%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Undirected_Graph_Theory/0018_Graph_Theory_Relations/prob_00231_007464__13361450_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1547 ( 646 unt; 272 typ;   0 def)
%            Number of atoms       : 3486 (1241 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 9298 ( 381   ~;  35   |; 236   &;7181   @)
%                                         (   0 <=>;1465  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :   52 (  51 usr)
%            Number of type conns  :  438 ( 438   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  222 ( 221 usr;  48 con; 0-4 aty)
%            Number of variables   : 3020 ( 191   ^;2748   !;  81   ?;3020   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:34:13.240
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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    undire2554140024507503526stem_a: set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
    undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Nat__Onat,type,
    undire7858122600432113898nt_nat: nat > set_nat > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    undire3369688177417741453od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Product____Type__Ounit,type,
    undire5866035466353400179t_unit: product_unit > set_Product_unit > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
    undire2320338297334612420_set_a: set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
    undire1521409233611534436dent_a: a > set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
    undire3231912044278729248dges_a: set_set_a > a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
    undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Nat__Onat,type,
    undire4970100481470743719et_nat: produc7248412053542808358at_nat > set_nat ).

thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    undire2459242765783757584od_a_a: produc8857593507947890343od_a_a > set_Product_prod_a_a ).

thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Set__Oset_Itf__a_J,type,
    undire4638465864238448455_set_a: produc3364680560414100336_set_a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001tf__a,type,
    undire8536760333753235943_set_a: produc4044097585999906000od_a_a > set_a ).

thf(sy_c_Undirected__Graph__Basics_Osgraph_001tf__a,type,
    undire3507641187627840796raph_a: set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Osgraph_Ocomplement__edges_001tf__a,type,
    undire4625228487420481630dges_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Osgraph_Ois__complement_001tf__a,type,
    undire8013100667316154652ment_a: set_a > set_set_a > produc7943277765024757383_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Osgraph_Ois__complete__n__graph_001tf__a,type,
    undire6087271738840788937raph_a: set_a > set_set_a > nat > $o ).

thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
    undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
    undire7251896706689453996raph_a: set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001tf__a,type,
    undire8383842906760478443ween_a: set_set_a > set_a > set_a > set_Product_prod_a_a ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
    undire8867928226783802224gree_a: set_set_a > a > nat ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
    undire297304480579013331sity_a: set_set_a > set_a > set_a > real ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
    undire3617971648856834880loop_a: set_set_a > a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
    undire4753905205749729249oops_a: set_set_a > a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
    undire1270416042309875431dges_a: set_set_a > a > set_set_a ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Nat__Onat,type,
    undire6814325412647357297en_nat: set_nat > set_nat > set_nat > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    undire7011261089604658374od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Product____Type__Ounit,type,
    undire7069873054131797420t_unit: set_Product_unit > set_Product_unit > set_Product_unit > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Set__Oset_Itf__a_J,type,
    undire2578756059399487229_set_a: set_set_a > set_set_a > set_set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
    undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
    undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
    undire2905028936066782638loop_a: set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
    undire4917966558017083288edge_a: set_a > $o ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
    undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).

thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
    undire397441198561214472_adj_a: set_set_a > a > a > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
    member7821138191923116944t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    member268004040519299248_set_a: produc7943277765024757383_set_a > set_Pr4256959165342167655_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: product_unit > set_Product_unit > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    member_set_set_a: set_set_a > set_set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_edges,type,
    edges: set_set_a ).

thf(sy_v_vertices,type,
    vertices: set_a ).

% Relevant facts (1274)
thf(fact_0_rel__edges__is,axiom,
    ( ( graph_8122095853558514513tion_a @ edges )
    = ( graph_8122095853558514513tion_a @ edges ) ) ).

% rel_edges_is
thf(fact_1_edge__adj__inE,axiom,
    ! [E1: set_a,E2: set_a] :
      ( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
     => ( ( member_set_a @ E1 @ edges )
        & ( member_set_a @ E2 @ edges ) ) ) ).

% edge_adj_inE
thf(fact_2_ulgraph_Oadj__relation_Ocong,axiom,
    graph_8122095853558514513tion_a = graph_8122095853558514513tion_a ).

% ulgraph.adj_relation.cong
thf(fact_3_edges__rel__is,axiom,
    ( edges
    = ( graph_9096688302331494522_set_a @ ( graph_8122095853558514513tion_a @ edges ) ) ) ).

% edges_rel_is
thf(fact_4_sym__adj,axiom,
    sym_on_a @ top_top_set_a @ ( graph_8122095853558514513tion_a @ edges ) ).

% sym_adj
thf(fact_5_UNIV__I,axiom,
    ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).

% UNIV_I
thf(fact_6_UNIV__I,axiom,
    ! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).

% UNIV_I
thf(fact_7_UNIV__I,axiom,
    ! [X: a] : ( member_a @ X @ top_top_set_a ) ).

% UNIV_I
thf(fact_8_UNIV__I,axiom,
    ! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).

% UNIV_I
thf(fact_9_UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
    ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).

% iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
    ! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).

% iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
    ! [X: a] : ( member_a @ X @ top_top_set_a ) ).

% iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
    ! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).

% iso_tuple_UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_15_is__rel__irrefl__alt,axiom,
    ! [U: a,V: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) )
     => ( U != V ) ) ).

% is_rel_irrefl_alt
thf(fact_16_rel__item__is,axiom,
    ! [U: a,V: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) ) ) ).

% rel_item_is
thf(fact_17_sym__alt,axiom,
    ! [U: a,V: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ U ) @ ( graph_8122095853558514513tion_a @ edges ) ) ) ).

% sym_alt
thf(fact_18_edge__density__commute,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
      = ( undire297304480579013331sity_a @ edges @ Y @ X2 ) ) ).

% edge_density_commute
thf(fact_19_sgraph__rel__axioms_Ointro,axiom,
    ! [Adj_rel: set_Product_prod_a_a] :
      ( ( irrefl_on_a @ top_top_set_a @ Adj_rel )
     => ( graph_8743372715275275884ioms_a @ Adj_rel ) ) ).

% sgraph_rel_axioms.intro
thf(fact_20_sgraph__rel__axioms_Ointro,axiom,
    ! [Adj_rel: set_Pr5094982260447487303t_unit] :
      ( ( irrefl6292445964007751171t_unit @ top_to1996260823553986621t_unit @ Adj_rel )
     => ( graph_6428585086700507003t_unit @ Adj_rel ) ) ).

% sgraph_rel_axioms.intro
thf(fact_21_sgraph__rel__axioms_Ointro,axiom,
    ! [Adj_rel: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ Adj_rel )
     => ( graph_3780608497661675746ms_nat @ Adj_rel ) ) ).

% sgraph_rel_axioms.intro
thf(fact_22_sgraph__rel__axioms_Ointro,axiom,
    ! [Adj_rel: set_Pr8600417178894128327od_a_a] :
      ( ( irrefl3954896097174259997od_a_a @ top_to8063371432257647191od_a_a @ Adj_rel )
     => ( graph_6846383398958086805od_a_a @ Adj_rel ) ) ).

% sgraph_rel_axioms.intro
thf(fact_23_sgraph__rel__axioms_Ointro,axiom,
    ! [Adj_rel: set_Pr5845495582615845127_set_a] :
      ( ( irrefl_on_set_a @ top_top_set_set_a @ Adj_rel )
     => ( graph_1670365429040299980_set_a @ Adj_rel ) ) ).

% sgraph_rel_axioms.intro
thf(fact_24_sgraph__rel__axioms__def,axiom,
    ( graph_8743372715275275884ioms_a
    = ( irrefl_on_a @ top_top_set_a ) ) ).

% sgraph_rel_axioms_def
thf(fact_25_sgraph__rel__axioms__def,axiom,
    ( graph_6428585086700507003t_unit
    = ( irrefl6292445964007751171t_unit @ top_to1996260823553986621t_unit ) ) ).

% sgraph_rel_axioms_def
thf(fact_26_sgraph__rel__axioms__def,axiom,
    ( graph_3780608497661675746ms_nat
    = ( irrefl_on_nat @ top_top_set_nat ) ) ).

% sgraph_rel_axioms_def
thf(fact_27_sgraph__rel__axioms__def,axiom,
    ( graph_6846383398958086805od_a_a
    = ( irrefl3954896097174259997od_a_a @ top_to8063371432257647191od_a_a ) ) ).

% sgraph_rel_axioms_def
thf(fact_28_sgraph__rel__axioms__def,axiom,
    ( graph_1670365429040299980_set_a
    = ( irrefl_on_set_a @ top_top_set_set_a ) ) ).

% sgraph_rel_axioms_def
thf(fact_29_empty__not__edge,axiom,
    ~ ( member_set_a @ bot_bot_set_a @ edges ) ).

% empty_not_edge
thf(fact_30_empty__iff,axiom,
    ! [C: product_unit] :
      ~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).

% empty_iff
thf(fact_31_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_32_empty__iff,axiom,
    ! [C: set_a] :
      ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).

% empty_iff
thf(fact_33_empty__iff,axiom,
    ! [C: product_prod_a_a] :
      ~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).

% empty_iff
thf(fact_34_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_35_all__not__in__conv,axiom,
    ! [A: set_Product_unit] :
      ( ( ! [X3: product_unit] :
            ~ ( member_Product_unit @ X3 @ A ) )
      = ( A = bot_bo3957492148770167129t_unit ) ) ).

% all_not_in_conv
thf(fact_36_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X3: a] :
            ~ ( member_a @ X3 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_37_all__not__in__conv,axiom,
    ! [A: set_set_a] :
      ( ( ! [X3: set_a] :
            ~ ( member_set_a @ X3 @ A ) )
      = ( A = bot_bot_set_set_a ) ) ).

% all_not_in_conv
thf(fact_38_all__not__in__conv,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( ! [X3: product_prod_a_a] :
            ~ ( member1426531477525435216od_a_a @ X3 @ A ) )
      = ( A = bot_bo3357376287454694259od_a_a ) ) ).

% all_not_in_conv
thf(fact_39_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_40_Collect__empty__eq,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( ! [X3: a] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_41_Collect__empty__eq,axiom,
    ! [P: set_a > $o] :
      ( ( ( collect_set_a @ P )
        = bot_bot_set_set_a )
      = ( ! [X3: set_a] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_42_Collect__empty__eq,axiom,
    ! [P: product_prod_a_a > $o] :
      ( ( ( collec3336397797384452498od_a_a @ P )
        = bot_bo3357376287454694259od_a_a )
      = ( ! [X3: product_prod_a_a] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_43_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_44_empty__Collect__eq,axiom,
    ! [P: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P ) )
      = ( ! [X3: a] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_45_empty__Collect__eq,axiom,
    ! [P: set_a > $o] :
      ( ( bot_bot_set_set_a
        = ( collect_set_a @ P ) )
      = ( ! [X3: set_a] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_46_empty__Collect__eq,axiom,
    ! [P: product_prod_a_a > $o] :
      ( ( bot_bo3357376287454694259od_a_a
        = ( collec3336397797384452498od_a_a @ P ) )
      = ( ! [X3: product_prod_a_a] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_47_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_48_ulgraph__rel_Oedge__set_Ocong,axiom,
    graph_9096688302331494522_set_a = graph_9096688302331494522_set_a ).

% ulgraph_rel.edge_set.cong
thf(fact_49_emptyE,axiom,
    ! [A2: product_unit] :
      ~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).

% emptyE
thf(fact_50_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_51_emptyE,axiom,
    ! [A2: set_a] :
      ~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).

% emptyE
thf(fact_52_emptyE,axiom,
    ! [A2: product_prod_a_a] :
      ~ ( member1426531477525435216od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ).

% emptyE
thf(fact_53_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_54_equals0D,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( ( A = bot_bo3957492148770167129t_unit )
     => ~ ( member_Product_unit @ A2 @ A ) ) ).

% equals0D
thf(fact_55_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_56_equals0D,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( A = bot_bot_set_set_a )
     => ~ ( member_set_a @ A2 @ A ) ) ).

% equals0D
thf(fact_57_equals0D,axiom,
    ! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( A = bot_bo3357376287454694259od_a_a )
     => ~ ( member1426531477525435216od_a_a @ A2 @ A ) ) ).

% equals0D
thf(fact_58_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_59_equals0I,axiom,
    ! [A: set_Product_unit] :
      ( ! [Y2: product_unit] :
          ~ ( member_Product_unit @ Y2 @ A )
     => ( A = bot_bo3957492148770167129t_unit ) ) ).

% equals0I
thf(fact_60_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y2: a] :
          ~ ( member_a @ Y2 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_61_equals0I,axiom,
    ! [A: set_set_a] :
      ( ! [Y2: set_a] :
          ~ ( member_set_a @ Y2 @ A )
     => ( A = bot_bot_set_set_a ) ) ).

% equals0I
thf(fact_62_equals0I,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ! [Y2: product_prod_a_a] :
          ~ ( member1426531477525435216od_a_a @ Y2 @ A )
     => ( A = bot_bo3357376287454694259od_a_a ) ) ).

% equals0I
thf(fact_63_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y2: nat] :
          ~ ( member_nat @ Y2 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_64_ex__in__conv,axiom,
    ! [A: set_Product_unit] :
      ( ( ? [X3: product_unit] : ( member_Product_unit @ X3 @ A ) )
      = ( A != bot_bo3957492148770167129t_unit ) ) ).

% ex_in_conv
thf(fact_65_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X3: a] : ( member_a @ X3 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_66_ex__in__conv,axiom,
    ! [A: set_set_a] :
      ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
      = ( A != bot_bot_set_set_a ) ) ).

% ex_in_conv
thf(fact_67_ex__in__conv,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( ? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
      = ( A != bot_bo3357376287454694259od_a_a ) ) ).

% ex_in_conv
thf(fact_68_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_69_top__set__def,axiom,
    ( top_top_set_a
    = ( collect_a @ top_top_a_o ) ) ).

% top_set_def
thf(fact_70_top__set__def,axiom,
    ( top_to1996260823553986621t_unit
    = ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).

% top_set_def
thf(fact_71_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_72_top__set__def,axiom,
    ( top_to8063371432257647191od_a_a
    = ( collec3336397797384452498od_a_a @ top_to8687885267596698950_a_a_o ) ) ).

% top_set_def
thf(fact_73_top__set__def,axiom,
    ( top_top_set_set_a
    = ( collect_set_a @ top_top_set_a_o ) ) ).

% top_set_def
thf(fact_74_empty__not__UNIV,axiom,
    bot_bot_set_a != top_top_set_a ).

% empty_not_UNIV
thf(fact_75_empty__not__UNIV,axiom,
    bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).

% empty_not_UNIV
thf(fact_76_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_77_empty__not__UNIV,axiom,
    bot_bo3357376287454694259od_a_a != top_to8063371432257647191od_a_a ).

% empty_not_UNIV
thf(fact_78_empty__not__UNIV,axiom,
    bot_bot_set_set_a != top_top_set_set_a ).

% empty_not_UNIV
thf(fact_79_UNIV__witness,axiom,
    ? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).

% UNIV_witness
thf(fact_80_UNIV__witness,axiom,
    ? [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ).

% UNIV_witness
thf(fact_81_UNIV__witness,axiom,
    ? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_82_UNIV__witness,axiom,
    ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ top_to8063371432257647191od_a_a ) ).

% UNIV_witness
thf(fact_83_UNIV__witness,axiom,
    ? [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ).

% UNIV_witness
thf(fact_84_UNIV__eq__I,axiom,
    ! [A: set_a] :
      ( ! [X4: a] : ( member_a @ X4 @ A )
     => ( top_top_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_85_UNIV__eq__I,axiom,
    ! [A: set_Product_unit] :
      ( ! [X4: product_unit] : ( member_Product_unit @ X4 @ A )
     => ( top_to1996260823553986621t_unit = A ) ) ).

% UNIV_eq_I
thf(fact_86_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X4: nat] : ( member_nat @ X4 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_87_UNIV__eq__I,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ! [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A )
     => ( top_to8063371432257647191od_a_a = A ) ) ).

% UNIV_eq_I
thf(fact_88_UNIV__eq__I,axiom,
    ! [A: set_set_a] :
      ( ! [X4: set_a] : ( member_set_a @ X4 @ A )
     => ( top_top_set_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_89_obtain__edge__pair__adj,axiom,
    ! [E: set_a] :
      ( ( member_set_a @ E @ edges )
     => ~ ! [U2: a,V2: a] :
            ( ( E
              = ( insert_a @ U2 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V2 ) @ ( graph_8122095853558514513tion_a @ edges ) ) ) ) ).

% obtain_edge_pair_adj
thf(fact_90_edge__density__zero,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( Y = bot_bot_set_a )
     => ( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
        = zero_zero_real ) ) ).

% edge_density_zero
thf(fact_91_rel__vert__adj__iff,axiom,
    ! [U: a,V: a] :
      ( ( undire397441198561214472_adj_a @ edges @ U @ V )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) ) ) ).

% rel_vert_adj_iff
thf(fact_92_edge__adj__def,axiom,
    ! [E1: set_a,E2: set_a] :
      ( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
      = ( ( ( inf_inf_set_a @ E1 @ E2 )
         != bot_bot_set_a )
        & ( member_set_a @ E1 @ edges )
        & ( member_set_a @ E2 @ edges ) ) ) ).

% edge_adj_def
thf(fact_93_irreflI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [A3: a] :
          ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R )
     => ( irrefl_on_a @ top_top_set_a @ R ) ) ).

% irreflI
thf(fact_94_irreflI,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ! [A3: product_unit] :
          ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A3 @ A3 ) @ R )
     => ( irrefl6292445964007751171t_unit @ top_to1996260823553986621t_unit @ R ) ) ).

% irreflI
thf(fact_95_irreflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
     => ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% irreflI
thf(fact_96_irreflI,axiom,
    ! [R: set_Pr8600417178894128327od_a_a] :
      ( ! [A3: product_prod_a_a] :
          ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ R )
     => ( irrefl3954896097174259997od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).

% irreflI
thf(fact_97_irreflI,axiom,
    ! [R: set_Pr5845495582615845127_set_a] :
      ( ! [A3: set_a] :
          ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ A3 ) @ R )
     => ( irrefl_on_set_a @ top_top_set_set_a @ R ) ) ).

% irreflI
thf(fact_98_irreflD,axiom,
    ! [R: set_Product_prod_a_a,X: a] :
      ( ( irrefl_on_a @ top_top_set_a @ R )
     => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_99_irreflD,axiom,
    ! [R: set_Pr5094982260447487303t_unit,X: product_unit] :
      ( ( irrefl6292445964007751171t_unit @ top_to1996260823553986621t_unit @ R )
     => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_100_irreflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_101_irreflD,axiom,
    ! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a] :
      ( ( irrefl3954896097174259997od_a_a @ top_to8063371432257647191od_a_a @ R )
     => ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_102_irreflD,axiom,
    ! [R: set_Pr5845495582615845127_set_a,X: set_a] :
      ( ( irrefl_on_set_a @ top_top_set_set_a @ R )
     => ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_103_symD,axiom,
    ! [R: set_Product_prod_a_a,X: a,Y3: a] :
      ( ( sym_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) ).

% symD
thf(fact_104_symD,axiom,
    ! [R: set_Pr5094982260447487303t_unit,X: product_unit,Y3: product_unit] :
      ( ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ R )
     => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y3 ) @ R )
       => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X ) @ R ) ) ) ).

% symD
thf(fact_105_symD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) ) ) ).

% symD
thf(fact_106_symD,axiom,
    ! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y3: product_prod_a_a] :
      ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
     => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R )
       => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X ) @ R ) ) ) ).

% symD
thf(fact_107_symD,axiom,
    ! [R: set_Pr5845495582615845127_set_a,X: set_a,Y3: set_a] :
      ( ( sym_on_set_a @ top_top_set_set_a @ R )
     => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X @ Y3 ) @ R )
       => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ X ) @ R ) ) ) ).

% symD
thf(fact_108_symE,axiom,
    ! [R: set_Product_prod_a_a,B: a,A2: a] :
      ( ( sym_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A2 ) @ R )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R ) ) ) ).

% symE
thf(fact_109_symE,axiom,
    ! [R: set_Pr5094982260447487303t_unit,B: product_unit,A2: product_unit] :
      ( ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ R )
     => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R )
       => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R ) ) ) ).

% symE
thf(fact_110_symE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B: nat,A2: nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R ) ) ) ).

% symE
thf(fact_111_symE,axiom,
    ! [R: set_Pr8600417178894128327od_a_a,B: product_prod_a_a,A2: product_prod_a_a] :
      ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
     => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B @ A2 ) @ R )
       => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ R ) ) ) ).

% symE
thf(fact_112_symE,axiom,
    ! [R: set_Pr5845495582615845127_set_a,B: set_a,A2: set_a] :
      ( ( sym_on_set_a @ top_top_set_set_a @ R )
     => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ B @ A2 ) @ R )
       => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ B ) @ R ) ) ) ).

% symE
thf(fact_113_symI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X4: a,Y2: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X4 ) @ R ) )
     => ( sym_on_a @ top_top_set_a @ R ) ) ).

% symI
thf(fact_114_symI,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ! [X4: product_unit,Y2: product_unit] :
          ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X4 @ Y2 ) @ R )
         => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ X4 ) @ R ) )
     => ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ R ) ) ).

% symI
thf(fact_115_symI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y2 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X4 ) @ R ) )
     => ( sym_on_nat @ top_top_set_nat @ R ) ) ).

% symI
thf(fact_116_symI,axiom,
    ! [R: set_Pr8600417178894128327od_a_a] :
      ( ! [X4: product_prod_a_a,Y2: product_prod_a_a] :
          ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y2 ) @ R )
         => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y2 @ X4 ) @ R ) )
     => ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).

% symI
thf(fact_117_symI,axiom,
    ! [R: set_Pr5845495582615845127_set_a] :
      ( ! [X4: set_a,Y2: set_a] :
          ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y2 ) @ R )
         => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y2 @ X4 ) @ R ) )
     => ( sym_on_set_a @ top_top_set_set_a @ R ) ) ).

% symI
thf(fact_118_prod_Oinject,axiom,
    ! [X1: a,X22: a,Y1: a,Y22: a] :
      ( ( ( product_Pair_a_a @ X1 @ X22 )
        = ( product_Pair_a_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_119_prod_Oinject,axiom,
    ! [X1: set_a,X22: set_set_a,Y1: set_a,Y22: set_set_a] :
      ( ( ( produc2116933609460601975_set_a @ X1 @ X22 )
        = ( produc2116933609460601975_set_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_120_old_Oprod_Oinject,axiom,
    ! [A2: a,B: a,A4: a,B2: a] :
      ( ( ( product_Pair_a_a @ A2 @ B )
        = ( product_Pair_a_a @ A4 @ B2 ) )
      = ( ( A2 = A4 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_121_old_Oprod_Oinject,axiom,
    ! [A2: set_a,B: set_set_a,A4: set_a,B2: set_set_a] :
      ( ( ( produc2116933609460601975_set_a @ A2 @ B )
        = ( produc2116933609460601975_set_a @ A4 @ B2 ) )
      = ( ( A2 = A4 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_122_singleton__not__edge,axiom,
    ! [X: a] :
      ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ edges ) ).

% singleton_not_edge
thf(fact_123_vert__adj__sym,axiom,
    ! [V1: a,V22: a] :
      ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
      = ( undire397441198561214472_adj_a @ edges @ V22 @ V1 ) ) ).

