) = (image_2025567968iple_a @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_131_Compr__image__eq, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ (image_1577193952iple_a @ F @ A2))) & ((P @ X))))) = (image_1577193952iple_a @ F @ (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_132_image__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G : pname > hoare_1678595023iple_a, A2 : set_pname]: ((image_1577193952iple_a @ F @ (image_2025567968iple_a @ G @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G @ X))) @ A2))))). % image_image
thf(fact_133_image__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, G : pname > pname, A2 : set_pname]: ((image_2025567968iple_a @ F @ (image_pname_pname @ G @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G @ X))) @ A2))))). % image_image
thf(fact_134_imageE, axiom,
    ((![B : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ B @ (image_pname_pname @ F @ A2)) => (~ ((![X5 : pname]: ((B = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_135_imageE, axiom,
    ((![B : pname, F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a]: ((member_pname @ B @ (image_243573216_pname @ F @ A2)) => (~ ((![X5 : hoare_1678595023iple_a]: ((B = (F @ X5)) => (~ ((member1332298086iple_a @ X5 @ A2))))))))))). % imageE
thf(fact_136_imageE, axiom,
    ((![B : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ B @ (image_2025567968iple_a @ F @ A2)) => (~ ((![X5 : pname]: ((B = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_137_imageE, axiom,
    ((![B : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ B @ (image_1577193952iple_a @ F @ A2)) => (~ ((![X5 : hoare_1678595023iple_a]: ((B = (F @ X5)) => (~ ((member1332298086iple_a @ X5 @ A2))))))))))). % imageE
thf(fact_138_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_139_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X : pname]: $false))))). % empty_def
thf(fact_140_Collect__subset, axiom,
    ((![A2 : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A2)) & ((P @ X))))) @ A2)))). % Collect_subset
thf(fact_141_Collect__subset, axiom,
    ((![A2 : set_pname, P : pname > $o]: (ord_le865024672_pname @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P @ X))))) @ A2)))). % Collect_subset
thf(fact_142_insert__Collect, axiom,
    ((![A : pname, P : pname > $o]: ((insert_pname @ A @ (collect_pname @ P)) = (collect_pname @ (^[U : pname]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_143_insert__Collect, axiom,
    ((![A : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((insert1477804543iple_a @ A @ (collec1600235172iple_a @ P)) = (collec1600235172iple_a @ (^[U : hoare_1678595023iple_a]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_144_insert__compr, axiom,
    ((insert_pname = (^[A4 : pname]: (^[B3 : set_pname]: (collect_pname @ (^[X : pname]: (((X = A4)) | ((member_pname @ X @ B3)))))))))). % insert_compr
thf(fact_145_insert__compr, axiom,
    ((insert1477804543iple_a = (^[A4 : hoare_1678595023iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A4)) | ((member1332298086iple_a @ X @ B3)))))))))). % insert_compr
thf(fact_146_subset__image__iff, axiom,
    ((![B2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ (image_1577193952iple_a @ F @ A2)) = (?[AA : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ AA @ A2)) & ((B2 = (image_1577193952iple_a @ F @ AA))))))))). % subset_image_iff
thf(fact_147_subset__image__iff, axiom,
    ((![B2 : set_Ho137910533iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((ord_le1221261669iple_a @ B2 @ (image_2025567968iple_a @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B2 = (image_2025567968iple_a @ F @ AA))))))))). % subset_image_iff
thf(fact_148_subset__image__iff, axiom,
    ((![B2 : set_pname, F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a]: ((ord_le865024672_pname @ B2 @ (image_243573216_pname @ F @ A2)) = (?[AA : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ AA @ A2)) & ((B2 = (image_243573216_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_149_subset__image__iff, axiom,
    ((![B2 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B2 = (image_pname_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_150_image__subset__iff, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A2) @ B2) = (![X : pname]: (((member_pname @ X @ A2)) => ((member1332298086iple_a @ (F @ X) @ B2)))))))). % image_subset_iff
thf(fact_151_subset__imageE, axiom,
    ((![B2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ (image_1577193952iple_a @ F @ A2)) => (~ ((![C4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C4 @ A2) => (~ ((B2 = (image_1577193952iple_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_152_subset__imageE, axiom,
