% 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_184__3251236_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:34.448

% Could-be-implicit typings (8)
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__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $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 (39)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__a, type,
    hoare_129598474rivs_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Com__Opname_M_Eo_J, type,
    sup_sup_pname_o : (pname > $o) > (pname > $o) > pname > $o).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    sup_su335701908le_a_o : (hoare_1678595023iple_a > $o) > (hoare_1678595023iple_a > $o) > hoare_1678595023iple_a > $o).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Com__Opname_J, type,
    sup_sup_set_pname : set_pname > set_pname > set_pname).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    sup_su1000235569iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
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_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_Obind_001t__Com__Opname_001t__Com__Opname, type,
    bind_pname_pname : set_pname > (pname > set_pname) > set_pname).
thf(sy_c_Set_Obind_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    bind_p599804078iple_a : set_pname > (pname > set_Ho137910533iple_a) > set_Ho137910533iple_a).
thf(sy_c_Set_Obind_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Com__Opname, type,
    bind_H965292974_pname : set_Ho137910533iple_a > (hoare_1678595023iple_a > set_pname) > set_pname).
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_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__empty_001t__Com__Opname, type,
    is_empty_pname : set_pname > $o).
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__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_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : a > state > $o).
thf(sy_v_Q, type,
    q : a > state > $o).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (234)
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_Body1, axiom,
    ((![G : set_Ho137910533iple_a, P : pname > a > state > $o, Q : pname > a > state > $o, Procs : set_pname, Pn : pname]: ((hoare_129598474rivs_a @ (sup_su1000235569iple_a @ G @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => ((member_pname @ Pn @ Procs) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn)) @ bot_bo1298296729iple_a))))))). % Body1
thf(fact_2_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P3 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (P3 @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_3_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q2 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_4_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P3 : a > state > $o, C : com, Q2 : a > state > $o, P : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_5_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_6_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_7_cut, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((hoare_129598474rivs_a @ G @ G2) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_8_hoare__derivs_OBody, axiom,
    ((![G : set_Ho137910533iple_a, P : pname > a > state > $o, Q : pname > a > state > $o, Procs : set_pname]: ((hoare_129598474rivs_a @ (sup_su1000235569iple_a @ G @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => (hoare_129598474rivs_a @ G @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)))))). % hoare_derivs.Body
thf(fact_9_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_10_conseq, axiom,
    ((![P : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P @ Z @ S) => (?[P4 : a > state > $o, Q3 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q3) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P4 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_11_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ Ts) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_12_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_13_singleton__conv, axiom,
    ((![A : pname]: ((collect_pname @ (^[X : pname]: (X = A))) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv
thf(fact_14_singleton__conv2, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ ((^[Y4 : hoare_1678595023iple_a]: (^[Z3 : hoare_1678595023iple_a]: (Y4 = Z3))) @ A)) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv2
thf(fact_15_singleton__conv2, axiom,
    ((![A : pname]: ((collect_pname @ ((^[Y4 : pname]: (^[Z3 : pname]: (Y4 = Z3))) @ A)) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_16_Un__insert__left, axiom,
    ((![A : pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ (insert_pname @ A @ B) @ C2) = (insert_pname @ A @ (sup_sup_set_pname @ B @ C2)))))). % Un_insert_left
thf(fact_17_Un__insert__left, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (insert1477804543iple_a @ A @ B) @ C2) = (insert1477804543iple_a @ A @ (sup_su1000235569iple_a @ B @ C2)))))). % Un_insert_left
thf(fact_18_Un__insert__right, axiom,
    ((![A2 : set_pname, A : pname, B : set_pname]: ((sup_sup_set_pname @ A2 @ (insert_pname @ A @ B)) = (insert_pname @ A @ (sup_sup_set_pname @ A2 @ B)))))). % Un_insert_right