% vert_adj_sym
thf(fact_124_insertCI,axiom,
    ! [A2: set_a,B3: set_set_a,B: set_a] :
      ( ( ~ ( member_set_a @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_set_a @ A2 @ ( insert_set_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_125_insertCI,axiom,
    ! [A2: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( ~ ( member1426531477525435216od_a_a @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_126_insertCI,axiom,
    ! [A2: a,B3: set_a,B: a] :
      ( ( ~ ( member_a @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_127_insertCI,axiom,
    ! [A2: nat,B3: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).

% insertCI
thf(fact_128_insertCI,axiom,
    ! [A2: product_unit,B3: set_Product_unit,B: product_unit] :
      ( ( ~ ( member_Product_unit @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B3 ) ) ) ).

% insertCI
thf(fact_129_insert__iff,axiom,
    ! [A2: set_a,B: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member_set_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_130_insert__iff,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_131_insert__iff,axiom,
    ! [A2: a,B: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_132_insert__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
      = ( ( A2 = B )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_133_insert__iff,axiom,
    ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
      = ( ( A2 = B )
        | ( member_Product_unit @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_134_mem__Collect__eq,axiom,
    ! [A2: set_a,P: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
    ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
    ! [A2: product_unit,P: product_unit > $o] :
      ( ( member_Product_unit @ A2 @ ( collect_Product_unit @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A: set_set_a] :
      ( ( collect_set_a
        @ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
    ! [A: set_Product_unit] :
      ( ( collect_Product_unit
        @ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_144_insert__absorb2,axiom,
    ! [X: a,A: set_a] :
      ( ( insert_a @ X @ ( insert_a @ X @ A ) )
      = ( insert_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_145_insert__absorb2,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ( insert_set_a @ X @ ( insert_set_a @ X @ A ) )
      = ( insert_set_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_146_IntI,axiom,
    ! [C: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ A )
     => ( ( member1426531477525435216od_a_a @ C @ B3 )
       => ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_147_IntI,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A )
     => ( ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_148_IntI,axiom,
    ! [C: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ C @ A )
     => ( ( member_Product_unit @ C @ B3 )
       => ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_149_IntI,axiom,
    ! [C: a,A: set_a,B3: set_a] :
      ( ( member_a @ C @ A )
     => ( ( member_a @ C @ B3 )
       => ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_150_IntI,axiom,
    ! [C: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ C @ A )
     => ( ( member_set_a @ C @ B3 )
       => ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_151_Int__iff,axiom,
    ! [C: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) )
      = ( ( member1426531477525435216od_a_a @ C @ A )
        & ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_152_Int__iff,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
      = ( ( member_nat @ C @ A )
        & ( member_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_153_Int__iff,axiom,
    ! [C: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B3 ) )
      = ( ( member_Product_unit @ C @ A )
        & ( member_Product_unit @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_154_Int__iff,axiom,
    ! [C: a,A: set_a,B3: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
      = ( ( member_a @ C @ A )
        & ( member_a @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_155_Int__iff,axiom,
    ! [C: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B3 ) )
      = ( ( member_set_a @ C @ A )
        & ( member_set_a @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_156_not__vert__adj,axiom,
    ! [V: a,U: a] :
      ( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
     => ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).

% not_vert_adj
thf(fact_157_vert__adj__def,axiom,
    ! [V1: a,V22: a] :
      ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
      = ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ).

% vert_adj_def
thf(fact_158_is__edge__between__def,axiom,
    ( undire8544646567961481629ween_a
    = ( ^ [X5: set_a,Y4: set_a,E3: set_a] :
        ? [X3: a,Y5: a] :
          ( ( E3
            = ( insert_a @ X3 @ ( insert_a @ Y5 @ bot_bot_set_a ) ) )
          & ( member_a @ X3 @ X5 )
          & ( member_a @ Y5 @ Y4 ) ) ) ) ).

% is_edge_between_def
thf(fact_159_singletonI,axiom,
    ! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ).

% singletonI
thf(fact_160_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_161_singletonI,axiom,
    ! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).

% singletonI
thf(fact_162_singletonI,axiom,
    ! [A2: product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).

% singletonI
thf(fact_163_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_164_Int__UNIV,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ( inf_inf_set_a @ A @ B3 )
        = top_top_set_a )
      = ( ( A = top_top_set_a )
        & ( B3 = top_top_set_a ) ) ) ).

% Int_UNIV
thf(fact_165_Int__UNIV,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A @ B3 )
        = top_to1996260823553986621t_unit )
      = ( ( A = top_to1996260823553986621t_unit )
        & ( B3 = top_to1996260823553986621t_unit ) ) ) ).

% Int_UNIV
thf(fact_166_Int__UNIV,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B3 )
        = top_top_set_nat )
      = ( ( A = top_top_set_nat )
        & ( B3 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_167_Int__UNIV,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = top_to8063371432257647191od_a_a )
      = ( ( A = top_to8063371432257647191od_a_a )
        & ( B3 = top_to8063371432257647191od_a_a ) ) ) ).

% Int_UNIV
thf(fact_168_Int__UNIV,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ( inf_inf_set_set_a @ A @ B3 )
        = top_top_set_set_a )
      = ( ( A = top_top_set_set_a )
        & ( B3 = top_top_set_set_a ) ) ) ).

% Int_UNIV
thf(fact_169_Int__insert__left__if0,axiom,
    ! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
     => ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) @ C2 )
        = ( inf_in8905007599844390133od_a_a @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_170_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
        = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_171_Int__insert__left__if0,axiom,
    ! [A2: product_unit,C2: set_Product_unit,B3: set_Product_unit] :
      ( ~ ( member_Product_unit @ A2 @ C2 )
     => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B3 ) @ C2 )
        = ( inf_in4660618365625256667t_unit @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_172_Int__insert__left__if0,axiom,
    ! [A2: a,C2: set_a,B3: set_a] :
      ( ~ ( member_a @ A2 @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
        = ( inf_inf_set_a @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_173_Int__insert__left__if0,axiom,
    ! [A2: set_a,C2: set_set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ A2 @ C2 )
     => ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B3 ) @ C2 )
        = ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_174_Int__insert__left__if1,axiom,
    ! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ C2 )
     => ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) @ C2 )
        = ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_175_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_176_Int__insert__left__if1,axiom,
    ! [A2: product_unit,C2: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ C2 )
     => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B3 ) @ C2 )
        = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_177_Int__insert__left__if1,axiom,
    ! [A2: a,C2: set_a,B3: set_a] :
      ( ( member_a @ A2 @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_178_Int__insert__left__if1,axiom,
    ! [A2: set_a,C2: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ A2 @ C2 )
     => ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B3 ) @ C2 )
        = ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_179_insert__inter__insert,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B3 ) )
      = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_180_insert__inter__insert,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ ( insert_set_a @ A2 @ B3 ) )
      = ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_181_Int__insert__right__if0,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) )
        = ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_182_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
        = ( inf_inf_set_nat @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_183_Int__insert__right__if0,axiom,
    ! [A2: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ~ ( member_Product_unit @ A2 @ A )
     => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B3 ) )
        = ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_184_Int__insert__right__if0,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B3 ) )
        = ( inf_inf_set_a @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_185_Int__insert__right__if0,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ A2 @ A )
     => ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B3 ) )
        = ( inf_inf_set_set_a @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_186_Int__insert__right__if1,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) )
        = ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_187_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_188_Int__insert__right__if1,axiom,
    ! [A2: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ A )
     => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B3 ) )
        = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_189_Int__insert__right__if1,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B3 ) )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_190_Int__insert__right__if1,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B3 ) )
        = ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_191_insert__disjoint_I1_J,axiom,
    ! [A2: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B3 )
        = bot_bo3957492148770167129t_unit )
      = ( ~ ( member_Product_unit @ A2 @ B3 )
        & ( ( inf_in4660618365625256667t_unit @ A @ B3 )
          = bot_bo3957492148770167129t_unit ) ) ) ).

% insert_disjoint(1)
thf(fact_192_insert__disjoint_I1_J,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B3 )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B3 )
        & ( ( inf_inf_set_a @ A @ B3 )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_193_insert__disjoint_I1_J,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B3 )
        = bot_bot_set_set_a )
      = ( ~ ( member_set_a @ A2 @ B3 )
        & ( ( inf_inf_set_set_a @ A @ B3 )
          = bot_bot_set_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_194_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B3 )
        = bot_bo3357376287454694259od_a_a )
      = ( ~ ( member1426531477525435216od_a_a @ A2 @ B3 )
        & ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
          = bot_bo3357376287454694259od_a_a ) ) ) ).

% insert_disjoint(1)
thf(fact_195_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( ( inf_inf_set_nat @ A @ B3 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_196_insert__disjoint_I2_J,axiom,
    ! [A2: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit
        = ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_Product_unit @ A2 @ B3 )
        & ( bot_bo3957492148770167129t_unit
          = ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_197_insert__disjoint_I2_J,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_a @ A2 @ B3 )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_198_insert__disjoint_I2_J,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ( bot_bot_set_set_a
        = ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_set_a @ A2 @ B3 )
        & ( bot_bot_set_set_a
          = ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_199_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( bot_bo3357376287454694259od_a_a
        = ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B3 ) )
      = ( ~ ( member1426531477525435216od_a_a @ A2 @ B3 )
        & ( bot_bo3357376287454694259od_a_a
          = ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_200_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_201_disjoint__insert_I1_J,axiom,
    ! [B3: set_Product_unit,A2: product_unit,A: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ B3 @ ( insert_Product_unit @ A2 @ A ) )
        = bot_bo3957492148770167129t_unit )
      = ( ~ ( member_Product_unit @ A2 @ B3 )
        & ( ( inf_in4660618365625256667t_unit @ B3 @ A )
          = bot_bo3957492148770167129t_unit ) ) ) ).

% disjoint_insert(1)
thf(fact_202_disjoint__insert_I1_J,axiom,
    ! [B3: set_a,A2: a,A: set_a] :
      ( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A2 @ A ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B3 )
        & ( ( inf_inf_set_a @ B3 @ A )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_203_disjoint__insert_I1_J,axiom,
    ! [B3: set_set_a,A2: set_a,A: set_set_a] :
      ( ( ( inf_inf_set_set_a @ B3 @ ( insert_set_a @ A2 @ A ) )
        = bot_bot_set_set_a )
      = ( ~ ( member_set_a @ A2 @ B3 )
        & ( ( inf_inf_set_set_a @ B3 @ A )
          = bot_bot_set_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_204_disjoint__insert_I1_J,axiom,
    ! [B3: set_Product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
        = bot_bo3357376287454694259od_a_a )
      = ( ~ ( member1426531477525435216od_a_a @ A2 @ B3 )
        & ( ( inf_in8905007599844390133od_a_a @ B3 @ A )
          = bot_bo3357376287454694259od_a_a ) ) ) ).

% disjoint_insert(1)
thf(fact_205_disjoint__insert_I1_J,axiom,
    ! [B3: set_nat,A2: nat,A: set_nat] :
      ( ( ( inf_inf_set_nat @ B3 @ ( insert_nat @ A2 @ A ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( ( inf_inf_set_nat @ B3 @ A )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_206_disjoint__insert_I2_J,axiom,
    ! [A: set_Product_unit,B: product_unit,B3: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit
        = ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ B @ B3 ) ) )
      = ( ~ ( member_Product_unit @ B @ A )
        & ( bot_bo3957492148770167129t_unit
          = ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_207_disjoint__insert_I2_J,axiom,
    ! [A: set_a,B: a,B3: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A @ ( insert_a @ B @ B3 ) ) )
      = ( ~ ( member_a @ B @ A )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_208_disjoint__insert_I2_J,axiom,
    ! [A: set_set_a,B: set_a,B3: set_set_a] :
      ( ( bot_bot_set_set_a
        = ( inf_inf_set_set_a @ A @ ( insert_set_a @ B @ B3 ) ) )
      = ( ~ ( member_set_a @ B @ A )
        & ( bot_bot_set_set_a
          = ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_209_disjoint__insert_I2_J,axiom,
    ! [A: set_Product_prod_a_a,B: product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( bot_bo3357376287454694259od_a_a
        = ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B3 ) ) )
      = ( ~ ( member1426531477525435216od_a_a @ B @ A )
        & ( bot_bo3357376287454694259od_a_a
          = ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_210_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B: nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B3 ) ) )
      = ( ~ ( member_nat @ B @ A )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_211_IntE,axiom,
    ! [C: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) )
     => ~ ( ( member1426531477525435216od_a_a @ C @ A )
         => ~ ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% IntE
thf(fact_212_IntE,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ~ ( ( member_nat @ C @ A )
         => ~ ( member_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_213_IntE,axiom,
    ! [C: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B3 ) )
     => ~ ( ( member_Product_unit @ C @ A )
         => ~ ( member_Product_unit @ C @ B3 ) ) ) ).

% IntE
thf(fact_214_IntE,axiom,
    ! [C: a,A: set_a,B3: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
     => ~ ( ( member_a @ C @ A )
         => ~ ( member_a @ C @ B3 ) ) ) ).

% IntE
thf(fact_215_IntE,axiom,
    ! [C: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B3 ) )
     => ~ ( ( member_set_a @ C @ A )
         => ~ ( member_set_a @ C @ B3 ) ) ) ).

% IntE
thf(fact_216_IntD1,axiom,
    ! [C: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) )
     => ( member1426531477525435216od_a_a @ C @ A ) ) ).

% IntD1
thf(fact_217_IntD1,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ( member_nat @ C @ A ) ) ).

% IntD1
thf(fact_218_IntD1,axiom,
    ! [C: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B3 ) )
     => ( member_Product_unit @ C @ A ) ) ).

% IntD1
thf(fact_219_IntD1,axiom,
    ! [C: a,A: set_a,B3: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
     => ( member_a @ C @ A ) ) ).

% IntD1
thf(fact_220_IntD1,axiom,
    ! [C: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B3 ) )
     => ( member_set_a @ C @ A ) ) ).

% IntD1
thf(fact_221_IntD2,axiom,
    ! [C: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) )
     => ( member1426531477525435216od_a_a @ C @ B3 ) ) ).

% IntD2
thf(fact_222_IntD2,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ( member_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_223_IntD2,axiom,
    ! [C: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B3 ) )
     => ( member_Product_unit @ C @ B3 ) ) ).

% IntD2
thf(fact_224_IntD2,axiom,
    ! [C: a,A: set_a,B3: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
     => ( member_a @ C @ B3 ) ) ).

% IntD2
thf(fact_225_IntD2,axiom,
    ! [C: set_a,A: set_set_a,B3: set_set_a] :
      ( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B3 ) )
     => ( member_set_a @ C @ B3 ) ) ).

% IntD2
thf(fact_226_insertE,axiom,
    ! [A2: set_a,B: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member_set_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_227_insertE,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_228_insertE,axiom,
    ! [A2: a,B: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_229_insertE,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
     => ( ( A2 != B )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_230_insertE,axiom,
    ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
     => ( ( A2 != B )
       => ( member_Product_unit @ A2 @ A ) ) ) ).

% insertE
thf(fact_231_insertI1,axiom,
    ! [A2: set_a,B3: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B3 ) ) ).

% insertI1
thf(fact_232_insertI1,axiom,
    ! [A2: product_prod_a_a,B3: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) ) ).

% insertI1
thf(fact_233_insertI1,axiom,
    ! [A2: a,B3: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B3 ) ) ).

% insertI1
thf(fact_234_insertI1,axiom,
    ! [A2: nat,B3: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B3 ) ) ).

% insertI1
thf(fact_235_insertI1,axiom,
    ! [A2: product_unit,B3: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ B3 ) ) ).

% insertI1
thf(fact_236_insertI2,axiom,
    ! [A2: set_a,B3: set_set_a,B: set_a] :
      ( ( member_set_a @ A2 @ B3 )
     => ( member_set_a @ A2 @ ( insert_set_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_237_insertI2,axiom,
    ! [A2: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ B3 )
     => ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_238_insertI2,axiom,
    ! [A2: a,B3: set_a,B: a] :
      ( ( member_a @ A2 @ B3 )
     => ( member_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_239_insertI2,axiom,
    ! [A2: nat,B3: set_nat,B: nat] :
      ( ( member_nat @ A2 @ B3 )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).

% insertI2
thf(fact_240_insertI2,axiom,
    ! [A2: product_unit,B3: set_Product_unit,B: product_unit] :
      ( ( member_Product_unit @ A2 @ B3 )
     => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B3 ) ) ) ).

% insertI2
thf(fact_241_Int__assoc,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C2 )
      = ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).

% Int_assoc
thf(fact_242_Int__assoc,axiom,
    ! [A: set_set_a,B3: set_set_a,C2: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B3 ) @ C2 )
      = ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ).

% Int_assoc
thf(fact_243_Int__absorb,axiom,
    ! [A: set_a] :
      ( ( inf_inf_set_a @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_244_Int__absorb,axiom,
    ! [A: set_set_a] :
      ( ( inf_inf_set_set_a @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_245_Set_Oset__insert,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ( member_set_a @ X @ A )
     => ~ ! [B4: set_set_a] :
            ( ( A
              = ( insert_set_a @ X @ B4 ) )
           => ( member_set_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_246_Set_Oset__insert,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ A )
     => ~ ! [B4: set_Product_prod_a_a] :
            ( ( A
              = ( insert4534936382041156343od_a_a @ X @ B4 ) )
           => ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_247_Set_Oset__insert,axiom,
    ! [X: a,A: set_a] :
      ( ( member_a @ X @ A )
     => ~ ! [B4: set_a] :
            ( ( A
              = ( insert_a @ X @ B4 ) )
           => ( member_a @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_248_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ~ ! [B4: set_nat] :
            ( ( A
              = ( insert_nat @ X @ B4 ) )
           => ( member_nat @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_249_Set_Oset__insert,axiom,
    ! [X: product_unit,A: set_Product_unit] :
      ( ( member_Product_unit @ X @ A )
     => ~ ! [B4: set_Product_unit] :
            ( ( A
              = ( insert_Product_unit @ X @ B4 ) )
           => ( member_Product_unit @ X @ B4 ) ) ) ).

% Set.set_insert
thf(fact_250_Int__commute,axiom,
    ( inf_inf_set_a
    = ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_251_Int__commute,axiom,
    ( inf_inf_set_set_a
    = ( ^ [A5: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_252_insert__ident,axiom,
    ! [X: set_a,A: set_set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ X @ A )
     => ( ~ ( member_set_a @ X @ B3 )
       => ( ( ( insert_set_a @ X @ A )
            = ( insert_set_a @ X @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_253_insert__ident,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X @ A )
     => ( ~ ( member1426531477525435216od_a_a @ X @ B3 )
       => ( ( ( insert4534936382041156343od_a_a @ X @ A )
            = ( insert4534936382041156343od_a_a @ X @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_254_insert__ident,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ~ ( member_a @ X @ B3 )
       => ( ( ( insert_a @ X @ A )
            = ( insert_a @ X @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_255_insert__ident,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ~ ( member_nat @ X @ B3 )
       => ( ( ( insert_nat @ X @ A )
            = ( insert_nat @ X @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_256_insert__ident,axiom,
    ! [X: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ~ ( member_Product_unit @ X @ A )
     => ( ~ ( member_Product_unit @ X @ B3 )
       => ( ( ( insert_Product_unit @ X @ A )
            = ( insert_Product_unit @ X @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_257_insert__absorb,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ( ( insert_set_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_258_insert__absorb,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ( insert4534936382041156343od_a_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_259_insert__absorb,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_260_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_261_insert__absorb,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ A )
     => ( ( insert_Product_unit @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_262_insert__eq__iff,axiom,
    ! [A2: set_a,A: set_set_a,B: set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ A2 @ A )
     => ( ~ ( member_set_a @ B @ B3 )
       => ( ( ( insert_set_a @ A2 @ A )
            = ( insert_set_a @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_set_a] :
                  ( ( A
                    = ( insert_set_a @ B @ C3 ) )
                  & ~ ( member_set_a @ B @ C3 )
                  & ( B3
                    = ( insert_set_a @ A2 @ C3 ) )
                  & ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_263_insert__eq__iff,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ~ ( member1426531477525435216od_a_a @ B @ B3 )
       => ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
            = ( insert4534936382041156343od_a_a @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_Product_prod_a_a] :
                  ( ( A
                    = ( insert4534936382041156343od_a_a @ B @ C3 ) )
                  & ~ ( member1426531477525435216od_a_a @ B @ C3 )
                  & ( B3
                    = ( insert4534936382041156343od_a_a @ A2 @ C3 ) )
                  & ~ ( member1426531477525435216od_a_a @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_264_insert__eq__iff,axiom,
    ! [A2: a,A: set_a,B: a,B3: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ~ ( member_a @ B @ B3 )
       => ( ( ( insert_a @ A2 @ A )
            = ( insert_a @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_a] :
                  ( ( A
                    = ( insert_a @ B @ C3 ) )
                  & ~ ( member_a @ B @ C3 )
                  & ( B3
                    = ( insert_a @ A2 @ C3 ) )
                  & ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_265_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B: nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B @ B3 )
       => ( ( ( insert_nat @ A2 @ A )
            = ( insert_nat @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_nat] :
                  ( ( A
                    = ( insert_nat @ B @ C3 ) )
                  & ~ ( member_nat @ B @ C3 )
                  & ( B3
                    = ( insert_nat @ A2 @ C3 ) )
                  & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_266_insert__eq__iff,axiom,
    ! [A2: product_unit,A: set_Product_unit,B: product_unit,B3: set_Product_unit] :
      ( ~ ( member_Product_unit @ A2 @ A )
     => ( ~ ( member_Product_unit @ B @ B3 )
       => ( ( ( insert_Product_unit @ A2 @ A )
            = ( insert_Product_unit @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_Product_unit] :
                  ( ( A
                    = ( insert_Product_unit @ B @ C3 ) )
                  & ~ ( member_Product_unit @ B @ C3 )
                  & ( B3
                    = ( insert_Product_unit @ A2 @ C3 ) )
                  & ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_267_insert__commute,axiom,
    ! [X: a,Y3: a,A: set_a] :
      ( ( insert_a @ X @ ( insert_a @ Y3 @ A ) )
      = ( insert_a @ Y3 @ ( insert_a @ X @ A ) ) ) ).

% insert_commute
thf(fact_268_insert__commute,axiom,
    ! [X: set_a,Y3: set_a,A: set_set_a] :
      ( ( insert_set_a @ X @ ( insert_set_a @ Y3 @ A ) )
      = ( insert_set_a @ Y3 @ ( insert_set_a @ X @ A ) ) ) ).

% insert_commute
thf(fact_269_Int__insert__left,axiom,
    ! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( member1426531477525435216od_a_a @ A2 @ C2 )
       => ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) @ C2 )
          = ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B3 @ C2 ) ) ) )
      & ( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
       => ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) @ C2 )
          = ( inf_in8905007599844390133od_a_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_270_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ( ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
          = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_271_Int__insert__left,axiom,
    ! [A2: product_unit,C2: set_Product_unit,B3: set_Product_unit] :
      ( ( ( member_Product_unit @ A2 @ C2 )
       => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B3 ) @ C2 )
          = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B3 @ C2 ) ) ) )
      & ( ~ ( member_Product_unit @ A2 @ C2 )
       => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B3 ) @ C2 )
          = ( inf_in4660618365625256667t_unit @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_272_Int__insert__left,axiom,
    ! [A2: a,C2: set_a,B3: set_a] :
      ( ( ( member_a @ A2 @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
          = ( insert_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) )
      & ( ~ ( member_a @ A2 @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
          = ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_273_Int__insert__left,axiom,
    ! [A2: set_a,C2: set_set_a,B3: set_set_a] :
      ( ( ( member_set_a @ A2 @ C2 )
       => ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B3 ) @ C2 )
          = ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) )
      & ( ~ ( member_set_a @ A2 @ C2 )
       => ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B3 ) @ C2 )
          = ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_274_Int__left__absorb,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B3 ) )
      = ( inf_inf_set_a @ A @ B3 ) ) ).