    ((![B2 : set_Ho137910533iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((ord_le1221261669iple_a @ B2 @ (image_2025567968iple_a @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B2 = (image_2025567968iple_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_153_subset__imageE, axiom,
    ((![B2 : set_pname, F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a]: ((ord_le865024672_pname @ B2 @ (image_243573216_pname @ F @ A2)) => (~ ((![C4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C4 @ A2) => (~ ((B2 = (image_243573216_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_154_subset__imageE, axiom,
    ((![B2 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B2 = (image_pname_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_155_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : pname]: ((member_pname @ X5 @ A2) => (member1332298086iple_a @ (F @ X5) @ B2))) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A2) @ B2))))). % image_subsetI
thf(fact_156_image__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => (member1332298086iple_a @ (F @ X5) @ B2))) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ A2) @ B2))))). % image_subsetI
thf(fact_157_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > pname, B2 : set_pname]: ((![X5 : pname]: ((member_pname @ X5 @ A2) => (member_pname @ (F @ X5) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F @ A2) @ B2))))). % image_subsetI
thf(fact_158_image__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > pname, B2 : set_pname]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => (member_pname @ (F @ X5) @ B2))) => (ord_le865024672_pname @ (image_243573216_pname @ F @ A2) @ B2))))). % image_subsetI
thf(fact_159_image__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ A2) @ (image_1577193952iple_a @ F @ B2)))))). % image_mono
thf(fact_160_image__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > pname]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le865024672_pname @ (image_243573216_pname @ F @ A2) @ (image_243573216_pname @ F @ B2)))))). % image_mono
thf(fact_161_image__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > hoare_1678595023iple_a]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A2) @ (image_2025567968iple_a @ F @ B2)))))). % image_mono
thf(fact_162_image__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > pname]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le865024672_pname @ (image_pname_pname @ F @ A2) @ (image_pname_pname @ F @ B2)))))). % image_mono
thf(fact_163_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_164_singleton__inject, axiom,
    ((![A : pname, B : pname]: (((insert_pname @ A @ bot_bot_set_pname) = (insert_pname @ B @ bot_bot_set_pname)) => (A = B))))). % singleton_inject
thf(fact_165_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_166_insert__not__empty, axiom,
    ((![A : pname, A2 : set_pname]: (~ (((insert_pname @ A @ A2) = bot_bot_set_pname)))))). % insert_not_empty
thf(fact_167_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_168_doubleton__eq__iff, axiom,
    ((![A : pname, B : pname, C : pname, D : pname]: (((insert_pname @ A @ (insert_pname @ B @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D @ bot_bot_set_pname))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_169_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_170_singleton__iff, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) = (B = A))))). % singleton_iff
thf(fact_171_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B = A))))). % singletonD
thf(fact_172_singletonD, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) => (B = A))))). % singletonD
thf(fact_173_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_174_subset__insertI2, axiom,
    ((![A2 : set_pname, B2 : set_pname, B : pname]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le865024672_pname @ A2 @ (insert_pname @ B @ B2)))))). % subset_insertI2
thf(fact_175_subset__insertI, axiom,
    ((![B2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B2 @ (insert1477804543iple_a @ A @ B2))))). % subset_insertI
thf(fact_176_subset__insertI, axiom,
    ((![B2 : set_pname, A : pname]: (ord_le865024672_pname @ B2 @ (insert_pname @ A @ B2))))). % subset_insertI
thf(fact_177_subset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X4 @ B2)) = (ord_le1221261669iple_a @ A2 @ B2)))))). % subset_insert
thf(fact_178_subset__insert, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : set_pname]: ((~ ((member_pname @ X4 @ A2))) => ((ord_le865024672_pname @ A2 @ (insert_pname @ X4 @ B2)) = (ord_le865024672_pname @ A2 @ B2)))))). % subset_insert
thf(fact_179_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_180_insert__mono, axiom,
    ((![C2 : set_pname, D2 : set_pname, A : pname]: ((ord_le865024672_pname @ C2 @ D2) => (ord_le865024672_pname @ (insert_pname @ A @ C2) @ (insert_pname @ A @ D2)))))). % insert_mono
thf(fact_181_Collect__conv__if2, axiom,
