% 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/FFT/prob_50__3222696_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:08:56.396

% Could-be-implicit typings (3)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (19)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001tf__a, type,
    groups1145913330_nat_a : (nat > a) > set_nat > a).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat, type,
    sup_sup_nat : nat > nat > nat).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J, type,
    sup_sup_set_nat : set_nat > set_nat > set_nat).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J, type,
    bot_bot_nat_o : nat > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_set_nat : set_nat > set_nat > $o).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_Oinsert_001t__Nat__Onat, type,
    insert_nat : nat > set_nat > set_nat).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat, type,
    is_singleton_nat : set_nat > $o).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat, type,
    the_elem_nat : set_nat > nat).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat, type,
    set_or1544565540an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_f, type,
    f : nat > a).
thf(sy_v_m, type,
    m : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (143)
thf(fact_0_less, axiom,
    ((ord_less_nat @ m @ n))). % less
thf(fact_1_Un__insert__left, axiom,
    ((![A : nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ (insert_nat @ A @ B) @ C) = (insert_nat @ A @ (sup_sup_set_nat @ B @ C)))))). % Un_insert_left
thf(fact_2_Un__insert__right, axiom,
    ((![A2 : set_nat, A : nat, B : set_nat]: ((sup_sup_set_nat @ A2 @ (insert_nat @ A @ B)) = (insert_nat @ A @ (sup_sup_set_nat @ A2 @ B)))))). % Un_insert_right
thf(fact_3_Un__empty, axiom,
    ((![A2 : set_nat, B : set_nat]: (((sup_sup_set_nat @ A2 @ B) = bot_bot_set_nat) = (((A2 = bot_bot_set_nat)) & ((B = bot_bot_set_nat))))))). % Un_empty
thf(fact_4_singletonI, axiom,
    ((![A : nat]: (member_nat @ A @ (insert_nat @ A @ bot_bot_set_nat))))). % singletonI
thf(fact_5_sup__bot__left, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ X) = X)))). % sup_bot_left
thf(fact_6_sup__bot__right, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ X @ bot_bot_set_nat) = X)))). % sup_bot_right
thf(fact_7_bot__eq__sup__iff, axiom,
    ((![X : set_nat, Y : set_nat]: ((bot_bot_set_nat = (sup_sup_set_nat @ X @ Y)) = (((X = bot_bot_set_nat)) & ((Y = bot_bot_set_nat))))))). % bot_eq_sup_iff
thf(fact_8_sup__eq__bot__iff, axiom,
    ((![X : set_nat, Y : set_nat]: (((sup_sup_set_nat @ X @ Y) = bot_bot_set_nat) = (((X = bot_bot_set_nat)) & ((Y = bot_bot_set_nat))))))). % sup_eq_bot_iff
thf(fact_9_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_nat, B2 : set_nat]: (((sup_sup_set_nat @ A @ B2) = bot_bot_set_nat) = (((A = bot_bot_set_nat)) & ((B2 = bot_bot_set_nat))))))). % sup_bot.eq_neutr_iff
thf(fact_10_sup__bot_Oleft__neutral, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ A) = A)))). % sup_bot.left_neutral
thf(fact_11_sup__bot_Oneutr__eq__iff, axiom,
