% 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_220__3251624_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:51.282

% 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 (27)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > 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__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_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__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__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_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_Ga, type,
    ga : set_Ho137910533iple_a).
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).

% Relevant facts (139)
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_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_4_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_5_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_6_weaken, axiom,
    ((![G : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts2) => ((ord_le1221261669iple_a @ Ts @ Ts2) => (hoare_129598474rivs_a @ G @ Ts)))))). % weaken
thf(fact_7_Un__subset__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A @ B) @ C) = (((ord_le1221261669iple_a @ A @ C)) & ((ord_le1221261669iple_a @ B @ C))))))). % Un_subset_iff
thf(fact_8_le__sup__iff, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ X @ Y) @ Z) = (((ord_le1221261669iple_a @ X @ Z)) & ((ord_le1221261669iple_a @ Y @ Z))))))). % le_sup_iff
thf(fact_9_sup_Obounded__iff, axiom,
    ((![B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ B2 @ C2) @ A2) = (((ord_le1221261669iple_a @ B2 @ A2)) & ((ord_le1221261669iple_a @ C2 @ A2))))))). % sup.bounded_iff
thf(fact_10_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_11_UnCI, axiom,
    ((![C2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: (((~ ((member1332298086iple_a @ C2 @ B))) => (member1332298086iple_a @ C2 @ A)) => (member1332298086iple_a @ C2 @ (sup_su1000235569iple_a @ A @ B)))))). % UnCI
thf(fact_12_Un__iff, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ (sup_su1000235569iple_a @ A @ B)) = (((member1332298086iple_a @ C2 @ A)) | ((member1332298086iple_a @ C2 @ B))))))). % Un_iff
thf(fact_13_sup_Oidem, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ A2) = A2)))). % sup.idem
thf(fact_14_sup__idem, axiom,
    ((![X : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ X) = X)))). % sup_idem
thf(fact_15_sup_Oleft__idem, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ A2 @ B2)) = (sup_su1000235569iple_a @ A2 @ B2))))). % sup.left_idem
thf(fact_16_sup__left__idem, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ X @ Y)) = (sup_su1000235569iple_a @ X @ Y))))). % sup_left_idem
thf(fact_17_image__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X : pname, A : set_pname]: ((B2 = (F @ X)) => ((member_pname @ X @ A) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A))))))). % image_eqI
thf(fact_18_subset__antisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_19_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A) => (member1332298086iple_a @ X4 @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_20_sup_Oright__idem, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ B2) = (sup_su1000235569iple_a @ A2 @ B2))))). % sup.right_idem
thf(fact_21_sup__set__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (sup_su335701908le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A3)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ B3))))))))). % sup_set_def
thf(fact_22_less__eq__set__def, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (ord_le1593108832le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A3)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ B3)))))))). % less_eq_set_def
thf(fact_23_rev__image__eqI, axiom,
    ((![X : pname, A : set_pname, B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member_pname @ X @ A) => ((B2 = (F @ X)) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A))))))). % rev_image_eqI
thf(fact_24_ball__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ (image_2025567968iple_a @ F @ A)) => (P @ X4))) => (![X6 : pname]: ((member_pname @ X6 @ A) => (P @ (F @ X6)))))))). % ball_imageD
thf(fact_25_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a]: ((M = N) => ((![X4 : pname]: ((member_pname @ X4 @ N) => ((F @ X4) = (G3 @ X4)))) => ((image_2025567968iple_a @ F @ M) = (image_2025567968iple_a @ G3 @ N))))))). % image_cong