thf(fact_19_Un__insert__right, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ (insert1477804543iple_a @ A @ B)) = (insert1477804543iple_a @ A @ (sup_su1000235569iple_a @ A2 @ B)))))). % Un_insert_right
thf(fact_20_Un__empty, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ A2 @ B) = bot_bo1298296729iple_a) = (((A2 = bot_bo1298296729iple_a)) & ((B = bot_bo1298296729iple_a))))))). % Un_empty
thf(fact_21_Un__empty, axiom,
    ((![A2 : set_pname, B : set_pname]: (((sup_sup_set_pname @ A2 @ B) = bot_bot_set_pname) = (((A2 = bot_bot_set_pname)) & ((B = bot_bot_set_pname))))))). % Un_empty
thf(fact_22_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_23_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_24_sup__bot__left, axiom,
    ((![X4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ X4) = X4)))). % sup_bot_left
thf(fact_25_sup__bot__left, axiom,
    ((![X4 : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ X4) = X4)))). % sup_bot_left
thf(fact_26_sup__bot__right, axiom,
    ((![X4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ bot_bo1298296729iple_a) = X4)))). % sup_bot_right
thf(fact_27_sup__bot__right, axiom,
    ((![X4 : set_pname]: ((sup_sup_set_pname @ X4 @ bot_bot_set_pname) = X4)))). % sup_bot_right
thf(fact_28_bot__eq__sup__iff, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (sup_su1000235569iple_a @ X4 @ Y)) = (((X4 = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % bot_eq_sup_iff
thf(fact_29_bot__eq__sup__iff, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ X4 @ Y)) = (((X4 = bot_bot_set_pname)) & ((Y = bot_bot_set_pname))))))). % bot_eq_sup_iff
thf(fact_30_sup__eq__bot__iff, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ X4 @ Y) = bot_bo1298296729iple_a) = (((X4 = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % sup_eq_bot_iff
thf(fact_31_sup__eq__bot__iff, axiom,
    ((![X4 : set_pname, Y : set_pname]: (((sup_sup_set_pname @ X4 @ Y) = bot_bot_set_pname) = (((X4 = bot_bot_set_pname)) & ((Y = bot_bot_set_pname))))))). % sup_eq_bot_iff
thf(fact_32_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ A @ B2) = bot_bo1298296729iple_a) = (((A = bot_bo1298296729iple_a)) & ((B2 = bot_bo1298296729iple_a))))))). % sup_bot.eq_neutr_iff
thf(fact_33_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_pname, B2 : set_pname]: (((sup_sup_set_pname @ A @ B2) = bot_bot_set_pname) = (((A = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.eq_neutr_iff
thf(fact_34_sup__bot_Oleft__neutral, axiom,
    ((![A : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ A) = A)))). % sup_bot.left_neutral
thf(fact_35_sup__bot_Oleft__neutral, axiom,
    ((![A : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ A) = A)))). % sup_bot.left_neutral
thf(fact_36_image__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X4 : pname, A2 : set_pname]: ((B2 = (F @ X4)) => ((member_pname @ X4 @ A2) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2))))))). % image_eqI
thf(fact_37_image__eqI, axiom,
    ((![B2 : pname, F : pname > pname, X4 : pname, A2 : set_pname]: ((B2 = (F @ X4)) => ((member_pname @ X4 @ A2) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_38_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_39_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X : pname]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_40_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_41_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X : pname]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_42_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_43_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_44_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_45_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_46_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_47_insert__absorb2, axiom,
    ((![X4 : pname, A2 : set_pname]: ((insert_pname @ X4 @ (insert_pname @ X4 @ A2)) = (insert_pname @ X4 @ A2))))). % insert_absorb2
thf(fact_48_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) = (((A = B2)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_49_insert__iff, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) = (((A = B2)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_50_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B))) => (A = B2)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertCI
thf(fact_51_insertCI, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: (((~ ((member_pname @ A @ B))) => (A = B2)) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertCI
thf(fact_52_sup_Oright__idem, axiom,
    ((![A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A @ B2) @ B2) = (sup_su1000235569iple_a @ A @ B2))))). % sup.right_idem
thf(fact_53_sup__left__idem, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ X4 @ Y)) = (sup_su1000235569iple_a @ X4 @ Y))))). % sup_left_idem
thf(fact_54_sup_Oleft__idem, axiom,
    ((![A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ A @ B2)) = (sup_su1000235569iple_a @ A @ B2))))). % sup.left_idem
thf(fact_55_sup__idem, axiom,
    ((![X4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ X4) = X4)))). % sup_idem
thf(fact_56_sup_Oidem, axiom,