% Int_left_absorb
thf(fact_275_Int__left__absorb,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B3 ) )
      = ( inf_inf_set_set_a @ A @ B3 ) ) ).

% Int_left_absorb
thf(fact_276_Int__insert__right,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( member1426531477525435216od_a_a @ A2 @ A )
       => ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) )
          = ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) )
      & ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
       => ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) )
          = ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_277_Int__insert__right,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B3 ) ) ) )
      & ( ~ ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_278_Int__insert__right,axiom,
    ! [A2: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( ( member_Product_unit @ A2 @ A )
       => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B3 ) )
          = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) )
      & ( ~ ( member_Product_unit @ A2 @ A )
       => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B3 ) )
          = ( inf_in4660618365625256667t_unit @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_279_Int__insert__right,axiom,
    ! [A2: a,A: set_a,B3: set_a] :
      ( ( ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B3 ) )
          = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B3 ) ) ) )
      & ( ~ ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B3 ) )
          = ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_280_Int__insert__right,axiom,
    ! [A2: set_a,A: set_set_a,B3: set_set_a] :
      ( ( ( member_set_a @ A2 @ A )
       => ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B3 ) )
          = ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B3 ) ) ) )
      & ( ~ ( member_set_a @ A2 @ A )
       => ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B3 ) )
          = ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_281_Int__left__commute,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) )
      = ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_282_Int__left__commute,axiom,
    ! [A: set_set_a,B3: set_set_a,C2: set_set_a] :
      ( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B3 @ C2 ) )
      = ( inf_inf_set_set_a @ B3 @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_283_mk__disjoint__insert,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ? [B4: set_set_a] :
          ( ( A
            = ( insert_set_a @ A2 @ B4 ) )
          & ~ ( member_set_a @ A2 @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_284_mk__disjoint__insert,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ A )
     => ? [B4: set_Product_prod_a_a] :
          ( ( A
            = ( insert4534936382041156343od_a_a @ A2 @ B4 ) )
          & ~ ( member1426531477525435216od_a_a @ A2 @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_285_mk__disjoint__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ? [B4: set_a] :
          ( ( A
            = ( insert_a @ A2 @ B4 ) )
          & ~ ( member_a @ A2 @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_286_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B4: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B4 ) )
          & ~ ( member_nat @ A2 @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_287_mk__disjoint__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( ( member_Product_unit @ A2 @ A )
     => ? [B4: set_Product_unit] :
          ( ( A
            = ( insert_Product_unit @ A2 @ B4 ) )
          & ~ ( member_Product_unit @ A2 @ B4 ) ) ) ).

% mk_disjoint_insert
thf(fact_288_bot__empty__eq,axiom,
    ( bot_bo4642748612307482820unit_o
    = ( ^ [X3: product_unit] : ( member_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).

% bot_empty_eq
thf(fact_289_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_290_bot__empty__eq,axiom,
    ( bot_bot_set_a_o
    = ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).

% bot_empty_eq
thf(fact_291_bot__empty__eq,axiom,
    ( bot_bo4160289986317612842_a_a_o
    = ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ).

% bot_empty_eq
thf(fact_292_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_293_bot__set__def,axiom,
    ( bot_bot_set_a
    = ( collect_a @ bot_bot_a_o ) ) ).

% bot_set_def
thf(fact_294_bot__set__def,axiom,
    ( bot_bot_set_set_a
    = ( collect_set_a @ bot_bot_set_a_o ) ) ).

% bot_set_def
thf(fact_295_bot__set__def,axiom,
    ( bot_bo3357376287454694259od_a_a
    = ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).

% bot_set_def
thf(fact_296_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_297_Int__UNIV__left,axiom,
    ! [B3: set_a] :
      ( ( inf_inf_set_a @ top_top_set_a @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_298_Int__UNIV__left,axiom,
    ! [B3: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_299_Int__UNIV__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_300_Int__UNIV__left,axiom,
    ! [B3: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_301_Int__UNIV__left,axiom,
    ! [B3: set_set_a] :
      ( ( inf_inf_set_set_a @ top_top_set_set_a @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_302_Int__UNIV__right,axiom,
    ! [A: set_a] :
      ( ( inf_inf_set_a @ A @ top_top_set_a )
      = A ) ).

% Int_UNIV_right
thf(fact_303_Int__UNIV__right,axiom,
    ! [A: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
      = A ) ).

% Int_UNIV_right
thf(fact_304_Int__UNIV__right,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ top_top_set_nat )
      = A ) ).

% Int_UNIV_right
thf(fact_305_Int__UNIV__right,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ A @ top_to8063371432257647191od_a_a )
      = A ) ).

% Int_UNIV_right
thf(fact_306_Int__UNIV__right,axiom,
    ! [A: set_set_a] :
      ( ( inf_inf_set_set_a @ A @ top_top_set_set_a )
      = A ) ).

% Int_UNIV_right
thf(fact_307_disjoint__iff__not__equal,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ( inf_inf_set_a @ A @ B3 )
        = bot_bot_set_a )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y5: a] :
                ( ( member_a @ Y5 @ B3 )
               => ( X3 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_308_disjoint__iff__not__equal,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ( inf_inf_set_set_a @ A @ B3 )
        = bot_bot_set_set_a )
      = ( ! [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
           => ! [Y5: set_a] :
                ( ( member_set_a @ Y5 @ B3 )
               => ( X3 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_309_disjoint__iff__not__equal,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = bot_bo3357376287454694259od_a_a )
      = ( ! [X3: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X3 @ A )
           => ! [Y5: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ Y5 @ B3 )
               => ( X3 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_310_disjoint__iff__not__equal,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B3 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y5: nat] :
                ( ( member_nat @ Y5 @ B3 )
               => ( X3 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_311_Int__empty__right,axiom,
    ! [A: set_a] :
      ( ( inf_inf_set_a @ A @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% Int_empty_right
thf(fact_312_Int__empty__right,axiom,
    ! [A: set_set_a] :
      ( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
      = bot_bot_set_set_a ) ).

% Int_empty_right
thf(fact_313_Int__empty__right,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ A @ bot_bo3357376287454694259od_a_a )
      = bot_bo3357376287454694259od_a_a ) ).

% Int_empty_right
thf(fact_314_Int__empty__right,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_315_Int__empty__left,axiom,
    ! [B3: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
      = bot_bot_set_a ) ).

% Int_empty_left
thf(fact_316_Int__empty__left,axiom,
    ! [B3: set_set_a] :
      ( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B3 )
      = bot_bot_set_set_a ) ).

% Int_empty_left
thf(fact_317_Int__empty__left,axiom,
    ! [B3: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B3 )
      = bot_bo3357376287454694259od_a_a ) ).

% Int_empty_left
thf(fact_318_Int__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_319_disjoint__iff,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A @ B3 )
        = bot_bo3957492148770167129t_unit )
      = ( ! [X3: product_unit] :
            ( ( member_Product_unit @ X3 @ A )
           => ~ ( member_Product_unit @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_320_disjoint__iff,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ( inf_inf_set_a @ A @ B3 )
        = bot_bot_set_a )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ~ ( member_a @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_321_disjoint__iff,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ( inf_inf_set_set_a @ A @ B3 )
        = bot_bot_set_set_a )
      = ( ! [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
           => ~ ( member_set_a @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_322_disjoint__iff,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = bot_bo3357376287454694259od_a_a )
      = ( ! [X3: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X3 @ A )
           => ~ ( member1426531477525435216od_a_a @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_323_disjoint__iff,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B3 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ~ ( member_nat @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_324_Int__emptyI,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ! [X4: product_unit] :
          ( ( member_Product_unit @ X4 @ A )
         => ~ ( member_Product_unit @ X4 @ B3 ) )
     => ( ( inf_in4660618365625256667t_unit @ A @ B3 )
        = bot_bo3957492148770167129t_unit ) ) ).

% Int_emptyI
thf(fact_325_Int__emptyI,axiom,
    ! [A: set_a,B3: set_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ~ ( member_a @ X4 @ B3 ) )
     => ( ( inf_inf_set_a @ A @ B3 )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_326_Int__emptyI,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A )
         => ~ ( member_set_a @ X4 @ B3 ) )
     => ( ( inf_inf_set_set_a @ A @ B3 )
        = bot_bot_set_set_a ) ) ).

% Int_emptyI
thf(fact_327_Int__emptyI,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A )
         => ~ ( member1426531477525435216od_a_a @ X4 @ B3 ) )
     => ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = bot_bo3357376287454694259od_a_a ) ) ).

% Int_emptyI
thf(fact_328_Int__emptyI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A )
         => ~ ( member_nat @ X4 @ B3 ) )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_329_insert__UNIV,axiom,
    ! [X: a] :
      ( ( insert_a @ X @ top_top_set_a )
      = top_top_set_a ) ).

% insert_UNIV
thf(fact_330_insert__UNIV,axiom,
    ! [X: product_unit] :
      ( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
      = top_to1996260823553986621t_unit ) ).

% insert_UNIV
thf(fact_331_insert__UNIV,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_332_insert__UNIV,axiom,
    ! [X: product_prod_a_a] :
      ( ( insert4534936382041156343od_a_a @ X @ top_to8063371432257647191od_a_a )
      = top_to8063371432257647191od_a_a ) ).

% insert_UNIV
thf(fact_333_insert__UNIV,axiom,
    ! [X: set_a] :
      ( ( insert_set_a @ X @ top_top_set_set_a )
      = top_top_set_set_a ) ).

% insert_UNIV
thf(fact_334_singleton__inject,axiom,
    ! [A2: a,B: a] :
      ( ( ( insert_a @ A2 @ bot_bot_set_a )
        = ( insert_a @ B @ bot_bot_set_a ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_335_singleton__inject,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( insert_set_a @ A2 @ bot_bot_set_set_a )
        = ( insert_set_a @ B @ bot_bot_set_set_a ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_336_singleton__inject,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a] :
      ( ( ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
        = ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_337_singleton__inject,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_338_insert__not__empty,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a @ A2 @ A )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_339_insert__not__empty,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( insert_set_a @ A2 @ A )
     != bot_bot_set_set_a ) ).

% insert_not_empty
thf(fact_340_insert__not__empty,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( insert4534936382041156343od_a_a @ A2 @ A )
     != bot_bo3357376287454694259od_a_a ) ).

% insert_not_empty
thf(fact_341_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_342_doubleton__eq__iff,axiom,
    ! [A2: a,B: a,C: a,D: a] :
      ( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
        = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_343_doubleton__eq__iff,axiom,
    ! [A2: set_a,B: set_a,C: set_a,D: set_a] :
      ( ( ( insert_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
        = ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_344_doubleton__eq__iff,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
      ( ( ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
        = ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D @ bot_bo3357376287454694259od_a_a ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_345_doubleton__eq__iff,axiom,
    ! [A2: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_346_singleton__iff,axiom,
    ! [B: product_unit,A2: product_unit] :
      ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_347_singleton__iff,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_348_singleton__iff,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_349_singleton__iff,axiom,
    ! [B: product_prod_a_a,A2: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_350_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_351_singletonD,axiom,
    ! [B: product_unit,A2: product_unit] :
      ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_352_singletonD,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_353_singletonD,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_354_singletonD,axiom,
    ! [B: product_prod_a_a,A2: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_355_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_356_Pair__inject,axiom,
    ! [A2: a,B: a,A4: a,B2: a] :
      ( ( ( product_Pair_a_a @ A2 @ B )
        = ( product_Pair_a_a @ A4 @ B2 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_357_Pair__inject,axiom,
    ! [A2: set_a,B: set_set_a,A4: set_a,B2: set_set_a] :
      ( ( ( produc2116933609460601975_set_a @ A2 @ B )
        = ( produc2116933609460601975_set_a @ A4 @ B2 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_358_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
      ( ! [A3: a,B6: a] : ( P @ ( product_Pair_a_a @ A3 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_359_prod__cases,axiom,
    ! [P: produc7943277765024757383_set_a > $o,P2: produc7943277765024757383_set_a] :
      ( ! [A3: set_a,B6: set_set_a] : ( P @ ( produc2116933609460601975_set_a @ A3 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_360_surj__pair,axiom,
    ! [P2: product_prod_a_a] :
    ? [X4: a,Y2: a] :
      ( P2
      = ( product_Pair_a_a @ X4 @ Y2 ) ) ).

% surj_pair
thf(fact_361_surj__pair,axiom,
    ! [P2: produc7943277765024757383_set_a] :
    ? [X4: set_a,Y2: set_set_a] :
      ( P2
      = ( produc2116933609460601975_set_a @ X4 @ Y2 ) ) ).

% surj_pair
thf(fact_362_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_a_a] :
      ~ ! [A3: a,B6: a] :
          ( Y3
         != ( product_Pair_a_a @ A3 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_363_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7943277765024757383_set_a] :
      ~ ! [A3: set_a,B6: set_set_a] :
          ( Y3
         != ( produc2116933609460601975_set_a @ A3 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_364_top__empty__eq,axiom,
    ( top_top_a_o
    = ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).

% top_empty_eq
thf(fact_365_top__empty__eq,axiom,
    ( top_to2465898995584390880unit_o
    = ( ^ [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ) ) ).

% top_empty_eq
thf(fact_366_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_367_top__empty__eq,axiom,
    ( top_to8687885267596698950_a_a_o
    = ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ) ) ).

% top_empty_eq
thf(fact_368_top__empty__eq,axiom,
    ( top_top_set_a_o
    = ( ^ [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ) ) ).

% top_empty_eq
thf(fact_369_sym__on__def,axiom,
    ( sym_on_a
    = ( ^ [A5: set_a,R2: set_Product_prod_a_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A5 )
         => ! [Y5: a] :
              ( ( member_a @ Y5 @ A5 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ R2 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X3 ) @ R2 ) ) ) ) ) ) ).

% sym_on_def
thf(fact_370_sym__onI,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a] :
      ( ! [X4: set_a,Y2: set_a] :
          ( ( member_set_a @ X4 @ A )
         => ( ( member_set_a @ Y2 @ A )
           => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y2 ) @ R )
             => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y2 @ X4 ) @ R ) ) ) )
     => ( sym_on_set_a @ A @ R ) ) ).

% sym_onI
thf(fact_371_sym__onI,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
      ( ! [X4: product_prod_a_a,Y2: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A )
         => ( ( member1426531477525435216od_a_a @ Y2 @ A )
           => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y2 ) @ R )
             => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y2 @ X4 ) @ R ) ) ) )
     => ( sym_on5631557199876295240od_a_a @ A @ R ) ) ).

% sym_onI
thf(fact_372_sym__onI,axiom,
    ! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y2: nat] :
          ( ( member_nat @ X4 @ A )
         => ( ( member_nat @ Y2 @ A )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y2 ) @ R )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X4 ) @ R ) ) ) )
     => ( sym_on_nat @ A @ R ) ) ).

% sym_onI
thf(fact_373_sym__onI,axiom,
    ! [A: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( ! [X4: product_unit,Y2: product_unit] :
          ( ( member_Product_unit @ X4 @ A )
         => ( ( member_Product_unit @ Y2 @ A )
           => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X4 @ Y2 ) @ R )
             => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ X4 ) @ R ) ) ) )
     => ( sym_on_Product_unit @ A @ R ) ) ).

% sym_onI
thf(fact_374_sym__onI,axiom,
    ! [A: set_a,R: set_Product_prod_a_a] :
      ( ! [X4: a,Y2: a] :
          ( ( member_a @ X4 @ A )
         => ( ( member_a @ Y2 @ A )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ R )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X4 ) @ R ) ) ) )
     => ( sym_on_a @ A @ R ) ) ).

% sym_onI
thf(fact_375_sym__onD,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a,X: set_a,Y3: set_a] :
      ( ( sym_on_set_a @ A @ R )
     => ( ( member_set_a @ X @ A )
       => ( ( member_set_a @ Y3 @ A )
         => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X @ Y3 ) @ R )
           => ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ X ) @ R ) ) ) ) ) ).

% sym_onD
thf(fact_376_sym__onD,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y3: product_prod_a_a] :
      ( ( sym_on5631557199876295240od_a_a @ A @ R )
     => ( ( member1426531477525435216od_a_a @ X @ A )
       => ( ( member1426531477525435216od_a_a @ Y3 @ A )
         => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R )
           => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X ) @ R ) ) ) ) ) ).

% sym_onD
thf(fact_377_sym__onD,axiom,
    ! [A: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
      ( ( sym_on_nat @ A @ R )
     => ( ( member_nat @ X @ A )
       => ( ( member_nat @ Y3 @ A )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) ) ) ) ) ).

% sym_onD
thf(fact_378_sym__onD,axiom,
    ! [A: set_Product_unit,R: set_Pr5094982260447487303t_unit,X: product_unit,Y3: product_unit] :
      ( ( sym_on_Product_unit @ A @ R )
     => ( ( member_Product_unit @ X @ A )
       => ( ( member_Product_unit @ Y3 @ A )
         => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y3 ) @ R )
           => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X ) @ R ) ) ) ) ) ).

% sym_onD
thf(fact_379_sym__onD,axiom,
    ! [A: set_a,R: set_Product_prod_a_a,X: a,Y3: a] :
      ( ( sym_on_a @ A @ R )
     => ( ( member_a @ X @ A )
       => ( ( member_a @ Y3 @ A )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) ) ) ).

% sym_onD
thf(fact_380_irrefl__onD,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a,A2: set_a] :
      ( ( irrefl_on_set_a @ A @ R )
     => ( ( member_set_a @ A2 @ A )
       => ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ A2 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_381_irrefl__onD,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
      ( ( irrefl3954896097174259997od_a_a @ A @ R )
     => ( ( member1426531477525435216od_a_a @ A2 @ A )
       => ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ A2 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_382_irrefl__onD,axiom,
    ! [A: set_nat,R: set_Pr1261947904930325089at_nat,A2: nat] :
      ( ( irrefl_on_nat @ A @ R )
     => ( ( member_nat @ A2 @ A )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_383_irrefl__onD,axiom,
    ! [A: set_Product_unit,R: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( ( irrefl6292445964007751171t_unit @ A @ R )
     => ( ( member_Product_unit @ A2 @ A )
       => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_384_irrefl__onD,axiom,
    ! [A: set_a,R: set_Product_prod_a_a,A2: a] :
      ( ( irrefl_on_a @ A @ R )
     => ( ( member_a @ A2 @ A )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_385_irrefl__onI,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a] :
      ( ! [A3: set_a] :
          ( ( member_set_a @ A3 @ A )
         => ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ A3 ) @ R ) )
     => ( irrefl_on_set_a @ A @ R ) ) ).

% irrefl_onI
thf(fact_386_irrefl__onI,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
      ( ! [A3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ A3 @ A )
         => ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ R ) )
     => ( irrefl3954896097174259997od_a_a @ A @ R ) ) ).

% irrefl_onI
thf(fact_387_irrefl__onI,axiom,
    ! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R ) )
     => ( irrefl_on_nat @ A @ R ) ) ).

% irrefl_onI
thf(fact_388_irrefl__onI,axiom,
    ! [A: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( ! [A3: product_unit] :
          ( ( member_Product_unit @ A3 @ A )
         => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A3 @ A3 ) @ R ) )
     => ( irrefl6292445964007751171t_unit @ A @ R ) ) ).

% irrefl_onI
thf(fact_389_irrefl__onI,axiom,
    ! [A: set_a,R: set_Product_prod_a_a] :
      ( ! [A3: a] :
          ( ( member_a @ A3 @ A )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R ) )
     => ( irrefl_on_a @ A @ R ) ) ).

% irrefl_onI
thf(fact_390_irrefl__on__def,axiom,
    ( irrefl_on_a
    = ( ^ [A5: set_a,R2: set_Product_prod_a_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A5 )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_391_vert__adj__inc__edge__iff,axiom,
    ! [V1: a,V22: a] :
      ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
      = ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
        & ( undire1521409233611534436dent_a @ V22 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
        & ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ) ).

% vert_adj_inc_edge_iff
thf(fact_392_has__loop__def,axiom,
    ! [V: a] :
      ( ( undire3617971648856834880loop_a @ edges @ V )
      = ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).

% has_loop_def
thf(fact_393_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% boolean_algebra.conj_zero_right
thf(fact_394_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
      = bot_bot_set_set_a ) ).

% boolean_algebra.conj_zero_right
thf(fact_395_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
      = bot_bo3357376287454694259od_a_a ) ).

% boolean_algebra.conj_zero_right
thf(fact_396_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_397_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ X )
      = bot_bot_set_a ) ).

% boolean_algebra.conj_zero_left
thf(fact_398_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
      = bot_bot_set_set_a ) ).

% boolean_algebra.conj_zero_left
thf(fact_399_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
      = bot_bo3357376287454694259od_a_a ) ).

% boolean_algebra.conj_zero_left
thf(fact_400_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_401_inf__bot__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% inf_bot_right
thf(fact_402_inf__bot__right,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
      = bot_bot_set_set_a ) ).

% inf_bot_right
thf(fact_403_inf__bot__right,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
      = bot_bo3357376287454694259od_a_a ) ).

% inf_bot_right
thf(fact_404_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_405_inf__bot__left,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ X )
      = bot_bot_set_a ) ).

% inf_bot_left
thf(fact_406_inf__bot__left,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
      = bot_bot_set_set_a ) ).

% inf_bot_left
thf(fact_407_inf__bot__left,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
      = bot_bo3357376287454694259od_a_a ) ).

% inf_bot_left
thf(fact_408_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_409_inf__top__left,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ top_top_set_a @ X )
      = X ) ).

% inf_top_left
thf(fact_410_inf__top__left,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
      = X ) ).

% inf_top_left
thf(fact_411_inf__top__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ X )
      = X ) ).

% inf_top_left
thf(fact_412_inf__top__left,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ X )
      = X ) ).

% inf_top_left
thf(fact_413_inf__top__left,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ top_top_set_set_a @ X )
      = X ) ).

% inf_top_left
thf(fact_414_inf__top__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ top_top_set_a )
      = X ) ).

% inf_top_right
thf(fact_415_inf__top__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% inf_top_right
thf(fact_416_inf__top__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% inf_top_right
thf(fact_417_inf__top__right,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ X @ top_to8063371432257647191od_a_a )
      = X ) ).

% inf_top_right
thf(fact_418_inf__top__right,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
      = X ) ).

% inf_top_right
thf(fact_419_inf__eq__top__iff,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( ( inf_inf_set_a @ X @ Y3 )
        = top_top_set_a )
      = ( ( X = top_top_set_a )
        & ( Y3 = top_top_set_a ) ) ) ).

% inf_eq_top_iff
thf(fact_420_inf__eq__top__iff,axiom,
    ! [X: set_Product_unit,Y3: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ X @ Y3 )
        = top_to1996260823553986621t_unit )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y3 = top_to1996260823553986621t_unit ) ) ) ).

% inf_eq_top_iff
thf(fact_421_inf__eq__top__iff,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y3 )
        = top_top_set_nat )
      = ( ( X = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_422_inf__eq__top__iff,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
        = top_to8063371432257647191od_a_a )
      = ( ( X = top_to8063371432257647191od_a_a )
        & ( Y3 = top_to8063371432257647191od_a_a ) ) ) ).

