TPTP Problem File: SLH0889^1.p

%------------------------------------------------------------------------------
% 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    : Knights_Tour/0000_KnightsTour/prob_00672_026219__5797486_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1686 ( 400 unt; 409 typ;   0 def)
%            Number of atoms       : 4350 (1119 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 12882 ( 278   ~;  17   |; 539   &;10040   @)
%                                         (   0 <=>;2008  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   7 avg)
%            Number of types       :   69 (  68 usr)
%            Number of type conns  : 1999 (1999   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  343 ( 341 usr;  40 con; 0-3 aty)
%            Number of variables   : 4301 ( 631   ^;3380   !; 290   ?;4301   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:00:44.686
%------------------------------------------------------------------------------
% Could-be-implicit typings (68)
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thf(sy_c_Wfrec_Osame__fst_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    same_f5472592420709775776nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int ) > set_Pr4708930517165415495nt_int ).

thf(sy_c_Wfrec_Osame__fst_001t__Int__Oint_001t__Int__Oint,type,
    same_fst_int_int: ( int > $o ) > ( int > set_Pr958786334691620121nt_int ) > set_Pr2560585780119916871nt_int ).

thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    same_f2440470920016040620nt_int: ( product_prod_int_int > $o ) > ( product_prod_int_int > set_Pr2560585780119916871nt_int ) > set_Pr5872125604998073543nt_int ).

thf(sy_c_member_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int: int > set_int > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member4957952664650131097nt_int: produc759720530913461378nt_int > set_Pr5025984505184017080nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $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__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member1920110580185997301nt_int: produc6945250483304103390nt_int > set_Pr6123067485828015636nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member4085533954029916580nt_int: produc2501202720802129403nt_int > set_Pr8634505666381077339nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member6582457606847315088nt_int: produc2432570611225516007nt_int > set_Pr4708930517165415495nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    member1729483249812673067nt_int: produc6754623152930779156nt_int > set_Pr53225590989397322nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J,type,
    member5907334269321869775nt_nat: produc1709102135585200056nt_nat > set_Pr2543308735921124078nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member8053542592415931152nt_int: produc340838079399958759nt_int > set_Pr5872125604998073543nt_int > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_n,type,
    n: nat ).

% Relevant facts (1276)
thf(fact_0_transpose__board__correct,axiom,
    ( member5262025264175285858nt_int
    = ( ^ [S_i: product_prod_int_int,B: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ ( transp5086951854761112574nt_int @ S_i ) @ ( transpose_board @ B ) ) ) ) ).

% transpose_board_correct
thf(fact_1_board__exec__correct,axiom,
    board = board_exec ).

% board_exec_correct
thf(fact_2_board__finite,axiom,
    ! [N: nat,M: nat] : ( finite2998713641127702882nt_int @ ( board @ N @ M ) ) ).

% board_finite
thf(fact_3_transpose2,axiom,
    ! [S_i2: product_prod_int_int] :
      ( ( transp5086951854761112574nt_int @ ( transp5086951854761112574nt_int @ S_i2 ) )
      = S_i2 ) ).

% transpose2
thf(fact_4_knights__circuit__exec__simp,axiom,
    ! [N: nat,M: nat,Ps: list_P5707943133018811711nt_int] :
      ( ( knights_circuit @ ( board @ N @ M ) @ Ps )
      = ( circuit_checker @ ( board_exec @ N @ M ) @ Ps ) ) ).

% knights_circuit_exec_simp
thf(fact_5_board__leq__subset,axiom,
    ! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
      ( ( ( ord_less_eq_nat @ N_1 @ N_2 )
        & ( ord_less_eq_nat @ M_1 @ M_2 ) )
     => ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ).

% board_leq_subset
thf(fact_6_card__board,axiom,
    ! [N: nat,M: nat] :
      ( ( finite6756421564338198497nt_int @ ( board @ N @ M ) )
      = ( times_times_nat @ N @ M ) ) ).

% card_board
thf(fact_7_knights__path__exec__simp,axiom,
    ! [N: nat,M: nat,Ps: list_P5707943133018811711nt_int] :
      ( ( knights_path @ ( board @ N @ M ) @ Ps )
      = ( path_checker @ ( board_exec @ N @ M ) @ Ps ) ) ).

% knights_path_exec_simp
thf(fact_8_transpose__board__def,axiom,
    ( transpose_board
    = ( ^ [B: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [I: int,J: int] :
              ( ( Uu
                = ( product_Pair_int_int @ J @ I ) )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ B ) ) ) ) ) ).

% transpose_board_def
thf(fact_9_finite__card__set__comp,axiom,
    ! [M2: set_int,K: int] :
      ( ( finite_finite_int @ M2 )
     => ( ( finite6756421564338198497nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [Uu: product_prod_int_int] :
              ? [J: int] :
                ( ( Uu
                  = ( product_Pair_int_int @ K @ J ) )
                & ( member_int @ J @ M2 ) ) ) )
        = ( finite_card_int @ M2 ) ) ) ).

% finite_card_set_comp
thf(fact_10_finite__card__set__comp,axiom,
    ! [M2: set_Pr958786334691620121nt_int,K: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ M2 )
     => ( ( finite1566866390884520591nt_int
          @ ( collec7447955092554649554nt_int
            @ ^ [Uu: produc1219242969750017639nt_int] :
              ? [J: product_prod_int_int] :
                ( ( Uu
                  = ( produc3646306378393792727nt_int @ K @ J ) )
                & ( member5262025264175285858nt_int @ J @ M2 ) ) ) )
        = ( finite6756421564338198497nt_int @ M2 ) ) ) ).

% finite_card_set_comp
thf(fact_11_finite__card__set__comp,axiom,
    ! [M2: set_Pr958786334691620121nt_int,K: int > option6357759511663192854e_term] :
      ( ( finite2998713641127702882nt_int @ M2 )
     => ( ( finite1470244919013085797nt_int
          @ ( collec506566255779805410nt_int
            @ ^ [Uu: produc7773217078559923341nt_int] :
              ? [J: product_prod_int_int] :
                ( ( Uu
                  = ( produc4305682042979456191nt_int @ K @ J ) )
                & ( member5262025264175285858nt_int @ J @ M2 ) ) ) )
        = ( finite6756421564338198497nt_int @ M2 ) ) ) ).

% finite_card_set_comp
thf(fact_12_finite__card__set__comp,axiom,
    ! [M2: set_Pr958786334691620121nt_int,K: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( finite2998713641127702882nt_int @ M2 )
     => ( ( finite7179617391050183899nt_int
          @ ( collec1790188477890212312nt_int
            @ ^ [Uu: produc2285326912895808259nt_int] :
              ? [J: product_prod_int_int] :
                ( ( Uu
                  = ( produc5700946648718959541nt_int @ K @ J ) )
                & ( member5262025264175285858nt_int @ J @ M2 ) ) ) )
        = ( finite6756421564338198497nt_int @ M2 ) ) ) ).

% finite_card_set_comp
thf(fact_13_path__checker__correct,axiom,
    path_checker = knights_path ).

% path_checker_correct
thf(fact_14_circuit__checker__correct,axiom,
    circuit_checker = knights_circuit ).

% circuit_checker_correct
thf(fact_15_knights__path__board__unique,axiom,
    ! [B_1: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int] :
      ( ( knights_path @ B_1 @ Ps )
     => ( ( knights_path @ B_2 @ Ps )
       => ( B_1 = B_2 ) ) ) ).

% knights_path_board_unique
thf(fact_16_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_17_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: int > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_18_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: nat > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_19_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: int > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_20_finite__Collect__bounded__ex,axiom,
    ! [P: product_prod_int_int > $o,Q: nat > product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: product_prod_int_int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: product_prod_int_int] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_21_finite__Collect__bounded__ex,axiom,
    ! [P: product_prod_int_int > $o,Q: int > product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: product_prod_int_int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: product_prod_int_int] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_22_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: product_prod_int_int > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [X: product_prod_int_int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_23_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: product_prod_int_int > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [X: product_prod_int_int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_24_finite__Collect__bounded__ex,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
              ? [Y: product_prod_int_int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: product_prod_int_int] :
              ( ( P @ Y )
             => ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [X: product_prod_int_int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_25_finite__Collect__bounded__ex,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: nat > produc1219242969750017639nt_int > $o] :
      ( ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: produc1219242969750017639nt_int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: produc1219242969750017639nt_int] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_26_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_27_finite__Collect__subsets,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B2: set_int] : ( ord_less_eq_set_int @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_28_finite__Collect__subsets,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [B2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_29_card__mono,axiom,
    ! [B3: set_Product_unit,A: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ( ord_le3507040750410214029t_unit @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ).

% card_mono
thf(fact_30_card__mono,axiom,
    ! [B3: set_Pr9222295170931077689nt_int,A: set_Pr9222295170931077689nt_int] :
      ( ( finite1225166462400338330nt_int @ B3 )
     => ( ( ord_le8725513860283290265nt_int @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite7179617391050183899nt_int @ A ) @ ( finite7179617391050183899nt_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_31_card__mono,axiom,
    ! [B3: set_Pr1872883991513573699nt_int,A: set_Pr1872883991513573699nt_int] :
      ( ( finite1606273004500963236nt_int @ B3 )
     => ( ( ord_le135402666524580259nt_int @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite1470244919013085797nt_int @ A ) @ ( finite1470244919013085797nt_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_32_card__mono,axiom,
    ! [B3: set_Pr2560585780119916871nt_int,A: set_Pr2560585780119916871nt_int] :
      ( ( finite4752125741994067600nt_int @ B3 )
     => ( ( ord_le6090609446090860775nt_int @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ A ) @ ( finite1566866390884520591nt_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_33_card__mono,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_34_card__mono,axiom,
    ! [B3: set_int,A: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_35_card__mono,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
       => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite6756421564338198497nt_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_36_card__seteq,axiom,
    ! [B3: set_Product_unit,A: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ( ord_le3507040750410214029t_unit @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B3 ) @ ( finite410649719033368117t_unit @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_37_card__seteq,axiom,
    ! [B3: set_Pr9222295170931077689nt_int,A: set_Pr9222295170931077689nt_int] :
      ( ( finite1225166462400338330nt_int @ B3 )
     => ( ( ord_le8725513860283290265nt_int @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite7179617391050183899nt_int @ B3 ) @ ( finite7179617391050183899nt_int @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_38_card__seteq,axiom,
    ! [B3: set_Pr1872883991513573699nt_int,A: set_Pr1872883991513573699nt_int] :
      ( ( finite1606273004500963236nt_int @ B3 )
     => ( ( ord_le135402666524580259nt_int @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite1470244919013085797nt_int @ B3 ) @ ( finite1470244919013085797nt_int @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_39_card__seteq,axiom,
    ! [B3: set_Pr2560585780119916871nt_int,A: set_Pr2560585780119916871nt_int] :
      ( ( finite4752125741994067600nt_int @ B3 )
     => ( ( ord_le6090609446090860775nt_int @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ B3 ) @ ( finite1566866390884520591nt_int @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_40_card__seteq,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_41_card__seteq,axiom,
    ! [B3: set_int,A: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_int @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_42_card__seteq,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
       => ( ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ B3 ) @ ( finite6756421564338198497nt_int @ A ) )
         => ( A = B3 ) ) ) ) ).

