% 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_436__3254692_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:48.249

% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    set_se421025541_state : $tType).
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__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    set_set_pname : $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 (31)
thf(sy_c_Com_OWT__bodies, type,
    wT_bodies : $o).
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_Finite__Set_Ofinite_001t__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    finite1266962254_state : set_se421025541_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_Ostate__not__singleton, type,
    hoare_405891322gleton : $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_Oord__class_Oless__eq_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    ord_le1483368280tate_o : (hoare_958474565_state > $o) > (hoare_958474565_state > $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_It__Com__Ostate_J_J, type,
    ord_le1945819589_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    order_800490238_state : (set_Ho840737317_state > $o) > 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_OCollect_001t__Set__Oset_It__Com__Opname_J, type,
    collect_set_pname : (set_pname > $o) > set_set_pname).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    collec305218192_state : (set_Ho840737317_state > $o) > set_se421025541_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_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_c_member_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    member1959617614_state : set_Ho840737317_state > set_se421025541_state > $o).
thf(sy_v_F, type,
    f : set_Ho840737317_state).

% Relevant facts (164)
thf(fact_0_thin, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => ((ord_le1945819589_state @ G @ G2) => (hoare_604442164_state @ G2 @ Ts)))))). % thin
thf(fact_1_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_2_asm, axiom,
    ((![Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((ord_le1945819589_state @ Ts @ G2) => (hoare_604442164_state @ G2 @ Ts))))). % asm
thf(fact_3_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_4_weaken, axiom,
    ((![G2 : set_Ho840737317_state, Ts2 : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts2) => ((ord_le1945819589_state @ Ts @ Ts2) => (hoare_604442164_state @ G2 @ Ts)))))). % weaken
thf(fact_5_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_6_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_7_subsetI, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (member109514606_state @ X @ B))) => (ord_le1945819589_state @ A @ B))))). % subsetI
thf(fact_8_subset__antisym, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_9_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, X2 : pname, A : set_pname]: ((B2 = (F @ X2)) => ((member_pname @ X2 @ A) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % image_eqI
thf(fact_10_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((B2 = (F @ X2)) => ((member109514606_state @ X2 @ A) => (member109514606_state @ B2 @ (image_1849792670_state @ F @ A))))))). % image_eqI
thf(fact_11_order__refl, axiom,
    ((![X2 : set_Ho840737317_state]: (ord_le1945819589_state @ X2 @ X2)))). % order_refl
thf(fact_12_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_958474565_state, B : set_Ho840737317_state]: ((![X : pname]: ((P @ X) => (member109514606_state @ (F @ X) @ B))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ (collect_pname @ P)) @ B))))). % image_Collect_subsetI
thf(fact_13_image__mono, axiom,
    ((![A : set_pname, B : set_pname, F : pname > hoare_958474565_state]: ((ord_le865024672_pname @ A @ B) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ (image_2144627828_state @ F @ B)))))). % image_mono
thf(fact_14_image__mono, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ (image_1849792670_state @ F @ B)))))). % image_mono
thf(fact_15_image__subsetI, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state, B : set_Ho840737317_state]: ((![X : pname]: ((member_pname @ X @ A) => (member109514606_state @ (F @ X) @ B))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B))))). % image_subsetI
thf(fact_16_image__subsetI, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, B : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (member109514606_state @ (F @ X) @ B))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ B))))). % image_subsetI
thf(fact_17_subset__imageE, axiom,
    ((![B : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) => (~ ((![C : set_pname]: ((ord_le865024672_pname @ C @ A) => (~ ((B = (image_2144627828_state @ F @ C)))))))))))). % subset_imageE