    ((![A : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ A) = A)))). % sup.idem
thf(fact_57_Un__iff, axiom,
    ((![C : pname, A2 : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A2 @ B)) = (((member_pname @ C @ A2)) | ((member_pname @ C @ B))))))). % Un_iff
thf(fact_58_Un__iff, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A2 @ B)) = (((member1332298086iple_a @ C @ A2)) | ((member1332298086iple_a @ C @ B))))))). % Un_iff
thf(fact_59_UnCI, axiom,
    ((![C : pname, B : set_pname, A2 : set_pname]: (((~ ((member_pname @ C @ B))) => (member_pname @ C @ A2)) => (member_pname @ C @ (sup_sup_set_pname @ A2 @ B)))))). % UnCI
thf(fact_60_UnCI, axiom,
    ((![C : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: (((~ ((member1332298086iple_a @ C @ B))) => (member1332298086iple_a @ C @ A2)) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A2 @ B)))))). % UnCI
thf(fact_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((image_1577193952iple_a @ F @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_70_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > pname]: ((image_243573216_pname @ F @ bot_bo1298296729iple_a) = bot_bot_set_pname)))). % image_empty
thf(fact_71_image__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ F @ bot_bot_set_pname) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_72_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_73_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_74_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_75_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_76_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_77_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (insert1477804543iple_a @ A @ B)) = (insert1477804543iple_a @ (F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_insert
thf(fact_78_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > pname, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((image_243573216_pname @ F @ (insert1477804543iple_a @ A @ B)) = (insert_pname @ (F @ A) @ (image_243573216_pname @ F @ B)))))). % image_insert
thf(fact_79_image__insert, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : pname, B : set_pname]: ((image_2025567968iple_a @ F @ (insert_pname @ A @ B)) = (insert1477804543iple_a @ (F @ A) @ (image_2025567968iple_a @ F @ B)))))). % image_insert
thf(fact_80_image__insert, axiom,
    ((![F : pname > pname, A : pname, B : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A @ B)) = (insert_pname @ (F @ A) @ (image_pname_pname @ F @ B)))))). % image_insert
thf(fact_81_sup__bot_Oright__neutral, axiom,
    ((![A : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ bot_bo1298296729iple_a) = A)))). % sup_bot.right_neutral
thf(fact_82_sup__bot_Oright__neutral, axiom,
    ((![A : set_pname]: ((sup_sup_set_pname @ A @ bot_bot_set_pname) = A)))). % sup_bot.right_neutral
thf(fact_83_sup__bot_Oneutr__eq__iff, axiom,
    ((![A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (sup_su1000235569iple_a @ A @ B2)) = (((A = bot_bo1298296729iple_a)) & ((B2 = bot_bo1298296729iple_a))))))). % sup_bot.neutr_eq_iff
thf(fact_84_sup__bot_Oneutr__eq__iff, axiom,
    ((![A : set_pname, B2 : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ A @ B2)) = (((A = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.neutr_eq_iff
thf(fact_85_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_86_bot__set__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ bot_bot_pname_o)))). % bot_set_def
thf(fact_87_sup__set__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (sup_sup_pname_o @ (^[X : pname]: (member_pname @ X @ A3)) @ (^[X : pname]: (member_pname @ X @ B3))))))))). % sup_set_def
thf(fact_88_sup__set__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (sup_su335701908le_a_o @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A3)) @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ B3))))))))). % sup_set_def
thf(fact_89_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((B2 = (F @ X4)) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2))))))). % rev_image_eqI
thf(fact_90_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : pname, F : pname > pname]: ((member_pname @ X4 @ A2) => ((B2 = (F @ X4)) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_91_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_92_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F @ X5) = (G3 @ X5)))) => ((image_2025567968iple_a @ F @ M) = (image_2025567968iple_a @ G3 @ N))))))). % image_cong
thf(fact_93_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_94_image__iff, axiom,
    ((![Z4 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ Z4 @ (image_2025567968iple_a @ F @ A2)) = (?[X : pname]: (((member_pname @ X @ A2)) & ((Z4 = (F @ X))))))))). % image_iff