% card_seteq
thf(fact_43_exists__subset__between,axiom,
    ! [A: set_Product_unit,N: nat,C: set_Product_unit] :
      ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ C ) )
       => ( ( ord_le3507040750410214029t_unit @ A @ C )
         => ( ( finite4290736615968046902t_unit @ C )
           => ? [B4: set_Product_unit] :
                ( ( ord_le3507040750410214029t_unit @ A @ B4 )
                & ( ord_le3507040750410214029t_unit @ B4 @ C )
                & ( ( finite410649719033368117t_unit @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_44_exists__subset__between,axiom,
    ! [A: set_Pr9222295170931077689nt_int,N: nat,C: set_Pr9222295170931077689nt_int] :
      ( ( ord_less_eq_nat @ ( finite7179617391050183899nt_int @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite7179617391050183899nt_int @ C ) )
       => ( ( ord_le8725513860283290265nt_int @ A @ C )
         => ( ( finite1225166462400338330nt_int @ C )
           => ? [B4: set_Pr9222295170931077689nt_int] :
                ( ( ord_le8725513860283290265nt_int @ A @ B4 )
                & ( ord_le8725513860283290265nt_int @ B4 @ C )
                & ( ( finite7179617391050183899nt_int @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_45_exists__subset__between,axiom,
    ! [A: set_Pr1872883991513573699nt_int,N: nat,C: set_Pr1872883991513573699nt_int] :
      ( ( ord_less_eq_nat @ ( finite1470244919013085797nt_int @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite1470244919013085797nt_int @ C ) )
       => ( ( ord_le135402666524580259nt_int @ A @ C )
         => ( ( finite1606273004500963236nt_int @ C )
           => ? [B4: set_Pr1872883991513573699nt_int] :
                ( ( ord_le135402666524580259nt_int @ A @ B4 )
                & ( ord_le135402666524580259nt_int @ B4 @ C )
                & ( ( finite1470244919013085797nt_int @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_46_exists__subset__between,axiom,
    ! [A: set_Pr2560585780119916871nt_int,N: nat,C: set_Pr2560585780119916871nt_int] :
      ( ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite1566866390884520591nt_int @ C ) )
       => ( ( ord_le6090609446090860775nt_int @ A @ C )
         => ( ( finite4752125741994067600nt_int @ C )
           => ? [B4: set_Pr2560585780119916871nt_int] :
                ( ( ord_le6090609446090860775nt_int @ A @ B4 )
                & ( ord_le6090609446090860775nt_int @ B4 @ C )
                & ( ( finite1566866390884520591nt_int @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_47_exists__subset__between,axiom,
    ! [A: set_nat,N: nat,C: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
       => ( ( ord_less_eq_set_nat @ A @ C )
         => ( ( finite_finite_nat @ C )
           => ? [B4: set_nat] :
                ( ( ord_less_eq_set_nat @ A @ B4 )
                & ( ord_less_eq_set_nat @ B4 @ C )
                & ( ( finite_card_nat @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_48_exists__subset__between,axiom,
    ! [A: set_int,N: nat,C: set_int] :
      ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C ) )
       => ( ( ord_less_eq_set_int @ A @ C )
         => ( ( finite_finite_int @ C )
           => ? [B4: set_int] :
                ( ( ord_less_eq_set_int @ A @ B4 )
                & ( ord_less_eq_set_int @ B4 @ C )
                & ( ( finite_card_int @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_49_exists__subset__between,axiom,
    ! [A: set_Pr958786334691620121nt_int,N: nat,C: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite6756421564338198497nt_int @ C ) )
       => ( ( ord_le2843351958646193337nt_int @ A @ C )
         => ( ( finite2998713641127702882nt_int @ C )
           => ? [B4: set_Pr958786334691620121nt_int] :
                ( ( ord_le2843351958646193337nt_int @ A @ B4 )
                & ( ord_le2843351958646193337nt_int @ B4 @ C )
                & ( ( finite6756421564338198497nt_int @ B4 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_50_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Product_unit] :
      ( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ S ) )
     => ~ ! [T: set_Product_unit] :
            ( ( ord_le3507040750410214029t_unit @ T @ S )
           => ( ( ( finite410649719033368117t_unit @ T )
                = N )
             => ~ ( finite4290736615968046902t_unit @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_51_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr9222295170931077689nt_int] :
      ( ( ord_less_eq_nat @ N @ ( finite7179617391050183899nt_int @ S ) )
     => ~ ! [T: set_Pr9222295170931077689nt_int] :
            ( ( ord_le8725513860283290265nt_int @ T @ S )
           => ( ( ( finite7179617391050183899nt_int @ T )
                = N )
             => ~ ( finite1225166462400338330nt_int @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_52_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr1872883991513573699nt_int] :
      ( ( ord_less_eq_nat @ N @ ( finite1470244919013085797nt_int @ S ) )
     => ~ ! [T: set_Pr1872883991513573699nt_int] :
            ( ( ord_le135402666524580259nt_int @ T @ S )
           => ( ( ( finite1470244919013085797nt_int @ T )
                = N )
             => ~ ( finite1606273004500963236nt_int @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_53_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr2560585780119916871nt_int] :
      ( ( ord_less_eq_nat @ N @ ( finite1566866390884520591nt_int @ S ) )
     => ~ ! [T: set_Pr2560585780119916871nt_int] :
            ( ( ord_le6090609446090860775nt_int @ T @ S )
           => ( ( ( finite1566866390884520591nt_int @ T )
                = N )
             => ~ ( finite4752125741994067600nt_int @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_54_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
     => ~ ! [T: set_nat] :
            ( ( ord_less_eq_set_nat @ T @ S )
           => ( ( ( finite_card_nat @ T )
                = N )
             => ~ ( finite_finite_nat @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_55_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_int] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
     => ~ ! [T: set_int] :
            ( ( ord_less_eq_set_int @ T @ S )
           => ( ( ( finite_card_int @ T )
                = N )
             => ~ ( finite_finite_int @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_56_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ N @ ( finite6756421564338198497nt_int @ S ) )
     => ~ ! [T: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ T @ S )
           => ( ( ( finite6756421564338198497nt_int @ T )
                = N )
             => ~ ( finite2998713641127702882nt_int @ T ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_57_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_Product_unit,C: nat] :
      ( ! [G: set_Product_unit] :
          ( ( ord_le3507040750410214029t_unit @ G @ F )
         => ( ( finite4290736615968046902t_unit @ G )
           => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G ) @ C ) ) )
     => ( ( finite4290736615968046902t_unit @ F )
        & ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_58_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_Pr9222295170931077689nt_int,C: nat] :
      ( ! [G: set_Pr9222295170931077689nt_int] :
          ( ( ord_le8725513860283290265nt_int @ G @ F )
         => ( ( finite1225166462400338330nt_int @ G )
           => ( ord_less_eq_nat @ ( finite7179617391050183899nt_int @ G ) @ C ) ) )
     => ( ( finite1225166462400338330nt_int @ F )
        & ( ord_less_eq_nat @ ( finite7179617391050183899nt_int @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_59_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_Pr1872883991513573699nt_int,C: nat] :
      ( ! [G: set_Pr1872883991513573699nt_int] :
          ( ( ord_le135402666524580259nt_int @ G @ F )
         => ( ( finite1606273004500963236nt_int @ G )
           => ( ord_less_eq_nat @ ( finite1470244919013085797nt_int @ G ) @ C ) ) )
     => ( ( finite1606273004500963236nt_int @ F )
        & ( ord_less_eq_nat @ ( finite1470244919013085797nt_int @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_60_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_Pr2560585780119916871nt_int,C: nat] :
      ( ! [G: set_Pr2560585780119916871nt_int] :
          ( ( ord_le6090609446090860775nt_int @ G @ F )
         => ( ( finite4752125741994067600nt_int @ G )
           => ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ G ) @ C ) ) )
     => ( ( finite4752125741994067600nt_int @ F )
        & ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_61_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_nat,C: nat] :
      ( ! [G: set_nat] :
          ( ( ord_less_eq_set_nat @ G @ F )
         => ( ( finite_finite_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
     => ( ( finite_finite_nat @ F )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_62_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_int,C: nat] :
      ( ! [G: set_int] :
          ( ( ord_less_eq_set_int @ G @ F )
         => ( ( finite_finite_int @ G )
           => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C ) ) )
     => ( ( finite_finite_int @ F )
        & ( ord_less_eq_nat @ ( finite_card_int @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_63_finite__if__finite__subsets__card__bdd,axiom,
    ! [F: set_Pr958786334691620121nt_int,C: nat] :
      ( ! [G: set_Pr958786334691620121nt_int] :
          ( ( ord_le2843351958646193337nt_int @ G @ F )
         => ( ( finite2998713641127702882nt_int @ G )
           => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ G ) @ C ) ) )
     => ( ( finite2998713641127702882nt_int @ F )
        & ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ F ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_64_finite__Collect__conjI,axiom,
    ! [P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ( ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ P ) )
        | ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ Q ) ) )
     => ( finite1225166462400338330nt_int
        @ ( collec1790188477890212312nt_int
          @ ^ [X: produc2285326912895808259nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_65_finite__Collect__conjI,axiom,
    ! [P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ( ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ P ) )
        | ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ Q ) ) )
     => ( finite1606273004500963236nt_int
        @ ( collec506566255779805410nt_int
          @ ^ [X: produc7773217078559923341nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_66_finite__Collect__conjI,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ( ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
        | ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ Q ) ) )
     => ( finite4752125741994067600nt_int
        @ ( collec7447955092554649554nt_int
          @ ^ [X: produc1219242969750017639nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_67_finite__Collect__conjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_68_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_69_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_70_finite__Collect__disjI,axiom,
    ! [P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ( finite1225166462400338330nt_int
        @ ( collec1790188477890212312nt_int
          @ ^ [X: produc2285326912895808259nt_int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ P ) )
        & ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_71_finite__Collect__disjI,axiom,
    ! [P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ( finite1606273004500963236nt_int
        @ ( collec506566255779805410nt_int
          @ ^ [X: produc7773217078559923341nt_int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ P ) )
        & ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_72_finite__Collect__disjI,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ( finite4752125741994067600nt_int
        @ ( collec7447955092554649554nt_int
          @ ^ [X: produc1219242969750017639nt_int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
        & ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_73_finite__Collect__disjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_74_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_75_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_76_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Product_unit,A: set_nat,R: nat > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ? [B5: product_unit] :
                ( ( member_Product_unit @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: nat,A22: nat,B6: product_unit] :
              ( ( member_nat @ A1 @ A )
             => ( ( member_nat @ A22 @ A )
               => ( ( member_Product_unit @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_77_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Product_unit,A: set_int,R: int > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ? [B5: product_unit] :
                ( ( member_Product_unit @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: int,A22: int,B6: product_unit] :
              ( ( member_int @ A1 @ A )
             => ( ( member_int @ A22 @ A )
               => ( ( member_Product_unit @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_78_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Product_unit,A: set_Product_unit,R: product_unit > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ! [A2: product_unit] :
            ( ( member_Product_unit @ A2 @ A )
           => ? [B5: product_unit] :
                ( ( member_Product_unit @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: product_unit,A22: product_unit,B6: product_unit] :
              ( ( member_Product_unit @ A1 @ A )
             => ( ( member_Product_unit @ A22 @ A )
               => ( ( member_Product_unit @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_79_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: nat,A22: nat,B6: nat] :
              ( ( member_nat @ A1 @ A )
             => ( ( member_nat @ A22 @ A )
               => ( ( member_nat @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_80_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A: set_int,R: int > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: int,A22: int,B6: nat] :
              ( ( member_int @ A1 @ A )
             => ( ( member_int @ A22 @ A )
               => ( ( member_nat @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_81_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A: set_Product_unit,R: product_unit > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A2: product_unit] :
            ( ( member_Product_unit @ A2 @ A )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: product_unit,A22: product_unit,B6: nat] :
              ( ( member_Product_unit @ A1 @ A )
             => ( ( member_Product_unit @ A22 @ A )
               => ( ( member_nat @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_82_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A: set_nat,R: nat > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ? [B5: int] :
                ( ( member_int @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: nat,A22: nat,B6: int] :
              ( ( member_nat @ A1 @ A )
             => ( ( member_nat @ A22 @ A )
               => ( ( member_int @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_83_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ? [B5: int] :
                ( ( member_int @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: int,A22: int,B6: int] :
              ( ( member_int @ A1 @ A )
             => ( ( member_int @ A22 @ A )
               => ( ( member_int @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_84_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A: set_Product_unit,R: product_unit > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A2: product_unit] :
            ( ( member_Product_unit @ A2 @ A )
           => ? [B5: int] :
                ( ( member_int @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: product_unit,A22: product_unit,B6: int] :
              ( ( member_Product_unit @ A1 @ A )
             => ( ( member_Product_unit @ A22 @ A )
               => ( ( member_int @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_85_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Product_unit,A: set_Pr958786334691620121nt_int,R: product_prod_int_int > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ! [A2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ A2 @ A )
           => ? [B5: product_unit] :
                ( ( member_Product_unit @ B5 @ B3 )
                & ( R @ A2 @ B5 ) ) )
       => ( ! [A1: product_prod_int_int,A22: product_prod_int_int,B6: product_unit] :
              ( ( member5262025264175285858nt_int @ A1 @ A )
             => ( ( member5262025264175285858nt_int @ A22 @ A )
               => ( ( member_Product_unit @ B6 @ B3 )
                 => ( ( R @ A1 @ B6 )
                   => ( ( R @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_86_card__subset__eq,axiom,
    ! [B3: set_Product_unit,A: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ B3 )
     => ( ( ord_le3507040750410214029t_unit @ A @ B3 )
       => ( ( ( finite410649719033368117t_unit @ A )
            = ( finite410649719033368117t_unit @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_87_card__subset__eq,axiom,
    ! [B3: set_Pr9222295170931077689nt_int,A: set_Pr9222295170931077689nt_int] :
      ( ( finite1225166462400338330nt_int @ B3 )
     => ( ( ord_le8725513860283290265nt_int @ A @ B3 )
       => ( ( ( finite7179617391050183899nt_int @ A )
            = ( finite7179617391050183899nt_int @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_88_card__subset__eq,axiom,
    ! [B3: set_Pr1872883991513573699nt_int,A: set_Pr1872883991513573699nt_int] :
      ( ( finite1606273004500963236nt_int @ B3 )
     => ( ( ord_le135402666524580259nt_int @ A @ B3 )
       => ( ( ( finite1470244919013085797nt_int @ A )
            = ( finite1470244919013085797nt_int @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_89_card__subset__eq,axiom,
    ! [B3: set_Pr2560585780119916871nt_int,A: set_Pr2560585780119916871nt_int] :
      ( ( finite4752125741994067600nt_int @ B3 )
     => ( ( ord_le6090609446090860775nt_int @ A @ B3 )
       => ( ( ( finite1566866390884520591nt_int @ A )
            = ( finite1566866390884520591nt_int @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_90_card__subset__eq,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_91_card__subset__eq,axiom,
    ! [B3: set_int,A: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A @ B3 )
       => ( ( ( finite_card_int @ A )
            = ( finite_card_int @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_92_card__subset__eq,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
       => ( ( ( finite6756421564338198497nt_int @ A )
            = ( finite6756421564338198497nt_int @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_93_infinite__arbitrarily__large,axiom,
    ! [A: set_Product_unit,N: nat] :
      ( ~ ( finite4290736615968046902t_unit @ A )
     => ? [B4: set_Product_unit] :
          ( ( finite4290736615968046902t_unit @ B4 )
          & ( ( finite410649719033368117t_unit @ B4 )
            = N )
          & ( ord_le3507040750410214029t_unit @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_94_infinite__arbitrarily__large,axiom,
    ! [A: set_Pr9222295170931077689nt_int,N: nat] :
      ( ~ ( finite1225166462400338330nt_int @ A )
     => ? [B4: set_Pr9222295170931077689nt_int] :
          ( ( finite1225166462400338330nt_int @ B4 )
          & ( ( finite7179617391050183899nt_int @ B4 )
            = N )
          & ( ord_le8725513860283290265nt_int @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_95_infinite__arbitrarily__large,axiom,
    ! [A: set_Pr1872883991513573699nt_int,N: nat] :
      ( ~ ( finite1606273004500963236nt_int @ A )
     => ? [B4: set_Pr1872883991513573699nt_int] :
          ( ( finite1606273004500963236nt_int @ B4 )
          & ( ( finite1470244919013085797nt_int @ B4 )
            = N )
          & ( ord_le135402666524580259nt_int @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_96_infinite__arbitrarily__large,axiom,
    ! [A: set_Pr2560585780119916871nt_int,N: nat] :
      ( ~ ( finite4752125741994067600nt_int @ A )
     => ? [B4: set_Pr2560585780119916871nt_int] :
          ( ( finite4752125741994067600nt_int @ B4 )
          & ( ( finite1566866390884520591nt_int @ B4 )
            = N )
          & ( ord_le6090609446090860775nt_int @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_97_infinite__arbitrarily__large,axiom,
    ! [A: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [B4: set_nat] :
          ( ( finite_finite_nat @ B4 )
          & ( ( finite_card_nat @ B4 )
            = N )
          & ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_98_infinite__arbitrarily__large,axiom,
    ! [A: set_int,N: nat] :
      ( ~ ( finite_finite_int @ A )
     => ? [B4: set_int] :
          ( ( finite_finite_int @ B4 )
          & ( ( finite_card_int @ B4 )
            = N )
          & ( ord_less_eq_set_int @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_99_infinite__arbitrarily__large,axiom,
    ! [A: set_Pr958786334691620121nt_int,N: nat] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ? [B4: set_Pr958786334691620121nt_int] :
          ( ( finite2998713641127702882nt_int @ B4 )
          & ( ( finite6756421564338198497nt_int @ B4 )
            = N )
          & ( ord_le2843351958646193337nt_int @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_100_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_101_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N3: set_nat] :
        ? [M3: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N3 )
         => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_102_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F2 @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_103_step__checker_Ocases,axiom,
    ! [X2: produc1219242969750017639nt_int] :
      ~ ! [I2: int,J2: int,I3: int,J3: int] :
          ( X2
         != ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I2 @ J2 ) @ ( product_Pair_int_int @ I3 @ J3 ) ) ) ).

% step_checker.cases
thf(fact_104_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M2: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M2 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_105_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B3: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_106_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B3: set_int,R2: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_107_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B3: set_nat,R2: int > nat > $o] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_108_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B3: set_int,R2: int > int > $o] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_109_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_nat,R2: product_prod_int_int > nat > $o] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite2998713641127702882nt_int
                  @ ( collec213857154873943460nt_int
                    @ ^ [A3: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_110_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int,R2: product_prod_int_int > int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ A )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite2998713641127702882nt_int
                  @ ( collec213857154873943460nt_int
                    @ ^ [A3: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_111_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B3: set_Pr958786334691620121nt_int,R2: nat > product_prod_int_int > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa: product_prod_int_int] :
                  ( ( member5262025264175285858nt_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_112_pigeonhole__infinite__rel,axiom,
    ! [A: set_int,B3: set_Pr958786334691620121nt_int,R2: int > product_prod_int_int > $o] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A )
             => ? [Xa: product_prod_int_int] :
                  ( ( member5262025264175285858nt_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_113_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,R2: product_prod_int_int > product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( ! [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ A )
             => ? [Xa: product_prod_int_int] :
                  ( ( member5262025264175285858nt_int @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ B3 )
              & ~ ( finite2998713641127702882nt_int
                  @ ( collec213857154873943460nt_int
                    @ ^ [A3: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_114_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr2560585780119916871nt_int,B3: set_nat,R2: produc1219242969750017639nt_int > nat > $o] :
      ( ~ ( finite4752125741994067600nt_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: produc1219242969750017639nt_int] :
              ( ( member8566619992076573584nt_int @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite4752125741994067600nt_int
                  @ ( collec7447955092554649554nt_int
                    @ ^ [A3: produc1219242969750017639nt_int] :
                        ( ( member8566619992076573584nt_int @ A3 @ A )
                        & ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_115_not__finite__existsD,axiom,
    ! [P: produc2285326912895808259nt_int > $o] :
      ( ~ ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ P ) )
     => ? [X_1: produc2285326912895808259nt_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_116_not__finite__existsD,axiom,
    ! [P: produc7773217078559923341nt_int > $o] :
      ( ~ ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ P ) )
     => ? [X_1: produc7773217078559923341nt_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_117_not__finite__existsD,axiom,
    ! [P: produc1219242969750017639nt_int > $o] :
      ( ~ ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
     => ? [X_1: produc1219242969750017639nt_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_118_not__finite__existsD,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_119_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_120_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_121_finite__has__minimal2,axiom,
    ! [A: set_nat,A4: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A4 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ X3 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_122_finite__has__minimal2,axiom,
    ! [A: set_se6260736226359567993nt_int,A4: set_Pr958786334691620121nt_int] :
      ( ( finite8744585540193469122nt_int @ A )
     => ( ( member2340774599025711042nt_int @ A4 @ A )
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A )
            & ( ord_le2843351958646193337nt_int @ X3 @ A4 )
            & ! [Xa: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ Xa @ A )
               => ( ( ord_le2843351958646193337nt_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_123_finite__has__minimal2,axiom,
    ! [A: set_int,A4: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ A4 @ A )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ( ord_less_eq_int @ X3 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_124_finite__has__maximal2,axiom,
    ! [A: set_nat,A4: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A4 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ A4 @ X3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_125_finite__has__maximal2,axiom,
    ! [A: set_se6260736226359567993nt_int,A4: set_Pr958786334691620121nt_int] :
      ( ( finite8744585540193469122nt_int @ A )
     => ( ( member2340774599025711042nt_int @ A4 @ A )
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A )
            & ( ord_le2843351958646193337nt_int @ A4 @ X3 )
            & ! [Xa: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ Xa @ A )
               => ( ( ord_le2843351958646193337nt_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_126_finite__has__maximal2,axiom,
    ! [A: set_int,A4: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ A4 @ A )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ( ord_less_eq_int @ A4 @ X3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_127_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_128_rev__finite__subset,axiom,
    ! [B3: set_int,A: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A @ B3 )
       => ( finite_finite_int @ A ) ) ) ).

% rev_finite_subset
thf(fact_129_rev__finite__subset,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
       => ( finite2998713641127702882nt_int @ A ) ) ) ).

% rev_finite_subset
thf(fact_130_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_131_infinite__super,axiom,
    ! [S: set_int,T2: set_int] :
      ( ( ord_less_eq_set_int @ S @ T2 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ T2 ) ) ) ).

% infinite_super
thf(fact_132_infinite__super,axiom,
    ! [S: set_Pr958786334691620121nt_int,T2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ S @ T2 )
     => ( ~ ( finite2998713641127702882nt_int @ S )
       => ~ ( finite2998713641127702882nt_int @ T2 ) ) ) ).

% infinite_super
thf(fact_133_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_134_finite__subset,axiom,
    ! [A: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A @ B3 )
     => ( ( finite_finite_int @ B3 )
       => ( finite_finite_int @ A ) ) ) ).

