% 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_212__3251546_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:44.946

% Could-be-implicit typings (2)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).

% Explicit typings (17)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    minus_1852999390iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__a, type,
    hoare_129598474rivs_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    bot_bo431311916le_a_o : hoare_1678595023iple_a > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1356158809iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1048771374iple_a : ($o > set_Ho137910533iple_a) > ($o > set_Ho137910533iple_a) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1221261669iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_929906668iple_a : (set_Ho137910533iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_emp901906557iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    pairwi531237284iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a > $o) > set_Ho137910533iple_a > $o).
thf(sy_c_Set_Oremove_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    remove1512401578iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_G, type,
    g : set_Ho137910533iple_a).

% Relevant facts (124)
thf(fact_0_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_1_cut, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((hoare_129598474rivs_a @ G @ G2) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_2_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_3_weaken, axiom,
    ((![G : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts2) => ((ord_le1221261669iple_a @ Ts @ Ts2) => (hoare_129598474rivs_a @ G @ Ts)))))). % weaken
thf(fact_4_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_5_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_6_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A) => (member1332298086iple_a @ X @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_7_subset__antisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_8_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_9_all__not__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X2 @ A)))) = (A = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_10_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_11_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_12_order__refl, axiom,
    ((![X3 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X3 @ X3)))). % order_refl
thf(fact_13_subset__emptyI, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X @ A)))) => (ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_14_bot_Oextremum, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % bot.extremum
thf(fact_15_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_16_dual__order_Oantisym, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((ord_le1221261669iple_a @ A2 @ B2) => (A2 = B2)))))). % dual_order.antisym
thf(fact_17_dual__order_Oeq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ A3)) & ((ord_le1221261669iple_a @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_18_dual__order_Otrans, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((ord_le1221261669iple_a @ C @ B2) => (ord_le1221261669iple_a @ C @ A2)))))). % dual_order.trans
thf(fact_19_dual__order_Orefl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % dual_order.refl
thf(fact_20_order__trans, axiom,
    ((![X3 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a, Z2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y2) => ((ord_le1221261669iple_a @ Y2 @ Z2) => (ord_le1221261669iple_a @ X3 @ Z2)))))). % order_trans
thf(fact_21_order__class_Oorder_Oantisym, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ A2) => (A2 = B2)))))). % order_class.order.antisym
thf(fact_22_ord__le__eq__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((B2 = C) => (ord_le1221261669iple_a @ A2 @ C)))))). % ord_le_eq_trans
thf(fact_23_ord__eq__le__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A2 = B2) => ((ord_le1221261669iple_a @ B2 @ C) => (ord_le1221261669iple_a @ A2 @ C)))))). % ord_eq_le_trans
thf(fact_24_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_25_antisym__conv, axiom,
    ((![Y2 : set_Ho137910533iple_a, X3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y2 @ X3) => ((ord_le1221261669iple_a @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv
thf(fact_26_order_Otrans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ C) => (ord_le1221261669iple_a @ A2 @ C)))))). % order.trans
thf(fact_27_eq__refl, axiom,
    ((![X3 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((X3 = Y2) => (ord_le1221261669iple_a @ X3 @ Y2))))). % eq_refl