    ((![A : set_nat, B2 : set_nat]: ((bot_bot_set_nat = (sup_sup_set_nat @ A @ B2)) = (((A = bot_bot_set_nat)) & ((B2 = bot_bot_set_nat))))))). % sup_bot.neutr_eq_iff
thf(fact_12_sup__bot_Oright__neutral, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ A @ bot_bot_set_nat) = A)))). % sup_bot.right_neutral
thf(fact_13_empty__Collect__eq, axiom,
    ((![P : nat > $o]: ((bot_bot_set_nat = (collect_nat @ P)) = (![X2 : nat]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_14_Collect__empty__eq, axiom,
    ((![P : nat > $o]: (((collect_nat @ P) = bot_bot_set_nat) = (![X2 : nat]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_15_all__not__in__conv, axiom,
    ((![A2 : set_nat]: ((![X2 : nat]: (~ ((member_nat @ X2 @ A2)))) = (A2 = bot_bot_set_nat))))). % all_not_in_conv
thf(fact_16_empty__iff, axiom,
    ((![C2 : nat]: (~ ((member_nat @ C2 @ bot_bot_set_nat)))))). % empty_iff
thf(fact_17_insert__absorb2, axiom,
    ((![X : nat, A2 : set_nat]: ((insert_nat @ X @ (insert_nat @ X @ A2)) = (insert_nat @ X @ A2))))). % insert_absorb2
thf(fact_18_insert__iff, axiom,
    ((![A : nat, B2 : nat, A2 : set_nat]: ((member_nat @ A @ (insert_nat @ B2 @ A2)) = (((A = B2)) | ((member_nat @ A @ A2))))))). % insert_iff
thf(fact_19_insertCI, axiom,
    ((![A : nat, B : set_nat, B2 : nat]: (((~ ((member_nat @ A @ B))) => (A = B2)) => (member_nat @ A @ (insert_nat @ B2 @ B)))))). % insertCI
thf(fact_20_sup_Oright__idem, axiom,
    ((![A : set_nat, B2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A @ B2) @ B2) = (sup_sup_set_nat @ A @ B2))))). % sup.right_idem
thf(fact_21_sup__left__idem, axiom,
    ((![X : set_nat, Y : set_nat]: ((sup_sup_set_nat @ X @ (sup_sup_set_nat @ X @ Y)) = (sup_sup_set_nat @ X @ Y))))). % sup_left_idem
thf(fact_22_sup_Oleft__idem, axiom,
    ((![A : set_nat, B2 : set_nat]: ((sup_sup_set_nat @ A @ (sup_sup_set_nat @ A @ B2)) = (sup_sup_set_nat @ A @ B2))))). % sup.left_idem
thf(fact_23_sup__idem, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ X @ X) = X)))). % sup_idem
thf(fact_24_sup_Oidem, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ A @ A) = A)))). % sup.idem
thf(fact_25_Un__iff, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)) = (((member_nat @ C2 @ A2)) | ((member_nat @ C2 @ B))))))). % Un_iff
thf(fact_26_UnCI, axiom,
    ((![C2 : nat, B : set_nat, A2 : set_nat]: (((~ ((member_nat @ C2 @ B))) => (member_nat @ C2 @ A2)) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnCI
thf(fact_27_bot__set__def, axiom,
    ((bot_bot_set_nat = (collect_nat @ bot_bot_nat_o)))). % bot_set_def
thf(fact_28_sup_Ostrict__coboundedI2, axiom,
    ((![C2 : set_nat, B2 : set_nat, A : set_nat]: ((ord_less_set_nat @ C2 @ B2) => (ord_less_set_nat @ C2 @ (sup_sup_set_nat @ A @ B2)))))). % sup.strict_coboundedI2
thf(fact_29_sup_Ostrict__coboundedI2, axiom,
    ((![C2 : nat, B2 : nat, A : nat]: ((ord_less_nat @ C2 @ B2) => (ord_less_nat @ C2 @ (sup_sup_nat @ A @ B2)))))). % sup.strict_coboundedI2
thf(fact_30_sup_Ostrict__coboundedI1, axiom,
    ((![C2 : set_nat, A : set_nat, B2 : set_nat]: ((ord_less_set_nat @ C2 @ A) => (ord_less_set_nat @ C2 @ (sup_sup_set_nat @ A @ B2)))))). % sup.strict_coboundedI1