% finite_subset
thf(fact_135_finite__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_subset
thf(fact_136_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F2: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_137_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F2: nat > nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu: int] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_138_finite__image__set2,axiom,
    ! [P: nat > $o,Q: int > $o,F2: nat > int > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: int] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_139_finite__image__set2,axiom,
    ! [P: nat > $o,Q: int > $o,F2: nat > int > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu: int] :
              ? [X: nat,Y: int] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_140_finite__image__set2,axiom,
    ! [P: int > $o,Q: nat > $o,F2: int > nat > nat] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: int,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_141_finite__image__set2,axiom,
    ! [P: int > $o,Q: nat > $o,F2: int > nat > int] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu: int] :
              ? [X: int,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_142_finite__image__set2,axiom,
    ! [P: int > $o,Q: int > $o,F2: int > int > nat] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: int,Y: int] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_143_finite__image__set2,axiom,
    ! [P: int > $o,Q: int > $o,F2: int > int > int] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu: int] :
              ? [X: int,Y: int] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_144_finite__image__set2,axiom,
    ! [P: product_prod_int_int > $o,Q: nat > $o,F2: product_prod_int_int > nat > nat] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: product_prod_int_int,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_145_finite__image__set2,axiom,
    ! [P: product_prod_int_int > $o,Q: nat > $o,F2: product_prod_int_int > nat > int] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu: int] :
              ? [X: product_prod_int_int,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_146_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_147_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu: int] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_148_finite__image__set,axiom,
    ! [P: int > $o,F2: int > nat] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_149_finite__image__set,axiom,
    ! [P: int > $o,F2: int > int] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu: int] :
            ? [X: int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_150_finite__image__set,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: product_prod_int_int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_151_finite__image__set,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu: int] :
            ? [X: product_prod_int_int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_152_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > product_prod_int_int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_153_finite__image__set,axiom,
    ! [P: int > $o,F2: int > product_prod_int_int] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [X: int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_154_finite__image__set,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [X: product_prod_int_int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_155_finite__image__set,axiom,
    ! [P: produc1219242969750017639nt_int > $o,F2: produc1219242969750017639nt_int > nat] :
      ( ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: produc1219242969750017639nt_int] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_156_subset__antisym,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_157_subsetI,axiom,
    ! [A: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member_int @ X3 @ B3 ) )
     => ( ord_less_eq_set_int @ A @ B3 ) ) ).

% subsetI
thf(fact_158_subsetI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B3 ) )
     => ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% subsetI
thf(fact_159_subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A )
         => ( member5262025264175285858nt_int @ X3 @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ).

% subsetI
thf(fact_160_mem__Collect__eq,axiom,
    ! [A4: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A4 @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
    ! [A4: nat,P: nat > $o] :
      ( ( member_nat @ A4 @ ( collect_nat @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
    ! [A4: int,P: int > $o] :
      ( ( member_int @ A4 @ ( collect_int @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
    ! [A4: produc2285326912895808259nt_int,P: produc2285326912895808259nt_int > $o] :
      ( ( member7618704894036264090nt_int @ A4 @ ( collec1790188477890212312nt_int @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
    ! [A4: produc7773217078559923341nt_int,P: produc7773217078559923341nt_int > $o] :
      ( ( member7034335876925520548nt_int @ A4 @ ( collec506566255779805410nt_int @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
    ! [A4: produc1219242969750017639nt_int,P: produc1219242969750017639nt_int > $o] :
      ( ( member8566619992076573584nt_int @ A4 @ ( collec7447955092554649554nt_int @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_166_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
    ! [A: set_Pr9222295170931077689nt_int] :
      ( ( collec1790188477890212312nt_int
        @ ^ [X: produc2285326912895808259nt_int] : ( member7618704894036264090nt_int @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
    ! [A: set_Pr1872883991513573699nt_int] :
      ( ( collec506566255779805410nt_int
        @ ^ [X: produc7773217078559923341nt_int] : ( member7034335876925520548nt_int @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
    ! [A: set_Pr2560585780119916871nt_int] :
      ( ( collec7447955092554649554nt_int
        @ ^ [X: produc1219242969750017639nt_int] : ( member8566619992076573584nt_int @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_172_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_173_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_174_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_175_Collect__cong,axiom,
    ! [P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ! [X3: produc2285326912895808259nt_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec1790188477890212312nt_int @ P )
        = ( collec1790188477890212312nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_176_Collect__cong,axiom,
    ! [P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ! [X3: produc7773217078559923341nt_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec506566255779805410nt_int @ P )
        = ( collec506566255779805410nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_177_Collect__cong,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ! [X3: produc1219242969750017639nt_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec7447955092554649554nt_int @ P )
        = ( collec7447955092554649554nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_178_old_Oprod_Oinject,axiom,
    ! [A4: int,B7: int,A5: int,B8: int] :
      ( ( ( product_Pair_int_int @ A4 @ B7 )
        = ( product_Pair_int_int @ A5 @ B8 ) )
      = ( ( A4 = A5 )
        & ( B7 = B8 ) ) ) ).

% old.prod.inject
thf(fact_179_old_Oprod_Oinject,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A5: product_prod_int_int,B8: product_prod_int_int] :
      ( ( ( produc3646306378393792727nt_int @ A4 @ B7 )
        = ( produc3646306378393792727nt_int @ A5 @ B8 ) )
      = ( ( A4 = A5 )
        & ( B7 = B8 ) ) ) ).

% old.prod.inject
thf(fact_180_old_Oprod_Oinject,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A5: int > option6357759511663192854e_term,B8: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A4 @ B7 )
        = ( produc4305682042979456191nt_int @ A5 @ B8 ) )
      = ( ( A4 = A5 )
        & ( B7 = B8 ) ) ) ).

% old.prod.inject
thf(fact_181_old_Oprod_Oinject,axiom,
    ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B8: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A4 @ B7 )
        = ( produc5700946648718959541nt_int @ A5 @ B8 ) )
      = ( ( A4 = A5 )
        & ( B7 = B8 ) ) ) ).

% old.prod.inject
thf(fact_182_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X22 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_183_prod_Oinject,axiom,
    ! [X1: product_prod_int_int,X22: product_prod_int_int,Y1: product_prod_int_int,Y2: product_prod_int_int] :
      ( ( ( produc3646306378393792727nt_int @ X1 @ X22 )
        = ( produc3646306378393792727nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_184_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_185_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_186_dual__order_Orefl,axiom,
    ! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).

% dual_order.refl
thf(fact_187_dual__order_Orefl,axiom,
    ! [A4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A4 @ A4 ) ).

% dual_order.refl
thf(fact_188_dual__order_Orefl,axiom,
    ! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).

% dual_order.refl
thf(fact_189_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_190_order__refl,axiom,
    ! [X2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X2 @ X2 ) ).

% order_refl
thf(fact_191_order__refl,axiom,
    ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).

% order_refl
thf(fact_192_subrelI,axiom,
    ! [R: set_Pr2560585780119916871nt_int,S2: set_Pr2560585780119916871nt_int] :
      ( ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
          ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
         => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le6090609446090860775nt_int @ R @ S2 ) ) ).

% subrelI
thf(fact_193_subrelI,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le135402666524580259nt_int @ R @ S2 ) ) ).

% subrelI
thf(fact_194_subrelI,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le8725513860283290265nt_int @ R @ S2 ) ) ).

% subrelI
thf(fact_195_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).

% subrelI
thf(fact_196_mult__le__mono2,axiom,
    ! [I4: nat,J4: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J4 ) ) ) ).

% mult_le_mono2
thf(fact_197_mult__le__mono1,axiom,
    ! [I4: nat,J4: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J4 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J4 @ K ) ) ) ).

% mult_le_mono1
thf(fact_198_mult__le__mono,axiom,
    ! [I4: nat,J4: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I4 @ J4 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J4 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_199_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_200_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B2: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_201_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B2: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_202_less__eq__set__def,axiom,
    ( ord_le2843351958646193337nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
          ( ord_le8369615600986905444_int_o
          @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A6 )
          @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_203_pred__subset__eq,axiom,
    ! [R2: set_int,S: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R2 )
        @ ^ [X: int] : ( member_int @ X @ S ) )
      = ( ord_less_eq_set_int @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_204_pred__subset__eq,axiom,
    ! [R2: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_205_pred__subset__eq,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ord_le8369615600986905444_int_o
        @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ R2 )
        @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ S ) )
      = ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_206_pred__subset__eq2,axiom,
    ! [R2: set_Pr2560585780119916871nt_int,S: set_Pr2560585780119916871nt_int] :
      ( ( ord_le1598226405681992910_int_o
        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R2 )
        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ S ) )
      = ( ord_le6090609446090860775nt_int @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_207_pred__subset__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 )
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S ) )
      = ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_208_pred__subset__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 )
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S ) )
      = ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_209_pred__subset__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) )
      = ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_210_nle__le,axiom,
    ! [A4: nat,B7: nat] :
      ( ( ~ ( ord_less_eq_nat @ A4 @ B7 ) )
      = ( ( ord_less_eq_nat @ B7 @ A4 )
        & ( B7 != A4 ) ) ) ).

% nle_le
thf(fact_211_nle__le,axiom,
    ! [A4: int,B7: int] :
      ( ( ~ ( ord_less_eq_int @ A4 @ B7 ) )
      = ( ( ord_less_eq_int @ B7 @ A4 )
        & ( B7 != A4 ) ) ) ).

% nle_le
thf(fact_212_le__cases3,axiom,
    ! [X2: nat,Y4: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y4 )
       => ~ ( ord_less_eq_nat @ Y4 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y4 @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y4 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y4 )
             => ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y4 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_213_le__cases3,axiom,
    ! [X2: int,Y4: int,Z: int] :
      ( ( ( ord_less_eq_int @ X2 @ Y4 )
       => ~ ( ord_less_eq_int @ Y4 @ Z ) )
     => ( ( ( ord_less_eq_int @ Y4 @ X2 )
         => ~ ( ord_less_eq_int @ X2 @ Z ) )
       => ( ( ( ord_less_eq_int @ X2 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y4 ) )
         => ( ( ( ord_less_eq_int @ Z @ Y4 )
             => ~ ( ord_less_eq_int @ Y4 @ X2 ) )
           => ( ( ( ord_less_eq_int @ Y4 @ Z )
               => ~ ( ord_less_eq_int @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X2 )
                 => ~ ( ord_less_eq_int @ X2 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_214_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_215_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
    = ( ^ [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
          ( ( ord_le2843351958646193337nt_int @ X @ Y )
          & ( ord_le2843351958646193337nt_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_216_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_217_ord__eq__le__trans,axiom,
    ! [A4: nat,B7: nat,C2: nat] :
      ( ( A4 = B7 )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_218_ord__eq__le__trans,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( A4 = B7 )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ord_le2843351958646193337nt_int @ A4 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_219_ord__eq__le__trans,axiom,
    ! [A4: int,B7: int,C2: int] :
      ( ( A4 = B7 )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ord_less_eq_int @ A4 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_220_ord__le__eq__trans,axiom,
    ! [A4: nat,B7: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( B7 = C2 )
       => ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_221_ord__le__eq__trans,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( B7 = C2 )
       => ( ord_le2843351958646193337nt_int @ A4 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_222_ord__le__eq__trans,axiom,
    ! [A4: int,B7: int,C2: int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( B7 = C2 )
       => ( ord_less_eq_int @ A4 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_223_order__antisym,axiom,
    ! [X2: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ X2 )
       => ( X2 = Y4 ) ) ) ).

% order_antisym
thf(fact_224_order__antisym,axiom,
    ! [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
     => ( ( ord_le2843351958646193337nt_int @ Y4 @ X2 )
       => ( X2 = Y4 ) ) ) ).

% order_antisym
thf(fact_225_order__antisym,axiom,
    ! [X2: int,Y4: int] :
      ( ( ord_less_eq_int @ X2 @ Y4 )
     => ( ( ord_less_eq_int @ Y4 @ X2 )
       => ( X2 = Y4 ) ) ) ).

% order_antisym
thf(fact_226_order_Otrans,axiom,
    ! [A4: nat,B7: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).

% order.trans
thf(fact_227_order_Otrans,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ord_le2843351958646193337nt_int @ A4 @ C2 ) ) ) ).

% order.trans
thf(fact_228_order_Otrans,axiom,
    ! [A4: int,B7: int,C2: int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ord_less_eq_int @ A4 @ C2 ) ) ) ).

% order.trans
thf(fact_229_order__trans,axiom,
    ! [X2: nat,Y4: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ Z )
       => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_230_order__trans,axiom,
    ! [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
     => ( ( ord_le2843351958646193337nt_int @ Y4 @ Z )
       => ( ord_le2843351958646193337nt_int @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_231_order__trans,axiom,
    ! [X2: int,Y4: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y4 )
     => ( ( ord_less_eq_int @ Y4 @ Z )
       => ( ord_less_eq_int @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_232_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A4: nat,B7: nat] :
      ( ! [A2: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A2 @ B6 )
         => ( P @ A2 @ B6 ) )
     => ( ! [A2: nat,B6: nat] :
            ( ( P @ B6 @ A2 )
           => ( P @ A2 @ B6 ) )
       => ( P @ A4 @ B7 ) ) ) ).

% linorder_wlog
thf(fact_233_linorder__wlog,axiom,
    ! [P: int > int > $o,A4: int,B7: int] :
      ( ! [A2: int,B6: int] :
          ( ( ord_less_eq_int @ A2 @ B6 )
         => ( P @ A2 @ B6 ) )
     => ( ! [A2: int,B6: int] :
            ( ( P @ B6 @ A2 )
           => ( P @ A2 @ B6 ) )
       => ( P @ A4 @ B7 ) ) ) ).

% linorder_wlog
thf(fact_234_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [A3: nat,B: nat] :
          ( ( ord_less_eq_nat @ B @ A3 )
          & ( ord_less_eq_nat @ A3 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_235_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
    = ( ^ [A3: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
          ( ( ord_le2843351958646193337nt_int @ B @ A3 )
          & ( ord_le2843351958646193337nt_int @ A3 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_236_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
    = ( ^ [A3: int,B: int] :
          ( ( ord_less_eq_int @ B @ A3 )
          & ( ord_less_eq_int @ A3 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_237_dual__order_Oantisym,axiom,
    ! [B7: nat,A4: nat] :
      ( ( ord_less_eq_nat @ B7 @ A4 )
     => ( ( ord_less_eq_nat @ A4 @ B7 )
       => ( A4 = B7 ) ) ) ).

% dual_order.antisym
thf(fact_238_dual__order_Oantisym,axiom,
    ! [B7: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ B7 @ A4 )
     => ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
       => ( A4 = B7 ) ) ) ).

% dual_order.antisym
thf(fact_239_dual__order_Oantisym,axiom,
    ! [B7: int,A4: int] :
      ( ( ord_less_eq_int @ B7 @ A4 )
     => ( ( ord_less_eq_int @ A4 @ B7 )
       => ( A4 = B7 ) ) ) ).

% dual_order.antisym
thf(fact_240_dual__order_Otrans,axiom,
    ! [B7: nat,A4: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B7 @ A4 )
     => ( ( ord_less_eq_nat @ C2 @ B7 )
       => ( ord_less_eq_nat @ C2 @ A4 ) ) ) ).

% dual_order.trans
thf(fact_241_dual__order_Otrans,axiom,
    ! [B7: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ B7 @ A4 )
     => ( ( ord_le2843351958646193337nt_int @ C2 @ B7 )
       => ( ord_le2843351958646193337nt_int @ C2 @ A4 ) ) ) ).

% dual_order.trans
thf(fact_242_dual__order_Otrans,axiom,
    ! [B7: int,A4: int,C2: int] :
      ( ( ord_less_eq_int @ B7 @ A4 )
     => ( ( ord_less_eq_int @ C2 @ B7 )
       => ( ord_less_eq_int @ C2 @ A4 ) ) ) ).

% dual_order.trans
thf(fact_243_antisym,axiom,
    ! [A4: nat,B7: nat] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ord_less_eq_nat @ B7 @ A4 )
       => ( A4 = B7 ) ) ) ).

% antisym
thf(fact_244_antisym,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ A4 )
       => ( A4 = B7 ) ) ) ).

% antisym
thf(fact_245_antisym,axiom,
    ! [A4: int,B7: int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ord_less_eq_int @ B7 @ A4 )
       => ( A4 = B7 ) ) ) ).

% antisym
thf(fact_246_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [A3: nat,B: nat] :
          ( ( ord_less_eq_nat @ A3 @ B )
          & ( ord_less_eq_nat @ B @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_247_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
    = ( ^ [A3: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
          ( ( ord_le2843351958646193337nt_int @ A3 @ B )
          & ( ord_le2843351958646193337nt_int @ B @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_248_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
    = ( ^ [A3: int,B: int] :
          ( ( ord_less_eq_int @ A3 @ B )
          & ( ord_less_eq_int @ B @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_249_order__subst1,axiom,
    ! [A4: nat,F2: nat > nat,B7: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_250_order__subst1,axiom,
    ! [A4: nat,F2: set_Pr958786334691620121nt_int > nat,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_251_order__subst1,axiom,
    ! [A4: nat,F2: int > nat,B7: int,C2: int] :
      ( ( ord_less_eq_nat @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_252_order__subst1,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B7: nat,C2: nat] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_253_order__subst1,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_254_order__subst1,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B7: int,C2: int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_255_order__subst1,axiom,
    ! [A4: int,F2: nat > int,B7: nat,C2: nat] :
      ( ( ord_less_eq_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_256_order__subst1,axiom,
    ! [A4: int,F2: set_Pr958786334691620121nt_int > int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_257_order__subst1,axiom,
    ! [A4: int,F2: int > int,B7: int,C2: int] :
      ( ( ord_less_eq_int @ A4 @ ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_258_order__subst2,axiom,
    ! [A4: nat,B7: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ord_less_eq_nat @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_259_order__subst2,axiom,
    ! [A4: nat,B7: nat,F2: nat > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ord_le2843351958646193337nt_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_260_order__subst2,axiom,
    ! [A4: nat,B7: nat,F2: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ord_less_eq_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_261_order__subst2,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > nat,C2: nat] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ord_less_eq_nat @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_262_order__subst2,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ord_le2843351958646193337nt_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_263_order__subst2,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > int,C2: int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ord_less_eq_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_264_order__subst2,axiom,
    ! [A4: int,B7: int,F2: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ord_less_eq_nat @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_265_order__subst2,axiom,
    ! [A4: int,B7: int,F2: int > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ord_le2843351958646193337nt_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_266_order__subst2,axiom,
    ! [A4: int,B7: int,F2: int > int,C2: int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ord_less_eq_int @ ( F2 @ B7 ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_267_order__eq__refl,axiom,
    ! [X2: nat,Y4: nat] :
      ( ( X2 = Y4 )
     => ( ord_less_eq_nat @ X2 @ Y4 ) ) ).