thf(fact_18_subset__imageE, axiom,
    ((![B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ (image_1849792670_state @ F @ A)) => (~ ((![C : set_Ho840737317_state]: ((ord_le1945819589_state @ C @ A) => (~ ((B = (image_1849792670_state @ F @ C)))))))))))). % subset_imageE
thf(fact_19_image__subset__iff, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B) = (![X3 : pname]: (((member_pname @ X3 @ A)) => ((member109514606_state @ (F @ X3) @ B)))))))). % image_subset_iff
thf(fact_20_subset__image__iff, axiom,
    ((![B : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B = (image_2144627828_state @ F @ AA))))))))). % subset_image_iff
thf(fact_21_subset__image__iff, axiom,
    ((![B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ (image_1849792670_state @ F @ A)) = (?[AA : set_Ho840737317_state]: (((ord_le1945819589_state @ AA @ A)) & ((B = (image_1849792670_state @ F @ AA))))))))). % subset_image_iff
thf(fact_22_less__eq__set__def, axiom,
    ((ord_le1945819589_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (ord_le1483368280tate_o @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ A2)) @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ B3)))))))). % less_eq_set_def
thf(fact_23_dual__order_Oantisym, axiom,
    ((![B2 : set_Ho840737317_state, A3 : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ A3) => ((ord_le1945819589_state @ A3 @ B2) => (A3 = B2)))))). % dual_order.antisym
thf(fact_24_dual__order_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ B4 @ A4)) & ((ord_le1945819589_state @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_25_dual__order_Otrans, axiom,
    ((![B2 : set_Ho840737317_state, A3 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ A3) => ((ord_le1945819589_state @ C2 @ B2) => (ord_le1945819589_state @ C2 @ A3)))))). % dual_order.trans
thf(fact_26_dual__order_Orefl, axiom,
    ((![A3 : set_Ho840737317_state]: (ord_le1945819589_state @ A3 @ A3)))). % dual_order.refl
thf(fact_27_order__trans, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => ((ord_le1945819589_state @ Y2 @ Z2) => (ord_le1945819589_state @ X2 @ Z2)))))). % order_trans
thf(fact_28_order__class_Oorder_Oantisym, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ B2) => ((ord_le1945819589_state @ B2 @ A3) => (A3 = B2)))))). % order_class.order.antisym
thf(fact_29_ord__le__eq__trans, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ B2) => ((B2 = C2) => (ord_le1945819589_state @ A3 @ C2)))))). % ord_le_eq_trans
thf(fact_30_ord__eq__le__trans, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((A3 = B2) => ((ord_le1945819589_state @ B2 @ C2) => (ord_le1945819589_state @ A3 @ C2)))))). % ord_eq_le_trans
thf(fact_31_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ A4 @ B4)) & ((ord_le1945819589_state @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_32_antisym__conv, axiom,
    ((![Y2 : set_Ho840737317_state, X2 : set_Ho840737317_state]: ((ord_le1945819589_state @ Y2 @ X2) => ((ord_le1945819589_state @ X2 @ Y2) = (X2 = Y2)))))). % antisym_conv
thf(fact_33_order_Otrans, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ B2) => ((ord_le1945819589_state @ B2 @ C2) => (ord_le1945819589_state @ A3 @ C2)))))). % order.trans
thf(fact_34_eq__refl, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((X2 = Y2) => (ord_le1945819589_state @ X2 @ Y2))))). % eq_refl
thf(fact_35_antisym, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => ((ord_le1945819589_state @ Y2 @ X2) => (X2 = Y2)))))). % antisym
thf(fact_36_eq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[X3 : set_Ho840737317_state]: (^[Y3 : set_Ho840737317_state]: (((ord_le1945819589_state @ X3 @ Y3)) & ((ord_le1945819589_state @ Y3 @ X3)))))))). % eq_iff
thf(fact_37_ord__le__eq__subst, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ B2) => (((F @ B2) = C2) => ((![X : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y4) => (ord_le1945819589_state @ (F @ X) @ (F @ Y4)))) => (ord_le1945819589_state @ (F @ A3) @ C2))))))). % ord_le_eq_subst
thf(fact_38_ord__eq__le__subst, axiom,
    ((![A3 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((A3 = (F @ B2)) => ((ord_le1945819589_state @ B2 @ C2) => ((![X : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y4) => (ord_le1945819589_state @ (F @ X) @ (F @ Y4)))) => (ord_le1945819589_state @ A3 @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_39_order__subst2, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ B2) => ((ord_le1945819589_state @ (F @ B2) @ C2) => ((![X : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y4) => (ord_le1945819589_state @ (F @ X) @ (F @ Y4)))) => (ord_le1945819589_state @ (F @ A3) @ C2))))))). % order_subst2
thf(fact_40_order__subst1, axiom,