thf(fact_26_bex__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((?[X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ (image_2025567968iple_a @ F @ A)) & (P @ X6))) => (?[X4 : pname]: ((member_pname @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_27_image__iff, axiom,
    ((![Z : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((member1332298086iple_a @ Z @ (image_2025567968iple_a @ F @ A)) = (?[X5 : pname]: (((member_pname @ X5 @ A)) & ((Z = (F @ X5))))))))). % image_iff
thf(fact_28_imageI, axiom,
    ((![X : pname, A : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X @ A) => (member1332298086iple_a @ (F @ X) @ (image_2025567968iple_a @ F @ A)))))). % imageI
thf(fact_29_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X5 : hoare_1678595023iple_a]: (((P @ X5)) => ((Q @ X5)))))))). % Collect_mono_iff
thf(fact_30_set__eq__subset, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z2 : set_Ho137910533iple_a]: (Y4 = Z2))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_31_subset__trans, 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)))))). % subset_trans
thf(fact_32_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X4 : hoare_1678595023iple_a]: ((P @ X4) => (Q @ X4))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_33_subset__refl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % subset_refl
thf(fact_34_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_35_equalityD2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ B @ A))))). % equalityD2
thf(fact_36_equalityD1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ A @ B))))). % equalityD1
thf(fact_37_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A3)) => ((member1332298086iple_a @ X5 @ B3))))))))). % subset_eq
thf(fact_38_equalityE, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (~ (((ord_le1221261669iple_a @ A @ B) => (~ ((ord_le1221261669iple_a @ B @ A)))))))))). % equalityE
thf(fact_39_subsetD, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ C2 @ A) => (member1332298086iple_a @ C2 @ B)))))). % subsetD
thf(fact_40_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X @ A) => (member1332298086iple_a @ X @ B)))))). % in_mono
thf(fact_41_sup__left__commute, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ Y @ Z)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X @ Z)))))). % sup_left_commute
thf(fact_42_sup_Oleft__commute, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ B2 @ (sup_su1000235569iple_a @ A2 @ C2)) = (sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B2 @ C2)))))). % sup.left_commute
thf(fact_43_sup__commute, axiom,
    ((sup_su1000235569iple_a = (^[X5 : set_Ho137910533iple_a]: (^[Y5 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y5 @ X5)))))). % sup_commute
thf(fact_44_sup_Ocommute, axiom,
    ((sup_su1000235569iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B4 @ A4)))))). % sup.commute
thf(fact_45_sup__assoc, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X @ Y) @ Z) = (sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ Y @ Z)))))). % sup_assoc
thf(fact_46_sup_Oassoc, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ C2) = (sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B2 @ C2)))))). % sup.assoc
thf(fact_47_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_Ho137910533iple_a, K : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((B = (sup_su1000235569iple_a @ K @ B2)) => ((sup_su1000235569iple_a @ A2 @ B) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_48_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_Ho137910533iple_a, K : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A = (sup_su1000235569iple_a @ K @ A2)) => ((sup_su1000235569iple_a @ A @ B2) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_49_inf__sup__aci_I5_J, axiom,
    ((sup_su1000235569iple_a = (^[X5 : set_Ho137910533iple_a]: (^[Y5 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y5 @ X5)))))). % inf_sup_aci(5)
thf(fact_50_inf__sup__aci_I6_J, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X @ Y) @ Z) = (sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_51_inf__sup__aci_I7_J, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ Y @ Z)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X @ Z)))))). % inf_sup_aci(7)
thf(fact_52_inf__sup__aci_I8_J, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ (sup_su1000235569iple_a @ X @ Y)) = (sup_su1000235569iple_a @ X @ Y))))). % inf_sup_aci(8)