% inf_eq_top_iff
thf(fact_423_inf__eq__top__iff,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( ( inf_inf_set_set_a @ X @ Y3 )
        = top_top_set_set_a )
      = ( ( X = top_top_set_set_a )
        & ( Y3 = top_top_set_set_a ) ) ) ).

% inf_eq_top_iff
thf(fact_424_top__eq__inf__iff,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( top_top_set_a
        = ( inf_inf_set_a @ X @ Y3 ) )
      = ( ( X = top_top_set_a )
        & ( Y3 = top_top_set_a ) ) ) ).

% top_eq_inf_iff
thf(fact_425_top__eq__inf__iff,axiom,
    ! [X: set_Product_unit,Y3: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ X @ Y3 ) )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y3 = top_to1996260823553986621t_unit ) ) ) ).

% top_eq_inf_iff
thf(fact_426_top__eq__inf__iff,axiom,
    ! [X: set_nat,Y3: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ X @ Y3 ) )
      = ( ( X = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_427_top__eq__inf__iff,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ( top_to8063371432257647191od_a_a
        = ( inf_in8905007599844390133od_a_a @ X @ Y3 ) )
      = ( ( X = top_to8063371432257647191od_a_a )
        & ( Y3 = top_to8063371432257647191od_a_a ) ) ) ).

% top_eq_inf_iff
thf(fact_428_top__eq__inf__iff,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( top_top_set_set_a
        = ( inf_inf_set_set_a @ X @ Y3 ) )
      = ( ( X = top_top_set_set_a )
        & ( Y3 = top_top_set_set_a ) ) ) ).

% top_eq_inf_iff
thf(fact_429_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A2 @ B )
        = top_top_set_a )
      = ( ( A2 = top_top_set_a )
        & ( B = top_top_set_a ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_430_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A2 @ B )
        = top_to1996260823553986621t_unit )
      = ( ( A2 = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_431_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B )
        = top_top_set_nat )
      = ( ( A2 = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_432_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
        = top_to8063371432257647191od_a_a )
      = ( ( A2 = top_to8063371432257647191od_a_a )
        & ( B = top_to8063371432257647191od_a_a ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_433_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ( inf_inf_set_set_a @ A2 @ B )
        = top_top_set_set_a )
      = ( ( A2 = top_top_set_set_a )
        & ( B = top_top_set_set_a ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_434_incident__def,axiom,
    undire1521409233611534436dent_a = member_a ).

% incident_def
thf(fact_435_vert__adj__edge__iff2,axiom,
    ! [V1: a,V22: a] :
      ( ( V1 != V22 )
     => ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
        = ( ? [X3: set_a] :
              ( ( member_set_a @ X3 @ edges )
              & ( undire1521409233611534436dent_a @ V1 @ X3 )
              & ( undire1521409233611534436dent_a @ V22 @ X3 ) ) ) ) ) ).

% vert_adj_edge_iff2
thf(fact_436_inf_Oidem,axiom,
    ! [A2: set_a] :
      ( ( inf_inf_set_a @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_437_inf_Oidem,axiom,
    ! [A2: set_set_a] :
      ( ( inf_inf_set_set_a @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_438_inf__idem,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ X )
      = X ) ).

% inf_idem
thf(fact_439_inf__idem,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ X )
      = X ) ).

% inf_idem
thf(fact_440_inf_Oleft__idem,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
      = ( inf_inf_set_a @ A2 @ B ) ) ).

% inf.left_idem
thf(fact_441_inf_Oleft__idem,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B ) )
      = ( inf_inf_set_set_a @ A2 @ B ) ) ).

% inf.left_idem
thf(fact_442_inf__left__idem,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y3 ) )
      = ( inf_inf_set_a @ X @ Y3 ) ) ).

% inf_left_idem
thf(fact_443_inf__left__idem,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y3 ) )
      = ( inf_inf_set_set_a @ X @ Y3 ) ) ).

% inf_left_idem
thf(fact_444_inf_Oright__idem,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B )
      = ( inf_inf_set_a @ A2 @ B ) ) ).

% inf.right_idem
thf(fact_445_inf_Oright__idem,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B )
      = ( inf_inf_set_set_a @ A2 @ B ) ) ).

% inf.right_idem
thf(fact_446_inf__right__idem,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 )
      = ( inf_inf_set_a @ X @ Y3 ) ) ).

% inf_right_idem
thf(fact_447_inf__right__idem,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 )
      = ( inf_inf_set_set_a @ X @ Y3 ) ) ).

% inf_right_idem
thf(fact_448_inf__top_Oright__neutral,axiom,
    ! [A2: set_a] :
      ( ( inf_inf_set_a @ A2 @ top_top_set_a )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_449_inf__top_Oright__neutral,axiom,
    ! [A2: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ A2 @ top_to1996260823553986621t_unit )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_450_inf__top_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_451_inf__top_Oright__neutral,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ A2 @ top_to8063371432257647191od_a_a )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_452_inf__top_Oright__neutral,axiom,
    ! [A2: set_set_a] :
      ( ( inf_inf_set_set_a @ A2 @ top_top_set_set_a )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_453_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( top_top_set_a
        = ( inf_inf_set_a @ A2 @ B ) )
      = ( ( A2 = top_top_set_a )
        & ( B = top_top_set_a ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_454_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ A2 @ B ) )
      = ( ( A2 = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_455_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ A2 @ B ) )
      = ( ( A2 = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_456_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( top_to8063371432257647191od_a_a
        = ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
      = ( ( A2 = top_to8063371432257647191od_a_a )
        & ( B = top_to8063371432257647191od_a_a ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_457_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( top_top_set_set_a
        = ( inf_inf_set_set_a @ A2 @ B ) )
      = ( ( A2 = top_top_set_set_a )
        & ( B = top_top_set_set_a ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_458_inf__top_Oleft__neutral,axiom,
    ! [A2: set_a] :
      ( ( inf_inf_set_a @ top_top_set_a @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_459_inf__top_Oleft__neutral,axiom,
    ! [A2: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_460_inf__top_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_461_inf__top_Oleft__neutral,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_462_inf__top_Oleft__neutral,axiom,
    ! [A2: set_set_a] :
      ( ( inf_inf_set_set_a @ top_top_set_set_a @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_463_sym__Int,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( sym_on_a @ top_top_set_a @ R )
     => ( ( sym_on_a @ top_top_set_a @ S )
       => ( sym_on_a @ top_top_set_a @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_464_sym__Int,axiom,
    ! [R: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
      ( ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ R )
     => ( ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ S )
       => ( sym_on_Product_unit @ top_to1996260823553986621t_unit @ ( inf_in3413610056909388085t_unit @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_465_sym__Int,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( sym_on_nat @ top_top_set_nat @ S )
       => ( sym_on_nat @ top_top_set_nat @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_466_sym__Int,axiom,
    ! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
      ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
     => ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ S )
       => ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ ( inf_in4058781255473215669od_a_a @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_467_sym__Int,axiom,
    ! [R: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
      ( ( sym_on_set_a @ top_top_set_set_a @ R )
     => ( ( sym_on_set_a @ top_top_set_set_a @ S )
       => ( sym_on_set_a @ top_top_set_set_a @ ( inf_in1230022433524902133_set_a @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_468_inf__sup__aci_I4_J,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y3 ) )
      = ( inf_inf_set_a @ X @ Y3 ) ) ).

% inf_sup_aci(4)
thf(fact_469_inf__sup__aci_I4_J,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y3 ) )
      = ( inf_inf_set_set_a @ X @ Y3 ) ) ).

% inf_sup_aci(4)
thf(fact_470_inf__sup__aci_I3_J,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) )
      = ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_471_inf__sup__aci_I3_J,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) )
      = ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_472_inf__sup__aci_I2_J,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Z )
      = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_473_inf__sup__aci_I2_J,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Z )
      = ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_474_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_a
    = ( ^ [X3: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_475_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_set_a
    = ( ^ [X3: set_set_a,Y5: set_set_a] : ( inf_inf_set_set_a @ Y5 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_476_inf_Oassoc,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
      = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).

% inf.assoc
thf(fact_477_inf_Oassoc,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C )
      = ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).

% inf.assoc
thf(fact_478_inf__assoc,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Z )
      = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).

% inf_assoc
thf(fact_479_inf__assoc,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Z )
      = ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ).

% inf_assoc
thf(fact_480_inf_Ocommute,axiom,
    ( inf_inf_set_a
    = ( ^ [A6: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A6 ) ) ) ).

% inf.commute
thf(fact_481_inf_Ocommute,axiom,
    ( inf_inf_set_set_a
    = ( ^ [A6: set_set_a,B7: set_set_a] : ( inf_inf_set_set_a @ B7 @ A6 ) ) ) ).

% inf.commute
thf(fact_482_inf__commute,axiom,
    ( inf_inf_set_a
    = ( ^ [X3: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X3 ) ) ) ).

% inf_commute
thf(fact_483_inf__commute,axiom,
    ( inf_inf_set_set_a
    = ( ^ [X3: set_set_a,Y5: set_set_a] : ( inf_inf_set_set_a @ Y5 @ X3 ) ) ) ).

% inf_commute
thf(fact_484_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_a,K: set_a,A2: set_a,B: set_a] :
      ( ( A
        = ( inf_inf_set_a @ K @ A2 ) )
     => ( ( inf_inf_set_a @ A @ B )
        = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_485_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_set_a,K: set_set_a,A2: set_set_a,B: set_set_a] :
      ( ( A
        = ( inf_inf_set_set_a @ K @ A2 ) )
     => ( ( inf_inf_set_set_a @ A @ B )
        = ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_486_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_a,K: set_a,B: set_a,A2: set_a] :
      ( ( B3
        = ( inf_inf_set_a @ K @ B ) )
     => ( ( inf_inf_set_a @ A2 @ B3 )
        = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_487_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_set_a,K: set_set_a,B: set_set_a,A2: set_set_a] :
      ( ( B3
        = ( inf_inf_set_set_a @ K @ B ) )
     => ( ( inf_inf_set_set_a @ A2 @ B3 )
        = ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_488_inf_Oleft__commute,axiom,
    ! [B: set_a,A2: set_a,C: set_a] :
      ( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) )
      = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_489_inf_Oleft__commute,axiom,
    ! [B: set_set_a,A2: set_set_a,C: set_set_a] :
      ( ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A2 @ C ) )
      = ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_490_inf__left__commute,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) )
      = ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_491_inf__left__commute,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) )
      = ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_492_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ top_top_set_a )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_493_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_494_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_495_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Product_prod_a_a] :
      ( ( inf_in8905007599844390133od_a_a @ X @ top_to8063371432257647191od_a_a )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_496_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_set_a] :
      ( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_497_incident__loops__simp_I1_J,axiom,
    ! [V: a] :
      ( ( undire3617971648856834880loop_a @ edges @ V )
     => ( ( undire4753905205749729249oops_a @ edges @ V )
        = ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).

% incident_loops_simp(1)
thf(fact_498_incident__loops__simp_I2_J,axiom,
    ! [V: a] :
      ( ~ ( undire3617971648856834880loop_a @ edges @ V )
     => ( ( undire4753905205749729249oops_a @ edges @ V )
        = bot_bot_set_set_a ) ) ).

% incident_loops_simp(2)
thf(fact_499_comp__sgraph_Ois__edge__between__def,axiom,
    ( undire7069873054131797420t_unit
    = ( ^ [X5: set_Product_unit,Y4: set_Product_unit,E3: set_Product_unit] :
        ? [X3: product_unit,Y5: product_unit] :
          ( ( E3
            = ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y5 @ bot_bo3957492148770167129t_unit ) ) )
          & ( member_Product_unit @ X3 @ X5 )
          & ( member_Product_unit @ Y5 @ Y4 ) ) ) ) ).

% comp_sgraph.is_edge_between_def
thf(fact_500_comp__sgraph_Ois__edge__between__def,axiom,
    ( undire2578756059399487229_set_a
    = ( ^ [X5: set_set_a,Y4: set_set_a,E3: set_set_a] :
        ? [X3: set_a,Y5: set_a] :
          ( ( E3
            = ( insert_set_a @ X3 @ ( insert_set_a @ Y5 @ bot_bot_set_set_a ) ) )
          & ( member_set_a @ X3 @ X5 )
          & ( member_set_a @ Y5 @ Y4 ) ) ) ) ).

% comp_sgraph.is_edge_between_def
thf(fact_501_comp__sgraph_Ois__edge__between__def,axiom,
    ( undire7011261089604658374od_a_a
    = ( ^ [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,E3: set_Product_prod_a_a] :
        ? [X3: product_prod_a_a,Y5: product_prod_a_a] :
          ( ( E3
            = ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y5 @ bot_bo3357376287454694259od_a_a ) ) )
          & ( member1426531477525435216od_a_a @ X3 @ X5 )
          & ( member1426531477525435216od_a_a @ Y5 @ Y4 ) ) ) ) ).

% comp_sgraph.is_edge_between_def
thf(fact_502_comp__sgraph_Ois__edge__between__def,axiom,
    ( undire6814325412647357297en_nat
    = ( ^ [X5: set_nat,Y4: set_nat,E3: set_nat] :
        ? [X3: nat,Y5: nat] :
          ( ( E3
            = ( insert_nat @ X3 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
          & ( member_nat @ X3 @ X5 )
          & ( member_nat @ Y5 @ Y4 ) ) ) ) ).

% comp_sgraph.is_edge_between_def
thf(fact_503_comp__sgraph_Ois__edge__between__def,axiom,
    ( undire8544646567961481629ween_a
    = ( ^ [X5: set_a,Y4: set_a,E3: set_a] :
        ? [X3: a,Y5: a] :
          ( ( E3
            = ( insert_a @ X3 @ ( insert_a @ Y5 @ bot_bot_set_a ) ) )
          & ( member_a @ X3 @ X5 )
          & ( member_a @ Y5 @ Y4 ) ) ) ) ).

% comp_sgraph.is_edge_between_def
thf(fact_504_edge__density__ge0,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y ) ) ).

% edge_density_ge0
thf(fact_505_all__edges__betw__D3,axiom,
    ! [X: a,Y3: a,X2: set_a,Y: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
     => ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges ) ) ).

% all_edges_betw_D3
thf(fact_506_all__edges__betw__I,axiom,
    ! [X: a,X2: set_a,Y3: a,Y: set_a] :
      ( ( member_a @ X @ X2 )
     => ( ( member_a @ Y3 @ Y )
       => ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ) ) ) ).

% all_edges_betw_I
thf(fact_507_the__elem__eq,axiom,
    ! [X: a] :
      ( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
      = X ) ).

% the_elem_eq
thf(fact_508_the__elem__eq,axiom,
    ! [X: set_a] :
      ( ( the_elem_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
      = X ) ).

% the_elem_eq
thf(fact_509_the__elem__eq,axiom,
    ! [X: product_prod_a_a] :
      ( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
      = X ) ).

% the_elem_eq
thf(fact_510_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_511_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_512_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_513_order__refl,axiom,
    ! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).

% order_refl
thf(fact_514_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_515_order__refl,axiom,
    ! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).

% order_refl
thf(fact_516_dual__order_Orefl,axiom,
    ! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_517_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_518_dual__order_Orefl,axiom,
    ! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_519_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_520_dual__order_Orefl,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_521_inf_Obounded__iff,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
      = ( ( ord_less_eq_real @ A2 @ B )
        & ( ord_less_eq_real @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_522_inf_Obounded__iff,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
      = ( ( ord_less_eq_set_a @ A2 @ B )
        & ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_523_inf_Obounded__iff,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
      = ( ( ord_le746702958409616551od_a_a @ A2 @ B )
        & ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_524_inf_Obounded__iff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_525_inf_Obounded__iff,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) )
      = ( ( ord_le3724670747650509150_set_a @ A2 @ B )
        & ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_526_le__inf__iff,axiom,
    ! [X: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y3 @ Z ) )
      = ( ( ord_less_eq_real @ X @ Y3 )
        & ( ord_less_eq_real @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_527_le__inf__iff,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) )
      = ( ( ord_less_eq_set_a @ X @ Y3 )
        & ( ord_less_eq_set_a @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_528_le__inf__iff,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z ) )
      = ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
        & ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_529_le__inf__iff,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y3 @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y3 )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_530_le__inf__iff,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) )
      = ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
        & ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_531_all__edges__between__empty_I2_J,axiom,
    ! [Z2: set_a] :
      ( ( undire8383842906760478443ween_a @ edges @ Z2 @ bot_bot_set_a )
      = bot_bo3357376287454694259od_a_a ) ).

% all_edges_between_empty(2)
thf(fact_532_all__edges__between__empty_I1_J,axiom,
    ! [Z2: set_a] :
      ( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z2 )
      = bot_bo3357376287454694259od_a_a ) ).

% all_edges_between_empty(1)
thf(fact_533_nle__le,axiom,
    ! [A2: real,B: real] :
      ( ( ~ ( ord_less_eq_real @ A2 @ B ) )
      = ( ( ord_less_eq_real @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_534_nle__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
      = ( ( ord_less_eq_nat @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_535_le__cases3,axiom,
    ! [X: real,Y3: real,Z: real] :
      ( ( ( ord_less_eq_real @ X @ Y3 )
       => ~ ( ord_less_eq_real @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_real @ Y3 @ X )
         => ~ ( ord_less_eq_real @ X @ Z ) )
       => ( ( ( ord_less_eq_real @ X @ Z )
           => ~ ( ord_less_eq_real @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_real @ Z @ Y3 )
             => ~ ( ord_less_eq_real @ Y3 @ X ) )
           => ( ( ( ord_less_eq_real @ Y3 @ Z )
               => ~ ( ord_less_eq_real @ Z @ X ) )
             => ~ ( ( ord_less_eq_real @ Z @ X )
                 => ~ ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_536_le__cases3,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_537_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
    = ( ^ [X3: real,Y5: real] :
          ( ( ord_less_eq_real @ X3 @ Y5 )
          & ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_538_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
    = ( ^ [X3: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X3 @ Y5 )
          & ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_539_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
    = ( ^ [X3: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ X3 @ Y5 )
          & ( ord_le746702958409616551od_a_a @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_540_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_541_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
    = ( ^ [X3: set_set_a,Y5: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
          & ( ord_le3724670747650509150_set_a @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_542_ord__eq__le__trans,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( A2 = B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_543_ord__eq__le__trans,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( A2 = B )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_544_ord__eq__le__trans,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( A2 = B )
     => ( ( ord_le746702958409616551od_a_a @ B @ C )
       => ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_545_ord__eq__le__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_546_ord__eq__le__trans,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( A2 = B )
     => ( ( ord_le3724670747650509150_set_a @ B @ C )
       => ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_547_ord__le__eq__trans,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_548_ord__le__eq__trans,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_549_ord__le__eq__trans,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( ( B = C )
       => ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_550_ord__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_551_ord__le__eq__trans,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( B = C )
       => ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_552_order__antisym,axiom,
    ! [X: real,Y3: real] :
      ( ( ord_less_eq_real @ X @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_553_order__antisym,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y3 )
     => ( ( ord_less_eq_set_a @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_554_order__antisym,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
     => ( ( ord_le746702958409616551od_a_a @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_555_order__antisym,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_556_order__antisym,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
     => ( ( ord_le3724670747650509150_set_a @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_557_order_Otrans,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% order.trans
thf(fact_558_order_Otrans,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_559_order_Otrans,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ C )
       => ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_560_order_Otrans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_561_order_Otrans,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ C )
       => ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_562_order__trans,axiom,
    ! [X: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ Z )
       => ( ord_less_eq_real @ X @ Z ) ) ) ).

% order_trans
thf(fact_563_order__trans,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y3 )
     => ( ( ord_less_eq_set_a @ Y3 @ Z )
       => ( ord_less_eq_set_a @ X @ Z ) ) ) ).

% order_trans
thf(fact_564_order__trans,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
     => ( ( ord_le746702958409616551od_a_a @ Y3 @ Z )
       => ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).

% order_trans
thf(fact_565_order__trans,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_566_order__trans,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
     => ( ( ord_le3724670747650509150_set_a @ Y3 @ Z )
       => ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).

% order_trans
thf(fact_567_linorder__wlog,axiom,
    ! [P: real > real > $o,A2: real,B: real] :
      ( ! [A3: real,B6: real] :
          ( ( ord_less_eq_real @ A3 @ B6 )
         => ( P @ A3 @ B6 ) )
     => ( ! [A3: real,B6: real] :
            ( ( P @ B6 @ A3 )
           => ( P @ A3 @ B6 ) )
       => ( P @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_568_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A3: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A3 @ B6 )
         => ( P @ A3 @ B6 ) )
     => ( ! [A3: nat,B6: nat] :
            ( ( P @ B6 @ A3 )
           => ( P @ A3 @ B6 ) )
       => ( P @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_569_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
    = ( ^ [A6: real,B7: real] :
          ( ( ord_less_eq_real @ B7 @ A6 )
          & ( ord_less_eq_real @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_570_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_a,B7: set_a] :
          ( ( ord_less_eq_set_a @ B7 @ A6 )
          & ( ord_less_eq_set_a @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_571_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ B7 @ A6 )
          & ( ord_le746702958409616551od_a_a @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_572_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_573_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_set_a,B7: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ B7 @ A6 )
          & ( ord_le3724670747650509150_set_a @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_574_dual__order_Oantisym,axiom,
    ! [B: real,A2: real] :
      ( ( ord_less_eq_real @ B @ A2 )
     => ( ( ord_less_eq_real @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_575_dual__order_Oantisym,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_576_dual__order_Oantisym,axiom,
    ! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ A2 )
     => ( ( ord_le746702958409616551od_a_a @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_577_dual__order_Oantisym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_578_dual__order_Oantisym,axiom,
    ! [B: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_579_dual__order_Otrans,axiom,
    ! [B: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B @ A2 )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_580_dual__order_Otrans,axiom,
    ! [B: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( ord_less_eq_set_a @ C @ B )
       => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_581_dual__order_Otrans,axiom,
    ! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ A2 )
     => ( ( ord_le746702958409616551od_a_a @ C @ B )
       => ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_582_dual__order_Otrans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_583_dual__order_Otrans,axiom,
    ! [B: set_set_a,A2: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ C @ B )
       => ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_584_antisym,axiom,
    ! [A2: real,B: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_real @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_585_antisym,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_586_antisym,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_587_antisym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_588_antisym,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_589_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
    = ( ^ [A6: real,B7: real] :
          ( ( ord_less_eq_real @ A6 @ B7 )
          & ( ord_less_eq_real @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_590_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_a,B7: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B7 )
          & ( ord_less_eq_set_a @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_591_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ A6 @ B7 )
          & ( ord_le746702958409616551od_a_a @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_592_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_593_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_set_a,B7: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A6 @ B7 )
          & ( ord_le3724670747650509150_set_a @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_594_order__subst1,axiom,
    ! [A2: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_595_order__subst1,axiom,
    ! [A2: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_596_order__subst1,axiom,
    ! [A2: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_597_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_598_order__subst1,axiom,
    ! [A2: real,F: set_a > real,B: set_a,C: set_a] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_599_order__subst1,axiom,
    ! [A2: set_a,F: real > set_a,B: real,C: real] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_600_order__subst1,axiom,
    ! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_601_order__subst1,axiom,
    ! [A2: nat,F: set_a > nat,B: set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_602_order__subst1,axiom,
    ! [A2: real,F: set_set_a > real,B: set_set_a,C: set_set_a] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B ) )
     => ( ( ord_le3724670747650509150_set_a @ B @ C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_603_order__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_604_order__subst2,axiom,
    ! [A2: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_605_order__subst2,axiom,
    ! [A2: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_606_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_607_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_608_order__subst2,axiom,
    ! [A2: real,B: real,F: real > set_a,C: set_a] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_609_order__subst2,axiom,
    ! [A2: set_a,B: set_a,F: set_a > real,C: real] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_610_order__subst2,axiom,
    ! [A2: set_a,B: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_611_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_612_order__subst2,axiom,
    ! [A2: real,B: real,F: real > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_613_order__subst2,axiom,
    ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_614_order__eq__refl,axiom,
    ! [X: real,Y3: real] :
      ( ( X = Y3 )
     => ( ord_less_eq_real @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_615_order__eq__refl,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( X = Y3 )
     => ( ord_less_eq_set_a @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_616_order__eq__refl,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ( X = Y3 )
     => ( ord_le746702958409616551od_a_a @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_617_order__eq__refl,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X = Y3 )
     => ( ord_less_eq_nat @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_618_order__eq__refl,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( X = Y3 )
     => ( ord_le3724670747650509150_set_a @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_619_linorder__linear,axiom,
    ! [X: real,Y3: real] :
      ( ( ord_less_eq_real @ X @ Y3 )
      | ( ord_less_eq_real @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_620_linorder__linear,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X ) ) ).