    ((![P : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if2
thf(fact_182_Collect__conv__if2, axiom,
    ((![P : pname > $o, A : pname]: (((P @ A) => ((collect_pname @ (^[X : pname]: (((A = X)) & ((P @ X))))) = (insert_pname @ A @ bot_bot_set_pname))) & ((~ ((P @ A))) => ((collect_pname @ (^[X : pname]: (((A = X)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if2
thf(fact_183_Collect__conv__if, axiom,
    ((![P : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if
thf(fact_184_Collect__conv__if, axiom,
    ((![P : pname > $o, A : pname]: (((P @ A) => ((collect_pname @ (^[X : pname]: (((X = A)) & ((P @ X))))) = (insert_pname @ A @ bot_bot_set_pname))) & ((~ ((P @ A))) => ((collect_pname @ (^[X : pname]: (((X = A)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if
thf(fact_185_subset__singleton__iff, axiom,
    ((![X7 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X7 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (((X7 = bot_bo1298296729iple_a)) | ((X7 = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_186_subset__singleton__iff, axiom,
    ((![X7 : set_pname, A : pname]: ((ord_le865024672_pname @ X7 @ (insert_pname @ A @ bot_bot_set_pname)) = (((X7 = bot_bot_set_pname)) | ((X7 = (insert_pname @ A @ bot_bot_set_pname)))))))). % subset_singleton_iff
thf(fact_187_subset__singletonD, axiom,
    ((![A2 : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) => ((A2 = bot_bo1298296729iple_a) | (A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_188_subset__singletonD, axiom,
    ((![A2 : set_pname, X4 : pname]: ((ord_le865024672_pname @ A2 @ (insert_pname @ X4 @ bot_bot_set_pname)) => ((A2 = bot_bot_set_pname) | (A2 = (insert_pname @ X4 @ bot_bot_set_pname))))))). % subset_singletonD
thf(fact_189_image__constant__conv, axiom,
    ((![A2 : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: (((A2 = bot_bo1298296729iple_a) => ((image_1577193952iple_a @ (^[X : hoare_1678595023iple_a]: C) @ A2) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((image_1577193952iple_a @ (^[X : hoare_1678595023iple_a]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a))))))). % image_constant_conv
thf(fact_190_image__constant__conv, axiom,
    ((![A2 : set_Ho137910533iple_a, C : pname]: (((A2 = bot_bo1298296729iple_a) => ((image_243573216_pname @ (^[X : hoare_1678595023iple_a]: C) @ A2) = bot_bot_set_pname)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((image_243573216_pname @ (^[X : hoare_1678595023iple_a]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_191_image__constant__conv, axiom,
    ((![A2 : set_pname, C : hoare_1678595023iple_a]: (((A2 = bot_bot_set_pname) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a))))))). % image_constant_conv
thf(fact_192_image__constant__conv, axiom,
    ((![A2 : set_pname, C : pname]: (((A2 = bot_bot_set_pname) => ((image_pname_pname @ (^[X : pname]: C) @ A2) = bot_bot_set_pname)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_pname_pname @ (^[X : pname]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_193_image__constant, axiom,
    ((![X4 : pname, A2 : set_pname, C : hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a)))))). % image_constant
thf(fact_194_image__constant, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => ((image_1577193952iple_a @ (^[X : hoare_1678595023iple_a]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a)))))). % image_constant
thf(fact_195_image__constant, axiom,
    ((![X4 : pname, A2 : set_pname, C : pname]: ((member_pname @ X4 @ A2) => ((image_pname_pname @ (^[X : pname]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname)))))). % image_constant
thf(fact_196_image__constant, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, C : pname]: ((member1332298086iple_a @ X4 @ A2) => ((image_243573216_pname @ (^[X : hoare_1678595023iple_a]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname)))))). % image_constant
thf(fact_197_com_Oinject_I6_J, axiom,
    ((![X72 : pname, Y7 : pname]: (((body @ X72) = (body @ Y7)) = (X72 = Y7))))). % com.inject(6)
thf(fact_198_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_199_the__elem__eq, axiom,
    ((![X4 : pname]: ((the_elem_pname @ (insert_pname @ X4 @ bot_bot_set_pname)) = X4)))). % the_elem_eq
thf(fact_200_order__refl, axiom,
    ((![X4 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X4 @ X4)))). % order_refl
thf(fact_201_order__refl, axiom,
    ((![X4 : set_pname]: (ord_le865024672_pname @ X4 @ X4)))). % order_refl
thf(fact_202_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : pname]: ((P @ X5) => (member1332298086iple_a @ (F @ X5) @ B2))) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_203_image__Collect__subsetI, axiom,