thf(fact_53_Un__left__commute, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B @ C)) = (sup_su1000235569iple_a @ B @ (sup_su1000235569iple_a @ A @ C)))))). % Un_left_commute
thf(fact_54_Un__left__absorb, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ A @ B)) = (sup_su1000235569iple_a @ A @ B))))). % Un_left_absorb
thf(fact_55_Un__commute, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B3 @ A3)))))). % Un_commute
thf(fact_56_Un__absorb, axiom,
    ((![A : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ A) = A)))). % Un_absorb
thf(fact_57_Un__assoc, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A @ B) @ C) = (sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B @ C)))))). % Un_assoc
thf(fact_58_ball__Un, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ A @ B))) => ((P @ X5)))) = (((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) => ((P @ X5))))) & ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) => ((P @ X5)))))))))). % ball_Un
thf(fact_59_bex__Un, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ A @ B))) & ((P @ X5)))) = (((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) & ((P @ X5))))) | ((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) & ((P @ X5)))))))))). % bex_Un
thf(fact_60_UnI2, axiom,
    ((![C2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ B) => (member1332298086iple_a @ C2 @ (sup_su1000235569iple_a @ A @ B)))))). % UnI2
thf(fact_61_UnI1, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ A) => (member1332298086iple_a @ C2 @ (sup_su1000235569iple_a @ A @ B)))))). % UnI1
thf(fact_62_UnE, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ (sup_su1000235569iple_a @ A @ B)) => ((~ ((member1332298086iple_a @ C2 @ A))) => (member1332298086iple_a @ C2 @ B)))))). % UnE
thf(fact_63_Compr__image__eq, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (image_2025567968iple_a @ F @ A))) & ((P @ X5))))) = (image_2025567968iple_a @ F @ (collect_pname @ (^[X5 : pname]: (((member_pname @ X5 @ A)) & ((P @ (F @ X5))))))))))). % Compr_image_eq
thf(fact_64_image__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a, A : set_pname]: ((image_1577193952iple_a @ F @ (image_2025567968iple_a @ G3 @ A)) = (image_2025567968iple_a @ (^[X5 : pname]: (F @ (G3 @ X5))) @ A))))). % image_image
thf(fact_65_image__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, G3 : pname > pname, A : set_pname]: ((image_2025567968iple_a @ F @ (image_pname_pname @ G3 @ A)) = (image_2025567968iple_a @ (^[X5 : pname]: (F @ (G3 @ X5))) @ A))))). % image_image
thf(fact_66_imageE, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A)) => (~ ((![X4 : pname]: ((B2 = (F @ X4)) => (~ ((member_pname @ X4 @ A))))))))))). % imageE
thf(fact_67_Collect__subset, axiom,
    ((![A : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) & ((P @ X5))))) @ A)))). % Collect_subset
thf(fact_68_Collect__disj__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((P @ X5)) | ((Q @ X5))))) = (sup_su1000235569iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_disj_eq
thf(fact_69_Un__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A3)) | ((member1332298086iple_a @ X5 @ B3)))))))))). % Un_def
thf(fact_70_sup_OcoboundedI2, axiom,
    ((![C2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C2 @ B2) => (ord_le1221261669iple_a @ C2 @ (sup_su1000235569iple_a @ A2 @ B2)))))). % sup.coboundedI2
thf(fact_71_sup_OcoboundedI1, axiom,
    ((![C2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C2 @ A2) => (ord_le1221261669iple_a @ C2 @ (sup_su1000235569iple_a @ A2 @ B2)))))). % sup.coboundedI1
thf(fact_72_sup_Oabsorb__iff2, axiom,
    ((ord_le1221261669iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A4 @ B4) = B4)))))). % sup.absorb_iff2
thf(fact_73_sup_Oabsorb__iff1, axiom,
    ((ord_le1221261669iple_a = (^[B4 : set_Ho137910533iple_a]: (^[A4 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A4 @ B4) = A4)))))). % sup.absorb_iff1
thf(fact_74_sup_Ocobounded2, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ B2 @ (sup_su1000235569iple_a @ A2 @ B2))))). % sup.cobounded2
thf(fact_75_sup_Ocobounded1, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ (sup_su1000235569iple_a @ A2 @ B2))))). % sup.cobounded1
thf(fact_76_sup_Oorder__iff, axiom,
    ((ord_le1221261669iple_a = (^[B4 : set_Ho137910533iple_a]: (^[A4 : set_Ho137910533iple_a]: (A4 = (sup_su1000235569iple_a @ A4 @ B4))))))). % sup.order_iff
thf(fact_77_sup_OboundedI, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((ord_le1221261669iple_a @ C2 @ A2) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ B2 @ C2) @ A2)))))). % sup.boundedI