thf(fact_28_antisym, axiom,
    ((![X3 : set_Ho137910533iple_a, Y2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X3 @ Y2) => ((ord_le1221261669iple_a @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_29_eq__iff, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[X2 : set_Ho137910533iple_a]: (^[Y3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X2 @ Y3)) & ((ord_le1221261669iple_a @ Y3 @ X2)))))))). % eq_iff
thf(fact_30_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (((F @ B2) = C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_31_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A2 = (F @ B2)) => ((ord_le1221261669iple_a @ B2 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_32_order__subst2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ (F @ B2) @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A2) @ C))))))). % order_subst2
thf(fact_33_order__subst1, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (F @ B2)) => ((ord_le1221261669iple_a @ B2 @ C) => ((![X : set_Ho137910533iple_a, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y4) => (ord_le1221261669iple_a @ (F @ X) @ (F @ Y4)))) => (ord_le1221261669iple_a @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_34_ex__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((?[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A)) = (~ ((A = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_35_mem__Collect__eq, axiom,
    ((![A2 : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A2 @ (collec1600235172iple_a @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_36_Collect__mem__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A))) = A)))). % Collect_mem_eq
thf(fact_37_equals0I, axiom,
    ((![A : set_Ho137910533iple_a]: ((![Y4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y4 @ A)))) => (A = bot_bo1298296729iple_a))))). % equals0I
thf(fact_38_equals0D, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((A = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A2 @ A))))))). % equals0D
thf(fact_39_emptyE, axiom,
    ((![A2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A2 @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_40_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X2 : hoare_1678595023iple_a]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_41_set__eq__subset, axiom,
    (((^[Y : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A4 @ B4)) & ((ord_le1221261669iple_a @ B4 @ A4)))))))). % set_eq_subset
thf(fact_42_subset__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C2) => (ord_le1221261669iple_a @ A @ C2)))))). % subset_trans
thf(fact_43_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: ((P @ X) => (Q @ X))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_44_subset__refl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % subset_refl
thf(fact_45_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![T : hoare_1678595023iple_a]: (((member1332298086iple_a @ T @ A4)) => ((member1332298086iple_a @ T @ B4))))))))). % subset_iff
thf(fact_46_equalityD2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ B @ A))))). % equalityD2
thf(fact_47_equalityD1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ A @ B))))). % equalityD1
thf(fact_48_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ A4)) => ((member1332298086iple_a @ X2 @ B4))))))))). % subset_eq
thf(fact_49_equalityE, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (~ (((ord_le1221261669iple_a @ A @ B) => (~ ((ord_le1221261669iple_a @ B @ A)))))))))). % equalityE
thf(fact_50_subsetD, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ C @ A) => (member1332298086iple_a @ C @ B)))))). % subsetD
thf(fact_51_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X3 @ A) => (member1332298086iple_a @ X3 @ B)))))). % in_mono
thf(fact_52_bot_Oextremum__uniqueI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) => (A2 = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_53_bot_Oextremum__unique, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_54_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_55_Greatest__equality, axiom,
    ((![P : set_Ho137910533iple_a > $o, X3 : set_Ho137910533iple_a]: ((P @ X3) => ((![Y4 : set_Ho137910533iple_a]: ((P @ Y4) => (ord_le1221261669iple_a @ Y4 @ X3))) => ((order_929906668iple_a @ P) = X3)))))). % Greatest_equality
thf(fact_56_GreatestI2__order, axiom,
    ((![P : set_Ho137910533iple_a > $o, X3 : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P @ X3) => ((![Y4 : set_Ho137910533iple_a]: ((P @ Y4) => (ord_le1221261669iple_a @ Y4 @ X3))) => ((![X : set_Ho137910533iple_a]: ((P @ X) => ((![Y5 : set_Ho137910533iple_a]: ((P @ Y5) => (ord_le1221261669iple_a @ Y5 @ X))) => (Q @ X)))) => (Q @ (order_929906668iple_a @ P)))))))). % GreatestI2_order