% order_eq_refl
thf(fact_268_order__eq__refl,axiom,
    ! [X2: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
      ( ( X2 = Y4 )
     => ( ord_le2843351958646193337nt_int @ X2 @ Y4 ) ) ).

% order_eq_refl
thf(fact_269_order__eq__refl,axiom,
    ! [X2: int,Y4: int] :
      ( ( X2 = Y4 )
     => ( ord_less_eq_int @ X2 @ Y4 ) ) ).

% order_eq_refl
thf(fact_270_linorder__linear,axiom,
    ! [X2: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y4 )
      | ( ord_less_eq_nat @ Y4 @ X2 ) ) ).

% linorder_linear
thf(fact_271_linorder__linear,axiom,
    ! [X2: int,Y4: int] :
      ( ( ord_less_eq_int @ X2 @ Y4 )
      | ( ord_less_eq_int @ Y4 @ X2 ) ) ).

% linorder_linear
thf(fact_272_ord__eq__le__subst,axiom,
    ! [A4: nat,F2: nat > nat,B7: nat,C2: nat] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_273_ord__eq__le__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: nat > set_Pr958786334691620121nt_int,B7: nat,C2: nat] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_274_ord__eq__le__subst,axiom,
    ! [A4: int,F2: nat > int,B7: nat,C2: nat] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_275_ord__eq__le__subst,axiom,
    ! [A4: nat,F2: set_Pr958786334691620121nt_int > nat,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_276_ord__eq__le__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_277_ord__eq__le__subst,axiom,
    ! [A4: int,F2: set_Pr958786334691620121nt_int > int,B7: set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_le2843351958646193337nt_int @ B7 @ C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_278_ord__eq__le__subst,axiom,
    ! [A4: nat,F2: int > nat,B7: int,C2: int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_279_ord__eq__le__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F2: int > set_Pr958786334691620121nt_int,B7: int,C2: int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
    ! [A4: int,F2: int > int,B7: int,C2: int] :
      ( ( A4
        = ( F2 @ B7 ) )
     => ( ( ord_less_eq_int @ B7 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A4 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_281_ord__le__eq__subst,axiom,
    ! [A4: nat,B7: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_282_ord__le__eq__subst,axiom,
    ! [A4: nat,B7: nat,F2: nat > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_283_ord__le__eq__subst,axiom,
    ! [A4: nat,B7: nat,F2: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_284_ord__le__eq__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > nat,C2: nat] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_285_ord__le__eq__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_286_ord__le__eq__subst,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int,F2: set_Pr958786334691620121nt_int > int,C2: int] :
      ( ( ord_le2843351958646193337nt_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_287_ord__le__eq__subst,axiom,
    ! [A4: int,B7: int,F2: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_288_ord__le__eq__subst,axiom,
    ! [A4: int,B7: int,F2: int > set_Pr958786334691620121nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_le2843351958646193337nt_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2843351958646193337nt_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_289_ord__le__eq__subst,axiom,
    ! [A4: int,B7: int,F2: int > int,C2: int] :
      ( ( ord_less_eq_int @ A4 @ B7 )
     => ( ( ( F2 @ B7 )
          = C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A4 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_290_linorder__le__cases,axiom,
    ! [X2: nat,Y4: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y4 )
     => ( ord_less_eq_nat @ Y4 @ X2 ) ) ).

% linorder_le_cases
thf(fact_291_linorder__le__cases,axiom,
    ! [X2: int,Y4: int] :
      ( ~ ( ord_less_eq_int @ X2 @ Y4 )
     => ( ord_less_eq_int @ Y4 @ X2 ) ) ).

% linorder_le_cases
thf(fact_292_order__antisym__conv,axiom,
    ! [Y4: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y4 @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y4 )
        = ( X2 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_293_order__antisym__conv,axiom,
    ! [Y4: set_Pr958786334691620121nt_int,X2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ Y4 @ X2 )
     => ( ( ord_le2843351958646193337nt_int @ X2 @ Y4 )
        = ( X2 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_294_order__antisym__conv,axiom,
    ! [Y4: int,X2: int] :
      ( ( ord_less_eq_int @ Y4 @ X2 )
     => ( ( ord_less_eq_int @ X2 @ Y4 )
        = ( X2 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_295_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_int_int] :
      ~ ! [A2: int,B6: int] :
          ( Y4
         != ( product_Pair_int_int @ A2 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_296_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc1219242969750017639nt_int] :
      ~ ! [A2: product_prod_int_int,B6: product_prod_int_int] :
          ( Y4
         != ( produc3646306378393792727nt_int @ A2 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_297_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc7773217078559923341nt_int] :
      ~ ! [A2: int > option6357759511663192854e_term,B6: product_prod_int_int] :
          ( Y4
         != ( produc4305682042979456191nt_int @ A2 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_298_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc2285326912895808259nt_int] :
      ~ ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
          ( Y4
         != ( produc5700946648718959541nt_int @ A2 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_299_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P2
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_300_surj__pair,axiom,
    ! [P2: produc1219242969750017639nt_int] :
    ? [X3: product_prod_int_int,Y3: product_prod_int_int] :
      ( P2
      = ( produc3646306378393792727nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_301_surj__pair,axiom,
    ! [P2: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P2
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_302_surj__pair,axiom,
    ! [P2: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P2
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_303_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
      ( ! [A2: int,B6: int] : ( P @ ( product_Pair_int_int @ A2 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_304_prod__cases,axiom,
    ! [P: produc1219242969750017639nt_int > $o,P2: produc1219242969750017639nt_int] :
      ( ! [A2: product_prod_int_int,B6: product_prod_int_int] : ( P @ ( produc3646306378393792727nt_int @ A2 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_305_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P2: produc7773217078559923341nt_int] :
      ( ! [A2: int > option6357759511663192854e_term,B6: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A2 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_306_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P2: produc2285326912895808259nt_int] :
      ( ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A2 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_307_Pair__inject,axiom,
    ! [A4: int,B7: int,A5: int,B8: int] :
      ( ( ( product_Pair_int_int @ A4 @ B7 )
        = ( product_Pair_int_int @ A5 @ B8 ) )
     => ~ ( ( A4 = A5 )
         => ( B7 != B8 ) ) ) ).

% Pair_inject
thf(fact_308_Pair__inject,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A5: product_prod_int_int,B8: product_prod_int_int] :
      ( ( ( produc3646306378393792727nt_int @ A4 @ B7 )
        = ( produc3646306378393792727nt_int @ A5 @ B8 ) )
     => ~ ( ( A4 = A5 )
         => ( B7 != B8 ) ) ) ).

% Pair_inject
thf(fact_309_Pair__inject,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A5: int > option6357759511663192854e_term,B8: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A4 @ B7 )
        = ( produc4305682042979456191nt_int @ A5 @ B8 ) )
     => ~ ( ( A4 = A5 )
         => ( B7 != B8 ) ) ) ).

% Pair_inject
thf(fact_310_Pair__inject,axiom,
    ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B8: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A4 @ B7 )
        = ( produc5700946648718959541nt_int @ A5 @ B8 ) )
     => ~ ( ( A4 = A5 )
         => ( B7 != B8 ) ) ) ).

% Pair_inject
thf(fact_311_prod__cases3,axiom,
    ! [Y4: produc1219242969750017639nt_int] :
      ~ ! [A2: product_prod_int_int,B6: int,C3: int] :
          ( Y4
         != ( produc3646306378393792727nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) ) ).

% prod_cases3
thf(fact_312_prod__cases3,axiom,
    ! [Y4: produc7773217078559923341nt_int] :
      ~ ! [A2: int > option6357759511663192854e_term,B6: int,C3: int] :
          ( Y4
         != ( produc4305682042979456191nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) ) ).

% prod_cases3
thf(fact_313_prod__cases3,axiom,
    ! [Y4: produc2285326912895808259nt_int] :
      ~ ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B6: int,C3: int] :
          ( Y4
         != ( produc5700946648718959541nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) ) ).

% prod_cases3
thf(fact_314_prod__induct3,axiom,
    ! [P: produc1219242969750017639nt_int > $o,X2: produc1219242969750017639nt_int] :
      ( ! [A2: product_prod_int_int,B6: int,C3: int] : ( P @ ( produc3646306378393792727nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_315_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X2: produc7773217078559923341nt_int] :
      ( ! [A2: int > option6357759511663192854e_term,B6: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_316_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X2: produc2285326912895808259nt_int] :
      ( ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B6: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A2 @ ( product_Pair_int_int @ B6 @ C3 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_317_in__mono,axiom,
    ! [A: set_int,B3: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A @ B3 )
     => ( ( member_int @ X2 @ A )
       => ( member_int @ X2 @ B3 ) ) ) ).

% in_mono
thf(fact_318_in__mono,axiom,
    ! [A: set_nat,B3: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ X2 @ A )
       => ( member_nat @ X2 @ B3 ) ) ) ).

% in_mono
thf(fact_319_in__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,X2: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ( member5262025264175285858nt_int @ X2 @ A )
       => ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ).

% in_mono
thf(fact_320_subsetD,axiom,
    ! [A: set_int,B3: set_int,C2: int] :
      ( ( ord_less_eq_set_int @ A @ B3 )
     => ( ( member_int @ C2 @ A )
       => ( member_int @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_321_subsetD,axiom,
    ! [A: set_nat,B3: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ C2 @ A )
       => ( member_nat @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_322_subsetD,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C2: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ( member5262025264175285858nt_int @ C2 @ A )
       => ( member5262025264175285858nt_int @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_323_equalityE,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( A = B3 )
     => ~ ( ( ord_le2843351958646193337nt_int @ A @ B3 )
         => ~ ( ord_le2843351958646193337nt_int @ B3 @ A ) ) ) ).

% equalityE
thf(fact_324_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B2: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A6 )
         => ( member_int @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_325_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B2: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_326_subset__eq,axiom,
    ( ord_le2843351958646193337nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
        ! [X: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X @ A6 )
         => ( member5262025264175285858nt_int @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_327_equalityD1,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( A = B3 )
     => ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ).

% equalityD1
thf(fact_328_equalityD2,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( A = B3 )
     => ( ord_le2843351958646193337nt_int @ B3 @ A ) ) ).

% equalityD2
thf(fact_329_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B2: set_int] :
        ! [T3: int] :
          ( ( member_int @ T3 @ A6 )
         => ( member_int @ T3 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_330_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B2: set_nat] :
        ! [T3: nat] :
          ( ( member_nat @ T3 @ A6 )
         => ( member_nat @ T3 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_331_subset__iff,axiom,
    ( ord_le2843351958646193337nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
        ! [T3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ T3 @ A6 )
         => ( member5262025264175285858nt_int @ T3 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_332_subset__refl,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).

% subset_refl
thf(fact_333_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_334_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_335_Collect__mono,axiom,
    ! [P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ! [X3: produc2285326912895808259nt_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le8725513860283290265nt_int @ ( collec1790188477890212312nt_int @ P ) @ ( collec1790188477890212312nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_336_Collect__mono,axiom,
    ! [P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ! [X3: produc7773217078559923341nt_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le135402666524580259nt_int @ ( collec506566255779805410nt_int @ P ) @ ( collec506566255779805410nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_337_Collect__mono,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ! [X3: produc1219242969750017639nt_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6090609446090860775nt_int @ ( collec7447955092554649554nt_int @ P ) @ ( collec7447955092554649554nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_338_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_339_subset__trans,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ C )
       => ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).

% subset_trans
thf(fact_340_set__eq__subset,axiom,
    ( ( ^ [Y5: set_Pr958786334691620121nt_int,Z2: set_Pr958786334691620121nt_int] : ( Y5 = Z2 ) )
    = ( ^ [A6: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
          ( ( ord_le2843351958646193337nt_int @ A6 @ B2 )
          & ( ord_le2843351958646193337nt_int @ B2 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_341_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_342_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_343_Collect__mono__iff,axiom,
    ! [P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ( ord_le8725513860283290265nt_int @ ( collec1790188477890212312nt_int @ P ) @ ( collec1790188477890212312nt_int @ Q ) )
      = ( ! [X: produc2285326912895808259nt_int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_344_Collect__mono__iff,axiom,
    ! [P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ( ord_le135402666524580259nt_int @ ( collec506566255779805410nt_int @ P ) @ ( collec506566255779805410nt_int @ Q ) )
      = ( ! [X: produc7773217078559923341nt_int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_345_Collect__mono__iff,axiom,
    ! [P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ( ord_le6090609446090860775nt_int @ ( collec7447955092554649554nt_int @ P ) @ ( collec7447955092554649554nt_int @ Q ) )
      = ( ! [X: produc1219242969750017639nt_int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_346_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X: product_prod_int_int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_347_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_348_le__trans,axiom,
    ! [I4: nat,J4: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J4 )
     => ( ( ord_less_eq_nat @ J4 @ K )
       => ( ord_less_eq_nat @ I4 @ K ) ) ) ).

% le_trans
thf(fact_349_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_350_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_351_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_352_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B7: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B7 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_353_pred__equals__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_354_pred__equals__eq2,axiom,
    ! [R2: set_Pr2560585780119916871nt_int,S: set_Pr2560585780119916871nt_int] :
      ( ( ( ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_355_pred__equals__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_356_pred__equals__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_357_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_358_Collect__subset,axiom,
    ! [A: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_359_Collect__subset,axiom,
    ! [A: set_Pr9222295170931077689nt_int,P: produc2285326912895808259nt_int > $o] :
      ( ord_le8725513860283290265nt_int
      @ ( collec1790188477890212312nt_int
        @ ^ [X: produc2285326912895808259nt_int] :
            ( ( member7618704894036264090nt_int @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_360_Collect__subset,axiom,
    ! [A: set_Pr1872883991513573699nt_int,P: produc7773217078559923341nt_int > $o] :
      ( ord_le135402666524580259nt_int
      @ ( collec506566255779805410nt_int
        @ ^ [X: produc7773217078559923341nt_int] :
            ( ( member7034335876925520548nt_int @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_361_Collect__subset,axiom,
    ! [A: set_Pr2560585780119916871nt_int,P: produc1219242969750017639nt_int > $o] :
      ( ord_le6090609446090860775nt_int
      @ ( collec7447955092554649554nt_int
        @ ^ [X: produc1219242969750017639nt_int] :
            ( ( member8566619992076573584nt_int @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_362_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_363_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_364_card__board__exec__aux,axiom,
    ! [M2: set_int,K: nat] :
      ( ( finite_finite_int @ M2 )
     => ( ( finite6756421564338198497nt_int @ ( board_exec_aux @ K @ M2 ) )
        = ( times_times_nat @ K @ ( finite_card_int @ M2 ) ) ) ) ).