thf(fact_31_sup_Ostrict__coboundedI1, axiom,
    ((![C2 : nat, A : nat, B2 : nat]: ((ord_less_nat @ C2 @ A) => (ord_less_nat @ C2 @ (sup_sup_nat @ A @ B2)))))). % sup.strict_coboundedI1
thf(fact_32_sup_Ostrict__order__iff, axiom,
    ((ord_less_set_nat = (^[B3 : set_nat]: (^[A3 : set_nat]: (((A3 = (sup_sup_set_nat @ A3 @ B3))) & ((~ ((A3 = B3)))))))))). % sup.strict_order_iff
thf(fact_33_sup_Ostrict__order__iff, axiom,
    ((ord_less_nat = (^[B3 : nat]: (^[A3 : nat]: (((A3 = (sup_sup_nat @ A3 @ B3))) & ((~ ((A3 = B3)))))))))). % sup.strict_order_iff
thf(fact_34_sup_Ostrict__boundedE, axiom,
    ((![B2 : set_nat, C2 : set_nat, A : set_nat]: ((ord_less_set_nat @ (sup_sup_set_nat @ B2 @ C2) @ A) => (~ (((ord_less_set_nat @ B2 @ A) => (~ ((ord_less_set_nat @ C2 @ A)))))))))). % sup.strict_boundedE
thf(fact_35_sup_Ostrict__boundedE, axiom,
    ((![B2 : nat, C2 : nat, A : nat]: ((ord_less_nat @ (sup_sup_nat @ B2 @ C2) @ A) => (~ (((ord_less_nat @ B2 @ A) => (~ ((ord_less_nat @ C2 @ A)))))))))). % sup.strict_boundedE
thf(fact_36_less__supI2, axiom,
    ((![X : set_nat, B2 : set_nat, A : set_nat]: ((ord_less_set_nat @ X @ B2) => (ord_less_set_nat @ X @ (sup_sup_set_nat @ A @ B2)))))). % less_supI2
thf(fact_37_less__supI2, axiom,
    ((![X : nat, B2 : nat, A : nat]: ((ord_less_nat @ X @ B2) => (ord_less_nat @ X @ (sup_sup_nat @ A @ B2)))))). % less_supI2
thf(fact_38_less__supI1, axiom,
    ((![X : set_nat, A : set_nat, B2 : set_nat]: ((ord_less_set_nat @ X @ A) => (ord_less_set_nat @ X @ (sup_sup_set_nat @ A @ B2)))))). % less_supI1
thf(fact_39_less__supI1, axiom,
    ((![X : nat, A : nat, B2 : nat]: ((ord_less_nat @ X @ A) => (ord_less_nat @ X @ (sup_sup_nat @ A @ B2)))))). % less_supI1
thf(fact_40_ex__in__conv, axiom,
    ((![A2 : set_nat]: ((?[X2 : nat]: (member_nat @ X2 @ A2)) = (~ ((A2 = bot_bot_set_nat))))))). % ex_in_conv
thf(fact_41_equals0I, axiom,
    ((![A2 : set_nat]: ((![Y2 : nat]: (~ ((member_nat @ Y2 @ A2)))) => (A2 = bot_bot_set_nat))))). % equals0I
thf(fact_42_equals0D, axiom,
    ((![A2 : set_nat, A : nat]: ((A2 = bot_bot_set_nat) => (~ ((member_nat @ A @ A2))))))). % equals0D
thf(fact_43_emptyE, axiom,
    ((![A : nat]: (~ ((member_nat @ A @ bot_bot_set_nat)))))). % emptyE
thf(fact_44_mk__disjoint__insert, axiom,
    ((![A : nat, A2 : set_nat]: ((member_nat @ A @ A2) => (?[B4 : set_nat]: ((A2 = (insert_nat @ A @ B4)) & (~ ((member_nat @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_45_insert__commute, axiom,
    ((![X : nat, Y : nat, A2 : set_nat]: ((insert_nat @ X @ (insert_nat @ Y @ A2)) = (insert_nat @ Y @ (insert_nat @ X @ A2)))))). % insert_commute
thf(fact_46_insert__eq__iff, axiom,
    ((![A : nat, A2 : set_nat, B2 : nat, B : set_nat]: ((~ ((member_nat @ A @ A2))) => ((~ ((member_nat @ B2 @ B))) => (((insert_nat @ A @ A2) = (insert_nat @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_nat]: (((A2 = (insert_nat @ B2 @ C3))) & ((((~ ((member_nat @ B2 @ C3)))) & ((((B = (insert_nat @ A @ C3))) & ((~ ((member_nat @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_47_insert__absorb, axiom,
