% 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_423__3254604_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:41.683

% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $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).

% Explicit typings (32)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite1986656878_state : set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_OMGT, type,
    hoare_Mirabelle_MGT : com > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001t__Com__Ostate, type,
    hoare_604442164_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001t__Com__Ostate, type,
    hoare_318887606_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001t__Com__Ostate, type,
    hoare_1659279548_state : (state > state > $o) > com > (state > state > $o) > hoare_958474565_state).
thf(sy_c_If_001t__Com__Opname, type,
    if_pname : $o > pname > pname > pname).
thf(sy_c_If_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    if_Hoa720680971_state : $o > hoare_958474565_state > hoare_958474565_state > hoare_958474565_state).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Com__Opname_J, type,
    inf_inf_set_pname : set_pname > set_pname > set_pname).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    inf_in1042203155_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
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_It__Com__Ostate_J_M_Eo_J, type,
    sup_su2044190244tate_o : (hoare_958474565_state > $o) > (hoare_958474565_state > $o) > hoare_958474565_state > $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_It__Com__Ostate_J_J, type,
    sup_su737426425_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
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_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    image_2144627828_state : (pname > hoare_958474565_state) > set_pname > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Opname, type,
    image_871391498_pname : (hoare_958474565_state > pname) > set_Ho840737317_state > set_pname).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_v_G, type,
    g : set_Ho840737317_state).
thf(sy_v_Procs, type,
    procs : set_pname).

% Relevant facts (241)
thf(fact_0_cut, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => ((hoare_604442164_state @ G2 @ G) => (hoare_604442164_state @ G2 @ Ts)))))). % cut
thf(fact_1_finite__Un, axiom,
    ((![F : set_pname, G2 : set_pname]: ((finite_finite_pname @ (sup_sup_set_pname @ F @ G2)) = (((finite_finite_pname @ F)) & ((finite_finite_pname @ G2))))))). % finite_Un
thf(fact_2_finite__Un, axiom,
    ((![F : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ (sup_su737426425_state @ F @ G2)) = (((finite1986656878_state @ F)) & ((finite1986656878_state @ G2))))))). % finite_Un
thf(fact_3_finite__imageI, axiom,
    ((![F : set_Ho840737317_state, H : hoare_958474565_state > pname]: ((finite1986656878_state @ F) => (finite_finite_pname @ (image_871391498_pname @ H @ F)))))). % finite_imageI
thf(fact_4_finite__imageI, axiom,
    ((![F : set_Ho840737317_state, H : hoare_958474565_state > hoare_958474565_state]: ((finite1986656878_state @ F) => (finite1986656878_state @ (image_1849792670_state @ H @ F)))))). % finite_imageI
thf(fact_5_finite__imageI, axiom,
    ((![F : set_pname, H : pname > hoare_958474565_state]: ((finite_finite_pname @ F) => (finite1986656878_state @ (image_2144627828_state @ H @ F)))))). % finite_imageI
thf(fact_6_finite__imageI, axiom,
    ((![F : set_pname, H : pname > pname]: ((finite_finite_pname @ F) => (finite_finite_pname @ (image_pname_pname @ H @ F)))))). % finite_imageI
thf(fact_7_finite__Collect__conjI, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: (((finite1986656878_state @ (collec305460656_state @ P)) | (finite1986656878_state @ (collec305460656_state @ Q))) => (finite1986656878_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_8_finite__Collect__conjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: (((finite_finite_pname @ (collect_pname @ P)) | (finite_finite_pname @ (collect_pname @ Q))) => (finite_finite_pname @ (collect_pname @ (^[X : pname]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_9_finite__Collect__disjI, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((finite1986656878_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) | ((Q @ X)))))) = (((finite1986656878_state @ (collec305460656_state @ P))) & ((finite1986656878_state @ (collec305460656_state @ Q)))))))). % finite_Collect_disjI
thf(fact_10_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X : pname]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_11_image__ident, axiom,
    ((![Y : set_pname]: ((image_pname_pname @ (^[X : pname]: X) @ Y) = Y)))). % image_ident
thf(fact_12_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_13_UnCI, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: (((~ ((member_pname @ C @ B))) => (member_pname @ C @ A)) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnCI
thf(fact_14_UnCI, axiom,
    ((![C : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: (((~ ((member109514606_state @ C @ B))) => (member109514606_state @ C @ A)) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnCI
thf(fact_15_Un__iff, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) = (((member_pname @ C @ A)) | ((member_pname @ C @ B))))))). % Un_iff
thf(fact_16_Un__iff, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A @ B)) = (((member109514606_state @ C @ A)) | ((member109514606_state @ C @ B))))))). % Un_iff
thf(fact_17_sup_Oidem, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ A2) = A2)))). % sup.idem
thf(fact_18_sup_Oidem, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ A2 @ A2) = A2)))). % sup.idem
thf(fact_19_sup__idem, axiom,
    ((![X2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ X2) = X2)))). % sup_idem
thf(fact_20_sup__idem, axiom,
    ((![X2 : set_pname]: ((sup_sup_set_pname @ X2 @ X2) = X2)))). % sup_idem
thf(fact_21_sup_Oleft__idem, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ (sup_su737426425_state @ A2 @ B2)) = (sup_su737426425_state @ A2 @ B2))))). % sup.left_idem
thf(fact_22_sup_Oleft__idem, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ A2 @ B2)) = (sup_sup_set_pname @ A2 @ B2))))). % sup.left_idem