thf(fact_78_sup_OboundedE, axiom,
    ((![B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ B2 @ C2) @ A2) => (~ (((ord_le1221261669iple_a @ B2 @ A2) => (~ ((ord_le1221261669iple_a @ C2 @ A2)))))))))). % sup.boundedE
thf(fact_79_sup__absorb2, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => ((sup_su1000235569iple_a @ X @ Y) = Y))))). % sup_absorb2
thf(fact_80_sup__absorb1, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ X) => ((sup_su1000235569iple_a @ X @ Y) = X))))). % sup_absorb1
thf(fact_81_sup_Oabsorb2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((sup_su1000235569iple_a @ A2 @ B2) = B2))))). % sup.absorb2
thf(fact_82_sup_Oabsorb1, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((sup_su1000235569iple_a @ A2 @ B2) = A2))))). % sup.absorb1
thf(fact_83_sup__unique, axiom,
    ((![F : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a, X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X4 @ (F @ X4 @ Y6))) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ Y6 @ (F @ X4 @ Y6))) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a, Z3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y6 @ X4) => ((ord_le1221261669iple_a @ Z3 @ X4) => (ord_le1221261669iple_a @ (F @ Y6 @ Z3) @ X4)))) => ((sup_su1000235569iple_a @ X @ Y) = (F @ X @ Y)))))))). % sup_unique
thf(fact_84_sup_OorderI, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = (sup_su1000235569iple_a @ A2 @ B2)) => (ord_le1221261669iple_a @ B2 @ A2))))). % sup.orderI
thf(fact_85_sup_OorderE, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => (A2 = (sup_su1000235569iple_a @ A2 @ B2)))))). % sup.orderE
thf(fact_86_le__iff__sup, axiom,
    ((ord_le1221261669iple_a = (^[X5 : set_Ho137910533iple_a]: (^[Y5 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X5 @ Y5) = Y5)))))). % le_iff_sup
thf(fact_87_sup__least, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ X) => ((ord_le1221261669iple_a @ Z @ X) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ Y @ Z) @ X)))))). % sup_least
thf(fact_88_sup__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, D : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ C2) => ((ord_le1221261669iple_a @ B2 @ D) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ (sup_su1000235569iple_a @ C2 @ D))))))). % sup_mono
thf(fact_89_sup_Omono, axiom,
    ((![C2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, D : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C2 @ A2) => ((ord_le1221261669iple_a @ D @ B2) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ C2 @ D) @ (sup_su1000235569iple_a @ A2 @ B2))))))). % sup.mono
thf(fact_90_le__supI2, axiom,
    ((![X : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ B2) => (ord_le1221261669iple_a @ X @ (sup_su1000235569iple_a @ A2 @ B2)))))). % le_supI2
thf(fact_91_le__supI1, axiom,
    ((![X : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ A2) => (ord_le1221261669iple_a @ X @ (sup_su1000235569iple_a @ A2 @ B2)))))). % le_supI1
thf(fact_92_sup__ge2, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ Y @ (sup_su1000235569iple_a @ X @ Y))))). % sup_ge2
thf(fact_93_sup__ge1, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ (sup_su1000235569iple_a @ X @ Y))))). % sup_ge1
thf(fact_94_le__supI, axiom,
    ((![A2 : set_Ho137910533iple_a, X : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ X) => ((ord_le1221261669iple_a @ B2 @ X) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ X)))))). % le_supI
thf(fact_95_le__supE, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ X) => (~ (((ord_le1221261669iple_a @ A2 @ X) => (~ ((ord_le1221261669iple_a @ B2 @ X)))))))))). % le_supE
thf(fact_96_inf__sup__ord_I3_J, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ (sup_su1000235569iple_a @ X @ Y))))). % inf_sup_ord(3)
thf(fact_97_inf__sup__ord_I4_J, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ Y @ (sup_su1000235569iple_a @ X @ Y))))). % inf_sup_ord(4)
thf(fact_98_subset__image__iff, axiom,
    ((![B : set_Ho137910533iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((ord_le1221261669iple_a @ B @ (image_2025567968iple_a @ F @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B = (image_2025567968iple_a @ F @ AA))))))))). % subset_image_iff
thf(fact_99_subset__image__iff, axiom,
    ((![B : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ (image_1577193952iple_a @ F @ A)) = (?[AA : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ AA @ A)) & ((B = (image_1577193952iple_a @ F @ AA))))))))). % subset_image_iff