    ((![A : nat, A2 : set_nat]: ((member_nat @ A @ A2) => ((insert_nat @ A @ A2) = A2))))). % insert_absorb
thf(fact_48_insert__ident, axiom,
    ((![X : nat, A2 : set_nat, B : set_nat]: ((~ ((member_nat @ X @ A2))) => ((~ ((member_nat @ X @ B))) => (((insert_nat @ X @ A2) = (insert_nat @ X @ B)) = (A2 = B))))))). % insert_ident
thf(fact_49_Set_Oset__insert, axiom,
    ((![X : nat, A2 : set_nat]: ((member_nat @ X @ A2) => (~ ((![B4 : set_nat]: ((A2 = (insert_nat @ X @ B4)) => (member_nat @ X @ B4))))))))). % Set.set_insert
thf(fact_50_mem__Collect__eq, axiom,
    ((![A : nat, P : nat > $o]: ((member_nat @ A @ (collect_nat @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_51_Collect__mem__eq, axiom,
    ((![A2 : set_nat]: ((collect_nat @ (^[X2 : nat]: (member_nat @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_52_insertI2, axiom,
    ((![A : nat, B : set_nat, B2 : nat]: ((member_nat @ A @ B) => (member_nat @ A @ (insert_nat @ B2 @ B)))))). % insertI2
thf(fact_53_insertI1, axiom,
    ((![A : nat, B : set_nat]: (member_nat @ A @ (insert_nat @ A @ B))))). % insertI1
thf(fact_54_insertE, axiom,
    ((![A : nat, B2 : nat, A2 : set_nat]: ((member_nat @ A @ (insert_nat @ B2 @ A2)) => ((~ ((A = B2))) => (member_nat @ A @ A2)))))). % insertE
thf(fact_55_sup__left__commute, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((sup_sup_set_nat @ X @ (sup_sup_set_nat @ Y @ Z)) = (sup_sup_set_nat @ Y @ (sup_sup_set_nat @ X @ Z)))))). % sup_left_commute
thf(fact_56_sup_Oleft__commute, axiom,
    ((![B2 : set_nat, A : set_nat, C2 : set_nat]: ((sup_sup_set_nat @ B2 @ (sup_sup_set_nat @ A @ C2)) = (sup_sup_set_nat @ A @ (sup_sup_set_nat @ B2 @ C2)))))). % sup.left_commute
thf(fact_57_sup__commute, axiom,
    ((sup_sup_set_nat = (^[X2 : set_nat]: (^[Y3 : set_nat]: (sup_sup_set_nat @ Y3 @ X2)))))). % sup_commute
thf(fact_58_sup_Ocommute, axiom,
    ((sup_sup_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (sup_sup_set_nat @ B3 @ A3)))))). % sup.commute
thf(fact_59_sup__assoc, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ X @ Y) @ Z) = (sup_sup_set_nat @ X @ (sup_sup_set_nat @ Y @ Z)))))). % sup_assoc
thf(fact_60_sup_Oassoc, axiom,
    ((![A : set_nat, B2 : set_nat, C2 : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A @ B2) @ C2) = (sup_sup_set_nat @ A @ (sup_sup_set_nat @ B2 @ C2)))))). % sup.assoc
thf(fact_61_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_nat, K : set_nat, B2 : set_nat, A : set_nat]: ((B = (sup_sup_set_nat @ K @ B2)) => ((sup_sup_set_nat @ A @ B) = (sup_sup_set_nat @ K @ (sup_sup_set_nat @ A @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_62_boolean__algebra__cancel_Osup1, axiom,
    ((![A2 : set_nat, K : set_nat, A : set_nat, B2 : set_nat]: ((A2 = (sup_sup_set_nat @ K @ A)) => ((sup_sup_set_nat @ A2 @ B2) = (sup_sup_set_nat @ K @ (sup_sup_set_nat @ A @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_63_inf__sup__aci_I5_J, axiom,
    ((sup_sup_set_nat = (^[X2 : set_nat]: (^[Y3 : set_nat]: (sup_sup_set_nat @ Y3 @ X2)))))). % inf_sup_aci(5)