thf(fact_23_image__eqI, axiom,
    ((![B2 : pname, F2 : pname > pname, X2 : pname, A : set_pname]: ((B2 = (F2 @ X2)) => ((member_pname @ X2 @ A) => (member_pname @ B2 @ (image_pname_pname @ F2 @ A))))))). % image_eqI
thf(fact_24_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state, X2 : pname, A : set_pname]: ((B2 = (F2 @ X2)) => ((member_pname @ X2 @ A) => (member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A))))))). % image_eqI
thf(fact_25_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F2 : hoare_958474565_state > hoare_958474565_state, X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((B2 = (F2 @ X2)) => ((member109514606_state @ X2 @ A) => (member109514606_state @ B2 @ (image_1849792670_state @ F2 @ A))))))). % image_eqI
thf(fact_26_sup_Oright__idem, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A2 @ B2) @ B2) = (sup_su737426425_state @ A2 @ B2))))). % sup.right_idem
thf(fact_27_sup_Oright__idem, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A2 @ B2) @ B2) = (sup_sup_set_pname @ A2 @ B2))))). % sup.right_idem
thf(fact_28_sup__left__idem, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (sup_su737426425_state @ X2 @ Y2)) = (sup_su737426425_state @ X2 @ Y2))))). % sup_left_idem
thf(fact_29_sup__left__idem, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: ((sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ X2 @ Y2)) = (sup_sup_set_pname @ X2 @ Y2))))). % sup_left_idem
thf(fact_30_sup__set__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (sup_su2044190244tate_o @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A3)) @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ B3))))))))). % sup_set_def
thf(fact_31_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_32_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B2 : pname, F2 : pname > pname]: ((member_pname @ X2 @ A) => ((B2 = (F2 @ X2)) => (member_pname @ B2 @ (image_pname_pname @ F2 @ A))))))). % rev_image_eqI
thf(fact_33_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => ((B2 = (F2 @ X2)) => (member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A))))))). % rev_image_eqI
thf(fact_34_rev__image__eqI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state, F2 : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((B2 = (F2 @ X2)) => (member109514606_state @ B2 @ (image_1849792670_state @ F2 @ A))))))). % rev_image_eqI
thf(fact_35_ball__imageD, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ (image_2144627828_state @ F2 @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_36_ball__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((![X3 : pname]: ((member_pname @ X3 @ (image_pname_pname @ F2 @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_37_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F2 : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X3 : pname]: ((member_pname @ X3 @ N) => ((F2 @ X3) = (G3 @ X3)))) => ((image_2144627828_state @ F2 @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_38_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F2 : pname > pname, G3 : pname > pname]: ((M = N) => ((![X3 : pname]: ((member_pname @ X3 @ N) => ((F2 @ X3) = (G3 @ X3)))) => ((image_pname_pname @ F2 @ M) = (image_pname_pname @ G3 @ N))))))). % image_cong
thf(fact_39_bex__imageD, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_2144627828_state @ F2 @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_40_bex__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((?[X4 : pname]: ((member_pname @ X4 @ (image_pname_pname @ F2 @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_41_image__iff, axiom,
    ((![Z : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ Z @ (image_pname_pname @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_42_image__iff, axiom,
    ((![Z : hoare_958474565_state, F2 : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z @ (image_2144627828_state @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_43_imageI, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > pname]: ((member_pname @ X2 @ A) => (member_pname @ (F2 @ X2) @ (image_pname_pname @ F2 @ A)))))). % imageI
thf(fact_44_imageI, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => (member109514606_state @ (F2 @ X2) @ (image_2144627828_state @ F2 @ A)))))). % imageI
thf(fact_45_imageI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F2 : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => (member109514606_state @ (F2 @ X2) @ (image_1849792670_state @ F2 @ A)))))). % imageI
thf(fact_46_sup__left__commute, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z)) = (sup_su737426425_state @ Y2 @ (sup_su737426425_state @ X2 @ Z)))))). % sup_left_commute
thf(fact_47_sup__left__commute, axiom,
    ((![X2 : set_pname, Y2 : set_pname, Z : set_pname]: ((sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ Y2 @ Z)) = (sup_sup_set_pname @ Y2 @ (sup_sup_set_pname @ X2 @ Z)))))). % sup_left_commute
thf(fact_48_sup_Oleft__commute, axiom,
    ((![B2 : set_Ho840737317_state, A2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((sup_su737426425_state @ B2 @ (sup_su737426425_state @ A2 @ C)) = (sup_su737426425_state @ A2 @ (sup_su737426425_state @ B2 @ C)))))). % sup.left_commute
thf(fact_49_sup_Oleft__commute, axiom,
    ((![B2 : set_pname, A2 : set_pname, C : set_pname]: ((sup_sup_set_pname @ B2 @ (sup_sup_set_pname @ A2 @ C)) = (sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ B2 @ C)))))). % sup.left_commute
thf(fact_50_sup__commute, axiom,
    ((sup_su737426425_state = (^[X : set_Ho840737317_state]: (^[Y3 : set_Ho840737317_state]: (sup_su737426425_state @ Y3 @ X)))))). % sup_commute
thf(fact_51_sup__commute, axiom,
    ((sup_sup_set_pname = (^[X : set_pname]: (^[Y3 : set_pname]: (sup_sup_set_pname @ Y3 @ X)))))). % sup_commute
thf(fact_52_sup_Ocommute, axiom,
    ((sup_su737426425_state = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (sup_su737426425_state @ B4 @ A4)))))). % sup.commute
thf(fact_53_sup_Ocommute, axiom,
    ((sup_sup_set_pname = (^[A4 : set_pname]: (^[B4 : set_pname]: (sup_sup_set_pname @ B4 @ A4)))))). % sup.commute
thf(fact_54_sup__assoc, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X2 @ Y2) @ Z) = (sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z)))))). % sup_assoc
thf(fact_55_sup__assoc, axiom,