thf(fact_95_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_96_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_97_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_98_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X : pname]: (member_pname @ X @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_99_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_100_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y5 : pname]: (~ ((member_pname @ Y5 @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_101_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_102_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_103_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_104_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_105_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_106_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_107_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_108_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_109_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B2 @ B))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B2 @ C3))) & ((((~ ((member1332298086iple_a @ B2 @ C3)))) & ((((B = (insert1477804543iple_a @ A @ C3))) & ((~ ((member1332298086iple_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_110_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B2 : pname, B : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B2 @ B))) => (((insert_pname @ A @ A2) = (insert_pname @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_pname]: (((A2 = (insert_pname @ B2 @ C3))) & ((((~ ((member_pname @ B2 @ C3)))) & ((((B = (insert_pname @ A @ C3))) & ((~ ((member_pname @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_111_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_112_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_113_insert__ident, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((~ ((member1332298086iple_a @ X4 @ B))) => (((insert1477804543iple_a @ X4 @ A2) = (insert1477804543iple_a @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_114_insert__ident, axiom,
    ((![X4 : pname, A2 : set_pname, B : set_pname]: ((~ ((member_pname @ X4 @ A2))) => ((~ ((member_pname @ X4 @ B))) => (((insert_pname @ X4 @ A2) = (insert_pname @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_115_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_116_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_117_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertI2
thf(fact_118_insertI2, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: ((member_pname @ A @ B) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertI2
thf(fact_119_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_120_insertI1, axiom,
    ((![A : pname, B : set_pname]: (member_pname @ A @ (insert_pname @ A @ B))))). % insertI1
thf(fact_121_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) => ((~ ((A = B2))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_122_insertE, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) => ((~ ((A = B2))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_123_sup__left__commute, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ Y @ Z4)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X4 @ Z4)))))). % sup_left_commute
thf(fact_124_sup_Oleft__commute, axiom,
    ((![B2 : set_Ho137910533iple_a, A : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ B2 @ (sup_su1000235569iple_a @ A @ C)) = (sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B2 @ C)))))). % sup.left_commute
thf(fact_125_sup__commute, axiom,
    ((sup_su1000235569iple_a = (^[X : set_Ho137910533iple_a]: (^[Y6 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y6 @ X)))))). % sup_commute
thf(fact_126_sup_Ocommute, axiom,
    ((sup_su1000235569iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B5 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B5 @ A4)))))). % sup.commute
thf(fact_127_sup__assoc, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X4 @ Y) @ Z4) = (sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ Y @ Z4)))))). % sup_assoc
thf(fact_128_sup_Oassoc, axiom,
    ((![A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A @ B2) @ C) = (sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B2 @ C)))))). % sup.assoc
thf(fact_129_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_Ho137910533iple_a, K : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((B = (sup_su1000235569iple_a @ K @ B2)) => ((sup_su1000235569iple_a @ A @ B) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_130_boolean__algebra__cancel_Osup1, axiom,
    ((![A2 : set_Ho137910533iple_a, K : set_Ho137910533iple_a, A : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = (sup_su1000235569iple_a @ K @ A)) => ((sup_su1000235569iple_a @ A2 @ B2) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_131_inf__sup__aci_I5_J, axiom,
    ((sup_su1000235569iple_a = (^[X : set_Ho137910533iple_a]: (^[Y6 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y6 @ X)))))). % inf_sup_aci(5)
thf(fact_132_inf__sup__aci_I6_J, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X4 @ Y) @ Z4) = (sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ Y @ Z4)))))). % inf_sup_aci(6)
thf(fact_133_inf__sup__aci_I7_J, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ Y @ Z4)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X4 @ Z4)))))). % inf_sup_aci(7)
thf(fact_134_inf__sup__aci_I8_J, axiom,
    ((![X4 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X4 @ (sup_su1000235569iple_a @ X4 @ Y)) = (sup_su1000235569iple_a @ X4 @ Y))))). % inf_sup_aci(8)