thf(fact_64_inf__sup__aci_I6_J, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ X @ Y) @ Z) = (sup_sup_set_nat @ X @ (sup_sup_set_nat @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_65_inf__sup__aci_I7_J, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((sup_sup_set_nat @ X @ (sup_sup_set_nat @ Y @ Z)) = (sup_sup_set_nat @ Y @ (sup_sup_set_nat @ X @ Z)))))). % inf_sup_aci(7)
thf(fact_66_inf__sup__aci_I8_J, axiom,
    ((![X : set_nat, Y : set_nat]: ((sup_sup_set_nat @ X @ (sup_sup_set_nat @ X @ Y)) = (sup_sup_set_nat @ X @ Y))))). % inf_sup_aci(8)
thf(fact_67_Un__left__commute, axiom,
    ((![A2 : set_nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ B @ C)) = (sup_sup_set_nat @ B @ (sup_sup_set_nat @ A2 @ C)))))). % Un_left_commute
thf(fact_68_Un__left__absorb, axiom,
    ((![A2 : set_nat, B : set_nat]: ((sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ A2 @ B)) = (sup_sup_set_nat @ A2 @ B))))). % Un_left_absorb
thf(fact_69_Un__commute, axiom,
    ((sup_sup_set_nat = (^[A4 : set_nat]: (^[B5 : set_nat]: (sup_sup_set_nat @ B5 @ A4)))))). % Un_commute
thf(fact_70_Un__absorb, axiom,
    ((![A2 : set_nat]: ((sup_sup_set_nat @ A2 @ A2) = A2)))). % Un_absorb
thf(fact_71_Un__assoc, axiom,
    ((![A2 : set_nat, B : set_nat, C : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A2 @ B) @ C) = (sup_sup_set_nat @ A2 @ (sup_sup_set_nat @ B @ C)))))). % Un_assoc
thf(fact_72_ball__Un, axiom,
    ((![A2 : set_nat, B : set_nat, P : nat > $o]: ((![X2 : nat]: (((member_nat @ X2 @ (sup_sup_set_nat @ A2 @ B))) => ((P @ X2)))) = (((![X2 : nat]: (((member_nat @ X2 @ A2)) => ((P @ X2))))) & ((![X2 : nat]: (((member_nat @ X2 @ B)) => ((P @ X2)))))))))). % ball_Un
thf(fact_73_bex__Un, axiom,
    ((![A2 : set_nat, B : set_nat, P : nat > $o]: ((?[X2 : nat]: (((member_nat @ X2 @ (sup_sup_set_nat @ A2 @ B))) & ((P @ X2)))) = (((?[X2 : nat]: (((member_nat @ X2 @ A2)) & ((P @ X2))))) | ((?[X2 : nat]: (((member_nat @ X2 @ B)) & ((P @ X2)))))))))). % bex_Un
thf(fact_74_UnI2, axiom,
    ((![C2 : nat, B : set_nat, A2 : set_nat]: ((member_nat @ C2 @ B) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnI2
thf(fact_75_UnI1, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ A2) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)))))). % UnI1
thf(fact_76_UnE, axiom,
    ((![C2 : nat, A2 : set_nat, B : set_nat]: ((member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B)) => ((~ ((member_nat @ C2 @ A2))) => (member_nat @ C2 @ B)))))). % UnE
thf(fact_77_singleton__inject, axiom,
    ((![A : nat, B2 : nat]: (((insert_nat @ A @ bot_bot_set_nat) = (insert_nat @ B2 @ bot_bot_set_nat)) => (A = B2))))). % singleton_inject
thf(fact_78_insert__not__empty, axiom,
    ((![A : nat, A2 : set_nat]: (~ (((insert_nat @ A @ A2) = bot_bot_set_nat)))))). % insert_not_empty
thf(fact_79_doubleton__eq__iff, axiom,
    ((![A : nat, B2 : nat, C2 : nat, D : nat]: (((insert_nat @ A @ (insert_nat @ B2 @ bot_bot_set_nat)) = (insert_nat @ C2 @ (insert_nat @ D @ bot_bot_set_nat))) = (((((A = C2)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C2))))))))). % doubleton_eq_iff
thf(fact_80_singleton__iff, axiom,
    ((![B2 : nat, A : nat]: ((member_nat @ B2 @ (insert_nat @ A @ bot_bot_set_nat)) = (B2 = A))))). % singleton_iff