    ((![A3 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A3 @ (F @ B2)) => ((ord_le1945819589_state @ B2 @ C2) => ((![X : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y4) => (ord_le1945819589_state @ (F @ X) @ (F @ Y4)))) => (ord_le1945819589_state @ A3 @ (F @ C2)))))))). % order_subst1
thf(fact_41_mem__Collect__eq, axiom,
    ((![A3 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((member109514606_state @ A3 @ (collec305460656_state @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_42_Collect__mem__eq, axiom,
    ((![A : set_Ho840737317_state]: ((collec305460656_state @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ A))) = A)))). % Collect_mem_eq
thf(fact_43_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B2 : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => ((B2 = (F @ X2)) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % rev_image_eqI
thf(fact_44_rev__image__eqI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((B2 = (F @ X2)) => (member109514606_state @ B2 @ (image_1849792670_state @ F @ A))))))). % rev_image_eqI
thf(fact_45_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ (image_2144627828_state @ F @ A)) => (P @ X))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_46_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X : pname]: ((member_pname @ X @ N) => ((F @ X) = (G3 @ X)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_47_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_2144627828_state @ F @ A)) & (P @ X4))) => (?[X : pname]: ((member_pname @ X @ A) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_48_image__iff, axiom,
    ((![Z2 : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z2 @ (image_2144627828_state @ F @ A)) = (?[X3 : pname]: (((member_pname @ X3 @ A)) & ((Z2 = (F @ X3))))))))). % image_iff
thf(fact_49_imageI, axiom,
    ((![X2 : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => (member109514606_state @ (F @ X2) @ (image_2144627828_state @ F @ A)))))). % imageI
thf(fact_50_imageI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => (member109514606_state @ (F @ X2) @ (image_1849792670_state @ F @ A)))))). % imageI
thf(fact_51_Collect__mono__iff, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)) = (![X3 : hoare_958474565_state]: (((P @ X3)) => ((Q @ X3)))))))). % Collect_mono_iff
thf(fact_52_set__eq__subset, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ A2 @ B3)) & ((ord_le1945819589_state @ B3 @ A2)))))))). % set_eq_subset
thf(fact_53_subset__trans, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C3 : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ C3) => (ord_le1945819589_state @ A @ C3)))))). % subset_trans
thf(fact_54_Collect__mono, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((P @ X) => (Q @ X))) => (ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_mono
thf(fact_55_subset__refl, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ A @ A)))). % subset_refl
thf(fact_56_subset__iff, axiom,
    ((ord_le1945819589_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (![T2 : hoare_958474565_state]: (((member109514606_state @ T2 @ A2)) => ((member109514606_state @ T2 @ B3))))))))). % subset_iff
thf(fact_57_equalityD2, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (ord_le1945819589_state @ B @ A))))). % equalityD2
thf(fact_58_equalityD1, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (ord_le1945819589_state @ A @ B))))). % equalityD1
thf(fact_59_subset__eq, axiom,
    ((ord_le1945819589_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (![X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A2)) => ((member109514606_state @ X3 @ B3))))))))). % subset_eq
thf(fact_60_equalityE, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (~ (((ord_le1945819589_state @ A @ B) => (~ ((ord_le1945819589_state @ B @ A)))))))))). % equalityE
thf(fact_61_subsetD, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => ((member109514606_state @ C2 @ A) => (member109514606_state @ C2 @ B)))))). % subsetD
thf(fact_62_in__mono, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, X2 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => ((member109514606_state @ X2 @ A) => (member109514606_state @ X2 @ B)))))). % in_mono
thf(fact_63_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ (image_2144627828_state @ F @ A))) & ((P @ X3))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_64_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ (image_1849792670_state @ F @ A))) & ((P @ X3))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_65_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F @ (image_2144627828_state @ G3 @ A)) = (image_2144627828_state @ (^[X3 : pname]: (F @ (G3 @ X3))) @ A))))). % image_image