% card_board_exec_aux
thf(fact_365_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_366_conj__subset__def,axiom,
    ! [A: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_int @ A @ ( collect_int @ P ) )
        & ( ord_less_eq_set_int @ A @ ( collect_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_367_conj__subset__def,axiom,
    ! [A: set_Pr9222295170931077689nt_int,P: produc2285326912895808259nt_int > $o,Q: produc2285326912895808259nt_int > $o] :
      ( ( ord_le8725513860283290265nt_int @ A
        @ ( collec1790188477890212312nt_int
          @ ^ [X: produc2285326912895808259nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le8725513860283290265nt_int @ A @ ( collec1790188477890212312nt_int @ P ) )
        & ( ord_le8725513860283290265nt_int @ A @ ( collec1790188477890212312nt_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_368_conj__subset__def,axiom,
    ! [A: set_Pr1872883991513573699nt_int,P: produc7773217078559923341nt_int > $o,Q: produc7773217078559923341nt_int > $o] :
      ( ( ord_le135402666524580259nt_int @ A
        @ ( collec506566255779805410nt_int
          @ ^ [X: produc7773217078559923341nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le135402666524580259nt_int @ A @ ( collec506566255779805410nt_int @ P ) )
        & ( ord_le135402666524580259nt_int @ A @ ( collec506566255779805410nt_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_369_conj__subset__def,axiom,
    ! [A: set_Pr2560585780119916871nt_int,P: produc1219242969750017639nt_int > $o,Q: produc1219242969750017639nt_int > $o] :
      ( ( ord_le6090609446090860775nt_int @ A
        @ ( collec7447955092554649554nt_int
          @ ^ [X: produc1219242969750017639nt_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le6090609446090860775nt_int @ A @ ( collec7447955092554649554nt_int @ P ) )
        & ( ord_le6090609446090860775nt_int @ A @ ( collec7447955092554649554nt_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_370_conj__subset__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ A
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le2843351958646193337nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
        & ( ord_le2843351958646193337nt_int @ A @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_371_prop__restrict,axiom,
    ! [X2: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ Z3 )
     => ( ( ord_less_eq_set_nat @ Z3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_372_prop__restrict,axiom,
    ! [X2: int,Z3: set_int,X5: set_int,P: int > $o] :
      ( ( member_int @ X2 @ Z3 )
     => ( ( ord_less_eq_set_int @ Z3
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_373_prop__restrict,axiom,
    ! [X2: produc2285326912895808259nt_int,Z3: set_Pr9222295170931077689nt_int,X5: set_Pr9222295170931077689nt_int,P: produc2285326912895808259nt_int > $o] :
      ( ( member7618704894036264090nt_int @ X2 @ Z3 )
     => ( ( ord_le8725513860283290265nt_int @ Z3
          @ ( collec1790188477890212312nt_int
            @ ^ [X: produc2285326912895808259nt_int] :
                ( ( member7618704894036264090nt_int @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_374_prop__restrict,axiom,
    ! [X2: produc7773217078559923341nt_int,Z3: set_Pr1872883991513573699nt_int,X5: set_Pr1872883991513573699nt_int,P: produc7773217078559923341nt_int > $o] :
      ( ( member7034335876925520548nt_int @ X2 @ Z3 )
     => ( ( ord_le135402666524580259nt_int @ Z3
          @ ( collec506566255779805410nt_int
            @ ^ [X: produc7773217078559923341nt_int] :
                ( ( member7034335876925520548nt_int @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_375_prop__restrict,axiom,
    ! [X2: produc1219242969750017639nt_int,Z3: set_Pr2560585780119916871nt_int,X5: set_Pr2560585780119916871nt_int,P: produc1219242969750017639nt_int > $o] :
      ( ( member8566619992076573584nt_int @ X2 @ Z3 )
     => ( ( ord_le6090609446090860775nt_int @ Z3
          @ ( collec7447955092554649554nt_int
            @ ^ [X: produc1219242969750017639nt_int] :
                ( ( member8566619992076573584nt_int @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_376_prop__restrict,axiom,
    ! [X2: product_prod_int_int,Z3: set_Pr958786334691620121nt_int,X5: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ X2 @ Z3 )
     => ( ( ord_le2843351958646193337nt_int @ Z3
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_377_Collect__restrict,axiom,
    ! [X5: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_378_Collect__restrict,axiom,
    ! [X5: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_379_Collect__restrict,axiom,
    ! [X5: set_Pr9222295170931077689nt_int,P: produc2285326912895808259nt_int > $o] :
      ( ord_le8725513860283290265nt_int
      @ ( collec1790188477890212312nt_int
        @ ^ [X: produc2285326912895808259nt_int] :
            ( ( member7618704894036264090nt_int @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_380_Collect__restrict,axiom,
    ! [X5: set_Pr1872883991513573699nt_int,P: produc7773217078559923341nt_int > $o] :
      ( ord_le135402666524580259nt_int
      @ ( collec506566255779805410nt_int
        @ ^ [X: produc7773217078559923341nt_int] :
            ( ( member7034335876925520548nt_int @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_381_Collect__restrict,axiom,
    ! [X5: set_Pr2560585780119916871nt_int,P: produc1219242969750017639nt_int > $o] :
      ( ord_le6090609446090860775nt_int
      @ ( collec7447955092554649554nt_int
        @ ^ [X: produc1219242969750017639nt_int] :
            ( ( member8566619992076573584nt_int @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_382_Collect__restrict,axiom,
    ! [X5: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_383_subset__CollectI,axiom,
    ! [B3: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_384_subset__CollectI,axiom,
    ! [B3: set_int,A: set_int,Q: int > $o,P: int > $o] :
      ( ( ord_less_eq_set_int @ B3 @ A )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_385_subset__CollectI,axiom,
    ! [B3: set_Pr9222295170931077689nt_int,A: set_Pr9222295170931077689nt_int,Q: produc2285326912895808259nt_int > $o,P: produc2285326912895808259nt_int > $o] :
      ( ( ord_le8725513860283290265nt_int @ B3 @ A )
     => ( ! [X3: produc2285326912895808259nt_int] :
            ( ( member7618704894036264090nt_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le8725513860283290265nt_int
          @ ( collec1790188477890212312nt_int
            @ ^ [X: produc2285326912895808259nt_int] :
                ( ( member7618704894036264090nt_int @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec1790188477890212312nt_int
            @ ^ [X: produc2285326912895808259nt_int] :
                ( ( member7618704894036264090nt_int @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_386_subset__CollectI,axiom,
    ! [B3: set_Pr1872883991513573699nt_int,A: set_Pr1872883991513573699nt_int,Q: produc7773217078559923341nt_int > $o,P: produc7773217078559923341nt_int > $o] :
      ( ( ord_le135402666524580259nt_int @ B3 @ A )
     => ( ! [X3: produc7773217078559923341nt_int] :
            ( ( member7034335876925520548nt_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le135402666524580259nt_int
          @ ( collec506566255779805410nt_int
            @ ^ [X: produc7773217078559923341nt_int] :
                ( ( member7034335876925520548nt_int @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec506566255779805410nt_int
            @ ^ [X: produc7773217078559923341nt_int] :
                ( ( member7034335876925520548nt_int @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_387_subset__CollectI,axiom,
    ! [B3: set_Pr2560585780119916871nt_int,A: set_Pr2560585780119916871nt_int,Q: produc1219242969750017639nt_int > $o,P: produc1219242969750017639nt_int > $o] :
      ( ( ord_le6090609446090860775nt_int @ B3 @ A )
     => ( ! [X3: produc1219242969750017639nt_int] :
            ( ( member8566619992076573584nt_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6090609446090860775nt_int
          @ ( collec7447955092554649554nt_int
            @ ^ [X: produc1219242969750017639nt_int] :
                ( ( member8566619992076573584nt_int @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec7447955092554649554nt_int
            @ ^ [X: produc1219242969750017639nt_int] :
                ( ( member8566619992076573584nt_int @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_388_subset__CollectI,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,Q: product_prod_int_int > $o,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le2843351958646193337nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_389_subset__Collect__iff,axiom,
    ! [B3: set_nat,A: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A )
     => ( ( ord_less_eq_set_nat @ B3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_390_subset__Collect__iff,axiom,
    ! [B3: set_int,A: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ B3 @ A )
     => ( ( ord_less_eq_set_int @ B3
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_391_subset__Collect__iff,axiom,
    ! [B3: set_Pr9222295170931077689nt_int,A: set_Pr9222295170931077689nt_int,P: produc2285326912895808259nt_int > $o] :
      ( ( ord_le8725513860283290265nt_int @ B3 @ A )
     => ( ( ord_le8725513860283290265nt_int @ B3
          @ ( collec1790188477890212312nt_int
            @ ^ [X: produc2285326912895808259nt_int] :
                ( ( member7618704894036264090nt_int @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc2285326912895808259nt_int] :
              ( ( member7618704894036264090nt_int @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_392_subset__Collect__iff,axiom,
    ! [B3: set_Pr1872883991513573699nt_int,A: set_Pr1872883991513573699nt_int,P: produc7773217078559923341nt_int > $o] :
      ( ( ord_le135402666524580259nt_int @ B3 @ A )
     => ( ( ord_le135402666524580259nt_int @ B3
          @ ( collec506566255779805410nt_int
            @ ^ [X: produc7773217078559923341nt_int] :
                ( ( member7034335876925520548nt_int @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc7773217078559923341nt_int] :
              ( ( member7034335876925520548nt_int @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_393_subset__Collect__iff,axiom,
    ! [B3: set_Pr2560585780119916871nt_int,A: set_Pr2560585780119916871nt_int,P: produc1219242969750017639nt_int > $o] :
      ( ( ord_le6090609446090860775nt_int @ B3 @ A )
     => ( ( ord_le6090609446090860775nt_int @ B3
          @ ( collec7447955092554649554nt_int
            @ ^ [X: produc1219242969750017639nt_int] :
                ( ( member8566619992076573584nt_int @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc1219242969750017639nt_int] :
              ( ( member8566619992076573584nt_int @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_394_subset__Collect__iff,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_395_same__fstI,axiom,
    ! [P: int > $o,X2: int,Y7: int,Y4: int,R2: int > set_Pr958786334691620121nt_int] :
      ( ( P @ X2 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y7 @ Y4 ) @ ( R2 @ X2 ) )
       => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ X2 @ Y7 ) @ ( product_Pair_int_int @ X2 @ Y4 ) ) @ ( same_fst_int_int @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_396_same__fstI,axiom,
    ! [P: product_prod_int_int > $o,X2: product_prod_int_int,Y7: product_prod_int_int,Y4: product_prod_int_int,R2: product_prod_int_int > set_Pr2560585780119916871nt_int] :
      ( ( P @ X2 )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y7 @ Y4 ) @ ( R2 @ X2 ) )
       => ( member8053542592415931152nt_int @ ( produc1923601798490594135nt_int @ ( produc3646306378393792727nt_int @ X2 @ Y7 ) @ ( produc3646306378393792727nt_int @ X2 @ Y4 ) ) @ ( same_f2440470920016040620nt_int @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_397_same__fstI,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > $o,X2: int > option6357759511663192854e_term,Y7: product_prod_int_int,Y4: product_prod_int_int,R2: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X2 )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y7 @ Y4 ) @ ( R2 @ X2 ) )
       => ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y7 ) @ ( produc4305682042979456191nt_int @ X2 @ Y4 ) ) @ ( same_f6662725367016992042nt_int @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_398_same__fstI,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y7: product_prod_int_int,Y4: product_prod_int_int,R2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X2 )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y7 @ Y4 ) @ ( R2 @ X2 ) )
       => ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y7 ) @ ( produc5700946648718959541nt_int @ X2 @ Y4 ) ) @ ( same_f5472592420709775776nt_int @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_399_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A6: set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( ord_less_eq_set_nat @ X6 @ A6 )
              & ( finite_finite_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_400_Fpow__def,axiom,
    ( finite_Fpow_int
    = ( ^ [A6: set_int] :
          ( collect_set_int
          @ ^ [X6: set_int] :
              ( ( ord_less_eq_set_int @ X6 @ A6 )
              & ( finite_finite_int @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_401_Fpow__def,axiom,
    ( finite1421331698046337823nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X6: set_Pr958786334691620121nt_int] :
              ( ( ord_le2843351958646193337nt_int @ X6 @ A6 )
              & ( finite2998713641127702882nt_int @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_402_finite__board__exec__aux,axiom,
    ! [M2: set_int,N: nat] :
      ( ( finite_finite_int @ M2 )
     => ( finite2998713641127702882nt_int @ ( board_exec_aux @ N @ M2 ) ) ) ).

% finite_board_exec_aux
thf(fact_403_Fpow__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ord_le483042692224249369nt_int @ ( finite1421331698046337823nt_int @ A ) @ ( finite1421331698046337823nt_int @ B3 ) ) ) ).

% Fpow_mono
thf(fact_404_ssubst__Pair__rhs,axiom,
    ! [R: int,S2: int,R2: set_Pr958786334691620121nt_int,S3: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_405_ssubst__Pair__rhs,axiom,
    ! [R: product_prod_int_int,S2: product_prod_int_int,R2: set_Pr2560585780119916871nt_int,S3: product_prod_int_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ R @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ R @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_406_ssubst__Pair__rhs,axiom,
    ! [R: int > option6357759511663192854e_term,S2: product_prod_int_int,R2: set_Pr1872883991513573699nt_int,S3: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_407_ssubst__Pair__rhs,axiom,
    ! [R: produc8551481072490612790e_term > option6357759511663192854e_term,S2: product_prod_int_int,R2: set_Pr9222295170931077689nt_int,S3: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_408_finite__interval__int1,axiom,
    ! [A4: int,B7: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I: int] :
            ( ( ord_less_eq_int @ A4 @ I )
            & ( ord_less_eq_int @ I @ B7 ) ) ) ) ).

% finite_interval_int1
thf(fact_409_image2__def,axiom,
    ( bNF_Gr8686673574714534880nt_int
    = ( ^ [A6: set_int,F3: int > int,G2: int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A3: int] :
              ( ( Uu
                = ( product_Pair_int_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_410_image2__def,axiom,
    ( bNF_Gr8315020014477690244nt_int
    = ( ^ [A6: set_nat,F3: nat > int,G2: nat > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A3: nat] :
              ( ( Uu
                = ( product_Pair_int_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_nat @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_411_image2__def,axiom,
    ( bNF_Gr4426612819127333049nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,F3: product_prod_int_int > int,G2: product_prod_int_int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A3: product_prod_int_int] :
              ( ( Uu
                = ( product_Pair_int_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member5262025264175285858nt_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_412_image2__def,axiom,
    ( bNF_Gr3379968192745358150nt_int
    = ( ^ [A6: set_int,F3: int > product_prod_int_int,G2: int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A3: int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_413_image2__def,axiom,
    ( bNF_Gr4976608462507087594nt_int
    = ( ^ [A6: set_nat,F3: nat > product_prod_int_int,G2: nat > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A3: nat] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_nat @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_414_image2__def,axiom,
    ( bNF_Gr133173068567543455nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,F3: product_prod_int_int > product_prod_int_int,G2: product_prod_int_int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A3: product_prod_int_int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member5262025264175285858nt_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_415_image2__def,axiom,
    ( bNF_Gr8978185744112786704nt_int
    = ( ^ [A6: set_int,F3: int > int > option6357759511663192854e_term,G2: int > product_prod_int_int] :
          ( collec506566255779805410nt_int
          @ ^ [Uu: produc7773217078559923341nt_int] :
            ? [A3: int] :
              ( ( Uu
                = ( produc4305682042979456191nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_416_image2__def,axiom,
    ( bNF_Gr9055198396104480364nt_int
    = ( ^ [A6: set_nat,F3: nat > int > option6357759511663192854e_term,G2: nat > product_prod_int_int] :
          ( collec506566255779805410nt_int
          @ ^ [Uu: produc7773217078559923341nt_int] :
            ? [A3: nat] :
              ( ( Uu
                = ( produc4305682042979456191nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_nat @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_417_image2__def,axiom,
    ( bNF_Gr4342778003469861111nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,F3: product_prod_int_int > int > option6357759511663192854e_term,G2: product_prod_int_int > product_prod_int_int] :
          ( collec506566255779805410nt_int
          @ ^ [Uu: produc7773217078559923341nt_int] :
            ? [A3: product_prod_int_int] :
              ( ( Uu
                = ( produc4305682042979456191nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member5262025264175285858nt_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_418_image2__def,axiom,
    ( bNF_Gr556869251677793414nt_int
    = ( ^ [A6: set_int,F3: int > produc8551481072490612790e_term > option6357759511663192854e_term,G2: int > product_prod_int_int] :
          ( collec1790188477890212312nt_int
          @ ^ [Uu: produc2285326912895808259nt_int] :
            ? [A3: int] :
              ( ( Uu
                = ( produc5700946648718959541nt_int @ ( F3 @ A3 ) @ ( G2 @ A3 ) ) )
              & ( member_int @ A3 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_419_relImage__def,axiom,
    ( bNF_Gr4175811214798380094nt_int
    = ( ^ [R3: set_Pr958786334691620121nt_int,F3: int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A12: int,A23: int] :
              ( ( Uu
                = ( product_Pair_int_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ A23 ) @ R3 ) ) ) ) ) ).

% relImage_def
thf(fact_420_relImage__def,axiom,
    ( bNF_Gr8897276713083563265nt_int
    = ( ^ [R3: set_Pr2560585780119916871nt_int,F3: product_prod_int_int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A12: product_prod_int_int,A23: product_prod_int_int] :
              ( ( Uu
                = ( product_Pair_int_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) )
              & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A12 @ A23 ) @ R3 ) ) ) ) ) ).

% relImage_def
thf(fact_421_relImage__def,axiom,
    ( bNF_Gr337212181706678055nt_int
    = ( ^ [R3: set_Pr958786334691620121nt_int,F3: int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A12: int,A23: int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A12 @ A23 ) @ R3 ) ) ) ) ) ).

% relImage_def
thf(fact_422_relImage__def,axiom,
    ( bNF_Gr2689605748215148196nt_int
    = ( ^ [R3: set_Pr2560585780119916871nt_int,F3: product_prod_int_int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A12: product_prod_int_int,A23: product_prod_int_int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) )
              & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A12 @ A23 ) @ R3 ) ) ) ) ) ).

% relImage_def
thf(fact_423_relInvImage__def,axiom,
    ( bNF_Gr8017871217638807999at_int
    = ( ^ [A6: set_nat,R3: set_Pr958786334691620121nt_int,F3: nat > int] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: nat,A23: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A12 @ A23 ) )
              & ( member_nat @ A12 @ A6 )
              & ( member_nat @ A23 @ A6 )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_424_relInvImage__def,axiom,
    ( bNF_Gr4473764933443066982nt_int
    = ( ^ [A6: set_nat,R3: set_Pr2560585780119916871nt_int,F3: nat > product_prod_int_int] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: nat,A23: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A12 @ A23 ) )
              & ( member_nat @ A12 @ A6 )
              & ( member_nat @ A23 @ A6 )
              & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_425_relInvImage__def,axiom,
    ( bNF_Gr9017224929539862555nt_int
    = ( ^ [A6: set_int,R3: set_Pr958786334691620121nt_int,F3: int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A12: int,A23: int] :
              ( ( Uu
                = ( product_Pair_int_int @ A12 @ A23 ) )
              & ( member_int @ A12 @ A6 )
              & ( member_int @ A23 @ A6 )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_426_relInvImage__def,axiom,
    ( bNF_Gr7511607017907200778nt_int
    = ( ^ [A6: set_int,R3: set_Pr2560585780119916871nt_int,F3: int > product_prod_int_int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A12: int,A23: int] :
              ( ( Uu
                = ( product_Pair_int_int @ A12 @ A23 ) )
              & ( member_int @ A12 @ A6 )
              & ( member_int @ A23 @ A6 )
              & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_427_relInvImage__def,axiom,
    ( bNF_Gr6848299512429310180nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,R3: set_Pr958786334691620121nt_int,F3: product_prod_int_int > int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A12: product_prod_int_int,A23: product_prod_int_int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ A12 @ A23 ) )
              & ( member5262025264175285858nt_int @ A12 @ A6 )
              & ( member5262025264175285858nt_int @ A23 @ A6 )
              & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_428_relInvImage__def,axiom,
    ( bNF_Gr3973862455062356737nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,R3: set_Pr2560585780119916871nt_int,F3: product_prod_int_int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A12: product_prod_int_int,A23: product_prod_int_int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ A12 @ A23 ) )
              & ( member5262025264175285858nt_int @ A12 @ A6 )
              & ( member5262025264175285858nt_int @ A23 @ A6 )
              & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( F3 @ A12 ) @ ( F3 @ A23 ) ) @ R3 ) ) ) ) ) ).