    ((![X2 : set_pname, Y2 : set_pname, Z : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ X2 @ Y2) @ Z) = (sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ Y2 @ Z)))))). % sup_assoc
thf(fact_56_sup_Oassoc, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A2 @ B2) @ C) = (sup_su737426425_state @ A2 @ (sup_su737426425_state @ B2 @ C)))))). % sup.assoc
thf(fact_57_sup_Oassoc, axiom,
    ((![A2 : set_pname, B2 : set_pname, C : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A2 @ B2) @ C) = (sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ B2 @ C)))))). % sup.assoc
thf(fact_58_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_Ho840737317_state, K : set_Ho840737317_state, B2 : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((B = (sup_su737426425_state @ K @ B2)) => ((sup_su737426425_state @ A2 @ B) = (sup_su737426425_state @ K @ (sup_su737426425_state @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_59_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_pname, K : set_pname, B2 : set_pname, A2 : set_pname]: ((B = (sup_sup_set_pname @ K @ B2)) => ((sup_sup_set_pname @ A2 @ B) = (sup_sup_set_pname @ K @ (sup_sup_set_pname @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_60_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_Ho840737317_state, K : set_Ho840737317_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A = (sup_su737426425_state @ K @ A2)) => ((sup_su737426425_state @ A @ B2) = (sup_su737426425_state @ K @ (sup_su737426425_state @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_61_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_pname, K : set_pname, A2 : set_pname, B2 : set_pname]: ((A = (sup_sup_set_pname @ K @ A2)) => ((sup_sup_set_pname @ A @ B2) = (sup_sup_set_pname @ K @ (sup_sup_set_pname @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_62_inf__sup__aci_I5_J, axiom,
    ((sup_su737426425_state = (^[X : set_Ho840737317_state]: (^[Y3 : set_Ho840737317_state]: (sup_su737426425_state @ Y3 @ X)))))). % inf_sup_aci(5)
thf(fact_63_inf__sup__aci_I5_J, axiom,
    ((sup_sup_set_pname = (^[X : set_pname]: (^[Y3 : set_pname]: (sup_sup_set_pname @ Y3 @ X)))))). % inf_sup_aci(5)
thf(fact_64_inf__sup__aci_I6_J, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X2 @ Y2) @ Z) = (sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z)))))). % inf_sup_aci(6)
thf(fact_65_inf__sup__aci_I6_J, axiom,
    ((![X2 : set_pname, Y2 : set_pname, Z : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ X2 @ Y2) @ Z) = (sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ Y2 @ Z)))))). % inf_sup_aci(6)
thf(fact_66_inf__sup__aci_I7_J, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z)) = (sup_su737426425_state @ Y2 @ (sup_su737426425_state @ X2 @ Z)))))). % inf_sup_aci(7)
thf(fact_67_inf__sup__aci_I7_J, axiom,
    ((![X2 : set_pname, Y2 : set_pname, Z : set_pname]: ((sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ Y2 @ Z)) = (sup_sup_set_pname @ Y2 @ (sup_sup_set_pname @ X2 @ Z)))))). % inf_sup_aci(7)
thf(fact_68_inf__sup__aci_I8_J, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (sup_su737426425_state @ X2 @ Y2)) = (sup_su737426425_state @ X2 @ Y2))))). % inf_sup_aci(8)
thf(fact_69_inf__sup__aci_I8_J, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: ((sup_sup_set_pname @ X2 @ (sup_sup_set_pname @ X2 @ Y2)) = (sup_sup_set_pname @ X2 @ Y2))))). % inf_sup_aci(8)
thf(fact_70_Un__left__commute, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C2)) = (sup_su737426425_state @ B @ (sup_su737426425_state @ A @ C2)))))). % Un_left_commute
thf(fact_71_Un__left__commute, axiom,
    ((![A : set_pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ A @ (sup_sup_set_pname @ B @ C2)) = (sup_sup_set_pname @ B @ (sup_sup_set_pname @ A @ C2)))))). % Un_left_commute
thf(fact_72_Un__left__absorb, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ A @ B))))). % Un_left_absorb
thf(fact_73_Un__left__absorb, axiom,
    ((![A : set_pname, B : set_pname]: ((sup_sup_set_pname @ A @ (sup_sup_set_pname @ A @ B)) = (sup_sup_set_pname @ A @ B))))). % Un_left_absorb
thf(fact_74_Un__commute, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (sup_su737426425_state @ B3 @ A3)))))). % Un_commute
thf(fact_75_Un__commute, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (sup_sup_set_pname @ B3 @ A3)))))). % Un_commute
thf(fact_76_Un__absorb, axiom,
    ((![A : set_Ho840737317_state]: ((sup_su737426425_state @ A @ A) = A)))). % Un_absorb
thf(fact_77_Un__absorb, axiom,
    ((![A : set_pname]: ((sup_sup_set_pname @ A @ A) = A)))). % Un_absorb
thf(fact_78_Un__assoc, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A @ B) @ C2) = (sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C2)))))). % Un_assoc
thf(fact_79_Un__assoc, axiom,
    ((![A : set_pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A @ B) @ C2) = (sup_sup_set_pname @ A @ (sup_sup_set_pname @ B @ C2)))))). % Un_assoc
thf(fact_80_ball__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A @ B))) => ((P @ X)))) = (((![X : hoare_958474565_state]: (((member109514606_state @ X @ A)) => ((P @ X))))) & ((![X : hoare_958474565_state]: (((member109514606_state @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_81_ball__Un, axiom,
    ((![A : set_pname, B : set_pname, P : pname > $o]: ((![X : pname]: (((member_pname @ X @ (sup_sup_set_pname @ A @ B))) => ((P @ X)))) = (((![X : pname]: (((member_pname @ X @ A)) => ((P @ X))))) & ((![X : pname]: (((member_pname @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_82_mem__Collect__eq, axiom,
    ((![A2 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((member109514606_state @ A2 @ (collec305460656_state @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_83_mem__Collect__eq, axiom,
    ((![A2 : pname, P : pname > $o]: ((member_pname @ A2 @ (collect_pname @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_84_Collect__mem__eq, axiom,
    ((![A : set_Ho840737317_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_85_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_86_Collect__cong, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X3 : pname]: ((P @ X3) = (Q @ X3))) => ((collect_pname @ P) = (collect_pname @ Q)))))). % Collect_cong