    ((![P : hoare_1678595023iple_a > $o, F : hoare_1678595023iple_a > hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: ((P @ X5) => (member1332298086iple_a @ (F @ X5) @ B2))) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ (collec1600235172iple_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_204_image__Collect__subsetI, axiom,
    ((![P : hoare_1678595023iple_a > $o, F : hoare_1678595023iple_a > pname, B2 : set_pname]: ((![X5 : hoare_1678595023iple_a]: ((P @ X5) => (member_pname @ (F @ X5) @ B2))) => (ord_le865024672_pname @ (image_243573216_pname @ F @ (collec1600235172iple_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_205_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_206_is__singletonI, axiom,
    ((![X4 : pname]: (is_singleton_pname @ (insert_pname @ X4 @ bot_bot_set_pname))))). % is_singletonI
thf(fact_207_insert__subsetI, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, X7 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ A2) => ((ord_le1221261669iple_a @ X7 @ A2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ X7) @ A2)))))). % insert_subsetI
thf(fact_208_insert__subsetI, axiom,
    ((![X4 : pname, A2 : set_pname, X7 : set_pname]: ((member_pname @ X4 @ A2) => ((ord_le865024672_pname @ X7 @ A2) => (ord_le865024672_pname @ (insert_pname @ X4 @ X7) @ A2)))))). % insert_subsetI
thf(fact_209_subset__emptyI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X5 @ A2)))) => (ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_210_subset__emptyI, axiom,
    ((![A2 : set_pname]: ((![X5 : pname]: (~ ((member_pname @ X5 @ A2)))) => (ord_le865024672_pname @ A2 @ bot_bot_set_pname))))). % subset_emptyI
thf(fact_211_all__subset__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, P : set_Ho137910533iple_a > $o]: ((![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ (image_1577193952iple_a @ F @ A2))) => ((P @ B3)))) = (![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ A2)) => ((P @ (image_1577193952iple_a @ F @ B3))))))))). % all_subset_image
thf(fact_212_all__subset__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P : set_Ho137910533iple_a > $o]: ((![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ (image_2025567968iple_a @ F @ A2))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A2)) => ((P @ (image_2025567968iple_a @ F @ B3))))))))). % all_subset_image
thf(fact_213_all__subset__image, axiom,
    ((![F : hoare_1678595023iple_a > pname, A2 : set_Ho137910533iple_a, P : set_pname > $o]: ((![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ (image_243573216_pname @ F @ A2))) => ((P @ B3)))) = (![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ A2)) => ((P @ (image_243573216_pname @ F @ B3))))))))). % all_subset_image
thf(fact_214_all__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A2)) => ((P @ (image_pname_pname @ F @ B3))))))))). % all_subset_image
thf(fact_215_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_216_bot_Oextremum__uniqueI, axiom,
    ((![A : set_pname]: ((ord_le865024672_pname @ A @ bot_bot_set_pname) => (A = bot_bot_set_pname))))). % bot.extremum_uniqueI
thf(fact_217_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_218_bot__set__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ bot_bot_pname_o)))). % bot_set_def
thf(fact_219_less__eq__set__def, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (ord_le1593108832le_a_o @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A3)) @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ B3)))))))). % less_eq_set_def
thf(fact_220_less__eq__set__def, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (ord_less_eq_pname_o @ (^[X : pname]: (member_pname @ X @ A3)) @ (^[X : pname]: (member_pname @ X @ B3)))))))). % less_eq_set_def
thf(fact_221_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_222_is__singleton__the__elem, axiom,
    ((is_singleton_pname = (^[A3 : set_pname]: (A3 = (insert_pname @ (the_elem_pname @ A3) @ bot_bot_set_pname)))))). % is_singleton_the_elem
thf(fact_223_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X5 : hoare_1678595023iple_a, Y5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y5 @ A2) => (X5 = Y5)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_224_is__singletonI_H, axiom,
    ((![A2 : set_pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![X5 : pname, Y5 : pname]: ((member_pname @ X5 @ A2) => ((member_pname @ Y5 @ A2) => (X5 = Y5)))) => (is_singleton_pname @ A2)))))). % is_singletonI'
thf(fact_225_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) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % order_subst1
thf(fact_226_order__subst1, axiom,
    ((![A : set_Ho137910533iple_a, F : set_pname > set_Ho137910533iple_a, B : set_pname, C : set_pname]: ((ord_le1221261669iple_a @ A @ (F @ B)) => ((ord_le865024672_pname @ B @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % order_subst1
thf(fact_227_order__subst1, axiom,