thf(fact_81_singletonD, axiom,
    ((![B2 : nat, A : nat]: ((member_nat @ B2 @ (insert_nat @ A @ bot_bot_set_nat)) => (B2 = A))))). % singletonD
thf(fact_82_Un__empty__right, axiom,
    ((![A2 : set_nat]: ((sup_sup_set_nat @ A2 @ bot_bot_set_nat) = A2)))). % Un_empty_right
thf(fact_83_Un__empty__left, axiom,
    ((![B : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ B) = B)))). % Un_empty_left
thf(fact_84_singleton__Un__iff, axiom,
    ((![X : nat, A2 : set_nat, B : set_nat]: (((insert_nat @ X @ bot_bot_set_nat) = (sup_sup_set_nat @ A2 @ B)) = (((((A2 = bot_bot_set_nat)) & ((B = (insert_nat @ X @ bot_bot_set_nat))))) | ((((((A2 = (insert_nat @ X @ bot_bot_set_nat))) & ((B = bot_bot_set_nat)))) | ((((A2 = (insert_nat @ X @ bot_bot_set_nat))) & ((B = (insert_nat @ X @ bot_bot_set_nat)))))))))))). % singleton_Un_iff
thf(fact_85_Un__singleton__iff, axiom,
    ((![A2 : set_nat, B : set_nat, X : nat]: (((sup_sup_set_nat @ A2 @ B) = (insert_nat @ X @ bot_bot_set_nat)) = (((((A2 = bot_bot_set_nat)) & ((B = (insert_nat @ X @ bot_bot_set_nat))))) | ((((((A2 = (insert_nat @ X @ bot_bot_set_nat))) & ((B = bot_bot_set_nat)))) | ((((A2 = (insert_nat @ X @ bot_bot_set_nat))) & ((B = (insert_nat @ X @ bot_bot_set_nat)))))))))))). % Un_singleton_iff
thf(fact_86_insert__is__Un, axiom,
    ((insert_nat = (^[A3 : nat]: (sup_sup_set_nat @ (insert_nat @ A3 @ bot_bot_set_nat)))))). % insert_is_Un
thf(fact_87_greaterThanLessThan__iff, axiom,
    ((![I : nat, L : nat, U : nat]: ((member_nat @ I @ (set_or1544565540an_nat @ L @ U)) = (((ord_less_nat @ L @ I)) & ((ord_less_nat @ I @ U))))))). % greaterThanLessThan_iff
thf(fact_88_add__less__cancel__right, axiom,
    ((![A : nat, C2 : nat, B2 : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B2 @ C2)) = (ord_less_nat @ A @ B2))))). % add_less_cancel_right
thf(fact_89_add__less__cancel__left, axiom,
    ((![C2 : nat, A : nat, B2 : nat]: ((ord_less_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B2)) = (ord_less_nat @ A @ B2))))). % add_less_cancel_left
thf(fact_90_the__elem__eq, axiom,
    ((![X : nat]: ((the_elem_nat @ (insert_nat @ X @ bot_bot_set_nat)) = X)))). % the_elem_eq
thf(fact_91_is__singletonI, axiom,
    ((![X : nat]: (is_singleton_nat @ (insert_nat @ X @ bot_bot_set_nat))))). % is_singletonI
thf(fact_92_bot_Onot__eq__extremum, axiom,
    ((![A : set_nat]: ((~ ((A = bot_bot_set_nat))) = (ord_less_set_nat @ bot_bot_set_nat @ A))))). % bot.not_eq_extremum
thf(fact_93_bot_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = bot_bot_nat))) = (ord_less_nat @ bot_bot_nat @ A))))). % bot.not_eq_extremum
thf(fact_94_bot_Oextremum__strict, axiom,
    ((![A : set_nat]: (~ ((ord_less_set_nat @ A @ bot_bot_set_nat)))))). % bot.extremum_strict
thf(fact_95_bot_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ bot_bot_nat)))))). % bot.extremum_strict
thf(fact_96_not__psubset__empty, axiom,
    ((![A2 : set_nat]: (~ ((ord_less_set_nat @ A2 @ bot_bot_set_nat)))))). % not_psubset_empty
thf(fact_97_is__singleton__the__elem, axiom,