thf(fact_57_Collect__empty__eq__bot, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (P = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_58_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_59_le__rel__bool__arg__iff, axiom,
    ((ord_le1048771374iple_a = (^[X4 : $o > set_Ho137910533iple_a]: (^[Y6 : $o > set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ (X4 @ $false) @ (Y6 @ $false))) & ((ord_le1221261669iple_a @ (X4 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_60_is__singletonI_H, axiom,
    ((![A : set_Ho137910533iple_a]: ((~ ((A = bot_bo1298296729iple_a))) => ((![X : hoare_1678595023iple_a, Y4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A) => ((member1332298086iple_a @ Y4 @ A) => (X = Y4)))) => (is_sin1784037339iple_a @ A)))))). % is_singletonI'
thf(fact_61_Diff__eq__empty__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((minus_1852999390iple_a @ A @ B) = bot_bo1298296729iple_a) = (ord_le1221261669iple_a @ A @ B))))). % Diff_eq_empty_iff
thf(fact_62_DiffI, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ A) => ((~ ((member1332298086iple_a @ C @ B))) => (member1332298086iple_a @ C @ (minus_1852999390iple_a @ A @ B))))))). % DiffI
thf(fact_63_Diff__iff, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A @ B)) = (((member1332298086iple_a @ C @ A)) & ((~ ((member1332298086iple_a @ C @ B))))))))). % Diff_iff
thf(fact_64_Diff__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A @ bot_bo1298296729iple_a) = A)))). % Diff_empty
thf(fact_65_empty__Diff, axiom,
    ((![A : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_66_Diff__cancel, axiom,
    ((![A : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A @ A) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_67_DiffE, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A @ B)) => (~ (((member1332298086iple_a @ C @ A) => (member1332298086iple_a @ C @ B)))))))). % DiffE
thf(fact_68_DiffD1, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A @ B)) => (member1332298086iple_a @ C @ A))))). % DiffD1
thf(fact_69_DiffD2, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A @ B)) => (~ ((member1332298086iple_a @ C @ B))))))). % DiffD2
thf(fact_70_Diff__mono, axiom,
    ((![A : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a, D : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ C2) => ((ord_le1221261669iple_a @ D @ B) => (ord_le1221261669iple_a @ (minus_1852999390iple_a @ A @ B) @ (minus_1852999390iple_a @ C2 @ D))))))). % Diff_mono
thf(fact_71_Diff__subset, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ (minus_1852999390iple_a @ A @ B) @ A)))). % Diff_subset
thf(fact_72_double__diff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C2) => ((minus_1852999390iple_a @ B @ (minus_1852999390iple_a @ C2 @ A)) = A)))))). % double_diff
thf(fact_73_is__singletonI, axiom,
    ((![X3 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_74_Diff__single__insert, axiom,
    ((![A : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) @ B) => (ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ B)))))). % Diff_single_insert
thf(fact_75_insertCI, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A2 @ B))) => (A2 = B2)) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % insertCI
thf(fact_76_insert__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ A)) = (((A2 = B2)) | ((member1332298086iple_a @ A2 @ A))))))). % insert_iff
thf(fact_77_singletonI, axiom,
    ((![A2 : hoare_1678595023iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_78_insert__subset, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X3 @ A) @ B) = (((member1332298086iple_a @ X3 @ B)) & ((ord_le1221261669iple_a @ A @ B))))))). % insert_subset
thf(fact_79_insert__Diff1, axiom,
    ((![X3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X3 @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X3 @ A) @ B) = (minus_1852999390iple_a @ A @ B)))))). % insert_Diff1
thf(fact_80_Diff__insert0, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((minus_1852999390iple_a @ A @ (insert1477804543iple_a @ X3 @ B)) = (minus_1852999390iple_a @ A @ B)))))). % Diff_insert0
thf(fact_81_singleton__insert__inj__eq, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A2 @ A)) = (((A2 = B2)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_82_singleton__insert__inj__eq_H, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (((A2 = B2)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_83_insert__Diff__single, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((insert1477804543iple_a @ A2 @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))) = (insert1477804543iple_a @ A2 @ A))))). % insert_Diff_single