thf(fact_87_bex__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A @ B))) & ((P @ X)))) = (((?[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ X))))) | ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_88_bex__Un, axiom,
    ((![A : set_pname, B : set_pname, P : pname > $o]: ((?[X : pname]: (((member_pname @ X @ (sup_sup_set_pname @ A @ B))) & ((P @ X)))) = (((?[X : pname]: (((member_pname @ X @ A)) & ((P @ X))))) | ((?[X : pname]: (((member_pname @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_89_UnI2, axiom,
    ((![C : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: ((member109514606_state @ C @ B) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnI2
thf(fact_90_UnI2, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: ((member_pname @ C @ B) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI2
thf(fact_91_UnI1, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ A) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnI1
thf(fact_92_UnI1, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ A) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI1
thf(fact_93_UnE, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A @ B)) => ((~ ((member109514606_state @ C @ A))) => (member109514606_state @ C @ B)))))). % UnE
thf(fact_94_UnE, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) => ((~ ((member_pname @ C @ A))) => (member_pname @ C @ B)))))). % UnE
thf(fact_95_Compr__image__eq, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_1849792670_state @ F2 @ A))) & ((P @ X))))) = (image_1849792670_state @ F2 @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_96_Compr__image__eq, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_2144627828_state @ F2 @ A))) & ((P @ X))))) = (image_2144627828_state @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_97_Compr__image__eq, axiom,
    ((![F2 : hoare_958474565_state > pname, A : set_Ho840737317_state, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_871391498_pname @ F2 @ A))) & ((P @ X))))) = (image_871391498_pname @ F2 @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_98_Compr__image__eq, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F2 @ A))) & ((P @ X))))) = (image_pname_pname @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_99_image__image, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F2 @ (image_2144627828_state @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_100_image__image, axiom,
    ((![F2 : hoare_958474565_state > pname, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_871391498_pname @ F2 @ (image_2144627828_state @ G3 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_101_image__image, axiom,
    ((![F2 : pname > hoare_958474565_state, G3 : pname > pname, A : set_pname]: ((image_2144627828_state @ F2 @ (image_pname_pname @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_102_image__image, axiom,
    ((![F2 : pname > pname, G3 : pname > pname, A : set_pname]: ((image_pname_pname @ F2 @ (image_pname_pname @ G3 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_103_imageE, axiom,
    ((![B2 : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F2 @ A)) => (~ ((![X3 : pname]: ((B2 = (F2 @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_104_imageE, axiom,
    ((![B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A)) => (~ ((![X3 : pname]: ((B2 = (F2 @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_105_imageE, axiom,
    ((![B2 : hoare_958474565_state, F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ B2 @ (image_1849792670_state @ F2 @ A)) => (~ ((![X3 : hoare_958474565_state]: ((B2 = (F2 @ X3)) => (~ ((member109514606_state @ X3 @ A))))))))))). % imageE
thf(fact_106_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_107_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B : set_Ho840737317_state, R : pname > hoare_958474565_state > $o]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ B) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B) & (R @ X3 @ Xa))))) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_108_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B : set_pname, R : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ B) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_109_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, R : hoare_958474565_state > hoare_958474565_state > $o]: ((~ ((finite1986656878_state @ A))) => ((finite1986656878_state @ B) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B) & (R @ X3 @ Xa))))) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_110_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_111_not__finite__existsD, axiom,
    ((![P : hoare_958474565_state > $o]: ((~ ((finite1986656878_state @ (collec305460656_state @ P)))) => (?[X_1 : hoare_958474565_state]: (P @ X_1)))))). % not_finite_existsD
thf(fact_112_Collect__disj__eq, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) | ((Q @ X))))) = (sup_su737426425_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_disj_eq
thf(fact_113_Collect__disj__eq, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((collect_pname @ (^[X : pname]: (((P @ X)) | ((Q @ X))))) = (sup_sup_set_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_disj_eq
thf(fact_114_Un__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A3)) | ((member109514606_state @ X @ B3)))))))))). % Un_def
thf(fact_115_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_116_image__Un, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F2 @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ (image_1849792670_state @ F2 @ A) @ (image_1849792670_state @ F2 @ B)))))). % image_Un
thf(fact_117_image__Un, axiom,
    ((![F2 : hoare_958474565_state > pname, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_871391498_pname @ F2 @ (sup_su737426425_state @ A @ B)) = (sup_sup_set_pname @ (image_871391498_pname @ F2 @ A) @ (image_871391498_pname @ F2 @ B)))))). % image_Un
thf(fact_118_image__Un, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, B : set_pname]: ((image_2144627828_state @ F2 @ (sup_sup_set_pname @ A @ B)) = (sup_su737426425_state @ (image_2144627828_state @ F2 @ A) @ (image_2144627828_state @ F2 @ B)))))). % image_Un
thf(fact_119_image__Un, axiom,