% linorder_linear
thf(fact_621_ord__eq__le__subst,axiom,
    ! [A2: real,F: real > real,B: real,C: real] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_622_ord__eq__le__subst,axiom,
    ! [A2: nat,F: real > nat,B: real,C: real] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_623_ord__eq__le__subst,axiom,
    ! [A2: real,F: nat > real,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_624_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_625_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: real > set_a,B: real,C: real] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_626_ord__eq__le__subst,axiom,
    ! [A2: real,F: set_a > real,B: set_a,C: set_a] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_627_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_a > nat,B: set_a,C: set_a] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_628_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: nat > set_a,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_629_ord__eq__le__subst,axiom,
    ! [A2: set_set_a,F: real > set_set_a,B: real,C: real] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_630_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_631_ord__le__eq__subst,axiom,
    ! [A2: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_632_ord__le__eq__subst,axiom,
    ! [A2: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_633_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_634_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_635_ord__le__eq__subst,axiom,
    ! [A2: real,B: real,F: real > set_a,C: set_a] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_636_ord__le__eq__subst,axiom,
    ! [A2: set_a,B: set_a,F: set_a > real,C: real] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_637_ord__le__eq__subst,axiom,
    ! [A2: set_a,B: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_638_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_639_ord__le__eq__subst,axiom,
    ! [A2: real,B: real,F: real > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y2: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_640_ord__le__eq__subst,axiom,
    ! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_641_linorder__le__cases,axiom,
    ! [X: real,Y3: real] :
      ( ~ ( ord_less_eq_real @ X @ Y3 )
     => ( ord_less_eq_real @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_642_linorder__le__cases,axiom,
    ! [X: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X ) ) ).

% linorder_le_cases
thf(fact_643_order__antisym__conv,axiom,
    ! [Y3: real,X: real] :
      ( ( ord_less_eq_real @ Y3 @ X )
     => ( ( ord_less_eq_real @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_644_order__antisym__conv,axiom,
    ! [Y3: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y3 @ X )
     => ( ( ord_less_eq_set_a @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_645_order__antisym__conv,axiom,
    ! [Y3: set_Product_prod_a_a,X: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ Y3 @ X )
     => ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_646_order__antisym__conv,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ( ( ord_less_eq_nat @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_647_order__antisym__conv,axiom,
    ! [Y3: set_set_a,X: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ Y3 @ X )
     => ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
        = ( X = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_648_ulgraph_Oall__edges__between_Ocong,axiom,
    undire8383842906760478443ween_a = undire8383842906760478443ween_a ).

% ulgraph.all_edges_between.cong
thf(fact_649_ulgraph_Oincident__loops_Ocong,axiom,
    undire4753905205749729249oops_a = undire4753905205749729249oops_a ).

% ulgraph.incident_loops.cong
thf(fact_650_top__greatest,axiom,
    ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).

% top_greatest
thf(fact_651_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_652_top__greatest,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).

% top_greatest
thf(fact_653_top__greatest,axiom,
    ! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ top_to8063371432257647191od_a_a ) ).

% top_greatest
thf(fact_654_top__greatest,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ top_top_set_set_a ) ).

% top_greatest
thf(fact_655_top_Oextremum__unique,axiom,
    ! [A2: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
      = ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_unique
thf(fact_656_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_657_top_Oextremum__unique,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
      = ( A2 = top_top_set_a ) ) ).

% top.extremum_unique
thf(fact_658_top_Oextremum__unique,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A2 )
      = ( A2 = top_to8063371432257647191od_a_a ) ) ).

% top.extremum_unique
thf(fact_659_top_Oextremum__unique,axiom,
    ! [A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A2 )
      = ( A2 = top_top_set_set_a ) ) ).

% top.extremum_unique
thf(fact_660_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
     => ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_uniqueI
thf(fact_661_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_662_top_Oextremum__uniqueI,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
     => ( A2 = top_top_set_a ) ) ).

% top.extremum_uniqueI
thf(fact_663_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A2 )
     => ( A2 = top_to8063371432257647191od_a_a ) ) ).

% top.extremum_uniqueI
thf(fact_664_top_Oextremum__uniqueI,axiom,
    ! [A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A2 )
     => ( A2 = top_top_set_set_a ) ) ).

% top.extremum_uniqueI
thf(fact_665_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_666_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
     => ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_667_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
     => ( A2 = bot_bo3357376287454694259od_a_a ) ) ).

% bot.extremum_uniqueI
thf(fact_668_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_669_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
     => ( A2 = bot_bot_set_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_670_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_671_bot_Oextremum__unique,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
      = ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_unique
thf(fact_672_bot_Oextremum__unique,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
      = ( A2 = bot_bo3357376287454694259od_a_a ) ) ).

% bot.extremum_unique
thf(fact_673_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_674_bot_Oextremum__unique,axiom,
    ! [A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
      = ( A2 = bot_bot_set_set_a ) ) ).

% bot.extremum_unique
thf(fact_675_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_676_bot_Oextremum,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).

% bot.extremum
thf(fact_677_bot_Oextremum,axiom,
    ! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).

% bot.extremum
thf(fact_678_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_679_bot_Oextremum,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).

% bot.extremum
thf(fact_680_inf_OcoboundedI2,axiom,
    ! [B: real,C: real,A2: real] :
      ( ( ord_less_eq_real @ B @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_681_inf_OcoboundedI2,axiom,
    ! [B: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_682_inf_OcoboundedI2,axiom,
    ! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ C )
     => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_683_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_684_inf_OcoboundedI2,axiom,
    ! [B: set_set_a,C: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ C )
     => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_685_inf_OcoboundedI1,axiom,
    ! [A2: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A2 @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_686_inf_OcoboundedI1,axiom,
    ! [A2: set_a,C: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_687_inf_OcoboundedI1,axiom,
    ! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ C )
     => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_688_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_689_inf_OcoboundedI1,axiom,
    ! [A2: set_set_a,C: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ C )
     => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_690_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [B7: real,A6: real] :
          ( ( inf_inf_real @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_691_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B7: set_a,A6: set_a] :
          ( ( inf_inf_set_a @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_692_inf_Oabsorb__iff2,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [B7: set_Product_prod_a_a,A6: set_Product_prod_a_a] :
          ( ( inf_in8905007599844390133od_a_a @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_693_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( inf_inf_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_694_inf_Oabsorb__iff2,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [B7: set_set_a,A6: set_set_a] :
          ( ( inf_inf_set_set_a @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_695_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [A6: real,B7: real] :
          ( ( inf_inf_real @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_696_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B7: set_a] :
          ( ( inf_inf_set_a @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_697_inf_Oabsorb__iff1,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
          ( ( inf_in8905007599844390133od_a_a @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_698_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( inf_inf_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_699_inf_Oabsorb__iff1,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A6: set_set_a,B7: set_set_a] :
          ( ( inf_inf_set_set_a @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_700_inf_Ocobounded2,axiom,
    ! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_701_inf_Ocobounded2,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_702_inf_Ocobounded2,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_703_inf_Ocobounded2,axiom,
    ! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_704_inf_Ocobounded2,axiom,
    ! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_705_inf_Ocobounded1,axiom,
    ! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ A2 ) ).

% inf.cobounded1
thf(fact_706_inf_Ocobounded1,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).

% inf.cobounded1
thf(fact_707_inf_Ocobounded1,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ A2 ) ).

% inf.cobounded1
thf(fact_708_inf_Ocobounded1,axiom,
    ! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).

% inf.cobounded1
thf(fact_709_inf_Ocobounded1,axiom,
    ! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ A2 ) ).

% inf.cobounded1
thf(fact_710_inf_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [A6: real,B7: real] :
          ( A6
          = ( inf_inf_real @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_711_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B7: set_a] :
          ( A6
          = ( inf_inf_set_a @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_712_inf_Oorder__iff,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
          ( A6
          = ( inf_in8905007599844390133od_a_a @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_713_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( A6
          = ( inf_inf_nat @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_714_inf_Oorder__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A6: set_set_a,B7: set_set_a] :
          ( A6
          = ( inf_inf_set_set_a @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_715_inf__greatest,axiom,
    ! [X: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y3 )
     => ( ( ord_less_eq_real @ X @ Z )
       => ( ord_less_eq_real @ X @ ( inf_inf_real @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_716_inf__greatest,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y3 )
     => ( ( ord_less_eq_set_a @ X @ Z )
       => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_717_inf__greatest,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
     => ( ( ord_le746702958409616551od_a_a @ X @ Z )
       => ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_718_inf__greatest,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_719_inf__greatest,axiom,
    ! [X: set_set_a,Y3: set_set_a,Z: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
     => ( ( ord_le3724670747650509150_set_a @ X @ Z )
       => ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_720_inf_OboundedI,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( ord_less_eq_real @ A2 @ C )
       => ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_721_inf_OboundedI,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ A2 @ C )
       => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_722_inf_OboundedI,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( ( ord_le746702958409616551od_a_a @ A2 @ C )
       => ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_723_inf_OboundedI,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_724_inf_OboundedI,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ A2 @ C )
       => ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_725_inf_OboundedE,axiom,
    ! [A2: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
     => ~ ( ( ord_less_eq_real @ A2 @ B )
         => ~ ( ord_less_eq_real @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_726_inf_OboundedE,axiom,
    ! [A2: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B )
         => ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_727_inf_OboundedE,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
     => ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B )
         => ~ ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_728_inf_OboundedE,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_729_inf_OboundedE,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) )
     => ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
         => ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_730_inf__absorb2,axiom,
    ! [Y3: real,X: real] :
      ( ( ord_less_eq_real @ Y3 @ X )
     => ( ( inf_inf_real @ X @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_731_inf__absorb2,axiom,
    ! [Y3: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y3 @ X )
     => ( ( inf_inf_set_a @ X @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_732_inf__absorb2,axiom,
    ! [Y3: set_Product_prod_a_a,X: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ Y3 @ X )
     => ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_733_inf__absorb2,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X )
     => ( ( inf_inf_nat @ X @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_734_inf__absorb2,axiom,
    ! [Y3: set_set_a,X: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ Y3 @ X )
     => ( ( inf_inf_set_set_a @ X @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_735_inf__absorb1,axiom,
    ! [X: real,Y3: real] :
      ( ( ord_less_eq_real @ X @ Y3 )
     => ( ( inf_inf_real @ X @ Y3 )
        = X ) ) ).

% inf_absorb1
thf(fact_736_inf__absorb1,axiom,
    ! [X: set_a,Y3: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y3 )
     => ( ( inf_inf_set_a @ X @ Y3 )
        = X ) ) ).

% inf_absorb1
thf(fact_737_inf__absorb1,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y3 )
     => ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
        = X ) ) ).

% inf_absorb1
thf(fact_738_inf__absorb1,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( inf_inf_nat @ X @ Y3 )
        = X ) ) ).

% inf_absorb1
thf(fact_739_inf__absorb1,axiom,
    ! [X: set_set_a,Y3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
     => ( ( inf_inf_set_set_a @ X @ Y3 )
        = X ) ) ).

% inf_absorb1
thf(fact_740_inf_Oabsorb2,axiom,
    ! [B: real,A2: real] :
      ( ( ord_less_eq_real @ B @ A2 )
     => ( ( inf_inf_real @ A2 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_741_inf_Oabsorb2,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_742_inf_Oabsorb2,axiom,
    ! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ A2 )
     => ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_743_inf_Oabsorb2,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( inf_inf_nat @ A2 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_744_inf_Oabsorb2,axiom,
    ! [B: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ A2 )
     => ( ( inf_inf_set_set_a @ A2 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_745_inf_Oabsorb1,axiom,
    ! [A2: real,B: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( ( inf_inf_real @ A2 @ B )
        = A2 ) ) ).

% inf.absorb1
thf(fact_746_inf_Oabsorb1,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( inf_inf_set_a @ A2 @ B )
        = A2 ) ) ).

% inf.absorb1
thf(fact_747_inf_Oabsorb1,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
        = A2 ) ) ).

% inf.absorb1
thf(fact_748_inf_Oabsorb1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( inf_inf_nat @ A2 @ B )
        = A2 ) ) ).

% inf.absorb1
thf(fact_749_inf_Oabsorb1,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( inf_inf_set_set_a @ A2 @ B )
        = A2 ) ) ).

% inf.absorb1
thf(fact_750_le__iff__inf,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y5: real] :
          ( ( inf_inf_real @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_751_le__iff__inf,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y5: set_a] :
          ( ( inf_inf_set_a @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_752_le__iff__inf,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [X3: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
          ( ( inf_in8905007599844390133od_a_a @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_753_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y5: nat] :
          ( ( inf_inf_nat @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_754_le__iff__inf,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [X3: set_set_a,Y5: set_set_a] :
          ( ( inf_inf_set_set_a @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_755_inf__unique,axiom,
    ! [F: real > real > real,X: real,Y3: real] :
      ( ! [X4: real,Y2: real] : ( ord_less_eq_real @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: real,Y2: real] : ( ord_less_eq_real @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: real,Y2: real,Z4: real] :
              ( ( ord_less_eq_real @ X4 @ Y2 )
             => ( ( ord_less_eq_real @ X4 @ Z4 )
               => ( ord_less_eq_real @ X4 @ ( F @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_real @ X @ Y3 )
            = ( F @ X @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_756_inf__unique,axiom,
    ! [F: set_a > set_a > set_a,X: set_a,Y3: set_a] :
      ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: set_a,Y2: set_a,Z4: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ( ord_less_eq_set_a @ X4 @ Z4 )
               => ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_set_a @ X @ Y3 )
            = ( F @ X @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_757_inf__unique,axiom,
    ! [F: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
      ( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
              ( ( ord_le746702958409616551od_a_a @ X4 @ Y2 )
             => ( ( ord_le746702958409616551od_a_a @ X4 @ Z4 )
               => ( ord_le746702958409616551od_a_a @ X4 @ ( F @ Y2 @ Z4 ) ) ) )
         => ( ( inf_in8905007599844390133od_a_a @ X @ Y3 )
            = ( F @ X @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_758_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y3: nat] :
      ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: nat,Y2: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y2 )
             => ( ( ord_less_eq_nat @ X4 @ Z4 )
               => ( ord_less_eq_nat @ X4 @ ( F @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y3 )
            = ( F @ X @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_759_inf__unique,axiom,
    ! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y3: set_set_a] :
      ( ! [X4: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: set_set_a,Y2: set_set_a,Z4: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ( ord_le3724670747650509150_set_a @ X4 @ Z4 )
               => ( ord_le3724670747650509150_set_a @ X4 @ ( F @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_set_set_a @ X @ Y3 )
            = ( F @ X @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_760_inf_OorderI,axiom,
    ! [A2: real,B: real] :
      ( ( A2
        = ( inf_inf_real @ A2 @ B ) )
     => ( ord_less_eq_real @ A2 @ B ) ) ).

% inf.orderI
thf(fact_761_inf_OorderI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( A2
        = ( inf_inf_set_a @ A2 @ B ) )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% inf.orderI
thf(fact_762_inf_OorderI,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( A2
        = ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
     => ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).

% inf.orderI
thf(fact_763_inf_OorderI,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( inf_inf_nat @ A2 @ B ) )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% inf.orderI
thf(fact_764_inf_OorderI,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( A2
        = ( inf_inf_set_set_a @ A2 @ B ) )
     => ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).

% inf.orderI
thf(fact_765_inf_OorderE,axiom,
    ! [A2: real,B: real] :
      ( ( ord_less_eq_real @ A2 @ B )
     => ( A2
        = ( inf_inf_real @ A2 @ B ) ) ) ).

% inf.orderE
thf(fact_766_inf_OorderE,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( A2
        = ( inf_inf_set_a @ A2 @ B ) ) ) ).

% inf.orderE
thf(fact_767_inf_OorderE,axiom,
    ! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ B )
     => ( A2
        = ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ).

% inf.orderE
thf(fact_768_inf_OorderE,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( A2
        = ( inf_inf_nat @ A2 @ B ) ) ) ).

% inf.orderE
thf(fact_769_inf_OorderE,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( A2
        = ( inf_inf_set_set_a @ A2 @ B ) ) ) ).

% inf.orderE
thf(fact_770_le__infI2,axiom,
    ! [B: real,X: real,A2: real] :
      ( ( ord_less_eq_real @ B @ X )
     => ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X ) ) ).

% le_infI2
thf(fact_771_le__infI2,axiom,
    ! [B: set_a,X: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ X )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ X ) ) ).

% le_infI2
thf(fact_772_le__infI2,axiom,
    ! [B: set_Product_prod_a_a,X: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ X )
     => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ X ) ) ).

% le_infI2
thf(fact_773_le__infI2,axiom,
    ! [B: nat,X: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).

% le_infI2
thf(fact_774_le__infI2,axiom,
    ! [B: set_set_a,X: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ X )
     => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ X ) ) ).

% le_infI2
thf(fact_775_le__infI1,axiom,
    ! [A2: real,X: real,B: real] :
      ( ( ord_less_eq_real @ A2 @ X )
     => ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X ) ) ).

% le_infI1
thf(fact_776_le__infI1,axiom,
    ! [A2: set_a,X: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ X ) ) ).

% le_infI1
thf(fact_777_le__infI1,axiom,
    ! [A2: set_Product_prod_a_a,X: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ X )
     => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ X ) ) ).

% le_infI1
thf(fact_778_le__infI1,axiom,
    ! [A2: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).

% le_infI1
thf(fact_779_le__infI1,axiom,
    ! [A2: set_set_a,X: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ X )
     => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ X ) ) ).

% le_infI1
thf(fact_780_inf__mono,axiom,
    ! [A2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ C )
     => ( ( ord_less_eq_real @ B @ D )
       => ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ ( inf_inf_real @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_781_inf__mono,axiom,
    ! [A2: set_a,C: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_782_inf__mono,axiom,
    ! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A2 @ C )
     => ( ( ord_le746702958409616551od_a_a @ B @ D )
       => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ ( inf_in8905007599844390133od_a_a @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_783_inf__mono,axiom,
    ! [A2: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_784_inf__mono,axiom,
    ! [A2: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ C )
     => ( ( ord_le3724670747650509150_set_a @ B @ D )
       => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_785_le__infI,axiom,
    ! [X: real,A2: real,B: real] :
      ( ( ord_less_eq_real @ X @ A2 )
     => ( ( ord_less_eq_real @ X @ B )
       => ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B ) ) ) ) ).

% le_infI
thf(fact_786_le__infI,axiom,
    ! [X: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ X @ A2 )
     => ( ( ord_less_eq_set_a @ X @ B )
       => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% le_infI
thf(fact_787_le__infI,axiom,
    ! [X: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ A2 )
     => ( ( ord_le746702958409616551od_a_a @ X @ B )
       => ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).

% le_infI
thf(fact_788_le__infI,axiom,
    ! [X: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).

% le_infI
thf(fact_789_le__infI,axiom,
    ! [X: set_set_a,A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ X @ B )
       => ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).

% le_infI
thf(fact_790_le__infE,axiom,
    ! [X: real,A2: real,B: real] :
      ( ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B ) )
     => ~ ( ( ord_less_eq_real @ X @ A2 )
         => ~ ( ord_less_eq_real @ X @ B ) ) ) ).

% le_infE
thf(fact_791_le__infE,axiom,
    ! [X: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B ) )
     => ~ ( ( ord_less_eq_set_a @ X @ A2 )
         => ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).

% le_infE
thf(fact_792_le__infE,axiom,
    ! [X: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
     => ~ ( ( ord_le746702958409616551od_a_a @ X @ A2 )
         => ~ ( ord_le746702958409616551od_a_a @ X @ B ) ) ) ).

% le_infE
thf(fact_793_le__infE,axiom,
    ! [X: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A2 )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_794_le__infE,axiom,
    ! [X: set_set_a,A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B ) )
     => ~ ( ( ord_le3724670747650509150_set_a @ X @ A2 )
         => ~ ( ord_le3724670747650509150_set_a @ X @ B ) ) ) ).

% le_infE
thf(fact_795_inf__le2,axiom,
    ! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_796_inf__le2,axiom,
    ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_797_inf__le2,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_798_inf__le2,axiom,
    ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_799_inf__le2,axiom,
    ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_800_inf__le1,axiom,
    ! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ X ) ).

% inf_le1
thf(fact_801_inf__le1,axiom,
    ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).

% inf_le1
thf(fact_802_inf__le1,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ X ) ).

% inf_le1
thf(fact_803_inf__le1,axiom,
    ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y3 ) @ X ) ).

% inf_le1
thf(fact_804_inf__le1,axiom,
    ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).

% inf_le1
thf(fact_805_inf__sup__ord_I1_J,axiom,
    ! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_806_inf__sup__ord_I1_J,axiom,
    ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_807_inf__sup__ord_I1_J,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_808_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y3 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_809_inf__sup__ord_I1_J,axiom,
    ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_810_inf__sup__ord_I2_J,axiom,
    ! [X: real,Y3: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_811_inf__sup__ord_I2_J,axiom,
    ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_812_inf__sup__ord_I2_J,axiom,
    ! [X: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_813_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_814_inf__sup__ord_I2_J,axiom,
    ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_815_mk__edge_Ocases,axiom,
    ! [X: product_prod_a_a] :
      ~ ! [U2: a,V2: a] :
          ( X
         != ( product_Pair_a_a @ U2 @ V2 ) ) ).

% mk_edge.cases
thf(fact_816_ulgraph_Overt__adj_Ocong,axiom,
    undire397441198561214472_adj_a = undire397441198561214472_adj_a ).

% ulgraph.vert_adj.cong
thf(fact_817_ulgraph_Oedge__density_Ocong,axiom,
    undire297304480579013331sity_a = undire297304480579013331sity_a ).

% ulgraph.edge_density.cong
thf(fact_818_comp__sgraph_Oincident__def,axiom,
    undire2320338297334612420_set_a = member_set_a ).

% comp_sgraph.incident_def
thf(fact_819_comp__sgraph_Oincident__def,axiom,
    undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).

% comp_sgraph.incident_def
thf(fact_820_comp__sgraph_Oincident__def,axiom,
    undire7858122600432113898nt_nat = member_nat ).