thf(fact_66_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G3 : pname > pname, A : set_pname]: ((image_2144627828_state @ F @ (image_pname_pname @ G3 @ A)) = (image_2144627828_state @ (^[X3 : pname]: (F @ (G3 @ X3))) @ A))))). % image_image
thf(fact_67_imageE, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B2 @ (image_2144627828_state @ F @ A)) => (~ ((![X : pname]: ((B2 = (F @ X)) => (~ ((member_pname @ X @ A))))))))))). % imageE
thf(fact_68_imageE, axiom,
    ((![B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ B2 @ (image_1849792670_state @ F @ A)) => (~ ((![X : hoare_958474565_state]: ((B2 = (F @ X)) => (~ ((member109514606_state @ X @ A))))))))))). % imageE
thf(fact_69_Collect__restrict, axiom,
    ((![X5 : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ X5)) & ((P @ X3))))) @ X5)))). % Collect_restrict
thf(fact_70_prop__restrict, axiom,
    ((![X2 : hoare_958474565_state, Z3 : set_Ho840737317_state, X5 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((member109514606_state @ X2 @ Z3) => ((ord_le1945819589_state @ Z3 @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ X5)) & ((P @ X3)))))) => (P @ X2)))))). % prop_restrict
thf(fact_71_Collect__subset, axiom,
    ((![A : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) & ((P @ X3))))) @ A)))). % Collect_subset
thf(fact_72_all__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A)) => ((P @ (image_2144627828_state @ F @ B3))))))))). % all_subset_image
thf(fact_73_all__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A))) => ((P @ B3)))) = (![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ A)) => ((P @ (image_1849792670_state @ F @ B3))))))))). % all_subset_image
thf(fact_74_pred__subset__eq, axiom,
    ((![R : set_Ho840737317_state, S3 : set_Ho840737317_state]: ((ord_le1483368280tate_o @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ R)) @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ S3))) = (ord_le1945819589_state @ R @ S3))))). % pred_subset_eq
thf(fact_75_subset__Collect__iff, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((ord_le1945819589_state @ B @ A) => ((ord_le1945819589_state @ B @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) & ((P @ X3)))))) = (![X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ B)) => ((P @ X3))))))))). % subset_Collect_iff
thf(fact_76_subset__CollectI, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state, Q : hoare_958474565_state > $o, P : hoare_958474565_state > $o]: ((ord_le1945819589_state @ B @ A) => ((![X : hoare_958474565_state]: ((member109514606_state @ X @ B) => ((Q @ X) => (P @ X)))) => (ord_le1945819589_state @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ B)) & ((Q @ X3))))) @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) & ((P @ X3))))))))))). % subset_CollectI
thf(fact_77_conj__subset__def, axiom,
    ((![A : set_Ho840737317_state, P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((ord_le1945819589_state @ A @ (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((P @ X3)) & ((Q @ X3)))))) = (((ord_le1945819589_state @ A @ (collec305460656_state @ P))) & ((ord_le1945819589_state @ A @ (collec305460656_state @ Q)))))))). % conj_subset_def
thf(fact_78_MGT__Body, axiom,
    ((![G2 : set_Ho840737317_state, Procs : set_pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn))) @ Procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn)))) @ Procs)) => ((finite_finite_pname @ Procs) => (hoare_604442164_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn))) @ Procs))))))). % MGT_Body
thf(fact_79_Greatest__equality, axiom,
    ((![P : set_Ho840737317_state > $o, X2 : set_Ho840737317_state]: ((P @ X2) => ((![Y4 : set_Ho840737317_state]: ((P @ Y4) => (ord_le1945819589_state @ Y4 @ X2))) => ((order_800490238_state @ P) = X2)))))). % Greatest_equality
thf(fact_80_UnCI, axiom,
    ((![C2 : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: (((~ ((member109514606_state @ C2 @ B))) => (member109514606_state @ C2 @ A)) => (member109514606_state @ C2 @ (sup_su737426425_state @ A @ B)))))). % UnCI
thf(fact_81_Un__iff, axiom,
    ((![C2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C2 @ (sup_su737426425_state @ A @ B)) = (((member109514606_state @ C2 @ A)) | ((member109514606_state @ C2 @ B))))))). % Un_iff
thf(fact_82_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X3 : pname]: (((P @ X3)) | ((Q @ X3)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_83_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 @ (^[X3 : pname]: (((P @ X3)) & ((Q @ X3)))))))))). % finite_Collect_conjI
thf(fact_84_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > hoare_958474565_state]: ((finite_finite_pname @ F2) => (finite1986656878_state @ (image_2144627828_state @ H @ F2)))))). % finite_imageI