thf(fact_84_insert__Diff__if, axiom,
    ((![X3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: (((member1332298086iple_a @ X3 @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X3 @ A) @ B) = (minus_1852999390iple_a @ A @ B))) & ((~ ((member1332298086iple_a @ X3 @ B))) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X3 @ A) @ B) = (insert1477804543iple_a @ X3 @ (minus_1852999390iple_a @ A @ B)))))))). % insert_Diff_if
thf(fact_85_insertE, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ A)) => ((~ ((A2 = B2))) => (member1332298086iple_a @ A2 @ A)))))). % insertE
thf(fact_86_insertI1, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ B))))). % insertI1
thf(fact_87_insertI2, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ A2 @ B) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % insertI2
thf(fact_88_Set_Oset__insert, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X3 @ A) => (~ ((![B5 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ X3 @ B5)) => (member1332298086iple_a @ X3 @ B5))))))))). % Set.set_insert
thf(fact_89_insert__ident, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((~ ((member1332298086iple_a @ X3 @ B))) => (((insert1477804543iple_a @ X3 @ A) = (insert1477804543iple_a @ X3 @ B)) = (A = B))))))). % insert_ident
thf(fact_90_insert__absorb, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => ((insert1477804543iple_a @ A2 @ A) = A))))). % insert_absorb
thf(fact_91_insert__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A2 @ A))) => ((~ ((member1332298086iple_a @ B2 @ B))) => (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B2 @ B)) = (((((A2 = B2)) => ((A = B)))) & ((((~ ((A2 = B2)))) => ((?[C3 : set_Ho137910533iple_a]: (((A = (insert1477804543iple_a @ B2 @ C3))) & ((((~ ((member1332298086iple_a @ B2 @ C3)))) & ((((B = (insert1477804543iple_a @ A2 @ C3))) & ((~ ((member1332298086iple_a @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_92_mk__disjoint__insert, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => (?[B5 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ A2 @ B5)) & (~ ((member1332298086iple_a @ A2 @ B5))))))))). % mk_disjoint_insert
thf(fact_93_insert__subsetI, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, X5 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X3 @ A) => ((ord_le1221261669iple_a @ X5 @ A) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X3 @ X5) @ A)))))). % insert_subsetI
thf(fact_94_subset__insertI2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => (ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % subset_insertI2
thf(fact_95_subset__insertI, axiom,
    ((![B : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B @ (insert1477804543iple_a @ A2 @ B))))). % subset_insertI
thf(fact_96_subset__insert, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ B)) = (ord_le1221261669iple_a @ A @ B)))))). % subset_insert
thf(fact_97_insert__mono, axiom,
    ((![C2 : set_Ho137910533iple_a, D : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C2 @ D) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A2 @ C2) @ (insert1477804543iple_a @ A2 @ D)))))). % insert_mono
thf(fact_98_singleton__inject, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) => (A2 = B2))))). % singleton_inject
thf(fact_99_insert__not__empty, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A2 @ A) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_100_doubleton__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D2 @ bot_bo1298296729iple_a))) = (((((A2 = C)) & ((B2 = D2)))) | ((((A2 = D2)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_101_singleton__iff, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (B2 = A2))))). % singleton_iff
thf(fact_102_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) => (B2 = A2))))). % singletonD
thf(fact_103_subset__singletonD, axiom,
    ((![A : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) => ((A = bot_bo1298296729iple_a) | (A = (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_104_subset__singleton__iff, axiom,
    ((![X5 : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X5 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (((X5 = bot_bo1298296729iple_a)) | ((X5 = (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_105_Diff__insert, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ B)) = (minus_1852999390iple_a @ (minus_1852999390iple_a @ A @ B) @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)))))). % Diff_insert