% comp_sgraph.incident_def
thf(fact_821_comp__sgraph_Oincident__def,axiom,
    undire5866035466353400179t_unit = member_Product_unit ).

% comp_sgraph.incident_def
thf(fact_822_comp__sgraph_Oincident__def,axiom,
    undire1521409233611534436dent_a = member_a ).

% comp_sgraph.incident_def
thf(fact_823_ulgraph_Ohas__loop_Ocong,axiom,
    undire3617971648856834880loop_a = undire3617971648856834880loop_a ).

% ulgraph.has_loop.cong
thf(fact_824_graph__system_Oedge__adj_Ocong,axiom,
    undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).

% graph_system.edge_adj.cong
thf(fact_825_mk__triangle__set_Ocases,axiom,
    ! [X: produc4044097585999906000od_a_a] :
      ~ ! [X4: a,Y2: a,Z4: a] :
          ( X
         != ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Y2 @ Z4 ) ) ) ).

% mk_triangle_set.cases
thf(fact_826_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_827_edge__density__eq0,axiom,
    ! [A: set_a,B3: set_a,X2: set_a,Y: set_a] :
      ( ( ( undire8383842906760478443ween_a @ edges @ A @ B3 )
        = bot_bo3357376287454694259od_a_a )
     => ( ( ord_less_eq_set_a @ X2 @ A )
       => ( ( ord_less_eq_set_a @ Y @ B3 )
         => ( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
            = zero_zero_real ) ) ) ) ).

% edge_density_eq0
thf(fact_828_edge__density__le1,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y ) @ one_one_real ) ).

% edge_density_le1
thf(fact_829_mk__triangle__set_Osimps,axiom,
    ! [X: a,Y3: a,Z: a] :
      ( ( undire8536760333753235943_set_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y3 @ Z ) ) )
      = ( insert_a @ X @ ( insert_a @ Y3 @ ( insert_a @ Z @ bot_bot_set_a ) ) ) ) ).

% mk_triangle_set.simps
thf(fact_830_mk__triangle__set_Osimps,axiom,
    ! [X: set_a,Y3: set_a,Z: set_a] :
      ( ( undire4638465864238448455_set_a @ ( produc7299740244201487072_set_a @ X @ ( produc9088192753505129239_set_a @ Y3 @ Z ) ) )
      = ( insert_set_a @ X @ ( insert_set_a @ Y3 @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) ) ) ).

% mk_triangle_set.simps
thf(fact_831_mk__triangle__set_Osimps,axiom,
    ! [X: product_prod_a_a,Y3: product_prod_a_a,Z: product_prod_a_a] :
      ( ( undire2459242765783757584od_a_a @ ( produc4925843558922497303od_a_a @ X @ ( produc7886510207707329367od_a_a @ Y3 @ Z ) ) )
      = ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y3 @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) ) ) ).

% mk_triangle_set.simps
thf(fact_832_mk__triangle__set_Osimps,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( undire4970100481470743719et_nat @ ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y3 @ Z ) ) )
      = ( insert_nat @ X @ ( insert_nat @ Y3 @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ).

% mk_triangle_set.simps
thf(fact_833_mk__triangle__set_Oelims,axiom,
    ! [X: produc4044097585999906000od_a_a,Y3: set_a] :
      ( ( ( undire8536760333753235943_set_a @ X )
        = Y3 )
     => ~ ! [X4: a,Y2: a,Z4: a] :
            ( ( X
              = ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Y2 @ Z4 ) ) )
           => ( Y3
             != ( insert_a @ X4 @ ( insert_a @ Y2 @ ( insert_a @ Z4 @ bot_bot_set_a ) ) ) ) ) ) ).

% mk_triangle_set.elims
thf(fact_834_mk__triangle__set_Oelims,axiom,
    ! [X: produc3364680560414100336_set_a,Y3: set_set_a] :
      ( ( ( undire4638465864238448455_set_a @ X )
        = Y3 )
     => ~ ! [X4: set_a,Y2: set_a,Z4: set_a] :
            ( ( X
              = ( produc7299740244201487072_set_a @ X4 @ ( produc9088192753505129239_set_a @ Y2 @ Z4 ) ) )
           => ( Y3
             != ( insert_set_a @ X4 @ ( insert_set_a @ Y2 @ ( insert_set_a @ Z4 @ bot_bot_set_set_a ) ) ) ) ) ) ).

% mk_triangle_set.elims
thf(fact_835_mk__triangle__set_Oelims,axiom,
    ! [X: produc8857593507947890343od_a_a,Y3: set_Product_prod_a_a] :
      ( ( ( undire2459242765783757584od_a_a @ X )
        = Y3 )
     => ~ ! [X4: product_prod_a_a,Y2: product_prod_a_a,Z4: product_prod_a_a] :
            ( ( X
              = ( produc4925843558922497303od_a_a @ X4 @ ( produc7886510207707329367od_a_a @ Y2 @ Z4 ) ) )
           => ( Y3
             != ( insert4534936382041156343od_a_a @ X4 @ ( insert4534936382041156343od_a_a @ Y2 @ ( insert4534936382041156343od_a_a @ Z4 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).

% mk_triangle_set.elims
thf(fact_836_mk__triangle__set_Oelims,axiom,
    ! [X: produc7248412053542808358at_nat,Y3: set_nat] :
      ( ( ( undire4970100481470743719et_nat @ X )
        = Y3 )
     => ~ ! [X4: nat,Y2: nat,Z4: nat] :
            ( ( X
              = ( produc487386426758144856at_nat @ X4 @ ( product_Pair_nat_nat @ Y2 @ Z4 ) ) )
           => ( Y3
             != ( insert_nat @ X4 @ ( insert_nat @ Y2 @ ( insert_nat @ Z4 @ bot_bot_set_nat ) ) ) ) ) ) ).

% mk_triangle_set.elims
thf(fact_837_finite__incident__loops,axiom,
    ! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).

% finite_incident_loops
thf(fact_838_Collect__empty__eq__bot,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( P = bot_bot_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_839_Collect__empty__eq__bot,axiom,
    ! [P: set_a > $o] :
      ( ( ( collect_set_a @ P )
        = bot_bot_set_set_a )
      = ( P = bot_bot_set_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_840_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_a_a > $o] :
      ( ( ( collec3336397797384452498od_a_a @ P )
        = bot_bo3357376287454694259od_a_a )
      = ( P = bot_bo4160289986317612842_a_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_841_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_842_all__edges__between__mono2,axiom,
    ! [Y: set_a,Z2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ Z2 )
     => ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) ) ) ).

% all_edges_between_mono2
thf(fact_843_all__edges__between__mono1,axiom,
    ! [Y: set_a,Z2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ Z2 )
     => ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y @ X2 ) @ ( undire8383842906760478443ween_a @ edges @ Z2 @ X2 ) ) ) ).

% all_edges_between_mono1
thf(fact_844_subsetI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A )
         => ( member_nat @ X4 @ B3 ) )
     => ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% subsetI
thf(fact_845_subsetI,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ! [X4: product_unit] :
          ( ( member_Product_unit @ X4 @ A )
         => ( member_Product_unit @ X4 @ B3 ) )
     => ( ord_le3507040750410214029t_unit @ A @ B3 ) ) ).

% subsetI
thf(fact_846_subsetI,axiom,
    ! [A: set_a,B3: set_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ( member_a @ X4 @ B3 ) )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% subsetI
thf(fact_847_subsetI,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A )
         => ( member1426531477525435216od_a_a @ X4 @ B3 ) )
     => ( ord_le746702958409616551od_a_a @ A @ B3 ) ) ).

% subsetI
thf(fact_848_subsetI,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A )
         => ( member_set_a @ X4 @ B3 ) )
     => ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ).

% subsetI
thf(fact_849_subset__antisym,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_850_subset__antisym,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( ord_le746702958409616551od_a_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_851_subset__antisym,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_852_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_853_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_854_subset__empty,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
      = ( A = bot_bo3357376287454694259od_a_a ) ) ).

% subset_empty
thf(fact_855_subset__empty,axiom,
    ! [A: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
      = ( A = bot_bot_set_set_a ) ) ).

% subset_empty
thf(fact_856_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_857_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_858_empty__subsetI,axiom,
    ! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).

% empty_subsetI
thf(fact_859_empty__subsetI,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).

% empty_subsetI
thf(fact_860_insert__subset,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B3 )
      = ( ( member_nat @ X @ B3 )
        & ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_861_insert__subset,axiom,
    ! [X: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ A ) @ B3 )
      = ( ( member_Product_unit @ X @ B3 )
        & ( ord_le3507040750410214029t_unit @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_862_insert__subset,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B3 )
      = ( ( member_a @ X @ B3 )
        & ( ord_less_eq_set_a @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_863_insert__subset,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A ) @ B3 )
      = ( ( member1426531477525435216od_a_a @ X @ B3 )
        & ( ord_le746702958409616551od_a_a @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_864_insert__subset,axiom,
    ! [X: set_a,A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B3 )
      = ( ( member_set_a @ X @ B3 )
        & ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_865_Int__subset__iff,axiom,
    ! [C2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B3 ) )
      = ( ( ord_less_eq_set_a @ C2 @ A )
        & ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_866_Int__subset__iff,axiom,
    ! [C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) )
      = ( ( ord_le746702958409616551od_a_a @ C2 @ A )
        & ( ord_le746702958409616551od_a_a @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_867_Int__subset__iff,axiom,
    ! [C2: set_set_a,A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B3 ) )
      = ( ( ord_le3724670747650509150_set_a @ C2 @ A )
        & ( ord_le3724670747650509150_set_a @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_868_singleton__insert__inj__eq,axiom,
    ! [B: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_869_singleton__insert__inj__eq,axiom,
    ! [B: a,A2: a,A: set_a] :
      ( ( ( insert_a @ B @ bot_bot_set_a )
        = ( insert_a @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_870_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
        = ( insert4534936382041156343od_a_a @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_871_singleton__insert__inj__eq,axiom,
    ! [B: set_a,A2: set_a,A: set_set_a] :
      ( ( ( insert_set_a @ B @ bot_bot_set_set_a )
        = ( insert_set_a @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_872_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_873_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B: a] :
      ( ( ( insert_a @ A2 @ A )
        = ( insert_a @ B @ bot_bot_set_a ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_874_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
        = ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
      = ( ( A2 = B )
        & ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_875_singleton__insert__inj__eq_H,axiom,
    ! [A2: set_a,A: set_set_a,B: set_a] :
      ( ( ( insert_set_a @ A2 @ A )
        = ( insert_set_a @ B @ bot_bot_set_set_a ) )
      = ( ( A2 = B )
        & ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_876_in__mono,axiom,
    ! [A: set_nat,B3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_877_in__mono,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit,X: product_unit] :
      ( ( ord_le3507040750410214029t_unit @ A @ B3 )
     => ( ( member_Product_unit @ X @ A )
       => ( member_Product_unit @ X @ B3 ) ) ) ).

% in_mono
thf(fact_878_in__mono,axiom,
    ! [A: set_a,B3: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B3 ) ) ) ).

% in_mono
thf(fact_879_in__mono,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a,X: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( member1426531477525435216od_a_a @ X @ A )
       => ( member1426531477525435216od_a_a @ X @ B3 ) ) ) ).

% in_mono
thf(fact_880_in__mono,axiom,
    ! [A: set_set_a,B3: set_set_a,X: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( member_set_a @ X @ A )
       => ( member_set_a @ X @ B3 ) ) ) ).

% in_mono
thf(fact_881_subsetD,axiom,
    ! [A: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_882_subsetD,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit,C: product_unit] :
      ( ( ord_le3507040750410214029t_unit @ A @ B3 )
     => ( ( member_Product_unit @ C @ A )
       => ( member_Product_unit @ C @ B3 ) ) ) ).

% subsetD
thf(fact_883_subsetD,axiom,
    ! [A: set_a,B3: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_884_subsetD,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( member1426531477525435216od_a_a @ C @ A )
       => ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_885_subsetD,axiom,
    ! [A: set_set_a,B3: set_set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( member_set_a @ C @ A )
       => ( member_set_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_886_equalityE,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ~ ( ( ord_less_eq_set_a @ A @ B3 )
         => ~ ( ord_less_eq_set_a @ B3 @ A ) ) ) ).

% equalityE
thf(fact_887_equalityE,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( A = B3 )
     => ~ ( ( ord_le746702958409616551od_a_a @ A @ B3 )
         => ~ ( ord_le746702958409616551od_a_a @ B3 @ A ) ) ) ).

% equalityE
thf(fact_888_equalityE,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( A = B3 )
     => ~ ( ( ord_le3724670747650509150_set_a @ A @ B3 )
         => ~ ( ord_le3724670747650509150_set_a @ B3 @ A ) ) ) ).

% equalityE
thf(fact_889_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( member_nat @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_890_subset__eq,axiom,
    ( ord_le3507040750410214029t_unit
    = ( ^ [A5: set_Product_unit,B5: set_Product_unit] :
        ! [X3: product_unit] :
          ( ( member_Product_unit @ X3 @ A5 )
         => ( member_Product_unit @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_891_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A5 )
         => ( member_a @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_892_subset__eq,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
        ! [X3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_893_subset__eq,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A5: set_set_a,B5: set_set_a] :
        ! [X3: set_a] :
          ( ( member_set_a @ X3 @ A5 )
         => ( member_set_a @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_894_equalityD1,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% equalityD1
thf(fact_895_equalityD1,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( A = B3 )
     => ( ord_le746702958409616551od_a_a @ A @ B3 ) ) ).

% equalityD1
thf(fact_896_equalityD1,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( A = B3 )
     => ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ).

% equalityD1
thf(fact_897_equalityD2,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ( ord_less_eq_set_a @ B3 @ A ) ) ).

% equalityD2
thf(fact_898_equalityD2,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( A = B3 )
     => ( ord_le746702958409616551od_a_a @ B3 @ A ) ) ).

% equalityD2
thf(fact_899_equalityD2,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( A = B3 )
     => ( ord_le3724670747650509150_set_a @ B3 @ A ) ) ).

% equalityD2
thf(fact_900_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A5 )
         => ( member_nat @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_901_subset__iff,axiom,
    ( ord_le3507040750410214029t_unit
    = ( ^ [A5: set_Product_unit,B5: set_Product_unit] :
        ! [T: product_unit] :
          ( ( member_Product_unit @ T @ A5 )
         => ( member_Product_unit @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_902_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A5 )
         => ( member_a @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_903_subset__iff,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
        ! [T: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ T @ A5 )
         => ( member1426531477525435216od_a_a @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_904_subset__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A5: set_set_a,B5: set_set_a] :
        ! [T: set_a] :
          ( ( member_set_a @ T @ A5 )
         => ( member_set_a @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_905_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_906_subset__refl,axiom,
    ! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).

% subset_refl
thf(fact_907_subset__refl,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).

% subset_refl
thf(fact_908_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_909_Collect__mono,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ! [X4: product_prod_a_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).

% Collect_mono
thf(fact_910_Collect__mono,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ! [X4: set_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).

% Collect_mono
thf(fact_911_subset__trans,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_912_subset__trans,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( ord_le746702958409616551od_a_a @ B3 @ C2 )
       => ( ord_le746702958409616551od_a_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_913_subset__trans,axiom,
    ! [A: set_set_a,B3: set_set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
       => ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_914_set__eq__subset,axiom,
    ( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
    = ( ^ [A5: set_a,B5: set_a] :
          ( ( ord_less_eq_set_a @ A5 @ B5 )
          & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_915_set__eq__subset,axiom,
    ( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
          & ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_916_set__eq__subset,axiom,
    ( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
    = ( ^ [A5: set_set_a,B5: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
          & ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_917_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X3: a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_918_Collect__mono__iff,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
      = ( ! [X3: product_prod_a_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_919_Collect__mono__iff,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
      = ( ! [X3: set_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_920_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_921_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_922_subrelI,axiom,
    ! [R: set_Pr4256959165342167655_set_a,S: set_Pr4256959165342167655_set_a] :
      ( ! [X4: set_a,Y2: set_set_a] :
          ( ( member268004040519299248_set_a @ ( produc2116933609460601975_set_a @ X4 @ Y2 ) @ R )
         => ( member268004040519299248_set_a @ ( produc2116933609460601975_set_a @ X4 @ Y2 ) @ S ) )
     => ( ord_le3617455369407883783_set_a @ R @ S ) ) ).

% subrelI
thf(fact_923_subrelI,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ! [X4: a,Y2: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ S ) )
     => ( ord_le746702958409616551od_a_a @ R @ S ) ) ).

% subrelI
thf(fact_924_subset__UNIV,axiom,
    ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).

% subset_UNIV
thf(fact_925_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_926_subset__UNIV,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).

% subset_UNIV
thf(fact_927_subset__UNIV,axiom,
    ! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ top_to8063371432257647191od_a_a ) ).

% subset_UNIV
thf(fact_928_subset__UNIV,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).

% subset_UNIV
thf(fact_929_subset__insertI2,axiom,
    ! [A: set_a,B3: set_a,B: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ord_less_eq_set_a @ A @ ( insert_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_930_subset__insertI2,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_931_subset__insertI2,axiom,
    ! [A: set_set_a,B3: set_set_a,B: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_932_subset__insertI,axiom,
    ! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).

% subset_insertI
thf(fact_933_subset__insertI,axiom,
    ! [B3: set_Product_prod_a_a,A2: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ B3 ) ) ).

% subset_insertI
thf(fact_934_subset__insertI,axiom,
    ! [B3: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B3 @ ( insert_set_a @ A2 @ B3 ) ) ).

% subset_insertI
thf(fact_935_subset__insert,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B3 ) )
        = ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_936_subset__insert,axiom,
    ! [X: product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ~ ( member_Product_unit @ X @ A )
     => ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B3 ) )
        = ( ord_le3507040750410214029t_unit @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_937_subset__insert,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B3 ) )
        = ( ord_less_eq_set_a @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_938_subset__insert,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X @ A )
     => ( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X @ B3 ) )
        = ( ord_le746702958409616551od_a_a @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_939_subset__insert,axiom,
    ! [X: set_a,A: set_set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ X @ A )
     => ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B3 ) )
        = ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_940_insert__mono,axiom,
    ! [C2: set_a,D2: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ C2 @ D2 )
     => ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).

% insert_mono
thf(fact_941_insert__mono,axiom,
    ! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ C2 @ D2 )
     => ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ C2 ) @ ( insert4534936382041156343od_a_a @ A2 @ D2 ) ) ) ).

% insert_mono
thf(fact_942_insert__mono,axiom,
    ! [C2: set_set_a,D2: set_set_a,A2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
     => ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C2 ) @ ( insert_set_a @ A2 @ D2 ) ) ) ).

% insert_mono
thf(fact_943_Int__Collect__mono,axiom,
    ! [A: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_944_Int__Collect__mono,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit,P: product_unit > $o,Q: product_unit > $o] :
      ( ( ord_le3507040750410214029t_unit @ A @ B3 )
     => ( ! [X4: product_unit] :
            ( ( member_Product_unit @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le3507040750410214029t_unit @ ( inf_in4660618365625256667t_unit @ A @ ( collect_Product_unit @ P ) ) @ ( inf_in4660618365625256667t_unit @ B3 @ ( collect_Product_unit @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_945_Int__Collect__mono,axiom,
    ! [A: set_a,B3: set_a,P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_946_Int__Collect__mono,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ! [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B3 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_947_Int__Collect__mono,axiom,
    ! [A: set_set_a,B3: set_set_a,P: set_a > $o,Q: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_948_Int__greatest,axiom,
    ! [C2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ A )
     => ( ( ord_less_eq_set_a @ C2 @ B3 )
       => ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_949_Int__greatest,axiom,
    ! [C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ C2 @ A )
     => ( ( ord_le746702958409616551od_a_a @ C2 @ B3 )
       => ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_950_Int__greatest,axiom,
    ! [C2: set_set_a,A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ C2 @ A )
     => ( ( ord_le3724670747650509150_set_a @ C2 @ B3 )
       => ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_951_Int__absorb2,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( inf_inf_set_a @ A @ B3 )
        = A ) ) ).

% Int_absorb2
thf(fact_952_Int__absorb2,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = A ) ) ).

% Int_absorb2
thf(fact_953_Int__absorb2,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( inf_inf_set_set_a @ A @ B3 )
        = A ) ) ).

% Int_absorb2
thf(fact_954_Int__absorb1,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( inf_inf_set_a @ A @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_955_Int__absorb1,axiom,
    ! [B3: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B3 @ A )
     => ( ( inf_in8905007599844390133od_a_a @ A @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_956_Int__absorb1,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B3 @ A )
     => ( ( inf_inf_set_set_a @ A @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_957_Int__lower2,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_958_Int__lower2,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_959_Int__lower2,axiom,
    ! [A: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_960_Int__lower1,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ A ) ).

% Int_lower1
thf(fact_961_Int__lower1,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) @ A ) ).

% Int_lower1
thf(fact_962_Int__lower1,axiom,
    ! [A: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B3 ) @ A ) ).

% Int_lower1
thf(fact_963_Int__mono,axiom,
    ! [A: set_a,C2: set_a,B3: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B3 @ D2 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_964_Int__mono,axiom,
    ! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B3: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ C2 )
     => ( ( ord_le746702958409616551od_a_a @ B3 @ D2 )
       => ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B3 ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_965_Int__mono,axiom,
    ! [A: set_set_a,C2: set_set_a,B3: set_set_a,D2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ C2 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ D2 )
       => ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B3 ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_966_sym__on__subset,axiom,
    ! [A: set_a,R: set_Product_prod_a_a,B3: set_a] :
      ( ( sym_on_a @ A @ R )
     => ( ( ord_less_eq_set_a @ B3 @ A )
       => ( sym_on_a @ B3 @ R ) ) ) ).

% sym_on_subset
thf(fact_967_sym__on__subset,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,B3: set_Product_prod_a_a] :
      ( ( sym_on5631557199876295240od_a_a @ A @ R )
     => ( ( ord_le746702958409616551od_a_a @ B3 @ A )
       => ( sym_on5631557199876295240od_a_a @ B3 @ R ) ) ) ).

% sym_on_subset
thf(fact_968_sym__on__subset,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a,B3: set_set_a] :
      ( ( sym_on_set_a @ A @ R )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ A )
       => ( sym_on_set_a @ B3 @ R ) ) ) ).

% sym_on_subset
thf(fact_969_irrefl__on__subset,axiom,
    ! [A: set_a,R: set_Product_prod_a_a,B3: set_a] :
      ( ( irrefl_on_a @ A @ R )
     => ( ( ord_less_eq_set_a @ B3 @ A )
       => ( irrefl_on_a @ B3 @ R ) ) ) ).

% irrefl_on_subset
thf(fact_970_irrefl__on__subset,axiom,
    ! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,B3: set_Product_prod_a_a] :
      ( ( irrefl3954896097174259997od_a_a @ A @ R )
     => ( ( ord_le746702958409616551od_a_a @ B3 @ A )
       => ( irrefl3954896097174259997od_a_a @ B3 @ R ) ) ) ).

% irrefl_on_subset
thf(fact_971_irrefl__on__subset,axiom,
    ! [A: set_set_a,R: set_Pr5845495582615845127_set_a,B3: set_set_a] :
      ( ( irrefl_on_set_a @ A @ R )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ A )
       => ( irrefl_on_set_a @ B3 @ R ) ) ) ).