thf(fact_85_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > pname]: ((finite_finite_pname @ F2) => (finite_finite_pname @ (image_pname_pname @ H @ F2)))))). % finite_imageI
thf(fact_86_finite__Un, axiom,
    ((![F2 : set_pname, G2 : set_pname]: ((finite_finite_pname @ (sup_sup_set_pname @ F2 @ G2)) = (((finite_finite_pname @ F2)) & ((finite_finite_pname @ G2))))))). % finite_Un
thf(fact_87_finite__Un, axiom,
    ((![F2 : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ (sup_su737426425_state @ F2 @ G2)) = (((finite1986656878_state @ F2)) & ((finite1986656878_state @ G2))))))). % finite_Un
thf(fact_88_Un__subset__iff, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C3 : set_Ho840737317_state]: ((ord_le1945819589_state @ (sup_su737426425_state @ A @ B) @ C3) = (((ord_le1945819589_state @ A @ C3)) & ((ord_le1945819589_state @ B @ C3))))))). % Un_subset_iff
thf(fact_89_finite__Collect__subsets, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => (finite505202775_pname @ (collect_set_pname @ (^[B3 : set_pname]: (ord_le865024672_pname @ B3 @ A)))))))). % finite_Collect_subsets
thf(fact_90_finite__Collect__subsets, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => (finite1266962254_state @ (collec305218192_state @ (^[B3 : set_Ho840737317_state]: (ord_le1945819589_state @ B3 @ A)))))))). % finite_Collect_subsets
thf(fact_91_UnE, axiom,
    ((![C2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C2 @ (sup_su737426425_state @ A @ B)) => ((~ ((member109514606_state @ C2 @ A))) => (member109514606_state @ C2 @ B)))))). % UnE
thf(fact_92_UnI1, axiom,
    ((![C2 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C2 @ A) => (member109514606_state @ C2 @ (sup_su737426425_state @ A @ B)))))). % UnI1
thf(fact_93_UnI2, axiom,
    ((![C2 : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: ((member109514606_state @ C2 @ B) => (member109514606_state @ C2 @ (sup_su737426425_state @ A @ B)))))). % UnI2
thf(fact_94_Un__def, axiom,
    ((sup_su737426425_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A2)) | ((member109514606_state @ X3 @ B3)))))))))). % Un_def
thf(fact_95_bex__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ (sup_su737426425_state @ A @ B))) & ((P @ X3)))) = (((?[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) & ((P @ X3))))) | ((?[X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ B)) & ((P @ X3)))))))))). % bex_Un
thf(fact_96_ball__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ (sup_su737426425_state @ A @ B))) => ((P @ X3)))) = (((![X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ A)) => ((P @ X3))))) & ((![X3 : hoare_958474565_state]: (((member109514606_state @ X3 @ B)) => ((P @ X3)))))))))). % ball_Un
thf(fact_97_Un__assoc, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C3 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A @ B) @ C3) = (sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C3)))))). % Un_assoc
thf(fact_98_Un__absorb, axiom,
    ((![A : set_Ho840737317_state]: ((sup_su737426425_state @ A @ A) = A)))). % Un_absorb
thf(fact_99_Un__commute, axiom,
    ((sup_su737426425_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (sup_su737426425_state @ B3 @ A2)))))). % Un_commute
thf(fact_100_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_101_Collect__disj__eq, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X3 : hoare_958474565_state]: (((P @ X3)) | ((Q @ X3))))) = (sup_su737426425_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_disj_eq
thf(fact_102_Un__left__commute, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C3 : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C3)) = (sup_su737426425_state @ B @ (sup_su737426425_state @ A @ C3)))))). % Un_left_commute
thf(fact_103_finite__has__minimal2, axiom,
    ((![A : set_se421025541_state, A3 : set_Ho840737317_state]: ((finite1266962254_state @ A) => ((member1959617614_state @ A3 @ A) => (?[X : set_Ho840737317_state]: ((member1959617614_state @ X @ A) & ((ord_le1945819589_state @ X @ A3) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A) => ((ord_le1945819589_state @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_104_finite__has__maximal2, axiom,
    ((![A : set_se421025541_state, A3 : set_Ho840737317_state]: ((finite1266962254_state @ A) => ((member1959617614_state @ A3 @ A) => (?[X : set_Ho840737317_state]: ((member1959617614_state @ X @ A) & ((ord_le1945819589_state @ A3 @ X) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A) => ((ord_le1945819589_state @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_105_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B : set_pname, R : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ B) => ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