thf(fact_135_Un__left__commute, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B @ C2)) = (sup_su1000235569iple_a @ B @ (sup_su1000235569iple_a @ A2 @ C2)))))). % Un_left_commute
thf(fact_136_Un__left__absorb, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ A2 @ B)) = (sup_su1000235569iple_a @ A2 @ B))))). % Un_left_absorb
thf(fact_137_Un__commute, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B3 @ A3)))))). % Un_commute
thf(fact_138_Un__absorb, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ A2) = A2)))). % Un_absorb
thf(fact_139_Un__assoc, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A2 @ B) @ C2) = (sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B @ C2)))))). % Un_assoc
thf(fact_140_ball__Un, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ (sup_su1000235569iple_a @ A2 @ B))) => ((P @ X)))) = (((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A2)) => ((P @ X))))) & ((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_141_bex__Un, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((?[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ (sup_su1000235569iple_a @ A2 @ B))) & ((P @ X)))) = (((?[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A2)) & ((P @ X))))) | ((?[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_142_UnI2, axiom,
    ((![C : pname, B : set_pname, A2 : set_pname]: ((member_pname @ C @ B) => (member_pname @ C @ (sup_sup_set_pname @ A2 @ B)))))). % UnI2
thf(fact_143_UnI2, axiom,
    ((![C : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ B) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A2 @ B)))))). % UnI2
thf(fact_144_UnI1, axiom,
    ((![C : pname, A2 : set_pname, B : set_pname]: ((member_pname @ C @ A2) => (member_pname @ C @ (sup_sup_set_pname @ A2 @ B)))))). % UnI1
thf(fact_145_UnI1, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ A2) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A2 @ B)))))). % UnI1
thf(fact_146_UnE, axiom,
    ((![C : pname, A2 : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A2 @ B)) => ((~ ((member_pname @ C @ A2))) => (member_pname @ C @ B)))))). % UnE
thf(fact_147_UnE, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A2 @ B)) => ((~ ((member1332298086iple_a @ C @ A2))) => (member1332298086iple_a @ C @ B)))))). % UnE
thf(fact_148_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_149_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_150_image__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a, A2 : set_pname]: ((image_1577193952iple_a @ F @ (image_2025567968iple_a @ G3 @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_151_image__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, G3 : pname > pname, A2 : set_pname]: ((image_2025567968iple_a @ F @ (image_pname_pname @ G3 @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_152_imageE, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2)) => (~ ((![X5 : pname]: ((B2 = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_153_imageE, axiom,
    ((![B2 : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F @ A2)) => (~ ((![X5 : pname]: ((B2 = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_154_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_155_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X : pname]: $false))))). % empty_def
thf(fact_156_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_157_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_158_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_159_insert__compr, axiom,
    ((insert_pname = (^[A4 : pname]: (^[B3 : set_pname]: (collect_pname @ (^[X : pname]: (((X = A4)) | ((member_pname @ X @ B3)))))))))). % insert_compr
thf(fact_160_Collect__disj__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((P @ X)) | ((Q @ X))))) = (sup_su1000235569iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_disj_eq
thf(fact_161_Un__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (^[X : pname]: (((member_pname @ X @ A3)) | ((member_pname @ X @ B3)))))))))). % Un_def
thf(fact_162_Un__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A3)) | ((member1332298086iple_a @ X @ B3)))))))))). % Un_def
thf(fact_163_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) => (A = B2))))). % singleton_inject
thf(fact_164_singleton__inject, axiom,
    ((![A : pname, B2 : pname]: (((insert_pname @ A @ bot_bot_set_pname) = (insert_pname @ B2 @ bot_bot_set_pname)) => (A = B2))))). % 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, B2 : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_168_doubleton__eq__iff, axiom,
    ((![A : pname, B2 : pname, C : pname, D : pname]: (((insert_pname @ A @ (insert_pname @ B2 @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D @ bot_bot_set_pname))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_169_singleton__iff, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B2 = A))))). % singleton_iff
thf(fact_170_singleton__iff, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) = (B2 = A))))). % singleton_iff
thf(fact_171_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_172_singletonD, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) => (B2 = A))))). % singletonD