thf(fact_106_insert__Diff, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => ((insert1477804543iple_a @ A2 @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))) = A))))). % insert_Diff
thf(fact_107_Diff__insert2, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ B)) = (minus_1852999390iple_a @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) @ B))))). % Diff_insert2
thf(fact_108_Diff__insert__absorb, axiom,
    ((![X3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X3 @ A))) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X3 @ A) @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) = A))))). % Diff_insert_absorb
thf(fact_109_subset__Diff__insert, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ (minus_1852999390iple_a @ B @ (insert1477804543iple_a @ X3 @ C2))) = (((ord_le1221261669iple_a @ A @ (minus_1852999390iple_a @ B @ C2))) & ((~ ((member1332298086iple_a @ X3 @ A))))))))). % subset_Diff_insert
thf(fact_110_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho137910533iple_a, T2 : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T2 @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ Ts) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T2 @ Ts))))))). % hoare_derivs.insert
thf(fact_111_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (?[X2 : hoare_1678595023iple_a]: (A4 = (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_112_is__singletonE, axiom,
    ((![A : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A) => (~ ((![X : hoare_1678595023iple_a]: (~ ((A = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_113_subset__insert__iff, axiom,
    ((![A : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X3 @ B)) = (((((member1332298086iple_a @ X3 @ A)) => ((ord_le1221261669iple_a @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) @ B)))) & ((((~ ((member1332298086iple_a @ X3 @ A)))) => ((ord_le1221261669iple_a @ A @ B))))))))). % subset_insert_iff
thf(fact_114_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = (insert1477804543iple_a @ (the_el434698138iple_a @ A4) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_115_the__elem__eq, axiom,
    ((![X3 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) = X3)))). % the_elem_eq
thf(fact_116_remove__def, axiom,
    ((remove1512401578iple_a = (^[X2 : hoare_1678595023iple_a]: (^[A4 : set_Ho137910533iple_a]: (minus_1852999390iple_a @ A4 @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))). % remove_def
thf(fact_117_member__remove, axiom,
    ((![X3 : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X3 @ (remove1512401578iple_a @ Y2 @ A)) = (((member1332298086iple_a @ X3 @ A)) & ((~ ((X3 = Y2))))))))). % member_remove
thf(fact_118_pairwise__alt, axiom,
    ((pairwi531237284iple_a = (^[R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (^[S : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ S)) => ((![Y3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y3 @ (minus_1852999390iple_a @ S @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)))) => ((R @ X2 @ Y3)))))))))))). % pairwise_alt
thf(fact_119_psubset__insert__iff, axiom,
    ((![A : set_Ho137910533iple_a, X3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((ord_le1356158809iple_a @ A @ (insert1477804543iple_a @ X3 @ B)) = (((((member1332298086iple_a @ X3 @ B)) => ((ord_le1356158809iple_a @ A @ B)))) & ((((~ ((member1332298086iple_a @ X3 @ B)))) => ((((((member1332298086iple_a @ X3 @ A)) => ((ord_le1356158809iple_a @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a)) @ B)))) & ((((~ ((member1332298086iple_a @ X3 @ A)))) => ((ord_le1221261669iple_a @ A @ B))))))))))))). % psubset_insert_iff
thf(fact_120_psubsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((~ ((A = B))) => (ord_le1356158809iple_a @ A @ B)))))). % psubsetI
thf(fact_121_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X3 : hoare_1678595023iple_a, S2 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X3 @ S2)) = (((![Y3 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y3 @ S2)) & ((~ ((Y3 = X3)))))) => ((((R2 @ X3 @ Y3)) & ((R2 @ Y3 @ X3))))))) & ((pairwi531237284iple_a @ R2 @ S2))))))). % pairwise_insert
thf(fact_122_pairwise__singleton, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (pairwi531237284iple_a @ P @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % pairwise_singleton
thf(fact_123_psubset__imp__ex__mem, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1356158809iple_a @ A @ B) => (?[B6 : hoare_1678595023iple_a]: (member1332298086iple_a @ B6 @ (minus_1852999390iple_a @ B @ A))))))). % psubset_imp_ex_mem

% Conjectures (1)
thf(conj_0, conjecture,
    ((![G3 : set_Ho137910533iple_a]: ((~ ((ord_le1221261669iple_a @ g @ G3))) | (hoare_129598474rivs_a @ G3 @ bot_bo1298296729iple_a))))).