% relInvImage_def
thf(fact_429_board__exec_Osimps,axiom,
    ( board_exec
    = ( ^ [N2: nat,M3: nat] : ( board_exec_aux @ N2 @ ( row_exec @ M3 ) ) ) ) ).

% board_exec.simps
thf(fact_430_board__exec_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y4: set_Pr958786334691620121nt_int] :
      ( ( ( board_exec @ X2 @ Xa2 )
        = Y4 )
     => ( Y4
        = ( board_exec_aux @ X2 @ ( row_exec @ Xa2 ) ) ) ) ).

% board_exec.elims
thf(fact_431_relChain__def,axiom,
    ( bNF_Ca1968104039914474786nt_nat
    = ( ^ [R4: set_Pr958786334691620121nt_int,As: int > nat] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R4 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_432_relChain__def,axiom,
    ( bNF_Ca1643832818461744997nt_nat
    = ( ^ [R4: set_Pr2560585780119916871nt_int,As: product_prod_int_int > nat] :
        ! [I: product_prod_int_int,J: product_prod_int_int] :
          ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I @ J ) @ R4 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_433_relChain__def,axiom,
    ( bNF_Ca8719598144974034247nt_int
    = ( ^ [R4: set_Pr958786334691620121nt_int,As: int > set_Pr958786334691620121nt_int] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R4 )
         => ( ord_le2843351958646193337nt_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_434_relChain__def,axiom,
    ( bNF_Ca5742924509254848324nt_int
    = ( ^ [R4: set_Pr2560585780119916871nt_int,As: product_prod_int_int > set_Pr958786334691620121nt_int] :
        ! [I: product_prod_int_int,J: product_prod_int_int] :
          ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I @ J ) @ R4 )
         => ( ord_le2843351958646193337nt_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_435_relChain__def,axiom,
    ( bNF_Ca1965613569405424510nt_int
    = ( ^ [R4: set_Pr958786334691620121nt_int,As: int > int] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R4 )
         => ( ord_less_eq_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_436_relChain__def,axiom,
    ( bNF_Ca1641342347952694721nt_int
    = ( ^ [R4: set_Pr2560585780119916871nt_int,As: product_prod_int_int > int] :
        ! [I: product_prod_int_int,J: product_prod_int_int] :
          ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ I @ J ) @ R4 )
         => ( ord_less_eq_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_437_surj__card__le,axiom,
    ! [A: set_Product_unit,B3: set_int,F2: product_unit > int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_873079544045703924it_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).

% surj_card_le
thf(fact_438_surj__card__le,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit,F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ord_le3507040750410214029t_unit @ B3 @ ( image_405062704495631173t_unit @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B3 ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).

% surj_card_le
thf(fact_439_surj__card__le,axiom,
    ! [A: set_nat,B3: set_int,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).

% surj_card_le
thf(fact_440_surj__card__le,axiom,
    ! [A: set_nat,B3: set_Product_unit,F2: nat > product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le3507040750410214029t_unit @ B3 @ ( image_8730104196221521654t_unit @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).

% surj_card_le
thf(fact_441_surj__card__le,axiom,
    ! [A: set_int,B3: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_int @ A ) ) ) ) ).

% surj_card_le
thf(fact_442_surj__card__le,axiom,
    ! [A: set_int,B3: set_Product_unit,F2: int > product_unit] :
      ( ( finite_finite_int @ A )
     => ( ( ord_le3507040750410214029t_unit @ B3 @ ( image_5846526143423701914t_unit @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B3 ) @ ( finite_card_int @ A ) ) ) ) ).

% surj_card_le
thf(fact_443_surj__card__le,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_5042161079198086560nt_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite6756421564338198497nt_int @ A ) ) ) ) ).

% surj_card_le
thf(fact_444_surj__card__le,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Product_unit,F2: product_prod_int_int > product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_le3507040750410214029t_unit @ B3 @ ( image_2335562724295632793t_unit @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B3 ) @ ( finite6756421564338198497nt_int @ A ) ) ) ) ).

% surj_card_le
thf(fact_445_surj__card__le,axiom,
    ! [A: set_Product_unit,B3: set_Pr958786334691620121nt_int,F2: product_unit > product_prod_int_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_4103462098156874993nt_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ B3 ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).

% surj_card_le
thf(fact_446_surj__card__le,axiom,
    ! [A: set_nat,B3: set_Pr958786334691620121nt_int,F2: nat > product_prod_int_int] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2667626500211843362nt_int @ F2 @ A ) )
       => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).

% surj_card_le
thf(fact_447_image__eqI,axiom,
    ! [B7: product_prod_int_int,F2: product_prod_int_int > product_prod_int_int,X2: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member5262025264175285858nt_int @ X2 @ A )
       => ( member5262025264175285858nt_int @ B7 @ ( image_2653370878348428101nt_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_448_image__eqI,axiom,
    ! [B7: int,F2: product_prod_int_int > int,X2: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member5262025264175285858nt_int @ X2 @ A )
       => ( member_int @ B7 @ ( image_5042161079198086560nt_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_449_image__eqI,axiom,
    ! [B7: nat,F2: product_prod_int_int > nat,X2: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member5262025264175285858nt_int @ X2 @ A )
       => ( member_nat @ B7 @ ( image_5044651549707136836nt_nat @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_450_image__eqI,axiom,
    ! [B7: product_prod_int_int,F2: int > product_prod_int_int,X2: int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_int @ X2 @ A )
       => ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_451_image__eqI,axiom,
    ! [B7: int,F2: int > int,X2: int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_int @ X2 @ A )
       => ( member_int @ B7 @ ( image_int_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_452_image__eqI,axiom,
    ! [B7: nat,F2: int > nat,X2: int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_int @ X2 @ A )
       => ( member_nat @ B7 @ ( image_int_nat @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_453_image__eqI,axiom,
    ! [B7: product_prod_int_int,F2: nat > product_prod_int_int,X2: nat,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_454_image__eqI,axiom,
    ! [B7: int,F2: nat > int,X2: nat,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_int @ B7 @ ( image_nat_int @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_455_image__eqI,axiom,
    ! [B7: nat,F2: nat > nat,X2: nat,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_nat @ B7 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).

% image_eqI
thf(fact_456_finite__imageI,axiom,
    ! [F: set_Pr958786334691620121nt_int,H: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ F )
     => ( finite2998713641127702882nt_int @ ( image_2653370878348428101nt_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_457_finite__imageI,axiom,
    ! [F: set_Pr958786334691620121nt_int,H: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int @ F )
     => ( finite_finite_nat @ ( image_5044651549707136836nt_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_458_finite__imageI,axiom,
    ! [F: set_Pr958786334691620121nt_int,H: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ F )
     => ( finite_finite_int @ ( image_5042161079198086560nt_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_459_finite__imageI,axiom,
    ! [F: set_nat,H: nat > product_prod_int_int] :
      ( ( finite_finite_nat @ F )
     => ( finite2998713641127702882nt_int @ ( image_2667626500211843362nt_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_460_finite__imageI,axiom,
    ! [F: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_461_finite__imageI,axiom,
    ! [F: set_nat,H: nat > int] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_int @ ( image_nat_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_462_finite__imageI,axiom,
    ! [F: set_int,H: int > product_prod_int_int] :
      ( ( finite_finite_int @ F )
     => ( finite2998713641127702882nt_int @ ( image_5705468584675977158nt_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_463_finite__imageI,axiom,
    ! [F: set_int,H: int > nat] :
      ( ( finite_finite_int @ F )
     => ( finite_finite_nat @ ( image_int_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_464_finite__imageI,axiom,
    ! [F: set_int,H: int > int] :
      ( ( finite_finite_int @ F )
     => ( finite_finite_int @ ( image_int_int @ H @ F ) ) ) ).

% finite_imageI
thf(fact_465_imageE,axiom,
    ! [B7: product_prod_int_int,F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member5262025264175285858nt_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_466_imageE,axiom,
    ! [B7: product_prod_int_int,F2: int > product_prod_int_int,A: set_int] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ A ) )
     => ~ ! [X3: int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_467_imageE,axiom,
    ! [B7: product_prod_int_int,F2: nat > product_prod_int_int,A: set_nat] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_468_imageE,axiom,
    ! [B7: int,F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( member_int @ B7 @ ( image_5042161079198086560nt_int @ F2 @ A ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member5262025264175285858nt_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_469_imageE,axiom,
    ! [B7: int,F2: int > int,A: set_int] :
      ( ( member_int @ B7 @ ( image_int_int @ F2 @ A ) )
     => ~ ! [X3: int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_470_imageE,axiom,
    ! [B7: int,F2: nat > int,A: set_nat] :
      ( ( member_int @ B7 @ ( image_nat_int @ F2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_471_imageE,axiom,
    ! [B7: nat,F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( member_nat @ B7 @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member5262025264175285858nt_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_472_imageE,axiom,
    ! [B7: nat,F2: int > nat,A: set_int] :
      ( ( member_nat @ B7 @ ( image_int_nat @ F2 @ A ) )
     => ~ ! [X3: int] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_int @ X3 @ A ) ) ) ).

% imageE
thf(fact_473_imageE,axiom,
    ! [B7: nat,F2: nat > nat,A: set_nat] :
      ( ( member_nat @ B7 @ ( image_nat_nat @ F2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( B7
              = ( F2 @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_474_Compr__image__eq,axiom,
    ! [F2: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_475_Compr__image__eq,axiom,
    ! [F2: int > nat,A: set_int,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_int_nat @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_int_nat @ F2
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_476_Compr__image__eq,axiom,
    ! [F2: nat > int,A: set_nat,P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ ( image_nat_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_int @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_477_Compr__image__eq,axiom,
    ! [F2: int > int,A: set_int,P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ ( image_int_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_int_int @ F2
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_478_Compr__image__eq,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat,P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ ( image_2667626500211843362nt_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_2667626500211843362nt_int @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_479_Compr__image__eq,axiom,
    ! [F2: int > product_prod_int_int,A: set_int,P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ ( image_5705468584675977158nt_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_5705468584675977158nt_int @ F2
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_480_Compr__image__eq,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_5044651549707136836nt_nat @ F2
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_481_Compr__image__eq,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ ( image_5042161079198086560nt_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_5042161079198086560nt_int @ F2
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_482_Compr__image__eq,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X @ ( image_2653370878348428101nt_int @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_2653370878348428101nt_int @ F2
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_483_Compr__image__eq,axiom,
    ! [F2: produc1219242969750017639nt_int > nat,A: set_Pr2560585780119916871nt_int,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_871192306115182322nt_nat @ F2 @ A ) )
            & ( P @ X ) ) )
      = ( image_871192306115182322nt_nat @ F2
        @ ( collec7447955092554649554nt_int
          @ ^ [X: produc1219242969750017639nt_int] :
              ( ( member8566619992076573584nt_int @ X @ A )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_484_rev__image__eqI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: product_prod_int_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( image_2653370878348428101nt_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_485_rev__image__eqI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: int,F2: product_prod_int_int > int] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_int @ B7 @ ( image_5042161079198086560nt_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_486_rev__image__eqI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: nat,F2: product_prod_int_int > nat] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_nat @ B7 @ ( image_5044651549707136836nt_nat @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_487_rev__image__eqI,axiom,
    ! [X2: int,A: set_int,B7: product_prod_int_int,F2: int > product_prod_int_int] :
      ( ( member_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_488_rev__image__eqI,axiom,
    ! [X2: int,A: set_int,B7: int,F2: int > int] :
      ( ( member_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_int @ B7 @ ( image_int_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_489_rev__image__eqI,axiom,
    ! [X2: int,A: set_int,B7: nat,F2: int > nat] :
      ( ( member_int @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_nat @ B7 @ ( image_int_nat @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_490_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B7: product_prod_int_int,F2: nat > product_prod_int_int] :
      ( ( member_nat @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_491_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B7: int,F2: nat > int] :
      ( ( member_nat @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_int @ B7 @ ( image_nat_int @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_492_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B7: nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( ( B7
          = ( F2 @ X2 ) )
       => ( member_nat @ B7 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_493_imageI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_2653370878348428101nt_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_494_imageI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( member_int @ ( F2 @ X2 ) @ ( image_5042161079198086560nt_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_495_imageI,axiom,
    ! [X2: product_prod_int_int,A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > nat] :
      ( ( member5262025264175285858nt_int @ X2 @ A )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_5044651549707136836nt_nat @ F2 @ A ) ) ) ).

% imageI
thf(fact_496_imageI,axiom,
    ! [X2: int,A: set_int,F2: int > product_prod_int_int] :
      ( ( member_int @ X2 @ A )
     => ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_5705468584675977158nt_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_497_imageI,axiom,
    ! [X2: int,A: set_int,F2: int > int] :
      ( ( member_int @ X2 @ A )
     => ( member_int @ ( F2 @ X2 ) @ ( image_int_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_498_imageI,axiom,
    ! [X2: int,A: set_int,F2: int > nat] :
      ( ( member_int @ X2 @ A )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_int_nat @ F2 @ A ) ) ) ).

% imageI
thf(fact_499_imageI,axiom,
    ! [X2: nat,A: set_nat,F2: nat > product_prod_int_int] :
      ( ( member_nat @ X2 @ A )
     => ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_2667626500211843362nt_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_500_imageI,axiom,
    ! [X2: nat,A: set_nat,F2: nat > int] :
      ( ( member_nat @ X2 @ A )
     => ( member_int @ ( F2 @ X2 ) @ ( image_nat_int @ F2 @ A ) ) ) ).

% imageI
thf(fact_501_imageI,axiom,
    ! [X2: nat,A: set_nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_nat_nat @ F2 @ A ) ) ) ).

% imageI
thf(fact_502_image__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B3 )
     => ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) @ ( image_2653370878348428101nt_int @ F2 @ B3 ) ) ) ).