thf(fact_173_image__Un, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, B : set_pname]: ((image_2025567968iple_a @ F @ (sup_sup_set_pname @ A2 @ B)) = (sup_su1000235569iple_a @ (image_2025567968iple_a @ F @ A2) @ (image_2025567968iple_a @ F @ B)))))). % image_Un
thf(fact_174_image__Un, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (sup_su1000235569iple_a @ A2 @ B)) = (sup_su1000235569iple_a @ (image_1577193952iple_a @ F @ A2) @ (image_1577193952iple_a @ F @ B)))))). % image_Un
thf(fact_175_Un__empty__right, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Un_empty_right
thf(fact_176_Un__empty__right, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ A2 @ bot_bot_set_pname) = A2)))). % Un_empty_right
thf(fact_177_Un__empty__left, axiom,
    ((![B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ B) = B)))). % Un_empty_left
thf(fact_178_Un__empty__left, axiom,
    ((![B : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ B) = B)))). % Un_empty_left
thf(fact_179_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_180_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_181_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_182_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_183_insert__def, axiom,
    ((insert_pname = (^[A4 : pname]: (sup_sup_set_pname @ (collect_pname @ (^[X : pname]: (X = A4)))))))). % insert_def
thf(fact_184_insert__def, axiom,
    ((insert1477804543iple_a = (^[A4 : hoare_1678595023iple_a]: (sup_su1000235569iple_a @ (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (X = A4)))))))). % insert_def
thf(fact_185_singleton__Un__iff, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a) = (sup_su1000235569iple_a @ A2 @ B)) = (((((A2 = bot_bo1298296729iple_a)) & ((B = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))) | ((((((A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))) & ((B = bot_bo1298296729iple_a)))) | ((((A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))) & ((B = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)))))))))))). % singleton_Un_iff
thf(fact_186_singleton__Un__iff, axiom,
    ((![X4 : pname, A2 : set_pname, B : set_pname]: (((insert_pname @ X4 @ bot_bot_set_pname) = (sup_sup_set_pname @ A2 @ B)) = (((((A2 = bot_bot_set_pname)) & ((B = (insert_pname @ X4 @ bot_bot_set_pname))))) | ((((((A2 = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = bot_bot_set_pname)))) | ((((A2 = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = (insert_pname @ X4 @ bot_bot_set_pname)))))))))))). % singleton_Un_iff
thf(fact_187_Un__singleton__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: (((sup_su1000235569iple_a @ A2 @ B) = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = (((((A2 = bot_bo1298296729iple_a)) & ((B = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))) | ((((((A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))) & ((B = bot_bo1298296729iple_a)))) | ((((A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))) & ((B = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)))))))))))). % Un_singleton_iff
thf(fact_188_Un__singleton__iff, axiom,
    ((![A2 : set_pname, B : set_pname, X4 : pname]: (((sup_sup_set_pname @ A2 @ B) = (insert_pname @ X4 @ bot_bot_set_pname)) = (((((A2 = bot_bot_set_pname)) & ((B = (insert_pname @ X4 @ bot_bot_set_pname))))) | ((((((A2 = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = bot_bot_set_pname)))) | ((((A2 = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = (insert_pname @ X4 @ bot_bot_set_pname)))))))))))). % Un_singleton_iff
thf(fact_189_insert__is__Un, axiom,
    ((insert1477804543iple_a = (^[A4 : hoare_1678595023iple_a]: (sup_su1000235569iple_a @ (insert1477804543iple_a @ A4 @ bot_bo1298296729iple_a)))))). % insert_is_Un
thf(fact_190_insert__is__Un, axiom,
    ((insert_pname = (^[A4 : pname]: (sup_sup_set_pname @ (insert_pname @ A4 @ bot_bot_set_pname)))))). % insert_is_Un
thf(fact_191_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_192_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_193_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_194_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_195_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_196_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_197_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = 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_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_201_is__singletonI, axiom,
    ((![X4 : pname]: (is_singleton_pname @ (insert_pname @ X4 @ bot_bot_set_pname))))). % is_singletonI
thf(fact_202_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ skip @ P) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip
thf(fact_203_Comp, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, D : com, R : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ Q @ D @ R) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (semi @ C @ D) @ R) @ bot_bo1298296729iple_a))))))). % Comp
thf(fact_204_the__elem__image__unique, axiom,
    ((![A2 : set_pname, F : pname > hoare_1678595023iple_a, X4 : pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![Y5 : pname]: ((member_pname @ Y5 @ A2) => ((F @ Y5) = (F @ X4)))) => ((the_el434698138iple_a @ (image_2025567968iple_a @ F @ A2)) = (F @ X4))))))). % the_elem_image_unique