    ((![F2 : pname > pname, A : set_pname, B : set_pname]: ((image_pname_pname @ F2 @ (sup_sup_set_pname @ A @ B)) = (sup_sup_set_pname @ (image_pname_pname @ F2 @ A) @ (image_pname_pname @ F2 @ B)))))). % image_Un
thf(fact_120_infinite__Un, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((~ ((finite1986656878_state @ (sup_su737426425_state @ S @ T)))) = (((~ ((finite1986656878_state @ S)))) | ((~ ((finite1986656878_state @ T))))))))). % infinite_Un
thf(fact_121_infinite__Un, axiom,
    ((![S : set_pname, T : set_pname]: ((~ ((finite_finite_pname @ (sup_sup_set_pname @ S @ T)))) = (((~ ((finite_finite_pname @ S)))) | ((~ ((finite_finite_pname @ T))))))))). % infinite_Un
thf(fact_122_Un__infinite, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S))) => (~ ((finite1986656878_state @ (sup_su737426425_state @ S @ T)))))))). % Un_infinite
thf(fact_123_Un__infinite, axiom,
    ((![S : set_pname, T : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ (sup_sup_set_pname @ S @ T)))))))). % Un_infinite
thf(fact_124_finite__UnI, axiom,
    ((![F : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ F) => ((finite1986656878_state @ G2) => (finite1986656878_state @ (sup_su737426425_state @ F @ G2))))))). % finite_UnI
thf(fact_125_finite__UnI, axiom,
    ((![F : set_pname, G2 : set_pname]: ((finite_finite_pname @ F) => ((finite_finite_pname @ G2) => (finite_finite_pname @ (sup_sup_set_pname @ F @ G2))))))). % finite_UnI
thf(fact_126_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F2 @ A4) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_127_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > hoare_958474565_state]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ (image_2144627828_state @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F2 @ A4) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_128_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F2 : hoare_958474565_state > pname]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ (image_871391498_pname @ F2 @ A)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & (((F2 @ A4) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_129_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F2 : hoare_958474565_state > hoare_958474565_state]: ((~ ((finite1986656878_state @ A))) => ((finite1986656878_state @ (image_1849792670_state @ F2 @ A)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & (((F2 @ A4) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_130_hoare__derivs_OBody, axiom,
    ((![G2 : set_Ho840737317_state, P : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => (hoare_604442164_state @ G2 @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)))))). % hoare_derivs.Body
thf(fact_131_Body__sound__lemma, axiom,
    ((![G2 : set_Ho840737317_state, P : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname]: ((hoare_318887606_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn))) @ Procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (the_com @ (body @ Pn)) @ (Q @ Pn))) @ Procs)) => (hoare_318887606_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn))) @ Procs)))))). % Body_sound_lemma
thf(fact_132_sup__Un__eq, axiom,
    ((![R : set_Ho840737317_state, S : set_Ho840737317_state]: ((sup_su2044190244tate_o @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ R)) @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ S))) = (^[X : hoare_958474565_state]: (member109514606_state @ X @ (sup_su737426425_state @ R @ S))))))). % sup_Un_eq
thf(fact_133_sup__Un__eq, axiom,
    ((![R : set_pname, S : set_pname]: ((sup_sup_pname_o @ (^[X : pname]: (member_pname @ X @ R)) @ (^[X : pname]: (member_pname @ X @ S))) = (^[X : pname]: (member_pname @ X @ (sup_sup_set_pname @ R @ S))))))). % sup_Un_eq
thf(fact_134_Sup_OSUP__identity__eq, axiom,
    ((![Sup : set_pname > pname, A : set_pname]: ((Sup @ (image_pname_pname @ (^[X : pname]: X) @ A)) = (Sup @ A))))). % Sup.SUP_identity_eq
thf(fact_135_Inf_OINF__identity__eq, axiom,
    ((![Inf : set_pname > pname, A : set_pname]: ((Inf @ (image_pname_pname @ (^[X : pname]: X) @ A)) = (Inf @ A))))). % Inf.INF_identity_eq
thf(fact_136_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_137_MGT__BodyN, axiom,
    ((![Pn2 : pname, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn2)) @ G2) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn2)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_138_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X3 : pname]: ((member_pname @ X3 @ B) => ((C2 @ X3) = (D @ X3)))) => ((Sup @ (image_2144627828_state @ C2 @ A)) = (Sup @ (image_2144627828_state @ D @ B)))))))). % Sup.SUP_cong
thf(fact_139_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > pname, D : pname > pname, Sup : set_pname > pname]: ((A = B) => ((![X3 : pname]: ((member_pname @ X3 @ B) => ((C2 @ X3) = (D @ X3)))) => ((Sup @ (image_pname_pname @ C2 @ A)) = (Sup @ (image_pname_pname @ D @ B)))))))). % Sup.SUP_cong
thf(fact_140_Inf_OINF__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X3 : pname]: ((member_pname @ X3 @ B) => ((C2 @ X3) = (D @ X3)))) => ((Inf @ (image_2144627828_state @ C2 @ A)) = (Inf @ (image_2144627828_state @ D @ B)))))))). % Inf.INF_cong
thf(fact_141_Inf_OINF__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > pname, D : pname > pname, Inf : set_pname > pname]: ((A = B) => ((![X3 : pname]: ((member_pname @ X3 @ B) => ((C2 @ X3) = (D @ X3)))) => ((Inf @ (image_pname_pname @ C2 @ A)) = (Inf @ (image_pname_pname @ D @ B)))))))). % Inf.INF_cong
thf(fact_142_if__image__distrib, axiom,
    ((![P : pname > $o, F2 : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state, S : set_pname]: ((image_2144627828_state @ (^[X : pname]: (if_Hoa720680971_state @ (P @ X) @ (F2 @ X) @ (G3 @ X))) @ S) = (sup_su737426425_state @ (image_2144627828_state @ F2 @ (inf_inf_set_pname @ S @ (collect_pname @ P))) @ (image_2144627828_state @ G3 @ (inf_inf_set_pname @ S @ (collect_pname @ (^[X : pname]: (~ ((P @ X)))))))))))). % if_image_distrib
thf(fact_143_if__image__distrib, axiom,
    ((![P : pname > $o, F2 : pname > pname, G3 : pname > pname, S : set_pname]: ((image_pname_pname @ (^[X : pname]: (if_pname @ (P @ X) @ (F2 @ X) @ (G3 @ X))) @ S) = (sup_sup_set_pname @ (image_pname_pname @ F2 @ (inf_inf_set_pname @ S @ (collect_pname @ P))) @ (image_pname_pname @ G3 @ (inf_inf_set_pname @ S @ (collect_pname @ (^[X : pname]: (~ ((P @ X)))))))))))). % if_image_distrib