% image_mono
thf(fact_503_image__subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int,B3: set_int] :
      ( ! [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_504_image__subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > nat,B3: set_nat] :
      ( ! [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_505_image__subsetI,axiom,
    ! [A: set_int,F2: int > int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_506_image__subsetI,axiom,
    ! [A: set_int,F2: int > nat,B3: set_nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_507_image__subsetI,axiom,
    ! [A: set_nat,F2: nat > int,B3: set_int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_508_image__subsetI,axiom,
    ! [A: set_nat,F2: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_509_image__subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_510_image__subsetI,axiom,
    ! [A: set_int,F2: int > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_511_image__subsetI,axiom,
    ! [A: set_nat,F2: nat > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_512_subset__imageE,axiom,
    ! [B3: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
     => ~ ! [C4: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ C4 @ A )
           => ( B3
             != ( image_2653370878348428101nt_int @ F2 @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_513_subset__image__iff,axiom,
    ! [B3: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
      = ( ? [AA: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ AA @ A )
            & ( B3
              = ( image_2653370878348428101nt_int @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_514_all__subset__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ B2 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ B2 @ A )
           => ( P @ ( image_2653370878348428101nt_int @ F2 @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_515_pigeonhole__infinite,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_516_pigeonhole__infinite,axiom,
    ! [A: set_nat,F2: nat > int] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ ( image_nat_int @ F2 @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_517_pigeonhole__infinite,axiom,
    ! [A: set_int,F2: int > nat] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ ( image_int_nat @ F2 @ A ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A3: int] :
                      ( ( member_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_518_pigeonhole__infinite,axiom,
    ! [A: set_int,F2: int > int] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite_finite_int @ ( image_int_int @ F2 @ A ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A3: int] :
                      ( ( member_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_519_pigeonhole__infinite,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > nat] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
       => ? [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
            & ~ ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [A3: product_prod_int_int] :
                      ( ( member5262025264175285858nt_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_520_pigeonhole__infinite,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
       => ? [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
            & ~ ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [A3: product_prod_int_int] :
                      ( ( member5262025264175285858nt_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_521_pigeonhole__infinite,axiom,
    ! [A: set_nat,F2: nat > product_prod_int_int] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_522_pigeonhole__infinite,axiom,
    ! [A: set_int,F2: int > product_prod_int_int] :
      ( ~ ( finite_finite_int @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A3: int] :
                      ( ( member_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_523_pigeonhole__infinite,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) )
       => ? [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
            & ~ ( finite2998713641127702882nt_int
                @ ( collec213857154873943460nt_int
                  @ ^ [A3: product_prod_int_int] :
                      ( ( member5262025264175285858nt_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_524_pigeonhole__infinite,axiom,
    ! [A: set_Pr2560585780119916871nt_int,F2: produc1219242969750017639nt_int > nat] :
      ( ~ ( finite4752125741994067600nt_int @ A )
     => ( ( finite_finite_nat @ ( image_871192306115182322nt_nat @ F2 @ A ) )
       => ? [X3: produc1219242969750017639nt_int] :
            ( ( member8566619992076573584nt_int @ X3 @ A )
            & ~ ( finite4752125741994067600nt_int
                @ ( collec7447955092554649554nt_int
                  @ ^ [A3: produc1219242969750017639nt_int] :
                      ( ( member8566619992076573584nt_int @ A3 @ A )
                      & ( ( F2 @ A3 )
                        = ( F2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_525_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F2: nat > int,B3: set_int] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_526_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F2: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_527_image__Collect__subsetI,axiom,
    ! [P: int > $o,F2: int > int,B3: set_int] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_int_int @ F2 @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_528_image__Collect__subsetI,axiom,
    ! [P: int > $o,F2: int > nat,B3: set_nat] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_529_image__Collect__subsetI,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > int,B3: set_int] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F2 @ ( collec213857154873943460nt_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_530_image__Collect__subsetI,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > nat,B3: set_nat] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( member_nat @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F2 @ ( collec213857154873943460nt_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_531_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F2: nat > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_532_image__Collect__subsetI,axiom,
    ! [P: int > $o,F2: int > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F2 @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_533_image__Collect__subsetI,axiom,
    ! [P: product_prod_int_int > $o,F2: product_prod_int_int > product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( member5262025264175285858nt_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ ( collec213857154873943460nt_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_534_image__Collect__subsetI,axiom,
    ! [P: produc1219242969750017639nt_int > $o,F2: produc1219242969750017639nt_int > int,B3: set_int] :
      ( ! [X3: produc1219242969750017639nt_int] :
          ( ( P @ X3 )
         => ( member_int @ ( F2 @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_868701835606132046nt_int @ F2 @ ( collec7447955092554649554nt_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_535_Setcompr__eq__image,axiom,
    ! [F2: int > nat,A: set_int] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_int @ X @ A ) ) )
      = ( image_int_nat @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_536_Setcompr__eq__image,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_nat @ X @ A ) ) )
      = ( image_nat_nat @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_537_Setcompr__eq__image,axiom,
    ! [F2: int > int,A: set_int] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_int @ X @ A ) ) )
      = ( image_int_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_538_Setcompr__eq__image,axiom,
    ! [F2: nat > int,A: set_nat] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_nat @ X @ A ) ) )
      = ( image_nat_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_539_Setcompr__eq__image,axiom,
    ! [F2: int > product_prod_int_int,A: set_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_int @ X @ A ) ) )
      = ( image_5705468584675977158nt_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_540_Setcompr__eq__image,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_nat @ X @ A ) ) )
      = ( image_2667626500211843362nt_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_541_Setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member5262025264175285858nt_int @ X @ A ) ) )
      = ( image_5044651549707136836nt_nat @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_542_Setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member5262025264175285858nt_int @ X @ A ) ) )
      = ( image_5042161079198086560nt_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_543_Setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member5262025264175285858nt_int @ X @ A ) ) )
      = ( image_2653370878348428101nt_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_544_Setcompr__eq__image,axiom,
    ! [F2: int > produc1219242969750017639nt_int,A: set_int] :
      ( ( collec7447955092554649554nt_int
        @ ^ [Uu: produc1219242969750017639nt_int] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_int @ X @ A ) ) )
      = ( image_6390838258573006580nt_int @ F2 @ A ) ) ).

% Setcompr_eq_image
thf(fact_545_setcompr__eq__image,axiom,
    ! [F2: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_546_setcompr__eq__image,axiom,
    ! [F2: int > nat,P: int > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_int_nat @ F2 @ ( collect_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_547_setcompr__eq__image,axiom,
    ! [F2: nat > int,P: nat > $o] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_int @ F2 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_548_setcompr__eq__image,axiom,
    ! [F2: int > int,P: int > $o] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_int_int @ F2 @ ( collect_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_549_setcompr__eq__image,axiom,
    ! [F2: nat > product_prod_int_int,P: nat > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_2667626500211843362nt_int @ F2 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_550_setcompr__eq__image,axiom,
    ! [F2: int > product_prod_int_int,P: int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_5705468584675977158nt_int @ F2 @ ( collect_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_551_setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_5044651549707136836nt_nat @ F2 @ ( collec213857154873943460nt_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_552_setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > int,P: product_prod_int_int > $o] :
      ( ( collect_int
        @ ^ [Uu: int] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_5042161079198086560nt_int @ F2 @ ( collec213857154873943460nt_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_553_setcompr__eq__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [Uu: product_prod_int_int] :
          ? [X: product_prod_int_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_2653370878348428101nt_int @ F2 @ ( collec213857154873943460nt_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_554_setcompr__eq__image,axiom,
    ! [F2: produc1219242969750017639nt_int > nat,P: produc1219242969750017639nt_int > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: produc1219242969750017639nt_int] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_871192306115182322nt_nat @ F2 @ ( collec7447955092554649554nt_int @ P ) ) ) ).

% setcompr_eq_image
thf(fact_555_finite__surj,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_nat,F2: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_556_finite__surj,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_5042161079198086560nt_int @ F2 @ A ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_557_finite__surj,axiom,
    ! [A: set_nat,B3: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_558_finite__surj,axiom,
    ! [A: set_nat,B3: set_int,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F2 @ A ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_559_finite__surj,axiom,
    ! [A: set_int,B3: set_nat,F2: int > nat] :
      ( ( finite_finite_int @ A )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F2 @ A ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_560_finite__surj,axiom,
    ! [A: set_int,B3: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F2 @ A ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_561_finite__surj,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
       => ( finite2998713641127702882nt_int @ B3 ) ) ) ).

% finite_surj
thf(fact_562_finite__surj,axiom,
    ! [A: set_nat,B3: set_Pr958786334691620121nt_int,F2: nat > product_prod_int_int] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2667626500211843362nt_int @ F2 @ A ) )
       => ( finite2998713641127702882nt_int @ B3 ) ) ) ).

% finite_surj
thf(fact_563_finite__surj,axiom,
    ! [A: set_int,B3: set_Pr958786334691620121nt_int,F2: int > product_prod_int_int] :
      ( ( finite_finite_int @ A )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_5705468584675977158nt_int @ F2 @ A ) )
       => ( finite2998713641127702882nt_int @ B3 ) ) ) ).

% finite_surj
thf(fact_564_finite__subset__image,axiom,
    ! [B3: set_nat,F2: nat > nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A )
            & ( finite_finite_nat @ C4 )
            & ( B3
              = ( image_nat_nat @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_565_finite__subset__image,axiom,
    ! [B3: set_nat,F2: int > nat,A: set_int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F2 @ A ) )
       => ? [C4: set_int] :
            ( ( ord_less_eq_set_int @ C4 @ A )
            & ( finite_finite_int @ C4 )
            & ( B3
              = ( image_int_nat @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_566_finite__subset__image,axiom,
    ! [B3: set_int,F2: nat > int,A: set_nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F2 @ A ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A )
            & ( finite_finite_nat @ C4 )
            & ( B3
              = ( image_nat_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_567_finite__subset__image,axiom,
    ! [B3: set_int,F2: int > int,A: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F2 @ A ) )
       => ? [C4: set_int] :
            ( ( ord_less_eq_set_int @ C4 @ A )
            & ( finite_finite_int @ C4 )
            & ( B3
              = ( image_int_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_568_finite__subset__image,axiom,
    ! [B3: set_nat,F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
       => ? [C4: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ C4 @ A )
            & ( finite2998713641127702882nt_int @ C4 )
            & ( B3
              = ( image_5044651549707136836nt_nat @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_569_finite__subset__image,axiom,
    ! [B3: set_int,F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_5042161079198086560nt_int @ F2 @ A ) )
       => ? [C4: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ C4 @ A )
            & ( finite2998713641127702882nt_int @ C4 )
            & ( B3
              = ( image_5042161079198086560nt_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_570_finite__subset__image,axiom,
    ! [B3: set_Pr958786334691620121nt_int,F2: nat > product_prod_int_int,A: set_nat] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2667626500211843362nt_int @ F2 @ A ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A )
            & ( finite_finite_nat @ C4 )
            & ( B3
              = ( image_2667626500211843362nt_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_571_finite__subset__image,axiom,
    ! [B3: set_Pr958786334691620121nt_int,F2: int > product_prod_int_int,A: set_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_5705468584675977158nt_int @ F2 @ A ) )
       => ? [C4: set_int] :
            ( ( ord_less_eq_set_int @ C4 @ A )
            & ( finite_finite_int @ C4 )
            & ( B3
              = ( image_5705468584675977158nt_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_572_finite__subset__image,axiom,
    ! [B3: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ B3 )
     => ( ( ord_le2843351958646193337nt_int @ B3 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
       => ? [C4: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ C4 @ A )
            & ( finite2998713641127702882nt_int @ C4 )
            & ( B3
              = ( image_2653370878348428101nt_int @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_573_ex__finite__subset__image,axiom,
    ! [F2: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A )
            & ( P @ ( image_nat_nat @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_574_ex__finite__subset__image,axiom,
    ! [F2: int > nat,A: set_int,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ A )
            & ( P @ ( image_int_nat @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_575_ex__finite__subset__image,axiom,
    ! [F2: nat > int,A: set_nat,P: set_int > $o] :
      ( ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A )
            & ( P @ ( image_nat_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_576_ex__finite__subset__image,axiom,
    ! [F2: int > int,A: set_int,P: set_int > $o] :
      ( ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ A )
            & ( P @ ( image_int_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_577_ex__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ A )
            & ( P @ ( image_5044651549707136836nt_nat @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_578_ex__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: set_int > $o] :
      ( ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ ( image_5042161079198086560nt_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ A )
            & ( P @ ( image_5042161079198086560nt_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_579_ex__finite__subset__image,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ ( image_2667626500211843362nt_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A )
            & ( P @ ( image_2667626500211843362nt_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_580_ex__finite__subset__image,axiom,
    ! [F2: int > product_prod_int_int,A: set_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ ( image_5705468584675977158nt_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_int] :
            ( ( finite_finite_int @ B2 )
            & ( ord_less_eq_set_int @ B2 @ A )
            & ( P @ ( image_5705468584675977158nt_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_581_ex__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ ( image_2653370878348428101nt_int @ F2 @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_Pr958786334691620121nt_int] :
            ( ( finite2998713641127702882nt_int @ B2 )
            & ( ord_le2843351958646193337nt_int @ B2 @ A )
            & ( P @ ( image_2653370878348428101nt_int @ F2 @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_582_all__finite__subset__image,axiom,
    ! [F2: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A ) )
           => ( P @ ( image_nat_nat @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_583_all__finite__subset__image,axiom,
    ! [F2: int > nat,A: set_int,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ A ) )
           => ( P @ ( image_int_nat @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_584_all__finite__subset__image,axiom,
    ! [F2: nat > int,A: set_nat,P: set_int > $o] :
      ( ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A ) )
           => ( P @ ( image_nat_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_585_all__finite__subset__image,axiom,
    ! [F2: int > int,A: set_int,P: set_int > $o] :
      ( ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ A ) )
           => ( P @ ( image_int_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_586_all__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_5044651549707136836nt_nat @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ A ) )
           => ( P @ ( image_5044651549707136836nt_nat @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_587_all__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: set_int > $o] :
      ( ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ ( image_5042161079198086560nt_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ A ) )
           => ( P @ ( image_5042161079198086560nt_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_588_all__finite__subset__image,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ ( image_2667626500211843362nt_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A ) )
           => ( P @ ( image_2667626500211843362nt_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_589_all__finite__subset__image,axiom,
    ! [F2: int > product_prod_int_int,A: set_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ ( image_5705468584675977158nt_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_int] :
            ( ( ( finite_finite_int @ B2 )
              & ( ord_less_eq_set_int @ B2 @ A ) )
           => ( P @ ( image_5705468584675977158nt_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_590_all__finite__subset__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ ( image_2653370878348428101nt_int @ F2 @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_Pr958786334691620121nt_int] :
            ( ( ( finite2998713641127702882nt_int @ B2 )
              & ( ord_le2843351958646193337nt_int @ B2 @ A ) )
           => ( P @ ( image_2653370878348428101nt_int @ F2 @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_591_finite__row__exec,axiom,
    ! [M: nat] : ( finite_finite_int @ ( row_exec @ M ) ) ).

% finite_row_exec
thf(fact_592_relInvImage__mono,axiom,
    ! [R1: set_Pr958786334691620121nt_int,R22: set_Pr958786334691620121nt_int,A: set_int,F2: int > int] :
      ( ( ord_le2843351958646193337nt_int @ R1 @ R22 )
     => ( ord_le2843351958646193337nt_int @ ( bNF_Gr9017224929539862555nt_int @ A @ R1 @ F2 ) @ ( bNF_Gr9017224929539862555nt_int @ A @ R22 @ F2 ) ) ) ).

% relInvImage_mono
thf(fact_593_relImage__mono,axiom,
    ! [R1: set_Pr958786334691620121nt_int,R22: set_Pr958786334691620121nt_int,F2: int > int] :
      ( ( ord_le2843351958646193337nt_int @ R1 @ R22 )
     => ( ord_le2843351958646193337nt_int @ ( bNF_Gr4175811214798380094nt_int @ R1 @ F2 ) @ ( bNF_Gr4175811214798380094nt_int @ R22 @ F2 ) ) ) ).

% relImage_mono
thf(fact_594_card__row__exec,axiom,
    ! [M: nat] :
      ( ( finite_card_int @ ( row_exec @ M ) )
      = M ) ).

% card_row_exec
thf(fact_595_card__image__le,axiom,
    ! [A: set_Product_unit,F2: product_unit > int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_873079544045703924it_int @ F2 @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image_le
thf(fact_596_card__image__le,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image_le
thf(fact_597_card__image__le,axiom,
    ! [A: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F2 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).

% card_image_le
thf(fact_598_card__image__le,axiom,
    ! [A: set_nat,F2: nat > product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F2 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).

% card_image_le
thf(fact_599_card__image__le,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_int @ F2 @ A ) ) @ ( finite_card_int @ A ) ) ) ).

% card_image_le
thf(fact_600_card__image__le,axiom,
    ! [A: set_int,F2: int > product_unit] :
      ( ( finite_finite_int @ A )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) ) @ ( finite_card_int @ A ) ) ) ).

% card_image_le
thf(fact_601_card__image__le,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_prod_int_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ ( image_4103462098156874993nt_int @ F2 @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image_le
thf(fact_602_card__image__le,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_5042161079198086560nt_int @ F2 @ A ) ) @ ( finite6756421564338198497nt_int @ A ) ) ) ).

% card_image_le
thf(fact_603_card__image__le,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) ) @ ( finite6756421564338198497nt_int @ A ) ) ) ).

% card_image_le
thf(fact_604_card__image__le,axiom,
    ! [A: set_nat,F2: nat > product_prod_int_int] :
      ( ( finite_finite_nat @ A )
     => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).

% card_image_le
thf(fact_605_image2__eqI,axiom,
    ! [B7: int,F2: int > int,X2: int,C2: int,G3: int > int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_int @ X2 @ A )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B7 @ C2 ) @ ( bNF_Gr8686673574714534880nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_606_image2__eqI,axiom,
    ! [B7: int,F2: nat > int,X2: nat,C2: int,G3: nat > int,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_nat @ X2 @ A )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B7 @ C2 ) @ ( bNF_Gr8315020014477690244nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_607_image2__eqI,axiom,
    ! [B7: int,F2: product_prod_int_int > int,X2: product_prod_int_int,C2: int,G3: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B7 @ C2 ) @ ( bNF_Gr4426612819127333049nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_608_image2__eqI,axiom,
    ! [B7: product_prod_int_int,F2: int > product_prod_int_int,X2: int,C2: product_prod_int_int,G3: int > product_prod_int_int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_int @ X2 @ A )
         => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B7 @ C2 ) @ ( bNF_Gr3379968192745358150nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_609_image2__eqI,axiom,
    ! [B7: product_prod_int_int,F2: nat > product_prod_int_int,X2: nat,C2: product_prod_int_int,G3: nat > product_prod_int_int,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_nat @ X2 @ A )
         => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B7 @ C2 ) @ ( bNF_Gr4976608462507087594nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_610_image2__eqI,axiom,
    ! [B7: product_prod_int_int,F2: product_prod_int_int > product_prod_int_int,X2: product_prod_int_int,C2: product_prod_int_int,G3: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B7 @ C2 ) @ ( bNF_Gr133173068567543455nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_611_image2__eqI,axiom,
    ! [B7: int > option6357759511663192854e_term,F2: int > int > option6357759511663192854e_term,X2: int,C2: product_prod_int_int,G3: int > product_prod_int_int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_int @ X2 @ A )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B7 @ C2 ) @ ( bNF_Gr8978185744112786704nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_612_image2__eqI,axiom,
    ! [B7: int > option6357759511663192854e_term,F2: nat > int > option6357759511663192854e_term,X2: nat,C2: product_prod_int_int,G3: nat > product_prod_int_int,A: set_nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_nat @ X2 @ A )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B7 @ C2 ) @ ( bNF_Gr9055198396104480364nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_613_image2__eqI,axiom,
    ! [B7: int > option6357759511663192854e_term,F2: product_prod_int_int > int > option6357759511663192854e_term,X2: product_prod_int_int,C2: product_prod_int_int,G3: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B7 @ C2 ) @ ( bNF_Gr4342778003469861111nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_614_image2__eqI,axiom,
    ! [B7: produc8551481072490612790e_term > option6357759511663192854e_term,F2: int > produc8551481072490612790e_term > option6357759511663192854e_term,X2: int,C2: product_prod_int_int,G3: int > product_prod_int_int,A: set_int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G3 @ X2 ) )
       => ( ( member_int @ X2 @ A )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ B7 @ C2 ) @ ( bNF_Gr556869251677793414nt_int @ A @ F2 @ G3 ) ) ) ) ) ).