thf(fact_205_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_206_Set_Ois__empty__def, axiom,
    ((is_empty_pname = (^[A3 : set_pname]: (A3 = bot_bot_set_pname))))). % Set.is_empty_def
thf(fact_207_com_Oinject_I3_J, axiom,
    ((![X41 : com, X42 : com, Y41 : com, Y42 : com]: (((semi @ X41 @ X42) = (semi @ Y41 @ Y42)) = (((X41 = Y41)) & ((X42 = Y42))))))). % com.inject(3)
thf(fact_208_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_209_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body2 @ X7))))))). % com.distinct(41)
thf(fact_210_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body2 @ X7))))))). % com.distinct(11)
thf(fact_211_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_212_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_213_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_214_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_215_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X5 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_216_is__singletonE, axiom,
    ((![A2 : set_pname]: ((is_singleton_pname @ A2) => (~ ((![X5 : pname]: (~ ((A2 = (insert_pname @ X5 @ bot_bot_set_pname))))))))))). % is_singletonE
thf(fact_217_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_218_is__singleton__def, axiom,
    ((is_singleton_pname = (^[A3 : set_pname]: (?[X : pname]: (A3 = (insert_pname @ X @ bot_bot_set_pname))))))). % is_singleton_def
thf(fact_219_sup__Un__eq, axiom,
    ((![R : set_pname, S3 : set_pname]: ((sup_sup_pname_o @ (^[X : pname]: (member_pname @ X @ R)) @ (^[X : pname]: (member_pname @ X @ S3))) = (^[X : pname]: (member_pname @ X @ (sup_sup_set_pname @ R @ S3))))))). % sup_Un_eq
thf(fact_220_sup__Un__eq, axiom,
    ((![R : set_Ho137910533iple_a, S3 : set_Ho137910533iple_a]: ((sup_su335701908le_a_o @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ R)) @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ S3))) = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ (sup_su1000235569iple_a @ R @ S3))))))). % sup_Un_eq
thf(fact_221_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_222_bot__empty__eq, axiom,
    ((bot_bot_pname_o = (^[X : pname]: (member_pname @ X @ bot_bot_set_pname))))). % bot_empty_eq
thf(fact_223_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_224_Collect__empty__eq__bot, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (P = bot_bot_pname_o))))). % Collect_empty_eq_bot
thf(fact_225_bind__singleton__conv__image, axiom,
    ((![A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((bind_p599804078iple_a @ A2 @ (^[X : pname]: (insert1477804543iple_a @ (F @ X) @ bot_bo1298296729iple_a))) = (image_2025567968iple_a @ F @ A2))))). % bind_singleton_conv_image
thf(fact_226_empty__bind, axiom,
    ((![F : hoare_1678595023iple_a > set_Ho137910533iple_a]: ((bind_H209762450iple_a @ bot_bo1298296729iple_a @ F) = bot_bo1298296729iple_a)))). % empty_bind
thf(fact_227_empty__bind, axiom,
    ((![F : hoare_1678595023iple_a > set_pname]: ((bind_H965292974_pname @ bot_bo1298296729iple_a @ F) = bot_bot_set_pname)))). % empty_bind
thf(fact_228_empty__bind, axiom,
    ((![F : pname > set_Ho137910533iple_a]: ((bind_p599804078iple_a @ bot_bot_set_pname @ F) = bot_bo1298296729iple_a)))). % empty_bind
thf(fact_229_empty__bind, axiom,
    ((![F : pname > set_pname]: ((bind_pname_pname @ bot_bot_set_pname @ F) = bot_bot_set_pname)))). % empty_bind
thf(fact_230_bind__const, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((A2 = bot_bo1298296729iple_a) => ((bind_H209762450iple_a @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((bind_H209762450iple_a @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = B)))))). % bind_const
thf(fact_231_bind__const, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_pname]: (((A2 = bot_bo1298296729iple_a) => ((bind_H965292974_pname @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = bot_bot_set_pname)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((bind_H965292974_pname @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = B)))))). % bind_const
thf(fact_232_bind__const, axiom,
    ((![A2 : set_pname, B : set_Ho137910533iple_a]: (((A2 = bot_bot_set_pname) => ((bind_p599804078iple_a @ A2 @ (^[Uu : pname]: B)) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bot_set_pname))) => ((bind_p599804078iple_a @ A2 @ (^[Uu : pname]: B)) = B)))))). % bind_const
thf(fact_233_bind__const, axiom,
    ((![A2 : set_pname, B : set_pname]: (((A2 = bot_bot_set_pname) => ((bind_pname_pname @ A2 @ (^[Uu : pname]: B)) = bot_bot_set_pname)) & ((~ ((A2 = bot_bot_set_pname))) => ((bind_pname_pname @ A2 @ (^[Uu : pname]: B)) = B)))))). % bind_const

% Conjectures (2)
thf(conj_0, hypothesis,
    ((hoare_129598474rivs_a @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ (body2 @ pn) @ q) @ g) @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ (the_com @ (body @ pn)) @ q) @ bot_bo1298296729iple_a)))).
thf(conj_1, conjecture,
    ((hoare_129598474rivs_a @ (sup_su1000235569iple_a @ g @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ p @ (body2 @ P2) @ q)) @ (insert_pname @ pn @ bot_bot_set_pname))) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ p @ (the_com @ (body @ P2)) @ q)) @ (insert_pname @ pn @ bot_bot_set_pname))))).