thf(fact_144_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_145_all__not__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((![X : hoare_958474565_state]: (~ ((member109514606_state @ X @ A)))) = (A = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_146_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X : pname]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_147_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_148_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X : pname]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_149_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_150_insertCI, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: (((~ ((member109514606_state @ A2 @ B))) => (A2 = B2)) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertCI
thf(fact_151_insert__iff, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) = (((A2 = B2)) | ((member109514606_state @ A2 @ A))))))). % insert_iff
thf(fact_152_insert__absorb2, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X2 @ (insert776267541_state @ X2 @ A)) = (insert776267541_state @ X2 @ A))))). % insert_absorb2
thf(fact_153_IntI, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ A) => ((member109514606_state @ C @ B) => (member109514606_state @ C @ (inf_in1042203155_state @ A @ B))))))). % IntI
thf(fact_154_Int__iff, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (inf_in1042203155_state @ A @ B)) = (((member109514606_state @ C @ A)) & ((member109514606_state @ C @ B))))))). % Int_iff
thf(fact_155_image__empty, axiom,
    ((![F2 : pname > pname]: ((image_pname_pname @ F2 @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_156_image__empty, axiom,
    ((![F2 : pname > hoare_958474565_state]: ((image_2144627828_state @ F2 @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_157_image__empty, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F2 @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_158_empty__is__image, axiom,
    ((![F2 : pname > pname, A : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F2 @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_159_empty__is__image, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname]: ((bot_bo105666705_state = (image_2144627828_state @ F2 @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_160_empty__is__image, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((bot_bo105666705_state = (image_1849792670_state @ F2 @ A)) = (A = bot_bo105666705_state))))). % empty_is_image
thf(fact_161_image__is__empty, axiom,
    ((![F2 : pname > pname, A : set_pname]: (((image_pname_pname @ F2 @ A) = bot_bot_set_pname) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_162_image__is__empty, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname]: (((image_2144627828_state @ F2 @ A) = bot_bo105666705_state) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_163_image__is__empty, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: (((image_1849792670_state @ F2 @ A) = bot_bo105666705_state) = (A = bot_bo105666705_state))))). % image_is_empty
thf(fact_164_image__insert, axiom,
    ((![F2 : pname > pname, A2 : pname, B : set_pname]: ((image_pname_pname @ F2 @ (insert_pname @ A2 @ B)) = (insert_pname @ (F2 @ A2) @ (image_pname_pname @ F2 @ B)))))). % image_insert
thf(fact_165_image__insert, axiom,
    ((![F2 : pname > hoare_958474565_state, A2 : pname, B : set_pname]: ((image_2144627828_state @ F2 @ (insert_pname @ A2 @ B)) = (insert776267541_state @ (F2 @ A2) @ (image_2144627828_state @ F2 @ B)))))). % image_insert
thf(fact_166_image__insert, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A2 : hoare_958474565_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F2 @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ (F2 @ A2) @ (image_1849792670_state @ F2 @ B)))))). % image_insert
thf(fact_167_insert__image, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > pname]: ((member_pname @ X2 @ A) => ((insert_pname @ (F2 @ X2) @ (image_pname_pname @ F2 @ A)) = (image_pname_pname @ F2 @ A)))))). % insert_image
thf(fact_168_insert__image, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => ((insert776267541_state @ (F2 @ X2) @ (image_2144627828_state @ F2 @ A)) = (image_2144627828_state @ F2 @ A)))))). % insert_image
thf(fact_169_insert__image, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F2 : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((insert776267541_state @ (F2 @ X2) @ (image_1849792670_state @ F2 @ A)) = (image_1849792670_state @ F2 @ A)))))). % insert_image
thf(fact_170_inf__bot__left, axiom,
    ((![X2 : set_Ho840737317_state]: ((inf_in1042203155_state @ bot_bo105666705_state @ X2) = bot_bo105666705_state)))). % inf_bot_left
thf(fact_171_inf__bot__right, axiom,
    ((![X2 : set_Ho840737317_state]: ((inf_in1042203155_state @ X2 @ bot_bo105666705_state) = bot_bo105666705_state)))). % inf_bot_right
thf(fact_172_sup__bot__left, axiom,
    ((![X2 : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ X2) = X2)))). % sup_bot_left
thf(fact_173_sup__bot__left, axiom,
    ((![X2 : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ X2) = X2)))). % sup_bot_left
thf(fact_174_sup__bot__right, axiom,
    ((![X2 : set_pname]: ((sup_sup_set_pname @ X2 @ bot_bot_set_pname) = X2)))). % sup_bot_right
thf(fact_175_sup__bot__right, axiom,
    ((![X2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ bot_bo105666705_state) = X2)))). % sup_bot_right
thf(fact_176_bot__eq__sup__iff, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ X2 @ Y2)) = (((X2 = bot_bot_set_pname)) & ((Y2 = bot_bot_set_pname))))))). % bot_eq_sup_iff
thf(fact_177_bot__eq__sup__iff, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((bot_bo105666705_state = (sup_su737426425_state @ X2 @ Y2)) = (((X2 = bot_bo105666705_state)) & ((Y2 = bot_bo105666705_state))))))). % bot_eq_sup_iff
thf(fact_178_sup__eq__bot__iff, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: (((sup_sup_set_pname @ X2 @ Y2) = bot_bot_set_pname) = (((X2 = bot_bot_set_pname)) & ((Y2 = bot_bot_set_pname))))))). % sup_eq_bot_iff
thf(fact_179_sup__eq__bot__iff, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: (((sup_su737426425_state @ X2 @ Y2) = bot_bo105666705_state) = (((X2 = bot_bo105666705_state)) & ((Y2 = bot_bo105666705_state))))))). % sup_eq_bot_iff