% irrefl_on_subset
thf(fact_972_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_973_subset__singletonD,axiom,
    ! [A: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
     => ( ( A = bot_bot_set_a )
        | ( A
          = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_974_subset__singletonD,axiom,
    ! [A: set_Product_prod_a_a,X: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
     => ( ( A = bot_bo3357376287454694259od_a_a )
        | ( A
          = ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).

% subset_singletonD
thf(fact_975_subset__singletonD,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
     => ( ( A = bot_bot_set_set_a )
        | ( A
          = ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_976_subset__singleton__iff,axiom,
    ! [X2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        | ( X2
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_977_subset__singleton__iff,axiom,
    ! [X2: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ X2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( ( X2 = bot_bot_set_a )
        | ( X2
          = ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_978_subset__singleton__iff,axiom,
    ! [X2: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
      = ( ( X2 = bot_bo3357376287454694259od_a_a )
        | ( X2
          = ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_979_subset__singleton__iff,axiom,
    ! [X2: set_set_a,A2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ X2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
      = ( ( X2 = bot_bot_set_set_a )
        | ( X2
          = ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_980_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_981_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_982_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_983_finite__inc__sedges,axiom,
    ! [V: a] :
      ( ( finite_finite_set_a @ edges )
     => ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).

% finite_inc_sedges
thf(fact_984_finite__Int,axiom,
    ! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
      ( ( ( finite6544458595007987280od_a_a @ F2 )
        | ( finite6544458595007987280od_a_a @ G ) )
     => ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_985_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_986_finite__Int,axiom,
    ! [F2: set_a,G: set_a] :
      ( ( ( finite_finite_a @ F2 )
        | ( finite_finite_a @ G ) )
     => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_987_finite__Int,axiom,
    ! [F2: set_set_a,G: set_set_a] :
      ( ( ( finite_finite_set_a @ F2 )
        | ( finite_finite_set_a @ G ) )
     => ( finite_finite_set_a @ ( inf_inf_set_set_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_988_finite__insert,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) )
      = ( finite_finite_set_a @ A ) ) ).

% finite_insert
thf(fact_989_finite__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
      = ( finite_finite_a @ A ) ) ).

% finite_insert
thf(fact_990_finite__insert,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
      = ( finite6544458595007987280od_a_a @ A ) ) ).

% finite_insert
thf(fact_991_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_992_finite__option__UNIV,axiom,
    ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% finite_option_UNIV
thf(fact_993_finite__option__UNIV,axiom,
    ( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% finite_option_UNIV
thf(fact_994_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_995_finite__option__UNIV,axiom,
    ( ( finite1824108633307372438od_a_a @ top_to5085949387790111389od_a_a )
    = ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ).

% finite_option_UNIV
thf(fact_996_finite__option__UNIV,axiom,
    ( ( finite3831083272032232269_set_a @ top_to3949272007228979924_set_a )
    = ( finite_finite_set_a @ top_top_set_set_a ) ) ).

% finite_option_UNIV
thf(fact_997_finite__Plus__UNIV__iff,axiom,
    ( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_998_finite__Plus__UNIV__iff,axiom,
    ( ( finite2069262655233506379t_unit @ top_to1755696212014396186t_unit )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_999_finite__Plus__UNIV__iff,axiom,
    ( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1000_finite__Plus__UNIV__iff,axiom,
    ( ( finite1276461556078370925unit_a @ top_to5559247480540603964unit_a )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1001_finite__Plus__UNIV__iff,axiom,
    ( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1002_finite__Plus__UNIV__iff,axiom,
    ( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1003_finite__Plus__UNIV__iff,axiom,
    ( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1004_finite__Plus__UNIV__iff,axiom,
    ( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1005_finite__Plus__UNIV__iff,axiom,
    ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1006_finite__Plus__UNIV__iff,axiom,
    ( ( finite8606893039029803292_set_a @ top_to6571144673325243243_set_a )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_set_a @ top_top_set_set_a ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1007_finite__subset__induct_H,axiom,
    ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ F2 )
     => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
       => ( ( P @ bot_bo3957492148770167129t_unit )
         => ( ! [A3: product_unit,F3: set_Product_unit] :
                ( ( finite4290736615968046902t_unit @ F3 )
               => ( ( member_Product_unit @ A3 @ A )
                 => ( ( ord_le3507040750410214029t_unit @ F3 @ A )
                   => ( ~ ( member_Product_unit @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_Product_unit @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1008_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A )
                 => ( ( ord_less_eq_set_nat @ F3 @ A )
                   => ( ~ ( member_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1009_finite__subset__induct_H,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A3 @ A )
                 => ( ( ord_less_eq_set_a @ F3 @ A )
                   => ( ~ ( member_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1010_finite__subset__induct_H,axiom,
    ! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( ord_le746702958409616551od_a_a @ F2 @ A )
       => ( ( P @ bot_bo3357376287454694259od_a_a )
         => ( ! [A3: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( member1426531477525435216od_a_a @ A3 @ A )
                 => ( ( ord_le746702958409616551od_a_a @ F3 @ A )
                   => ( ~ ( member1426531477525435216od_a_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert4534936382041156343od_a_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1011_finite__subset__induct_H,axiom,
    ! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
      ( ( finite_finite_set_a @ F2 )
     => ( ( ord_le3724670747650509150_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_set_a )
         => ( ! [A3: set_a,F3: set_set_a] :
                ( ( finite_finite_set_a @ F3 )
               => ( ( member_set_a @ A3 @ A )
                 => ( ( ord_le3724670747650509150_set_a @ F3 @ A )
                   => ( ~ ( member_set_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_set_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1012_finite__subset__induct,axiom,
    ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ F2 )
     => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
       => ( ( P @ bot_bo3957492148770167129t_unit )
         => ( ! [A3: product_unit,F3: set_Product_unit] :
                ( ( finite4290736615968046902t_unit @ F3 )
               => ( ( member_Product_unit @ A3 @ A )
                 => ( ~ ( member_Product_unit @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_Product_unit @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1013_finite__subset__induct,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A )
                 => ( ~ ( member_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1014_finite__subset__induct,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A3 @ A )
                 => ( ~ ( member_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1015_finite__subset__induct,axiom,
    ! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( ord_le746702958409616551od_a_a @ F2 @ A )
       => ( ( P @ bot_bo3357376287454694259od_a_a )
         => ( ! [A3: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( member1426531477525435216od_a_a @ A3 @ A )
                 => ( ~ ( member1426531477525435216od_a_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert4534936382041156343od_a_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1016_finite__subset__induct,axiom,
    ! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
      ( ( finite_finite_set_a @ F2 )
     => ( ( ord_le3724670747650509150_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_set_a )
         => ( ! [A3: set_a,F3: set_set_a] :
                ( ( finite_finite_set_a @ F3 )
               => ( ( member_set_a @ A3 @ A )
                 => ( ~ ( member_set_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1017_finite__all__edges__between,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( finite_finite_a @ X2 )
     => ( ( finite_finite_a @ Y )
       => ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ) ) ).

% finite_all_edges_between
thf(fact_1018_ulgraph_Oincident__sedges_Ocong,axiom,
    undire1270416042309875431dges_a = undire1270416042309875431dges_a ).

% ulgraph.incident_sedges.cong
thf(fact_1019_finite__has__maximal2,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( member_Product_unit @ A2 @ A )
       => ? [X4: product_unit] :
            ( ( member_Product_unit @ X4 @ A )
            & ( ord_le3221252021190050221t_unit @ A2 @ X4 )
            & ! [Xa: product_unit] :
                ( ( member_Product_unit @ Xa @ A )
               => ( ( ord_le3221252021190050221t_unit @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1020_finite__has__maximal2,axiom,
    ! [A: set_real,A2: real] :
      ( ( finite_finite_real @ A )
     => ( ( member_real @ A2 @ A )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ( ord_less_eq_real @ A2 @ X4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1021_finite__has__maximal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( ord_less_eq_set_a @ A2 @ X4 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1022_finite__has__maximal2,axiom,
    ! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
      ( ( finite8717734299975451184od_a_a @ A )
     => ( ( member1816616512716248880od_a_a @ A2 @ A )
       => ? [X4: set_Product_prod_a_a] :
            ( ( member1816616512716248880od_a_a @ X4 @ A )
            & ( ord_le746702958409616551od_a_a @ A2 @ X4 )
            & ! [Xa: set_Product_prod_a_a] :
                ( ( member1816616512716248880od_a_a @ Xa @ A )
               => ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1023_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ A2 @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1024_finite__has__maximal2,axiom,
    ! [A: set_set_set_a,A2: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( member_set_set_a @ A2 @ A )
       => ? [X4: set_set_a] :
            ( ( member_set_set_a @ X4 @ A )
            & ( ord_le3724670747650509150_set_a @ A2 @ X4 )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1025_finite__has__minimal2,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( member_Product_unit @ A2 @ A )
       => ? [X4: product_unit] :
            ( ( member_Product_unit @ X4 @ A )
            & ( ord_le3221252021190050221t_unit @ X4 @ A2 )
            & ! [Xa: product_unit] :
                ( ( member_Product_unit @ Xa @ A )
               => ( ( ord_le3221252021190050221t_unit @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1026_finite__has__minimal2,axiom,
    ! [A: set_real,A2: real] :
      ( ( finite_finite_real @ A )
     => ( ( member_real @ A2 @ A )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ( ord_less_eq_real @ X4 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1027_finite__has__minimal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( ord_less_eq_set_a @ X4 @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1028_finite__has__minimal2,axiom,
    ! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
      ( ( finite8717734299975451184od_a_a @ A )
     => ( ( member1816616512716248880od_a_a @ A2 @ A )
       => ? [X4: set_Product_prod_a_a] :
            ( ( member1816616512716248880od_a_a @ X4 @ A )
            & ( ord_le746702958409616551od_a_a @ X4 @ A2 )
            & ! [Xa: set_Product_prod_a_a] :
                ( ( member1816616512716248880od_a_a @ Xa @ A )
               => ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1029_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1030_finite__has__minimal2,axiom,
    ! [A: set_set_set_a,A2: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( member_set_set_a @ A2 @ A )
       => ? [X4: set_set_a] :
            ( ( member_set_set_a @ X4 @ A )
            & ( ord_le3724670747650509150_set_a @ X4 @ A2 )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1031_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_1032_ex__new__if__finite,axiom,
    ! [A: set_a] :
      ( ~ ( finite_finite_a @ top_top_set_a )
     => ( ( finite_finite_a @ A )
       => ? [A3: a] :
            ~ ( member_a @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_1033_ex__new__if__finite,axiom,
    ! [A: set_Product_unit] :
      ( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( ( finite4290736615968046902t_unit @ A )
       => ? [A3: product_unit] :
            ~ ( member_Product_unit @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_1034_ex__new__if__finite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A )
       => ? [A3: nat] :
            ~ ( member_nat @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_1035_ex__new__if__finite,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ~ ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ? [A3: product_prod_a_a] :
            ~ ( member1426531477525435216od_a_a @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_1036_ex__new__if__finite,axiom,
    ! [A: set_set_a] :
      ( ~ ( finite_finite_set_a @ top_top_set_set_a )
     => ( ( finite_finite_set_a @ A )
       => ? [A3: set_a] :
            ~ ( member_set_a @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_1037_finite__UNIV,axiom,
    finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).

% finite_UNIV
thf(fact_1038_finite__Prod__UNIV,axiom,
    ( ( finite_finite_a @ top_top_set_a )
   => ( ( finite_finite_a @ top_top_set_a )
     => ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).

% finite_Prod_UNIV
thf(fact_1039_finite__Prod__UNIV,axiom,
    ( ( finite_finite_a @ top_top_set_a )
   => ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( finite1408885517383445215t_unit @ top_to6636102223169616742t_unit ) ) ) ).

% finite_Prod_UNIV
thf(fact_1040_finite__Prod__UNIV,axiom,
    ( ( finite_finite_a @ top_top_set_a )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1041_finite__Prod__UNIV,axiom,
    ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
   => ( ( finite_finite_a @ top_top_set_a )
     => ( finite616084418228309761unit_a @ top_to1216281454841048712unit_a ) ) ) ).

% finite_Prod_UNIV
thf(fact_1042_finite__Prod__UNIV,axiom,
    ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
   => ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).

% finite_Prod_UNIV
thf(fact_1043_finite__Prod__UNIV,axiom,
    ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1044_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_a @ top_top_set_a )
     => ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).

% finite_Prod_UNIV
thf(fact_1045_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).

% finite_Prod_UNIV
thf(fact_1046_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1047_finite__Prod__UNIV,axiom,
    ( ( finite_finite_a @ top_top_set_a )
   => ( ( finite_finite_set_a @ top_top_set_set_a )
     => ( finite2449716632925952944_set_a @ top_to1705781063600484279_set_a ) ) ) ).

% finite_Prod_UNIV
thf(fact_1048_finite__prod,axiom,
    ( ( finite1408885517383445215t_unit @ top_to6636102223169616742t_unit )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_1049_finite__prod,axiom,
    ( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_1050_finite__prod,axiom,
    ( ( finite616084418228309761unit_a @ top_to1216281454841048712unit_a )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_prod
thf(fact_1051_finite__prod,axiom,
    ( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_1052_finite__prod,axiom,
    ( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_1053_finite__prod,axiom,
    ( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_prod
thf(fact_1054_finite__prod,axiom,
    ( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_1055_finite__prod,axiom,
    ( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_1056_finite__prod,axiom,
    ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_a @ top_top_set_a ) ) ) ).

% finite_prod
thf(fact_1057_finite__prod,axiom,
    ( ( finite2449716632925952944_set_a @ top_to1705781063600484279_set_a )
    = ( ( finite_finite_a @ top_top_set_a )
      & ( finite_finite_set_a @ top_top_set_set_a ) ) ) ).

% finite_prod
thf(fact_1058_infinite__imp__nonempty,axiom,
    ! [S2: set_a] :
      ( ~ ( finite_finite_a @ S2 )
     => ( S2 != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_1059_infinite__imp__nonempty,axiom,
    ! [S2: set_set_a] :
      ( ~ ( finite_finite_set_a @ S2 )
     => ( S2 != bot_bot_set_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_1060_infinite__imp__nonempty,axiom,
    ! [S2: set_Product_prod_a_a] :
      ( ~ ( finite6544458595007987280od_a_a @ S2 )
     => ( S2 != bot_bo3357376287454694259od_a_a ) ) ).

% infinite_imp_nonempty
thf(fact_1061_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1062_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_1063_finite_OemptyI,axiom,
    finite_finite_set_a @ bot_bot_set_set_a ).

% finite.emptyI
thf(fact_1064_finite_OemptyI,axiom,
    finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).

% finite.emptyI
thf(fact_1065_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1066_finite__subset,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_1067_finite__subset,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( finite_finite_a @ B3 )
       => ( finite_finite_a @ A ) ) ) ).

% finite_subset
thf(fact_1068_finite__subset,axiom,
    ! [A: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B3 )
     => ( ( finite6544458595007987280od_a_a @ B3 )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_subset
thf(fact_1069_finite__subset,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( finite_finite_set_a @ B3 )
       => ( finite_finite_set_a @ A ) ) ) ).

% finite_subset
thf(fact_1070_infinite__super,axiom,
    ! [S2: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T2 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_1071_infinite__super,axiom,
    ! [S2: set_a,T2: set_a] :
      ( ( ord_less_eq_set_a @ S2 @ T2 )
     => ( ~ ( finite_finite_a @ S2 )
       => ~ ( finite_finite_a @ T2 ) ) ) ).

% infinite_super
thf(fact_1072_infinite__super,axiom,
    ! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ S2 @ T2 )
     => ( ~ ( finite6544458595007987280od_a_a @ S2 )
       => ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ).

% infinite_super
thf(fact_1073_infinite__super,axiom,
    ! [S2: set_set_a,T2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
     => ( ~ ( finite_finite_set_a @ S2 )
       => ~ ( finite_finite_set_a @ T2 ) ) ) ).

% infinite_super
thf(fact_1074_rev__finite__subset,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_1075_rev__finite__subset,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( finite_finite_a @ B3 )
     => ( ( ord_less_eq_set_a @ A @ B3 )
       => ( finite_finite_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_1076_rev__finite__subset,axiom,
    ! [B3: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B3 )
     => ( ( ord_le746702958409616551od_a_a @ A @ B3 )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_1077_rev__finite__subset,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ A @ B3 )
       => ( finite_finite_set_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_1078_finite_OinsertI,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1079_finite_OinsertI,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1080_finite_OinsertI,axiom,
    ! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1081_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1082_finite__has__maximal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1083_finite__has__maximal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1084_finite__has__maximal,axiom,
    ! [A: set_se5735800977113168103od_a_a] :
      ( ( finite8717734299975451184od_a_a @ A )
     => ( ( A != bot_bo777872063958040403od_a_a )
       => ? [X4: set_Product_prod_a_a] :
            ( ( member1816616512716248880od_a_a @ X4 @ A )
            & ! [Xa: set_Product_prod_a_a] :
                ( ( member1816616512716248880od_a_a @ Xa @ A )
               => ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1085_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1086_finite__has__maximal,axiom,
    ! [A: set_set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( A != bot_bo3380559777022489994_set_a )
       => ? [X4: set_set_a] :
            ( ( member_set_set_a @ X4 @ A )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1087_finite__has__minimal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1088_finite__has__minimal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1089_finite__has__minimal,axiom,
    ! [A: set_se5735800977113168103od_a_a] :
      ( ( finite8717734299975451184od_a_a @ A )
     => ( ( A != bot_bo777872063958040403od_a_a )
       => ? [X4: set_Product_prod_a_a] :
            ( ( member1816616512716248880od_a_a @ X4 @ A )
            & ! [Xa: set_Product_prod_a_a] :
                ( ( member1816616512716248880od_a_a @ Xa @ A )
               => ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1090_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1091_finite__has__minimal,axiom,
    ! [A: set_set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( A != bot_bo3380559777022489994_set_a )
       => ? [X4: set_set_a] :
            ( ( member_set_set_a @ X4 @ A )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1092_finite_Ocases,axiom,
    ! [A2: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( A2 != bot_bot_set_a )
       => ~ ! [A7: set_a] :
              ( ? [A3: a] :
                  ( A2
                  = ( insert_a @ A3 @ A7 ) )
             => ~ ( finite_finite_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1093_finite_Ocases,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ~ ! [A7: set_set_a] :
              ( ? [A3: set_a] :
                  ( A2
                  = ( insert_set_a @ A3 @ A7 ) )
             => ~ ( finite_finite_set_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1094_finite_Ocases,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A2 )
     => ( ( A2 != bot_bo3357376287454694259od_a_a )
       => ~ ! [A7: set_Product_prod_a_a] :
              ( ? [A3: product_prod_a_a] :
                  ( A2
                  = ( insert4534936382041156343od_a_a @ A3 @ A7 ) )
             => ~ ( finite6544458595007987280od_a_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1095_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A3: nat] :
                  ( A2
                  = ( insert_nat @ A3 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1096_finite_Osimps,axiom,
    ( finite_finite_a
    = ( ^ [A6: set_a] :
          ( ( A6 = bot_bot_set_a )
          | ? [A5: set_a,B7: a] :
              ( ( A6
                = ( insert_a @ B7 @ A5 ) )
              & ( finite_finite_a @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1097_finite_Osimps,axiom,
    ( finite_finite_set_a
    = ( ^ [A6: set_set_a] :
          ( ( A6 = bot_bot_set_set_a )
          | ? [A5: set_set_a,B7: set_a] :
              ( ( A6
                = ( insert_set_a @ B7 @ A5 ) )
              & ( finite_finite_set_a @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1098_finite_Osimps,axiom,
    ( finite6544458595007987280od_a_a
    = ( ^ [A6: set_Product_prod_a_a] :
          ( ( A6 = bot_bo3357376287454694259od_a_a )
          | ? [A5: set_Product_prod_a_a,B7: product_prod_a_a] :
              ( ( A6
                = ( insert4534936382041156343od_a_a @ B7 @ A5 ) )
              & ( finite6544458595007987280od_a_a @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1099_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A6: set_nat] :
          ( ( A6 = bot_bot_set_nat )
          | ? [A5: set_nat,B7: nat] :
              ( ( A6
                = ( insert_nat @ B7 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1100_finite__induct,axiom,
    ! [F2: set_Product_unit,P: set_Product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ F2 )
     => ( ( P @ bot_bo3957492148770167129t_unit )
       => ( ! [X4: product_unit,F3: set_Product_unit] :
              ( ( finite4290736615968046902t_unit @ F3 )
             => ( ~ ( member_Product_unit @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_Product_unit @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1101_finite__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X4: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1102_finite__induct,axiom,
    ! [F2: set_set_a,P: set_set_a > $o] :
      ( ( finite_finite_set_a @ F2 )
     => ( ( P @ bot_bot_set_set_a )
       => ( ! [X4: set_a,F3: set_set_a] :
              ( ( finite_finite_set_a @ F3 )
             => ( ~ ( member_set_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1103_finite__induct,axiom,
    ! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ F3 )
             => ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1104_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1105_finite__ne__induct,axiom,
    ! [F2: set_Product_unit,P: set_Product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ F2 )
     => ( ( F2 != bot_bo3957492148770167129t_unit )
       => ( ! [X4: product_unit] : ( P @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
         => ( ! [X4: product_unit,F3: set_Product_unit] :
                ( ( finite4290736615968046902t_unit @ F3 )
               => ( ( F3 != bot_bo3957492148770167129t_unit )
                 => ( ~ ( member_Product_unit @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_Product_unit @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1106_finite__ne__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( F2 != bot_bot_set_a )
       => ( ! [X4: a] : ( P @ ( insert_a @ X4 @ bot_bot_set_a ) )
         => ( ! [X4: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( F3 != bot_bot_set_a )
                 => ( ~ ( member_a @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1107_finite__ne__induct,axiom,
    ! [F2: set_set_a,P: set_set_a > $o] :
      ( ( finite_finite_set_a @ F2 )
     => ( ( F2 != bot_bot_set_set_a )
       => ( ! [X4: set_a] : ( P @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
         => ( ! [X4: set_a,F3: set_set_a] :
                ( ( finite_finite_set_a @ F3 )
               => ( ( F3 != bot_bot_set_set_a )
                 => ( ~ ( member_set_a @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1108_finite__ne__induct,axiom,
    ! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( F2 != bot_bo3357376287454694259od_a_a )
       => ( ! [X4: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) )
         => ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( F3 != bot_bo3357376287454694259od_a_a )
                 => ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1109_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1110_infinite__finite__induct,axiom,
    ! [P: set_Product_unit > $o,A: set_Product_unit] :
      ( ! [A7: set_Product_unit] :
          ( ~ ( finite4290736615968046902t_unit @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo3957492148770167129t_unit )
       => ( ! [X4: product_unit,F3: set_Product_unit] :
              ( ( finite4290736615968046902t_unit @ F3 )
             => ( ~ ( member_Product_unit @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_Product_unit @ X4 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1111_infinite__finite__induct,axiom,
    ! [P: set_a > $o,A: set_a] :
      ( ! [A7: set_a] :
          ( ~ ( finite_finite_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X4: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1112_infinite__finite__induct,axiom,
    ! [P: set_set_a > $o,A: set_set_a] :
      ( ! [A7: set_set_a] :
          ( ~ ( finite_finite_set_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_set_a )
       => ( ! [X4: set_a,F3: set_set_a] :
              ( ( finite_finite_set_a @ F3 )
             => ( ~ ( member_set_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1113_infinite__finite__induct,axiom,
    ! [P: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
      ( ! [A7: set_Product_prod_a_a] :
          ( ~ ( finite6544458595007987280od_a_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ F3 )
             => ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1114_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1115_Finite__Set_Ofinite__set,axiom,
    ( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% Finite_Set.finite_set
thf(fact_1116_Finite__Set_Ofinite__set,axiom,
    ( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% Finite_Set.finite_set
thf(fact_1117_Finite__Set_Ofinite__set,axiom,
    ( ( finite8717734299975451184od_a_a @ top_to1047947862415971895od_a_a )
    = ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ).