thf(fact_100_image__subset__iff, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A) @ B) = (![X5 : pname]: (((member_pname @ X5 @ A)) => ((member1332298086iple_a @ (F @ X5) @ B)))))))). % image_subset_iff
thf(fact_101_subset__imageE, axiom,
    ((![B : set_Ho137910533iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((ord_le1221261669iple_a @ B @ (image_2025567968iple_a @ F @ A)) => (~ ((![C3 : set_pname]: ((ord_le865024672_pname @ C3 @ A) => (~ ((B = (image_2025567968iple_a @ F @ C3)))))))))))). % subset_imageE
thf(fact_102_subset__imageE, axiom,
    ((![B : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ (image_1577193952iple_a @ F @ A)) => (~ ((![C3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C3 @ A) => (~ ((B = (image_1577193952iple_a @ F @ C3)))))))))))). % subset_imageE
thf(fact_103_image__subsetI, axiom,
    ((![A : set_pname, F : pname > hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((![X4 : pname]: ((member_pname @ X4 @ A) => (member1332298086iple_a @ (F @ X4) @ B))) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A) @ B))))). % image_subsetI
thf(fact_104_image__mono, axiom,
    ((![A : set_pname, B : set_pname, F : pname > hoare_1678595023iple_a]: ((ord_le865024672_pname @ A @ B) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ A) @ (image_2025567968iple_a @ F @ B)))))). % image_mono
thf(fact_105_image__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_mono
thf(fact_106_image__Un, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, B : set_pname]: ((image_2025567968iple_a @ F @ (sup_sup_set_pname @ A @ B)) = (sup_su1000235569iple_a @ (image_2025567968iple_a @ F @ A) @ (image_2025567968iple_a @ F @ B)))))). % image_Un
thf(fact_107_image__Un, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (sup_su1000235569iple_a @ A @ B)) = (sup_su1000235569iple_a @ (image_1577193952iple_a @ F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_Un
thf(fact_108_subset__Un__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A3 @ B3) = B3)))))). % subset_Un_eq
thf(fact_109_subset__UnE, axiom,
    ((![C : set_Ho137910533iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ C @ (sup_su1000235569iple_a @ A @ B)) => (~ ((![A5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A5 @ A) => (![B5 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B5 @ B) => (~ ((C = (sup_su1000235569iple_a @ A5 @ B5)))))))))))))). % subset_UnE
thf(fact_110_Un__absorb2, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ A) => ((sup_su1000235569iple_a @ A @ B) = A))))). % Un_absorb2
thf(fact_111_Un__absorb1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((sup_su1000235569iple_a @ A @ B) = B))))). % Un_absorb1
thf(fact_112_Un__upper2, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ B @ (sup_su1000235569iple_a @ A @ B))))). % Un_upper2
thf(fact_113_Un__upper1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ (sup_su1000235569iple_a @ A @ B))))). % Un_upper1
thf(fact_114_Un__least, axiom,
    ((![A : set_Ho137910533iple_a, C : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ C) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A @ B) @ C)))))). % Un_least
thf(fact_115_Un__mono, axiom,
    ((![A : set_Ho137910533iple_a, C : set_Ho137910533iple_a, B : set_Ho137910533iple_a, D2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ C) => ((ord_le1221261669iple_a @ B @ D2) => (ord_le1221261669iple_a @ (sup_su1000235569iple_a @ A @ B) @ (sup_su1000235569iple_a @ C @ D2))))))). % Un_mono
thf(fact_116_order__refl, axiom,
    ((![X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ X)))). % order_refl
thf(fact_117_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((![X4 : pname]: ((P @ X4) => (member1332298086iple_a @ (F @ X4) @ B))) => (ord_le1221261669iple_a @ (image_2025567968iple_a @ F @ (collect_pname @ P)) @ B))))). % image_Collect_subsetI
thf(fact_118_all__subset__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : set_Ho137910533iple_a > $o]: ((![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ (image_2025567968iple_a @ F @ A))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A)) => ((P @ (image_2025567968iple_a @ F @ B3))))))))). % all_subset_image
thf(fact_119_all__subset__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a, P : set_Ho137910533iple_a > $o]: ((![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ (image_1577193952iple_a @ F @ A))) => ((P @ B3)))) = (![B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ A)) => ((P @ (image_1577193952iple_a @ F @ B3))))))))). % all_subset_image