thf(fact_180_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_pname, B2 : set_pname]: (((sup_sup_set_pname @ A2 @ B2) = bot_bot_set_pname) = (((A2 = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.eq_neutr_iff
thf(fact_181_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: (((sup_su737426425_state @ A2 @ B2) = bot_bo105666705_state) = (((A2 = bot_bo105666705_state)) & ((B2 = bot_bo105666705_state))))))). % sup_bot.eq_neutr_iff
thf(fact_182_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_183_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_184_sup__bot_Oneutr__eq__iff, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ A2 @ B2)) = (((A2 = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.neutr_eq_iff
thf(fact_185_sup__bot_Oneutr__eq__iff, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((bot_bo105666705_state = (sup_su737426425_state @ A2 @ B2)) = (((A2 = bot_bo105666705_state)) & ((B2 = bot_bo105666705_state))))))). % sup_bot.neutr_eq_iff
thf(fact_186_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ A2 @ bot_bot_set_pname) = A2)))). % sup_bot.right_neutral
thf(fact_187_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ bot_bo105666705_state) = A2)))). % sup_bot.right_neutral
thf(fact_188_singletonI, axiom,
    ((![A2 : hoare_958474565_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singletonI
thf(fact_189_finite__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A2 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_190_finite__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A2 @ A)) = (finite1986656878_state @ A))))). % finite_insert
thf(fact_191_inf__sup__absorb, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((inf_in1042203155_state @ X2 @ (sup_su737426425_state @ X2 @ Y2)) = X2)))). % inf_sup_absorb
thf(fact_192_inf__sup__absorb, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: ((inf_inf_set_pname @ X2 @ (sup_sup_set_pname @ X2 @ Y2)) = X2)))). % inf_sup_absorb
thf(fact_193_sup__inf__absorb, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (inf_in1042203155_state @ X2 @ Y2)) = X2)))). % sup_inf_absorb
thf(fact_194_sup__inf__absorb, axiom,
    ((![X2 : set_pname, Y2 : set_pname]: ((sup_sup_set_pname @ X2 @ (inf_inf_set_pname @ X2 @ Y2)) = X2)))). % sup_inf_absorb
thf(fact_195_finite__Int, axiom,
    ((![F : set_pname, G2 : set_pname]: (((finite_finite_pname @ F) | (finite_finite_pname @ G2)) => (finite_finite_pname @ (inf_inf_set_pname @ F @ G2)))))). % finite_Int
thf(fact_196_finite__Int, axiom,
    ((![F : set_Ho840737317_state, G2 : set_Ho840737317_state]: (((finite1986656878_state @ F) | (finite1986656878_state @ G2)) => (finite1986656878_state @ (inf_in1042203155_state @ F @ G2)))))). % finite_Int
thf(fact_197_Un__empty, axiom,
    ((![A : set_pname, B : set_pname]: (((sup_sup_set_pname @ A @ B) = bot_bot_set_pname) = (((A = bot_bot_set_pname)) & ((B = bot_bot_set_pname))))))). % Un_empty
thf(fact_198_Un__empty, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: (((sup_su737426425_state @ A @ B) = bot_bo105666705_state) = (((A = bot_bo105666705_state)) & ((B = bot_bo105666705_state))))))). % Un_empty
thf(fact_199_Int__insert__left__if0, axiom,
    ((![A2 : hoare_958474565_state, C2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A2 @ C2))) => ((inf_in1042203155_state @ (insert776267541_state @ A2 @ B) @ C2) = (inf_in1042203155_state @ B @ C2)))))). % Int_insert_left_if0
thf(fact_200_Int__insert__left__if1, axiom,
    ((![A2 : hoare_958474565_state, C2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ A2 @ C2) => ((inf_in1042203155_state @ (insert776267541_state @ A2 @ B) @ C2) = (insert776267541_state @ A2 @ (inf_in1042203155_state @ B @ C2))))))). % Int_insert_left_if1
thf(fact_201_insert__inter__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((inf_in1042203155_state @ (insert776267541_state @ A2 @ A) @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ A2 @ (inf_in1042203155_state @ A @ B)))))). % insert_inter_insert
thf(fact_202_Int__insert__right__if0, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A2 @ A))) => ((inf_in1042203155_state @ A @ (insert776267541_state @ A2 @ B)) = (inf_in1042203155_state @ A @ B)))))). % Int_insert_right_if0
thf(fact_203_Int__insert__right__if1, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => ((inf_in1042203155_state @ A @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ A2 @ (inf_in1042203155_state @ A @ B))))))). % Int_insert_right_if1
thf(fact_204_Un__insert__left, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (insert776267541_state @ A2 @ B) @ C2) = (insert776267541_state @ A2 @ (sup_su737426425_state @ B @ C2)))))). % Un_insert_left
thf(fact_205_Un__insert__left, axiom,
    ((![A2 : pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ (insert_pname @ A2 @ B) @ C2) = (insert_pname @ A2 @ (sup_sup_set_pname @ B @ C2)))))). % Un_insert_left
thf(fact_206_Un__insert__right, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state, B : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ A2 @ (sup_su737426425_state @ A @ B)))))). % Un_insert_right
thf(fact_207_Un__insert__right, axiom,
    ((![A : set_pname, A2 : pname, B : set_pname]: ((sup_sup_set_pname @ A @ (insert_pname @ A2 @ B)) = (insert_pname @ A2 @ (sup_sup_set_pname @ A @ B)))))). % Un_insert_right
thf(fact_208_Int__Un__eq_I4_J, axiom,
    ((![T : set_Ho840737317_state, S : set_Ho840737317_state]: ((sup_su737426425_state @ T @ (inf_in1042203155_state @ S @ T)) = T)))). % Int_Un_eq(4)
thf(fact_209_Int__Un__eq_I4_J, axiom,
    ((![T : set_pname, S : set_pname]: ((sup_sup_set_pname @ T @ (inf_inf_set_pname @ S @ T)) = T)))). % Int_Un_eq(4)
thf(fact_210_Int__Un__eq_I3_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((sup_su737426425_state @ S @ (inf_in1042203155_state @ S @ T)) = S)))). % Int_Un_eq(3)
thf(fact_211_Int__Un__eq_I3_J, axiom,
    ((![S : set_pname, T : set_pname]: ((sup_sup_set_pname @ S @ (inf_inf_set_pname @ S @ T)) = S)))). % Int_Un_eq(3)
thf(fact_212_Int__Un__eq_I2_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((sup_su737426425_state @ (inf_in1042203155_state @ S @ T) @ T) = T)))). % Int_Un_eq(2)
thf(fact_213_Int__Un__eq_I2_J, axiom,