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) => ((![X : pname]: ((member_pname @ X @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_107_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_108_infinite__Un, axiom,
    ((![S3 : set_pname, T3 : set_pname]: ((~ ((finite_finite_pname @ (sup_sup_set_pname @ S3 @ T3)))) = (((~ ((finite_finite_pname @ S3)))) | ((~ ((finite_finite_pname @ T3))))))))). % infinite_Un
thf(fact_109_infinite__Un, axiom,
    ((![S3 : set_Ho840737317_state, T3 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ (sup_su737426425_state @ S3 @ T3)))) = (((~ ((finite1986656878_state @ S3)))) | ((~ ((finite1986656878_state @ T3))))))))). % infinite_Un
thf(fact_110_Un__infinite, axiom,
    ((![S3 : set_pname, T3 : set_pname]: ((~ ((finite_finite_pname @ S3))) => (~ ((finite_finite_pname @ (sup_sup_set_pname @ S3 @ T3)))))))). % Un_infinite
thf(fact_111_Un__infinite, axiom,
    ((![S3 : set_Ho840737317_state, T3 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S3))) => (~ ((finite1986656878_state @ (sup_su737426425_state @ S3 @ T3)))))))). % Un_infinite
thf(fact_112_finite__UnI, axiom,
    ((![F2 : set_pname, G2 : set_pname]: ((finite_finite_pname @ F2) => ((finite_finite_pname @ G2) => (finite_finite_pname @ (sup_sup_set_pname @ F2 @ G2))))))). % finite_UnI
thf(fact_113_finite__UnI, axiom,
    ((![F2 : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ F2) => ((finite1986656878_state @ G2) => (finite1986656878_state @ (sup_su737426425_state @ F2 @ G2))))))). % finite_UnI
thf(fact_114_finite__subset, axiom,
    ((![A : set_pname, B : set_pname]: ((ord_le865024672_pname @ A @ B) => ((finite_finite_pname @ B) => (finite_finite_pname @ A)))))). % finite_subset
thf(fact_115_finite__subset, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((finite1986656878_state @ B) => (finite1986656878_state @ A)))))). % finite_subset
thf(fact_116_infinite__super, axiom,
    ((![S3 : set_pname, T3 : set_pname]: ((ord_le865024672_pname @ S3 @ T3) => ((~ ((finite_finite_pname @ S3))) => (~ ((finite_finite_pname @ T3)))))))). % infinite_super
thf(fact_117_infinite__super, axiom,
    ((![S3 : set_Ho840737317_state, T3 : set_Ho840737317_state]: ((ord_le1945819589_state @ S3 @ T3) => ((~ ((finite1986656878_state @ S3))) => (~ ((finite1986656878_state @ T3)))))))). % infinite_super
thf(fact_118_rev__finite__subset, axiom,
    ((![B : set_pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ A @ B) => (finite_finite_pname @ A)))))). % rev_finite_subset
thf(fact_119_rev__finite__subset, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state]: ((finite1986656878_state @ B) => ((ord_le1945819589_state @ A @ B) => (finite1986656878_state @ A)))))). % rev_finite_subset
thf(fact_120_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > pname]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ (image_871391498_pname @ F @ A)) => (?[X : hoare_958474565_state]: ((member109514606_state @ X @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & (((F @ A4) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_121_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ (image_2144627828_state @ F @ A)) => (?[X : pname]: ((member_pname @ X @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F @ A4) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_122_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F @ A)) => (?[X : pname]: ((member_pname @ X @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F @ A4) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_123_image__Un, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B : set_pname]: ((image_2144627828_state @ F @ (sup_sup_set_pname @ A @ B)) = (sup_su737426425_state @ (image_2144627828_state @ F @ A) @ (image_2144627828_state @ F @ B)))))). % image_Un
thf(fact_124_image__Un, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ (image_1849792670_state @ F @ A) @ (image_1849792670_state @ F @ B)))))). % image_Un
thf(fact_125_Un__mono, axiom,
    ((![A : set_Ho840737317_state, C3 : set_Ho840737317_state, B : set_Ho840737317_state, D : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ C3) => ((ord_le1945819589_state @ B @ D) => (ord_le1945819589_state @ (sup_su737426425_state @ A @ B) @ (sup_su737426425_state @ C3 @ D))))))). % Un_mono
thf(fact_126_Un__least, axiom,
    ((![A : set_Ho840737317_state, C3 : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ C3) => ((ord_le1945819589_state @ B @ C3) => (ord_le1945819589_state @ (sup_su737426425_state @ A @ B) @ C3)))))). % Un_least
thf(fact_127_Un__upper1, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: (ord_le1945819589_state @ A @ (sup_su737426425_state @ A @ B))))). % Un_upper1
thf(fact_128_Un__upper2, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state]: (ord_le1945819589_state @ B @ (sup_su737426425_state @ A @ B))))). % Un_upper2