thf(fact_120_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_121_sup__Un__eq, axiom,
    ((![R : set_Ho137910533iple_a, S : set_Ho137910533iple_a]: ((sup_su335701908le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ R)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ S))) = (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ R @ S))))))). % sup_Un_eq
thf(fact_122_pred__subset__eq, axiom,
    ((![R : set_Ho137910533iple_a, S : set_Ho137910533iple_a]: ((ord_le1593108832le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ R)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ S))) = (ord_le1221261669iple_a @ R @ S))))). % pred_subset_eq
thf(fact_123_subset__Collect__iff, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ B @ A) => ((ord_le1221261669iple_a @ B @ (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) & ((P @ X5)))))) = (![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) => ((P @ X5))))))))). % subset_Collect_iff
thf(fact_124_subset__CollectI, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a, Q : hoare_1678595023iple_a > $o, P : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ B @ A) => ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ B) => ((Q @ X4) => (P @ X4)))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) & ((Q @ X5))))) @ (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) & ((P @ X5))))))))))). % subset_CollectI
thf(fact_125_Collect__restrict, axiom,
    ((![X8 : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ X8)) & ((P @ X5))))) @ X8)))). % Collect_restrict
thf(fact_126_image__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ F @ bot_bot_set_pname) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_127_empty__is__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname]: ((bot_bo1298296729iple_a = (image_2025567968iple_a @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_128_image__is__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname]: (((image_2025567968iple_a @ F @ A) = bot_bo1298296729iple_a) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_129_image__insert, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : pname, B : set_pname]: ((image_2025567968iple_a @ F @ (insert_pname @ A2 @ B)) = (insert1477804543iple_a @ (F @ A2) @ (image_2025567968iple_a @ F @ B)))))). % image_insert
thf(fact_130_insert__image, axiom,
    ((![X : pname, A : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X @ A) => ((insert1477804543iple_a @ (F @ X) @ (image_2025567968iple_a @ F @ A)) = (image_2025567968iple_a @ F @ A)))))). % insert_image
thf(fact_131_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_132_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_133_sup__bot__left, axiom,
    ((![X : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ X) = X)))). % sup_bot_left
thf(fact_134_sup__bot__right, axiom,
    ((![X : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X @ bot_bo1298296729iple_a) = X)))). % sup_bot_right
thf(fact_135_bot__eq__sup__iff, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (sup_su1000235569iple_a @ X @ Y)) = (((X = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % bot_eq_sup_iff
thf(fact_136_sup__eq__bot__iff, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ X @ Y) = bot_bo1298296729iple_a) = (((X = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % sup_eq_bot_iff
thf(fact_137_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ A2 @ B2) = bot_bo1298296729iple_a) = (((A2 = bot_bo1298296729iple_a)) & ((B2 = bot_bo1298296729iple_a))))))). % sup_bot.eq_neutr_iff
thf(fact_138_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ A2) = A2)))). % sup_bot.left_neutral

% Conjectures (4)
thf(conj_0, hypothesis,
    ((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)))).
thf(conj_1, hypothesis,
    ((![G4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (sup_su1000235569iple_a @ g @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (p @ P2) @ (body2 @ P2) @ (q @ P2))) @ procs)) @ G4) => (hoare_129598474rivs_a @ G4 @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (p @ P2) @ (the_com @ (body @ P2)) @ (q @ P2))) @ procs)))))).
thf(conj_2, hypothesis,
    ((ord_le1221261669iple_a @ g @ ga))).
thf(conj_3, conjecture,
    ((hoare_129598474rivs_a @ ga @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (p @ P2) @ (body2 @ P2) @ (q @ P2))) @ procs)))).