% Finite_Set.finite_set
thf(fact_1118_Finite__Set_Ofinite__set,axiom,
    ( ( finite7209287970140883943_set_a @ top_to4027821306633060462_set_a )
    = ( finite_finite_set_a @ top_top_set_set_a ) ) ).

% Finite_Set.finite_set
thf(fact_1119_Finite__Set_Ofinite__set,axiom,
    ( ( finite_finite_set_a @ top_top_set_set_a )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% Finite_Set.finite_set
thf(fact_1120_finite__ranking__induct,axiom,
    ! [S2: set_Product_unit,P: set_Product_unit > $o,F: product_unit > real] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( P @ bot_bo3957492148770167129t_unit )
       => ( ! [X4: product_unit,S3: set_Product_unit] :
              ( ( finite4290736615968046902t_unit @ S3 )
             => ( ! [Y7: product_unit] :
                    ( ( member_Product_unit @ Y7 @ S3 )
                   => ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_Product_unit @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1121_finite__ranking__induct,axiom,
    ! [S2: set_a,P: set_a > $o,F: a > real] :
      ( ( finite_finite_a @ S2 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X4: a,S3: set_a] :
              ( ( finite_finite_a @ S3 )
             => ( ! [Y7: a] :
                    ( ( member_a @ Y7 @ S3 )
                   => ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1122_finite__ranking__induct,axiom,
    ! [S2: set_set_a,P: set_set_a > $o,F: set_a > real] :
      ( ( finite_finite_set_a @ S2 )
     => ( ( P @ bot_bot_set_set_a )
       => ( ! [X4: set_a,S3: set_set_a] :
              ( ( finite_finite_set_a @ S3 )
             => ( ! [Y7: set_a] :
                    ( ( member_set_a @ Y7 @ S3 )
                   => ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_set_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1123_finite__ranking__induct,axiom,
    ! [S2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > real] :
      ( ( finite6544458595007987280od_a_a @ S2 )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X4: product_prod_a_a,S3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ S3 )
             => ( ! [Y7: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ Y7 @ S3 )
                   => ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1124_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S3: set_nat] :
              ( ( finite_finite_nat @ S3 )
             => ( ! [Y7: nat] :
                    ( ( member_nat @ Y7 @ S3 )
                   => ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_nat @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1125_finite__ranking__induct,axiom,
    ! [S2: set_Product_unit,P: set_Product_unit > $o,F: product_unit > nat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( P @ bot_bo3957492148770167129t_unit )
       => ( ! [X4: product_unit,S3: set_Product_unit] :
              ( ( finite4290736615968046902t_unit @ S3 )
             => ( ! [Y7: product_unit] :
                    ( ( member_Product_unit @ Y7 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_Product_unit @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1126_finite__ranking__induct,axiom,
    ! [S2: set_a,P: set_a > $o,F: a > nat] :
      ( ( finite_finite_a @ S2 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X4: a,S3: set_a] :
              ( ( finite_finite_a @ S3 )
             => ( ! [Y7: a] :
                    ( ( member_a @ Y7 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1127_finite__ranking__induct,axiom,
    ! [S2: set_set_a,P: set_set_a > $o,F: set_a > nat] :
      ( ( finite_finite_set_a @ S2 )
     => ( ( P @ bot_bot_set_set_a )
       => ( ! [X4: set_a,S3: set_set_a] :
              ( ( finite_finite_set_a @ S3 )
             => ( ! [Y7: set_a] :
                    ( ( member_set_a @ Y7 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_set_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1128_finite__ranking__induct,axiom,
    ! [S2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > nat] :
      ( ( finite6544458595007987280od_a_a @ S2 )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X4: product_prod_a_a,S3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ S3 )
             => ( ! [Y7: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ Y7 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1129_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S3: set_nat] :
              ( ( finite_finite_nat @ S3 )
             => ( ! [Y7: nat] :
                    ( ( member_nat @ Y7 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X4 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_nat @ X4 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1130_finite__incident__edges,axiom,
    ! [V: a] :
      ( ( finite_finite_set_a @ edges )
     => ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).

% finite_incident_edges
thf(fact_1131_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_1132_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_1133_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1134_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1135_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1136_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1137_card1__incident__imp__vert,axiom,
    ! [V: a,E: set_a] :
      ( ( ( undire1521409233611534436dent_a @ V @ E )
        & ( ( finite_card_a @ E )
          = one_one_nat ) )
     => ( E
        = ( insert_a @ V @ bot_bot_set_a ) ) ) ).

% card1_incident_imp_vert
thf(fact_1138_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1139_incident__edges__sedges,axiom,
    ! [V: a] :
      ( ~ ( undire3617971648856834880loop_a @ edges @ V )
     => ( ( undire3231912044278729248dges_a @ edges @ V )
        = ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).

% incident_edges_sedges
thf(fact_1140_degree0__inc__edges__empt__iff,axiom,
    ! [V: a] :
      ( ( finite_finite_set_a @ edges )
     => ( ( ( undire8867928226783802224gree_a @ edges @ V )
          = zero_zero_nat )
        = ( ( undire3231912044278729248dges_a @ edges @ V )
          = bot_bot_set_set_a ) ) ) ).

% degree0_inc_edges_empt_iff
thf(fact_1141_is__loop__def,axiom,
    ( undire2905028936066782638loop_a
    = ( ^ [E3: set_a] :
          ( ( finite_card_a @ E3 )
          = one_one_nat ) ) ) ).

% is_loop_def
thf(fact_1142_incident__edges__union,axiom,
    ! [V: a] :
      ( ( undire3231912044278729248dges_a @ edges @ V )
      = ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).

% incident_edges_union
thf(fact_1143_incident__loops__card,axiom,
    ! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) @ one_one_nat ) ).

% incident_loops_card
thf(fact_1144_adj__relation__wf,axiom,
    ! [U: a,V: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) )
     => ( ord_less_eq_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ vertices ) ) ).

% adj_relation_wf
thf(fact_1145_card__all__edges__between__commute,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
      = ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y @ X2 ) ) ) ).

% card_all_edges_between_commute
thf(fact_1146_wellformed,axiom,
    ! [E: set_a] :
      ( ( member_set_a @ E @ edges )
     => ( ord_less_eq_set_a @ E @ vertices ) ) ).

% wellformed
thf(fact_1147_vert__adj__imp__inV,axiom,
    ! [V1: a,V22: a] :
      ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
     => ( ( member_a @ V1 @ vertices )
        & ( member_a @ V22 @ vertices ) ) ) ).

% vert_adj_imp_inV
thf(fact_1148_incident__edge__in__wf,axiom,
    ! [E: set_a,V: a] :
      ( ( member_set_a @ E @ edges )
     => ( ( undire1521409233611534436dent_a @ V @ E )
       => ( member_a @ V @ vertices ) ) ) ).

% incident_edge_in_wf
thf(fact_1149_has__loop__in__verts,axiom,
    ! [V: a] :
      ( ( undire3617971648856834880loop_a @ edges @ V )
     => ( member_a @ V @ vertices ) ) ).

% has_loop_in_verts
thf(fact_1150_no__loops,axiom,
    ! [V: a] :
      ( ( member_a @ V @ vertices )
     => ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).

% no_loops
thf(fact_1151_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_1152_edge__adjacent__alt__def,axiom,
    ! [E1: set_a,E2: set_a] :
      ( ( member_set_a @ E1 @ edges )
     => ( ( member_set_a @ E2 @ edges )
       => ( ? [X6: a] :
              ( ( member_a @ X6 @ vertices )
              & ( member_a @ X6 @ E1 )
              & ( member_a @ X6 @ E2 ) )
         => ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).

% edge_adjacent_alt_def
thf(fact_1153_wellformed__alt__fst,axiom,
    ! [X: a,Y3: a] :
      ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
     => ( member_a @ X @ vertices ) ) ).

% wellformed_alt_fst
thf(fact_1154_wellformed__alt__snd,axiom,
    ! [X: a,Y3: a] :
      ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
     => ( member_a @ Y3 @ vertices ) ) ).

% wellformed_alt_snd
thf(fact_1155_local_Owf,axiom,
    ! [U: a,V: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) )
     => ( ( member_a @ U @ vertices )
        & ( member_a @ V @ vertices ) ) ) ).

% local.wf
thf(fact_1156_all__edges__between__rem__wf,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
      = ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X2 @ vertices ) @ ( inf_inf_set_a @ Y @ vertices ) ) ) ).

% all_edges_between_rem_wf
thf(fact_1157_incident__edges__empty,axiom,
    ! [V: a] :
      ( ~ ( member_a @ V @ vertices )
     => ( ( undire3231912044278729248dges_a @ edges @ V )
        = bot_bot_set_set_a ) ) ).

% incident_edges_empty
thf(fact_1158_sgraph__axioms,axiom,
    undire3507641187627840796raph_a @ vertices @ edges ).

% sgraph_axioms
thf(fact_1159_vert__adj__rel__iff,axiom,
    ! [U: a,V: a] :
      ( ( member_a @ U @ vertices )
     => ( ( member_a @ V @ vertices )
       => ( ( undire397441198561214472_adj_a @ edges @ U @ V )
          = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( graph_8122095853558514513tion_a @ edges ) ) ) ) ) ).

% vert_adj_rel_iff
thf(fact_1160_degree__none,axiom,
    ! [V: a] :
      ( ~ ( member_a @ V @ vertices )
     => ( ( undire8867928226783802224gree_a @ edges @ V )
        = zero_zero_nat ) ) ).

% degree_none
thf(fact_1161_alt__degree__def,axiom,
    ! [V: a] :
      ( ( undire8867928226783802224gree_a @ edges @ V )
      = ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).

% alt_degree_def
thf(fact_1162_incident__sedges__empty,axiom,
    ! [V: a] :
      ( ~ ( member_a @ V @ vertices )
     => ( ( undire1270416042309875431dges_a @ edges @ V )
        = bot_bot_set_set_a ) ) ).

% incident_sedges_empty
thf(fact_1163_degree__no__loops,axiom,
    ! [V: a] :
      ( ~ ( undire3617971648856834880loop_a @ edges @ V )
     => ( ( undire8867928226783802224gree_a @ edges @ V )
        = ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ) ).

% degree_no_loops
thf(fact_1164_graph__rel__axioms,axiom,
    graph_7976503222945204450_rel_a @ vertices @ ( graph_8122095853558514513tion_a @ edges ) ).

% graph_rel_axioms
thf(fact_1165_is__isolated__vertex__degree0,axiom,
    ! [V: a] :
      ( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
     => ( ( undire8867928226783802224gree_a @ edges @ V )
        = zero_zero_nat ) ) ).

% is_isolated_vertex_degree0
thf(fact_1166_is__edge__or__loop,axiom,
    ! [E: set_a] :
      ( ( member_set_a @ E @ edges )
     => ( ( undire2905028936066782638loop_a @ E )
        | ( undire4917966558017083288edge_a @ E ) ) ) ).

% is_edge_or_loop
thf(fact_1167_is__isolated__vertex__no__loop,axiom,
    ! [V: a] :
      ( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
     => ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).

% is_isolated_vertex_no_loop
thf(fact_1168_is__isolated__vertex__edge,axiom,
    ! [V: a,E: set_a] :
      ( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
     => ( ( member_set_a @ E @ edges )
       => ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).

% is_isolated_vertex_edge
thf(fact_1169_is__isolated__vertex__def,axiom,
    ! [V: a] :
      ( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
      = ( ( member_a @ V @ vertices )
        & ! [X3: a] :
            ( ( member_a @ X3 @ vertices )
           => ~ ( undire397441198561214472_adj_a @ edges @ X3 @ V ) ) ) ) ).

% is_isolated_vertex_def
thf(fact_1170_all__edges__between__Un1,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
      = ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) @ ( undire8383842906760478443ween_a @ edges @ Y @ Z2 ) ) ) ).

% all_edges_between_Un1
thf(fact_1171_all__edges__between__Un2,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( undire8383842906760478443ween_a @ edges @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) )
      = ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) ) ) ).

% all_edges_between_Un2
thf(fact_1172_degree0__neighborhood__empt__iff,axiom,
    ! [V: a] :
      ( ( finite_finite_set_a @ edges )
     => ( ( ( undire8867928226783802224gree_a @ edges @ V )
          = zero_zero_nat )
        = ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
          = bot_bot_set_a ) ) ) ).

% degree0_neighborhood_empt_iff
thf(fact_1173_card__incident__sedges__neighborhood,axiom,
    ! [V: a] :
      ( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) )
      = ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).

% card_incident_sedges_neighborhood
thf(fact_1174_neighborhood__incident,axiom,
    ! [U: a,V: a] :
      ( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
      = ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).

% neighborhood_incident
thf(fact_1175_ulgraph__rel__axioms,axiom,
    graph_6777131348317456619_rel_a @ vertices @ ( graph_8122095853558514513tion_a @ edges ) ).

% ulgraph_rel_axioms
thf(fact_1176_alt__deg__neighborhood,axiom,
    ! [V: a] :
      ( ( undire8867928226783802224gree_a @ edges @ V )
      = ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).

% alt_deg_neighborhood
thf(fact_1177_iso__vertex__empty__neighborhood,axiom,
    ! [V: a] :
      ( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
     => ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
        = bot_bot_set_a ) ) ).

% iso_vertex_empty_neighborhood
thf(fact_1178_max__all__edges__between,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( finite_finite_a @ X2 )
     => ( ( finite_finite_a @ Y )
       => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) @ ( times_times_nat @ ( finite_card_a @ X2 ) @ ( finite_card_a @ Y ) ) ) ) ) ).

% max_all_edges_between
thf(fact_1179_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1180_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_1181_inc_Oadd__point__def,axiom,
    ! [P2: a] :
      ( ( design2964366272795260673oint_a @ vertices @ P2 )
      = ( insert_a @ P2 @ vertices ) ) ).

% inc.add_point_def
thf(fact_1182_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1183_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1184_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1185_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1186_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1187_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1188_inc_Oadd__existing__point,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ vertices )
     => ( ( design2964366272795260673oint_a @ vertices @ P2 )
        = vertices ) ) ).

% inc.add_existing_point
thf(fact_1189_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1190_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_1191_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_1192_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_1193_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_1194_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_1195_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1196_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1197_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_1198_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1199_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y7: nat] :
                ( ( P @ Y7 )
               => ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1200_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_1201_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_1202_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1203_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_1204_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1205_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1206_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1207_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1208_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1209_induced__edges__ss,axiom,
    ! [V3: set_a] :
      ( ( ord_less_eq_set_a @ V3 @ vertices )
     => ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ edges ) ) ).

% induced_edges_ss
thf(fact_1210_inc_Oblock__complement__inter__empty,axiom,
    ! [Bl1: set_a,Bl2: set_a] :
      ( ( ( design6447616907850319326ment_a @ vertices @ Bl1 )
        = Bl2 )
     => ( ( inf_inf_set_a @ Bl1 @ Bl2 )
        = bot_bot_set_a ) ) ).

% inc.block_complement_inter_empty
thf(fact_1211_inc_Oadd__delete__point__inv,axiom,
    ! [P2: a] :
      ( ~ ( member_a @ P2 @ vertices )
     => ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ vertices @ P2 ) @ P2 )
        = vertices ) ) ).

% inc.add_delete_point_inv
thf(fact_1212_induced__edges__self,axiom,
    ( ( undire7777452895879145676dges_a @ edges @ vertices )
    = edges ) ).

% induced_edges_self
thf(fact_1213_inc_Odel__invalid__point,axiom,
    ! [P2: a] :
      ( ~ ( member_a @ P2 @ vertices )
     => ( ( design108908007054065099oint_a @ vertices @ P2 )
        = vertices ) ) ).

% inc.del_invalid_point
thf(fact_1214_inc_Oblock__comp__elem__alt__left,axiom,
    ! [X: a,Bl: set_a,Ps: set_a] :
      ( ( member_a @ X @ Bl )
     => ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
       => ~ ( member_a @ X @ Ps ) ) ) ).

% inc.block_comp_elem_alt_left
thf(fact_1215_inc_Oblock__comp__elem__alt__right,axiom,
    ! [Ps: set_a,Bl: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ vertices )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ Ps )
           => ~ ( member_a @ X4 @ Bl ) )
       => ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) ) ) ) ).

% inc.block_comp_elem_alt_right
thf(fact_1216_inc_Oblock__complement__elem__iff,axiom,
    ! [Ps: set_a,Bl: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ vertices )
     => ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
        = ( ! [X3: a] :
              ( ( member_a @ X3 @ Ps )
             => ~ ( member_a @ X3 @ Bl ) ) ) ) ) ).

% inc.block_complement_elem_iff
thf(fact_1217_inc_Oblock__complement__subset__points,axiom,
    ! [Ps: set_a,Bl: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ vertices @ Bl ) )
     => ( ord_less_eq_set_a @ Ps @ vertices ) ) ).

% inc.block_complement_subset_points
thf(fact_1218_induced__edges__union,axiom,
    ! [VH1: set_a,S2: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
      ( ( ord_less_eq_set_a @ VH1 @ S2 )
     => ( ( ord_less_eq_set_a @ VH2 @ T2 )
       => ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
         => ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
           => ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S2 @ T2 ) ) )
             => ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S2 ) ) ) ) ) ) ) ).

% induced_edges_union
thf(fact_1219_induced__is__subgraph,axiom,
    ! [V3: set_a] :
      ( ( ord_less_eq_set_a @ V3 @ vertices )
     => ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ vertices @ edges ) ) ).

% induced_is_subgraph
thf(fact_1220_inc_Odel__point__def,axiom,
    ! [P2: a] :
      ( ( design108908007054065099oint_a @ vertices @ P2 )
      = ( minus_minus_set_a @ vertices @ ( insert_a @ P2 @ bot_bot_set_a ) ) ) ).

% inc.del_point_def
thf(fact_1221_induced__is__graph__sys,axiom,
    ! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).

% induced_is_graph_sys
thf(fact_1222_inc_Oblock__complement__def,axiom,
    ! [B: set_a] :
      ( ( design6447616907850319326ment_a @ vertices @ B )
      = ( minus_minus_set_a @ vertices @ B ) ) ).

% inc.block_complement_def
thf(fact_1223_subgraph__refl,axiom,
    undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).

% subgraph_refl
thf(fact_1224_is__graph__system,axiom,
    undire2554140024507503526stem_a @ vertices @ edges ).

% is_graph_system
thf(fact_1225_induced__union__subgraph,axiom,
    ! [VH1: set_a,S2: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
      ( ( ord_less_eq_set_a @ VH1 @ S2 )
     => ( ( ord_less_eq_set_a @ VH2 @ T2 )
       => ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
         => ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
           => ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S2 @ ( undire7777452895879145676dges_a @ edges @ S2 ) )
                & ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
              = ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S2 @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S2 @ T2 ) ) ) ) ) ) ) ) ).

% induced_union_subgraph
thf(fact_1226_induced__edges__union__subgraph__single,axiom,
    ! [VH1: set_a,S2: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
      ( ( ord_less_eq_set_a @ VH1 @ S2 )
     => ( ( ord_less_eq_set_a @ VH2 @ T2 )
       => ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
         => ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
           => ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S2 @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S2 @ T2 ) ) )
             => ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S2 @ ( undire7777452895879145676dges_a @ edges @ S2 ) ) ) ) ) ) ) ).

% induced_edges_union_subgraph_single
thf(fact_1227_subgraph__complete,axiom,
    undire7103218114511261257raph_a @ vertices @ edges @ vertices @ ( undire2918257014606996450dges_a @ vertices ) ).

% subgraph_complete
thf(fact_1228_wellformed__all__edges,axiom,
    ord_le3724670747650509150_set_a @ edges @ ( undire2918257014606996450dges_a @ vertices ) ).

% wellformed_all_edges
thf(fact_1229_induced__edges__alt,axiom,
    ! [V3: set_a] :
      ( ( undire7777452895879145676dges_a @ edges @ V3 )
      = ( inf_inf_set_set_a @ edges @ ( undire2918257014606996450dges_a @ V3 ) ) ) ).

% induced_edges_alt
thf(fact_1230_e__in__all__edges,axiom,
    ! [E: set_a] :
      ( ( member_set_a @ E @ edges )
     => ( member_set_a @ E @ ( undire2918257014606996450dges_a @ vertices ) ) ) ).

% e_in_all_edges
thf(fact_1231_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1232_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1233_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1234_e__in__all__edges__ss,axiom,
    ! [E: set_a,V3: set_a] :
      ( ( member_set_a @ E @ edges )
     => ( ( ord_less_eq_set_a @ E @ V3 )
       => ( ( ord_less_eq_set_a @ V3 @ vertices )
         => ( member_set_a @ E @ ( undire2918257014606996450dges_a @ V3 ) ) ) ) ) ).

% e_in_all_edges_ss
thf(fact_1235_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1236_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1237_is__complete__n__graph__def,axiom,
    ! [N: nat] :
      ( ( undire6087271738840788937raph_a @ vertices @ edges @ N )
      = ( ( ( finite_card_a @ vertices )
          = N )
        & ( edges
          = ( undire2918257014606996450dges_a @ vertices ) ) ) ) ).

% is_complete_n_graph_def
thf(fact_1238_complement__edges__def,axiom,
    ( ( undire4625228487420481630dges_a @ vertices @ edges )
    = ( minus_5736297505244876581_set_a @ ( undire2918257014606996450dges_a @ vertices ) @ edges ) ) ).

% complement_edges_def
thf(fact_1239_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1240_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1241_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1242_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1243_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1244_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1245_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1246_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1247_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1248_is__complement__edges,axiom,
    ! [V3: set_a,E4: set_set_a] :
      ( ( undire8013100667316154652ment_a @ vertices @ edges @ ( produc2116933609460601975_set_a @ V3 @ E4 ) )
      = ( ( vertices = V3 )
        & ( ( undire4625228487420481630dges_a @ vertices @ edges )
          = E4 ) ) ) ).

% is_complement_edges
thf(fact_1249_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M2: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M2 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1250_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N2: set_nat] :
        ? [M4: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N2 )
         => ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1251_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1252_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M4: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M4 @ N3 )
            & ( member_nat @ N3 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1253_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1254_ulgraph__axioms,axiom,
    undire7251896706689453996raph_a @ vertices @ edges ).

% ulgraph_axioms
thf(fact_1255_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1256_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1257_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1258_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1259_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1260_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1261_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_1262_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1263_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1264_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X4: nat] : ( R3 @ X4 @ X4 )
       => ( ! [X4: nat,Y2: nat,Z4: nat] :
              ( ( R3 @ X4 @ Y2 )
             => ( ( R3 @ Y2 @ Z4 )
               => ( R3 @ X4 @ Z4 ) ) )
         => ( ! [N4: nat] : ( R3 @ N4 @ ( suc @ N4 ) )
           => ( R3 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1265_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ M @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1266_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
             => ( P @ M5 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1267_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1268_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1269_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1270_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_1271_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1272_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1273_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD

% Conjectures (1)
thf(conj_0,conjecture,
    irrefl_on_a @ top_top_set_a @ ( graph_8122095853558514513tion_a @ edges ) ).

%------------------------------------------------------------------------------