    ((is_singleton_nat = (^[A4 : set_nat]: (A4 = (insert_nat @ (the_elem_nat @ A4) @ bot_bot_set_nat)))))). % is_singleton_the_elem
thf(fact_98_is__singletonI_H, axiom,
    ((![A2 : set_nat]: ((~ ((A2 = bot_bot_set_nat))) => ((![X3 : nat, Y2 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ Y2 @ A2) => (X3 = Y2)))) => (is_singleton_nat @ A2)))))). % is_singletonI'
thf(fact_99_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_nat @ B2 @ A) => (~ ((A = B2))))))). % dual_order.strict_implies_not_eq
thf(fact_100_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((A = B2))))))). % order.strict_implies_not_eq
thf(fact_101_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (((ord_less_nat @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_102_dual__order_Ostrict__trans, axiom,
    ((![B2 : nat, A : nat, C2 : nat]: ((ord_less_nat @ B2 @ A) => ((ord_less_nat @ C2 @ B2) => (ord_less_nat @ C2 @ A)))))). % dual_order.strict_trans
thf(fact_103_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B2 : nat]: ((![A5 : nat, B6 : nat]: ((ord_less_nat @ A5 @ B6) => (P @ A5 @ B6))) => ((![A5 : nat]: (P @ A5 @ A5)) => ((![A5 : nat, B6 : nat]: ((P @ B6 @ A5) => (P @ A5 @ B6))) => (P @ A @ B2))))))). % linorder_less_wlog
thf(fact_104_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N : nat]: (((P3 @ N)) & ((![M : nat]: (((ord_less_nat @ M @ N)) => ((~ ((P3 @ M))))))))))))). % exists_least_iff
thf(fact_105_less__imp__not__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_imp_not_less
thf(fact_106_order_Ostrict__trans, axiom,
    ((![A : nat, B2 : nat, C2 : nat]: ((ord_less_nat @ A @ B2) => ((ord_less_nat @ B2 @ C2) => (ord_less_nat @ A @ C2)))))). % order.strict_trans
thf(fact_107_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_108_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_109_less__imp__triv, axiom,
    ((![X : nat, Y : nat, P : $o]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ X) => P))))). % less_imp_triv
thf(fact_110_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_111_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_112_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X3 : nat]: ((![Y4 : nat]: ((ord_less_nat @ Y4 @ X3) => (P @ Y4))) => (P @ X3))) => (P @ A))))). % less_induct
thf(fact_113_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_114_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_115_dual__order_Oasym, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_nat @ B2 @ A) => (~ ((ord_less_nat @ A @ B2))))))). % dual_order.asym
thf(fact_116_ord__less__eq__trans, axiom,
    ((![A : nat, B2 : nat, C2 : nat]: ((ord_less_nat @ A @ B2) => ((B2 = C2) => (ord_less_nat @ A @ C2)))))). % ord_less_eq_trans
thf(fact_117_ord__eq__less__trans, axiom,
    ((![A : nat, B2 : nat, C2 : nat]: ((A = B2) => ((ord_less_nat @ B2 @ C2) => (ord_less_nat @ A @ C2)))))). % ord_eq_less_trans
thf(fact_118_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_119_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_120_less__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X @ Z)))))). % less_trans
thf(fact_121_less__asym_H, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((ord_less_nat @ B2 @ A))))))). % less_asym'