    ((![A : set_pname, F : set_Ho137910533iple_a > set_pname, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le865024672_pname @ A @ (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ A @ (F @ C)))))))). % order_subst1
thf(fact_228_order__subst1, axiom,
    ((![A : set_pname, F : set_pname > set_pname, B : set_pname, C : set_pname]: ((ord_le865024672_pname @ A @ (F @ B)) => ((ord_le865024672_pname @ B @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ A @ (F @ C)))))))). % order_subst1
thf(fact_229_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) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % order_subst2
thf(fact_230_order__subst2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_pname, C : set_pname]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le865024672_pname @ (F @ B) @ C) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ (F @ A) @ C))))))). % order_subst2
thf(fact_231_order__subst2, axiom,
    ((![A : set_pname, B : set_pname, F : set_pname > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le865024672_pname @ A @ B) => ((ord_le1221261669iple_a @ (F @ B) @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % order_subst2
thf(fact_232_order__subst2, axiom,
    ((![A : set_pname, B : set_pname, F : set_pname > set_pname, C : set_pname]: ((ord_le865024672_pname @ A @ B) => ((ord_le865024672_pname @ (F @ B) @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ (F @ A) @ C))))))). % order_subst2
thf(fact_233_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) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_234_ord__eq__le__subst, axiom,
    ((![A : set_pname, F : set_Ho137910533iple_a > set_pname, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A = (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_235_ord__eq__le__subst, axiom,
    ((![A : set_Ho137910533iple_a, F : set_pname > set_Ho137910533iple_a, B : set_pname, C : set_pname]: ((A = (F @ B)) => ((ord_le865024672_pname @ B @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_236_ord__eq__le__subst, axiom,
    ((![A : set_pname, F : set_pname > set_pname, B : set_pname, C : set_pname]: ((A = (F @ B)) => ((ord_le865024672_pname @ B @ C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_237_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) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_238_ord__le__eq__subst, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_pname, C : set_pname]: ((ord_le1221261669iple_a @ A @ B) => (((F @ B) = C) => ((![X5 : set_Ho137910533iple_a, Y5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_239_ord__le__eq__subst, axiom,
    ((![A : set_pname, B : set_pname, F : set_pname > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le865024672_pname @ A @ B) => (((F @ B) = C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y5)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_240_ord__le__eq__subst, axiom,
    ((![A : set_pname, B : set_pname, F : set_pname > set_pname, C : set_pname]: ((ord_le865024672_pname @ A @ B) => (((F @ B) = C) => ((![X5 : set_pname, Y5 : set_pname]: ((ord_le865024672_pname @ X5 @ Y5) => (ord_le865024672_pname @ (F @ X5) @ (F @ Y5)))) => (ord_le865024672_pname @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_241_eq__iff, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[X : set_Ho137910533iple_a]: (^[Y6 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X @ Y6)) & ((ord_le1221261669iple_a @ Y6 @ X)))))))). % eq_iff
thf(fact_242_eq__iff, axiom,
    (((^[Y4 : set_pname]: (^[Z : set_pname]: (Y4 = Z))) = (^[X : set_pname]: (^[Y6 : set_pname]: (((ord_le865024672_pname @ X @ Y6)) & ((ord_le865024672_pname @ Y6 @ X)))))))). % eq_iff
thf(fact_243_antisym, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y) => ((ord_le1221261669iple_a @ Y @ X4) => (X4 = Y)))))). % antisym
thf(fact_244_antisym, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((ord_le865024672_pname @ X4 @ Y) => ((ord_le865024672_pname @ Y @ X4) => (X4 = Y)))))). % antisym
thf(fact_245_eq__refl, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((X4 = Y) => (ord_le1221261669iple_a @ X4 @ Y))))). % eq_refl
thf(fact_246_eq__refl, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((X4 = Y) => (ord_le865024672_pname @ X4 @ Y))))). % eq_refl
thf(fact_247_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_248_order_Otrans, axiom,
    ((![A : set_pname, B : set_pname, C : set_pname]: ((ord_le865024672_pname @ A @ B) => ((ord_le865024672_pname @ B @ C) => (ord_le865024672_pname @ A @ C)))))). % order.trans

% Conjectures (2)
thf(conj_0, hypothesis,
    ((member_pname @ pn @ procs))).
thf(conj_1, conjecture,
    ((ord_le1221261669iple_a @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (p @ pn) @ (body @ pn) @ (q @ pn)) @ bot_bo1298296729iple_a) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (p @ P2) @ (body @ P2) @ (q @ P2))) @ procs)))).