thf(fact_129_Un__absorb1, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((sup_su737426425_state @ A @ B) = B))))). % Un_absorb1
thf(fact_130_Un__absorb2, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ A) => ((sup_su737426425_state @ A @ B) = A))))). % Un_absorb2
thf(fact_131_subset__UnE, axiom,
    ((![C3 : set_Ho840737317_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ C3 @ (sup_su737426425_state @ A @ B)) => (~ ((![A5 : set_Ho840737317_state]: ((ord_le1945819589_state @ A5 @ A) => (![B5 : set_Ho840737317_state]: ((ord_le1945819589_state @ B5 @ B) => (~ ((C3 = (sup_su737426425_state @ A5 @ B5)))))))))))))). % subset_UnE
thf(fact_132_subset__Un__eq, axiom,
    ((ord_le1945819589_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ B3) = B3)))))). % subset_Un_eq
thf(fact_133_finite__surj, axiom,
    ((![A : set_pname, B : set_pname, F : pname > pname]: ((finite_finite_pname @ A) => ((ord_le865024672_pname @ B @ (image_pname_pname @ F @ A)) => (finite_finite_pname @ B)))))). % finite_surj
thf(fact_134_finite__surj, axiom,
    ((![A : set_pname, B : set_Ho840737317_state, F : pname > hoare_958474565_state]: ((finite_finite_pname @ A) => ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) => (finite1986656878_state @ B)))))). % finite_surj
thf(fact_135_finite__subset__image, axiom,
    ((![B : set_pname, F : pname > pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ B @ (image_pname_pname @ F @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B = (image_pname_pname @ F @ C)))))))))). % finite_subset_image
thf(fact_136_finite__subset__image, axiom,
    ((![B : set_pname, F : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ B @ (image_871391498_pname @ F @ A)) => (?[C : set_Ho840737317_state]: ((ord_le1945819589_state @ C @ A) & ((finite1986656878_state @ C) & (B = (image_871391498_pname @ F @ C)))))))))). % finite_subset_image
thf(fact_137_finite__subset__image, axiom,
    ((![B : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((finite1986656878_state @ B) => ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B = (image_2144627828_state @ F @ C)))))))))). % finite_subset_image
thf(fact_138_finite__subset__image, axiom,
    ((![B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((finite1986656878_state @ B) => ((ord_le1945819589_state @ B @ (image_1849792670_state @ F @ A)) => (?[C : set_Ho840737317_state]: ((ord_le1945819589_state @ C @ A) & ((finite1986656878_state @ C) & (B = (image_1849792670_state @ F @ C)))))))))). % finite_subset_image
thf(fact_139_ex__finite__subset__image, axiom,
    ((![F : pname > pname, A : set_pname, P : set_pname > $o]: ((?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A))) & ((P @ B3)))))) = (?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ A)) & ((P @ (image_pname_pname @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_140_ex__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state, P : set_pname > $o]: ((?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ (image_871391498_pname @ F @ A))) & ((P @ B3)))))) = (?[B3 : set_Ho840737317_state]: (((finite1986656878_state @ B3)) & ((((ord_le1945819589_state @ B3 @ A)) & ((P @ (image_871391498_pname @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_141_ex__finite__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : set_Ho840737317_state > $o]: ((?[B3 : set_Ho840737317_state]: (((finite1986656878_state @ B3)) & ((((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A))) & ((P @ B3)))))) = (?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ A)) & ((P @ (image_2144627828_state @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_142_ex__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((?[B3 : set_Ho840737317_state]: (((finite1986656878_state @ B3)) & ((((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A))) & ((P @ B3)))))) = (?[B3 : set_Ho840737317_state]: (((finite1986656878_state @ B3)) & ((((ord_le1945819589_state @ B3 @ A)) & ((P @ (image_1849792670_state @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_143_all__finite__subset__image, axiom,
    ((![F : pname > pname, A : set_pname, P : set_pname > $o]: ((![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A))))) => ((P @ B3)))) = (![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ A)))) => ((P @ (image_pname_pname @ F @ B3))))))))). % all_finite_subset_image
thf(fact_144_all__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state, P : set_pname > $o]: ((![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ (image_871391498_pname @ F @ A))))) => ((P @ B3)))) = (![B3 : set_Ho840737317_state]: (((((finite1986656878_state @ B3)) & ((ord_le1945819589_state @ B3 @ A)))) => ((P @ (image_871391498_pname @ F @ B3))))))))). % all_finite_subset_image