    ((![S : set_pname, T : set_pname]: ((sup_sup_set_pname @ (inf_inf_set_pname @ S @ T) @ T) = T)))). % Int_Un_eq(2)
thf(fact_214_Int__Un__eq_I1_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((sup_su737426425_state @ (inf_in1042203155_state @ S @ T) @ S) = S)))). % Int_Un_eq(1)
thf(fact_215_Int__Un__eq_I1_J, axiom,
    ((![S : set_pname, T : set_pname]: ((sup_sup_set_pname @ (inf_inf_set_pname @ S @ T) @ S) = S)))). % Int_Un_eq(1)
thf(fact_216_Un__Int__eq_I4_J, axiom,
    ((![T : set_Ho840737317_state, S : set_Ho840737317_state]: ((inf_in1042203155_state @ T @ (sup_su737426425_state @ S @ T)) = T)))). % Un_Int_eq(4)
thf(fact_217_Un__Int__eq_I4_J, axiom,
    ((![T : set_pname, S : set_pname]: ((inf_inf_set_pname @ T @ (sup_sup_set_pname @ S @ T)) = T)))). % Un_Int_eq(4)
thf(fact_218_Un__Int__eq_I3_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((inf_in1042203155_state @ S @ (sup_su737426425_state @ S @ T)) = S)))). % Un_Int_eq(3)
thf(fact_219_Un__Int__eq_I3_J, axiom,
    ((![S : set_pname, T : set_pname]: ((inf_inf_set_pname @ S @ (sup_sup_set_pname @ S @ T)) = S)))). % Un_Int_eq(3)
thf(fact_220_Un__Int__eq_I2_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((inf_in1042203155_state @ (sup_su737426425_state @ S @ T) @ T) = T)))). % Un_Int_eq(2)
thf(fact_221_Un__Int__eq_I2_J, axiom,
    ((![S : set_pname, T : set_pname]: ((inf_inf_set_pname @ (sup_sup_set_pname @ S @ T) @ T) = T)))). % Un_Int_eq(2)
thf(fact_222_Un__Int__eq_I1_J, axiom,
    ((![S : set_Ho840737317_state, T : set_Ho840737317_state]: ((inf_in1042203155_state @ (sup_su737426425_state @ S @ T) @ S) = S)))). % Un_Int_eq(1)
thf(fact_223_Un__Int__eq_I1_J, axiom,
    ((![S : set_pname, T : set_pname]: ((inf_inf_set_pname @ (sup_sup_set_pname @ S @ T) @ S) = S)))). % Un_Int_eq(1)
thf(fact_224_singleton__conv, axiom,
    ((![A2 : pname]: ((collect_pname @ (^[X : pname]: (X = A2))) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv
thf(fact_225_singleton__conv, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (X = A2))) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv
thf(fact_226_singleton__conv2, axiom,
    ((![A2 : pname]: ((collect_pname @ ((^[Y4 : pname]: (^[Z2 : pname]: (Y4 = Z2))) @ A2)) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_227_singleton__conv2, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ ((^[Y4 : hoare_958474565_state]: (^[Z2 : hoare_958474565_state]: (Y4 = Z2))) @ A2)) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_228_disjoint__insert_I2_J, axiom,
    ((![A : set_Ho840737317_state, B2 : hoare_958474565_state, B : set_Ho840737317_state]: ((bot_bo105666705_state = (inf_in1042203155_state @ A @ (insert776267541_state @ B2 @ B))) = (((~ ((member109514606_state @ B2 @ A)))) & ((bot_bo105666705_state = (inf_in1042203155_state @ A @ B)))))))). % disjoint_insert(2)
thf(fact_229_disjoint__insert_I1_J, axiom,
    ((![B : set_Ho840737317_state, A2 : hoare_958474565_state, A : set_Ho840737317_state]: (((inf_in1042203155_state @ B @ (insert776267541_state @ A2 @ A)) = bot_bo105666705_state) = (((~ ((member109514606_state @ A2 @ B)))) & (((inf_in1042203155_state @ B @ A) = bot_bo105666705_state))))))). % disjoint_insert(1)
thf(fact_230_insert__disjoint_I2_J, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((bot_bo105666705_state = (inf_in1042203155_state @ (insert776267541_state @ A2 @ A) @ B)) = (((~ ((member109514606_state @ A2 @ B)))) & ((bot_bo105666705_state = (inf_in1042203155_state @ A @ B)))))))). % insert_disjoint(2)
thf(fact_231_insert__disjoint_I1_J, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: (((inf_in1042203155_state @ (insert776267541_state @ A2 @ A) @ B) = bot_bo105666705_state) = (((~ ((member109514606_state @ A2 @ B)))) & (((inf_in1042203155_state @ A @ B) = bot_bo105666705_state))))))). % insert_disjoint(1)
thf(fact_232_IntE, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (inf_in1042203155_state @ A @ B)) => (~ (((member109514606_state @ C @ A) => (~ ((member109514606_state @ C @ B)))))))))). % IntE
thf(fact_233_IntD1, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (inf_in1042203155_state @ A @ B)) => (member109514606_state @ C @ A))))). % IntD1
thf(fact_234_IntD2, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (inf_in1042203155_state @ A @ B)) => (member109514606_state @ C @ B))))). % IntD2
thf(fact_235_emptyE, axiom,
    ((![A2 : hoare_958474565_state]: (~ ((member109514606_state @ A2 @ bot_bo105666705_state)))))). % emptyE
thf(fact_236_insertE, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) => ((~ ((A2 = B2))) => (member109514606_state @ A2 @ A)))))). % insertE
thf(fact_237_equals0D, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((A = bot_bo105666705_state) => (~ ((member109514606_state @ A2 @ A))))))). % equals0D
thf(fact_238_equals0I, axiom,
    ((![A : set_Ho840737317_state]: ((![Y5 : hoare_958474565_state]: (~ ((member109514606_state @ Y5 @ A)))) => (A = bot_bo105666705_state))))). % equals0I
thf(fact_239_insertI1, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ B))))). % insertI1
thf(fact_240_insertI2, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((member109514606_state @ A2 @ B) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertI2

% Helper facts (5)
thf(help_If_2_1_If_001t__Com__Opname_T, axiom,
    ((![X2 : pname, Y2 : pname]: ((if_pname @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Com__Opname_T, axiom,
    ((![X2 : pname, Y2 : pname]: ((if_pname @ $true @ X2 @ Y2) = X2)))).
thf(help_If_3_1_If_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_T, axiom,
    ((![X2 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((if_Hoa720680971_state @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_T, axiom,
    ((![X2 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((if_Hoa720680971_state @ $true @ X2 @ Y2) = X2)))).

% Conjectures (3)
thf(conj_0, hypothesis,
    ((hoare_604442164_state @ (sup_su737426425_state @ g @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn))) @ procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn)))) @ procs)))).
thf(conj_1, hypothesis,
    ((finite_finite_pname @ procs))).
thf(conj_2, conjecture,
    ((hoare_604442164_state @ g @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn))) @ procs)))).