thf(fact_122_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_123_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_124_order_Oasym, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_nat @ A @ B2) => (~ ((ord_less_nat @ B2 @ A))))))). % order.asym
thf(fact_125_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_126_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_127_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_128_order__less__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B2) => ((ord_less_nat @ (F @ B2) @ C2) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C2))))))). % order_less_subst2
thf(fact_129_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C2 : nat]: ((ord_less_nat @ A @ (F @ B2)) => ((ord_less_nat @ B2 @ C2) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C2)))))))). % order_less_subst1
thf(fact_130_ord__less__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B2) => (((F @ B2) = C2) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_131_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C2 : nat]: ((A = (F @ B2)) => ((ord_less_nat @ B2 @ C2) => ((![X3 : nat, Y2 : nat]: ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_132_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B2 : a, C2 : a]: ((plus_plus_a @ (plus_plus_a @ A @ B2) @ C2) = (plus_plus_a @ A @ (plus_plus_a @ B2 @ C2)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_133_group__cancel_Oadd1, axiom,
    ((![A2 : a, K : a, A : a, B2 : a]: ((A2 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A2 @ B2) = (plus_plus_a @ K @ (plus_plus_a @ A @ B2))))))). % group_cancel.add1
thf(fact_134_group__cancel_Oadd2, axiom,
    ((![B : a, K : a, B2 : a, A : a]: ((B = (plus_plus_a @ K @ B2)) => ((plus_plus_a @ A @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B2))))))). % group_cancel.add2
thf(fact_135_add_Oassoc, axiom,
    ((![A : a, B2 : a, C2 : a]: ((plus_plus_a @ (plus_plus_a @ A @ B2) @ C2) = (plus_plus_a @ A @ (plus_plus_a @ B2 @ C2)))))). % add.assoc
thf(fact_136_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_137_add_Oleft__commute, axiom,
    ((![B2 : a, A : a, C2 : a]: ((plus_plus_a @ B2 @ (plus_plus_a @ A @ C2)) = (plus_plus_a @ A @ (plus_plus_a @ B2 @ C2)))))). % add.left_commute
thf(fact_138_is__singletonE, axiom,
    ((![A2 : set_nat]: ((is_singleton_nat @ A2) => (~ ((![X3 : nat]: (~ ((A2 = (insert_nat @ X3 @ bot_bot_set_nat))))))))))). % is_singletonE
thf(fact_139_is__singleton__def, axiom,
    ((is_singleton_nat = (^[A4 : set_nat]: (?[X2 : nat]: (A4 = (insert_nat @ X2 @ bot_bot_set_nat))))))). % is_singleton_def
thf(fact_140_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_141_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_142_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)

% Conjectures (1)
thf(conj_0, conjecture,
    (((plus_plus_a @ (f @ m) @ (groups1145913330_nat_a @ f @ (set_or1544565540an_nat @ m @ n))) = (groups1145913330_nat_a @ f @ (sup_sup_set_nat @ (insert_nat @ m @ bot_bot_set_nat) @ (set_or1544565540an_nat @ m @ n)))))).