thf(fact_145_all__finite__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((((finite1986656878_state @ B3)) & ((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A))))) => ((P @ B3)))) = (![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ A)))) => ((P @ (image_2144627828_state @ F @ B3))))))))). % all_finite_subset_image
thf(fact_146_all__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((((finite1986656878_state @ B3)) & ((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A))))) => ((P @ B3)))) = (![B3 : set_Ho840737317_state]: (((((finite1986656878_state @ B3)) & ((ord_le1945819589_state @ B3 @ A)))) => ((P @ (image_1849792670_state @ F @ B3))))))))). % all_finite_subset_image
thf(fact_147_Inf_OINF__cong, axiom,
    ((![A : set_pname, B : set_pname, C3 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X : pname]: ((member_pname @ X @ B) => ((C3 @ X) = (D @ X)))) => ((Inf @ (image_2144627828_state @ C3 @ A)) = (Inf @ (image_2144627828_state @ D @ B)))))))). % Inf.INF_cong
thf(fact_148_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B : set_pname, C3 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X : pname]: ((member_pname @ X @ B) => ((C3 @ X) = (D @ X)))) => ((Sup @ (image_2144627828_state @ C3 @ A)) = (Sup @ (image_2144627828_state @ D @ B)))))))). % Sup.SUP_cong
thf(fact_149_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_150_GreatestI2__order, axiom,
    ((![P : set_Ho840737317_state > $o, X2 : set_Ho840737317_state, Q : set_Ho840737317_state > $o]: ((P @ X2) => ((![Y4 : set_Ho840737317_state]: ((P @ Y4) => (ord_le1945819589_state @ Y4 @ X2))) => ((![X : set_Ho840737317_state]: ((P @ X) => ((![Y5 : set_Ho840737317_state]: ((P @ Y5) => (ord_le1945819589_state @ Y5 @ X))) => (Q @ X)))) => (Q @ (order_800490238_state @ P)))))))). % GreatestI2_order
thf(fact_151_le__sup__iff, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (sup_su737426425_state @ X2 @ Y2) @ Z2) = (((ord_le1945819589_state @ X2 @ Z2)) & ((ord_le1945819589_state @ Y2 @ Z2))))))). % le_sup_iff
thf(fact_152_sup_Obounded__iff, axiom,
    ((![B2 : set_Ho840737317_state, C2 : set_Ho840737317_state, A3 : set_Ho840737317_state]: ((ord_le1945819589_state @ (sup_su737426425_state @ B2 @ C2) @ A3) = (((ord_le1945819589_state @ B2 @ A3)) & ((ord_le1945819589_state @ C2 @ A3))))))). % sup.bounded_iff
thf(fact_153_sup_Oright__idem, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A3 @ B2) @ B2) = (sup_su737426425_state @ A3 @ B2))))). % sup.right_idem
thf(fact_154_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_155_sup_Oidem, axiom,
    ((![A3 : set_Ho840737317_state]: ((sup_su737426425_state @ A3 @ A3) = A3)))). % sup.idem
thf(fact_156_sup__idem, axiom,
    ((![X2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ X2) = X2)))). % sup_idem
thf(fact_157_sup_Oleft__idem, axiom,
    ((![A3 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ A3 @ (sup_su737426425_state @ A3 @ B2)) = (sup_su737426425_state @ A3 @ B2))))). % sup.left_idem
thf(fact_158_sup__set__def, axiom,
    ((sup_su737426425_state = (^[A2 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (sup_su2044190244tate_o @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ A2)) @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ B3))))))))). % sup_set_def
thf(fact_159_sup__Un__eq, axiom,
    ((![R : set_Ho840737317_state, S3 : set_Ho840737317_state]: ((sup_su2044190244tate_o @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ R)) @ (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ S3))) = (^[X3 : hoare_958474565_state]: (member109514606_state @ X3 @ (sup_su737426425_state @ R @ S3))))))). % sup_Un_eq
thf(fact_160_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_161_inf__sup__aci_I7_J, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z2)) = (sup_su737426425_state @ Y2 @ (sup_su737426425_state @ X2 @ Z2)))))). % inf_sup_aci(7)
thf(fact_162_inf__sup__aci_I6_J, axiom,
    ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X2 @ Y2) @ Z2) = (sup_su737426425_state @ X2 @ (sup_su737426425_state @ Y2 @ Z2)))))). % inf_sup_aci(6)
thf(fact_163_inf__sup__aci_I5_J, axiom,
    ((sup_su737426425_state = (^[X3 : set_Ho840737317_state]: (^[Y3 : set_Ho840737317_state]: (sup_su737426425_state @ Y3 @ X3)))))). % inf_sup_aci(5)

% Conjectures (4)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, hypothesis,
    ((ord_le1945819589_state @ f @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn)))) @ (dom_pname_com @ body))))).
thf(conj_3, conjecture,
    ((hoare_604442164_state @ (image_2144627828_state @ (^[Pn : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn))) @ (dom_pname_com @ body)) @ f))).