% image2_eqI
thf(fact_615_card__le__inj,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) )
         => ? [F4: product_unit > product_unit] :
              ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F4 @ A ) @ B3 )
              & ( inj_on8151373323710067377t_unit @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_616_card__le__inj,axiom,
    ! [A: set_Product_unit,B3: set_nat] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B3 ) )
         => ? [F4: product_unit > nat] :
              ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F4 @ A ) @ B3 )
              & ( inj_on8430439091780834860it_nat @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_617_card__le__inj,axiom,
    ! [A: set_Product_unit,B3: set_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_int @ B3 ) )
         => ? [F4: product_unit > int] :
              ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F4 @ A ) @ B3 )
              & ( inj_on8427948621271784584it_int @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_618_card__le__inj,axiom,
    ! [A: set_nat,B3: set_Product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B3 ) )
         => ? [F4: nat > product_unit] :
              ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F4 @ A ) @ B3 )
              & ( inj_on7061601236592826506t_unit @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_619_card__le__inj,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
         => ? [F4: nat > nat] :
              ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ B3 )
              & ( inj_on_nat_nat @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_620_card__le__inj,axiom,
    ! [A: set_nat,B3: set_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) )
         => ? [F4: nat > int] :
              ( ( ord_less_eq_set_int @ ( image_nat_int @ F4 @ A ) @ B3 )
              & ( inj_on_nat_int @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_621_card__le__inj,axiom,
    ! [A: set_int,B3: set_Product_unit] :
      ( ( finite_finite_int @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) )
         => ? [F4: int > product_unit] :
              ( ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F4 @ A ) @ B3 )
              & ( inj_on4178023183795006766t_unit @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_622_card__le__inj,axiom,
    ! [A: set_int,B3: set_nat] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) )
         => ? [F4: int > nat] :
              ( ( ord_less_eq_set_nat @ ( image_int_nat @ F4 @ A ) @ B3 )
              & ( inj_on_int_nat @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_623_card__le__inj,axiom,
    ! [A: set_int,B3: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) )
         => ? [F4: int > int] :
              ( ( ord_less_eq_set_int @ ( image_int_int @ F4 @ A ) @ B3 )
              & ( inj_on_int_int @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_624_card__le__inj,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) )
         => ? [F4: product_prod_int_int > product_unit] :
              ( ( ord_le3507040750410214029t_unit @ ( image_2335562724295632793t_unit @ F4 @ A ) @ B3 )
              & ( inj_on4340349633117443333t_unit @ F4 @ A ) ) ) ) ) ).

% card_le_inj
thf(fact_625_card__inj__on__le,axiom,
    ! [F2: int > product_unit,A: set_int,B3: set_Product_unit] :
      ( ( inj_on4178023183795006766t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) @ B3 )
       => ( ( finite4290736615968046902t_unit @ B3 )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_626_card__inj__on__le,axiom,
    ! [F2: product_unit > product_unit,A: set_Product_unit,B3: set_Product_unit] :
      ( ( inj_on8151373323710067377t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) @ B3 )
       => ( ( finite4290736615968046902t_unit @ B3 )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_627_card__inj__on__le,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_628_card__inj__on__le,axiom,
    ! [F2: int > nat,A: set_int,B3: set_nat] :
      ( ( inj_on_int_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_629_card__inj__on__le,axiom,
    ! [F2: product_unit > nat,A: set_Product_unit,B3: set_nat] :
      ( ( inj_on8430439091780834860it_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_630_card__inj__on__le,axiom,
    ! [F2: int > int,A: set_int,B3: set_int] :
      ( ( inj_on_int_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_631_card__inj__on__le,axiom,
    ! [F2: product_unit > int,A: set_Product_unit,B3: set_int] :
      ( ( inj_on8427948621271784584it_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_632_card__inj__on__le,axiom,
    ! [F2: product_prod_int_int > product_unit,A: set_Pr958786334691620121nt_int,B3: set_Product_unit] :
      ( ( inj_on4340349633117443333t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) @ B3 )
       => ( ( finite4290736615968046902t_unit @ B3 )
         => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_633_card__inj__on__le,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int,B3: set_nat] :
      ( ( inj_on4736580515542835928nt_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_634_card__inj__on__le,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,B3: set_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_635_inj__on__iff__card__le,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ? [F3: product_unit > product_unit] :
                ( ( inj_on8151373323710067377t_unit @ F3 @ A )
                & ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_636_inj__on__iff__card__le,axiom,
    ! [A: set_Product_unit,B3: set_nat] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ? [F3: product_unit > nat] :
                ( ( inj_on8430439091780834860it_nat @ F3 @ A )
                & ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_637_inj__on__iff__card__le,axiom,
    ! [A: set_Product_unit,B3: set_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ? [F3: product_unit > int] :
                ( ( inj_on8427948621271784584it_int @ F3 @ A )
                & ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_638_inj__on__iff__card__le,axiom,
    ! [A: set_nat,B3: set_Product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ? [F3: nat > product_unit] :
                ( ( inj_on7061601236592826506t_unit @ F3 @ A )
                & ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_639_inj__on__iff__card__le,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ? [F3: nat > nat] :
                ( ( inj_on_nat_nat @ F3 @ A )
                & ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_640_inj__on__iff__card__le,axiom,
    ! [A: set_nat,B3: set_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ? [F3: nat > int] :
                ( ( inj_on_nat_int @ F3 @ A )
                & ( ord_less_eq_set_int @ ( image_nat_int @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_641_inj__on__iff__card__le,axiom,
    ! [A: set_int,B3: set_Product_unit] :
      ( ( finite_finite_int @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ? [F3: int > product_unit] :
                ( ( inj_on4178023183795006766t_unit @ F3 @ A )
                & ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_642_inj__on__iff__card__le,axiom,
    ! [A: set_int,B3: set_nat] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ? [F3: int > nat] :
                ( ( inj_on_int_nat @ F3 @ A )
                & ( ord_less_eq_set_nat @ ( image_int_nat @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_643_inj__on__iff__card__le,axiom,
    ! [A: set_int,B3: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( ( ? [F3: int > int] :
                ( ( inj_on_int_int @ F3 @ A )
                & ( ord_less_eq_set_int @ ( image_int_int @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_644_inj__on__iff__card__le,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite4290736615968046902t_unit @ B3 )
       => ( ( ? [F3: product_prod_int_int > product_unit] :
                ( ( inj_on4340349633117443333t_unit @ F3 @ A )
                & ( ord_le3507040750410214029t_unit @ ( image_2335562724295632793t_unit @ F3 @ A ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_645_small__lazy_H_Ocases,axiom,
    ! [X2: product_prod_int_int] :
      ~ ! [D: int,I2: int] :
          ( X2
         != ( product_Pair_int_int @ D @ I2 ) ) ).

% small_lazy'.cases
thf(fact_646_exhaustive__int_H_Ocases,axiom,
    ! [X2: produc7773217078559923341nt_int] :
      ~ ! [F4: int > option6357759511663192854e_term,D: int,I2: int] :
          ( X2
         != ( produc4305682042979456191nt_int @ F4 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).

% exhaustive_int'.cases
thf(fact_647_UNIV__I,axiom,
    ! [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ top_to4366644338036079209nt_int ) ).

% UNIV_I
thf(fact_648_UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_649_UNIV__I,axiom,
    ! [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).

% UNIV_I
thf(fact_650_UNIV__I,axiom,
    ! [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).

% UNIV_I
thf(fact_651_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_652_finite__Plus__UNIV__iff,axiom,
    ( ( finite2009855664264564338at_int @ top_to4171737849581180865at_int )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_653_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_654_finite__Plus__UNIV__iff,axiom,
    ( ( finite7187060395674815602nt_nat @ top_to8848742569205929409nt_nat )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_655_finite__Plus__UNIV__iff,axiom,
    ( ( finite3009209376165618894nt_int @ top_to6358659424274202653nt_int )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_656_finite__Plus__UNIV__iff,axiom,
    ( ( finite1443934553334965505t_unit @ top_to7020036278376945288t_unit )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_657_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_658_finite__Plus__UNIV__iff,axiom,
    ( ( finite224101892120063507it_int @ top_to404534460850747034it_int )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_659_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_660_finite__Plus__UNIV__iff,axiom,
    ( ( finite2360936127647704163nt_nat @ top_to1358739243333571306nt_nat )
    = ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_661_mem__Sigma__iff,axiom,
    ! [A4: int,B7: nat,A: set_int,B3: int > set_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A4 @ B7 ) @ ( produc1456381018704787142nt_nat @ A @ B3 ) )
      = ( ( member_int @ A4 @ A )
        & ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_662_mem__Sigma__iff,axiom,
    ! [A4: nat,B7: int,A: set_nat,B3: nat > set_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A4 @ B7 ) @ ( produc454536836294682310at_int @ A @ B3 ) )
      = ( ( member_nat @ A4 @ A )
        & ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_663_mem__Sigma__iff,axiom,
    ! [A4: nat,B7: nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B7 ) @ ( produc457027306803732586at_nat @ A @ B3 ) )
      = ( ( member_nat @ A4 @ A )
        & ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_664_mem__Sigma__iff,axiom,
    ! [A4: int,B7: int,A: set_int,B3: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B7 ) @ ( produc1453890548195736866nt_int @ A @ B3 ) )
      = ( ( member_int @ A4 @ A )
        & ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_665_mem__Sigma__iff,axiom,
    ! [A4: product_prod_int_int,B7: int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ A4 @ B7 ) @ ( produc1005374827195948829nt_int @ A @ B3 ) )
      = ( ( member5262025264175285858nt_int @ A4 @ A )
        & ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_666_mem__Sigma__iff,axiom,
    ! [A4: product_prod_int_int,B7: nat,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( member5907334269321869775nt_nat @ ( produc8909286205500071986nt_nat @ A4 @ B7 ) @ ( produc1007865297704999105nt_nat @ A @ B3 ) )
      = ( ( member5262025264175285858nt_int @ A4 @ A )
        & ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_667_mem__Sigma__iff,axiom,
    ! [A4: int,B7: product_prod_int_int,A: set_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ A4 @ B7 ) @ ( produc1668682332673839427nt_int @ A @ B3 ) )
      = ( ( member_int @ A4 @ A )
        & ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_668_mem__Sigma__iff,axiom,
    ! [A4: nat,B7: product_prod_int_int,A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( member1920110580185997301nt_int @ ( produc6532261156004778512nt_int @ A4 @ B7 ) @ ( produc7854212285064481439nt_int @ A @ B3 ) )
      = ( ( member_nat @ A4 @ A )
        & ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_669_mem__Sigma__iff,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A4 @ B7 ) @ ( produc245848125267896712nt_int @ A @ B3 ) )
      = ( ( member5262025264175285858nt_int @ A4 @ A )
        & ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_670_mem__Sigma__iff,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A: set_in3461395444621081367e_term,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B7 ) @ ( produc1318478031590598990nt_int @ A @ B3 ) )
      = ( ( member8845023287901829240e_term @ A4 @ A )
        & ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_671_SigmaI,axiom,
    ! [A4: int,A: set_int,B7: nat,B3: int > set_nat] :
      ( ( member_int @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A4 @ B7 ) @ ( produc1456381018704787142nt_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_672_SigmaI,axiom,
    ! [A4: nat,A: set_nat,B7: int,B3: nat > set_int] :
      ( ( member_nat @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A4 @ B7 ) @ ( produc454536836294682310at_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_673_SigmaI,axiom,
    ! [A4: nat,A: set_nat,B7: nat,B3: nat > set_nat] :
      ( ( member_nat @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B7 ) @ ( produc457027306803732586at_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_674_SigmaI,axiom,
    ! [A4: int,A: set_int,B7: int,B3: int > set_int] :
      ( ( member_int @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B7 ) @ ( produc1453890548195736866nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_675_SigmaI,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: int,B3: product_prod_int_int > set_int] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ A4 @ B7 ) @ ( produc1005374827195948829nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_676_SigmaI,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: nat,B3: product_prod_int_int > set_nat] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member5907334269321869775nt_nat @ ( produc8909286205500071986nt_nat @ A4 @ B7 ) @ ( produc1007865297704999105nt_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_677_SigmaI,axiom,
    ! [A4: int,A: set_int,B7: product_prod_int_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( member_int @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ A4 @ B7 ) @ ( produc1668682332673839427nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_678_SigmaI,axiom,
    ! [A4: nat,A: set_nat,B7: product_prod_int_int,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( member_nat @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member1920110580185997301nt_int @ ( produc6532261156004778512nt_int @ A4 @ B7 ) @ ( produc7854212285064481439nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_679_SigmaI,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: product_prod_int_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A4 @ B7 ) @ ( produc245848125267896712nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_680_SigmaI,axiom,
    ! [A4: int > option6357759511663192854e_term,A: set_in3461395444621081367e_term,B7: product_prod_int_int,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member8845023287901829240e_term @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B7 ) @ ( produc1318478031590598990nt_int @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_681_finite__Collect__not,axiom,
    ! [P: produc2285326912895808259nt_int > $o] :
      ( ( finite1225166462400338330nt_int @ ( collec1790188477890212312nt_int @ P ) )
     => ( ( finite1225166462400338330nt_int
          @ ( collec1790188477890212312nt_int
            @ ^ [X: produc2285326912895808259nt_int] :
                ~ ( P @ X ) ) )
        = ( finite1225166462400338330nt_int @ top_to3181862456159035625nt_int ) ) ) ).

% finite_Collect_not
thf(fact_682_finite__Collect__not,axiom,
    ! [P: produc7773217078559923341nt_int > $o] :
      ( ( finite1606273004500963236nt_int @ ( collec506566255779805410nt_int @ P ) )
     => ( ( finite1606273004500963236nt_int
          @ ( collec506566255779805410nt_int
            @ ^ [X: produc7773217078559923341nt_int] :
                ~ ( P @ X ) ) )
        = ( finite1606273004500963236nt_int @ top_to2069137843433766899nt_int ) ) ) ).

% finite_Collect_not
thf(fact_683_finite__Collect__not,axiom,
    ! [P: produc1219242969750017639nt_int > $o] :
      ( ( finite4752125741994067600nt_int @ ( collec7447955092554649554nt_int @ P ) )
     => ( ( finite4752125741994067600nt_int
          @ ( collec7447955092554649554nt_int
            @ ^ [X: produc1219242969750017639nt_int] :
                ~ ( P @ X ) ) )
        = ( finite4752125741994067600nt_int @ top_to2321149100101803671nt_int ) ) ) ).

% finite_Collect_not
thf(fact_684_finite__Collect__not,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ( ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ~ ( P @ X ) ) )
        = ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ) ).

% finite_Collect_not
thf(fact_685_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_686_finite__Collect__not,axiom,
    ! [P: int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ~ ( P @ X ) ) )
        = ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_Collect_not
thf(fact_687_finite__Collect__not,axiom,
    ! [P: product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
     => ( ( finite4290736615968046902t_unit
          @ ( collect_Product_unit
            @ ^ [X: product_unit] :
                ~ ( P @ X ) ) )
        = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_Collect_not
thf(fact_688_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_689_UNIV__Times__UNIV,axiom,
    ( ( produc454536836294682310at_int @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_int )
    = top_to2179722763343057421at_int ) ).

% UNIV_Times_UNIV
thf(fact_690_UNIV__Times__UNIV,axiom,
    ( ( produc5506539147108004339t_unit @ top_top_set_nat
      @ ^ [Uu: nat] : top_to1996260823553986621t_unit )
    = top_to8544742955230171288t_unit ) ).

% UNIV_Times_UNIV
thf(fact_691_UNIV__Times__UNIV,axiom,
    ( ( produc1456381018704787142nt_nat @ top_top_set_int
      @ ^ [Uu: int] : top_top_set_nat )
    = top_to6856727482967805965nt_nat ) ).

% UNIV_Times_UNIV
thf(fact_692_UNIV__Times__UNIV,axiom,
    ( ( produc1453890548195736866nt_int @ top_top_set_int
      @ ^ [Uu: int] : top_top_set_int )
    = top_to4366644338036079209nt_int ) ).

% UNIV_Times_UNIV
thf(fact_693_UNIV__Times__UNIV,axiom,
    ( ( produc2622961094310184599t_unit @ top_top_set_int
      @ ^ [Uu: int] : top_to1996260823553986621t_unit )
    = top_to876157113852465980t_unit ) ).

% UNIV_Times_UNIV
thf(fact_694_UNIV__Times__UNIV,axiom,
    ( ( produc6875377002296012693it_nat @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_top_set_nat )
    = top_to5974110478112770290it_nat ) ).

% UNIV_Times_UNIV
thf(fact_695_UNIV__Times__UNIV,axiom,
    ( ( produc6872886531786962417it_int @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_top_set_int )
    = top_to3484027333181043534it_int ) ).

% UNIV_Times_UNIV
thf(fact_696_UNIV__Times__UNIV,axiom,
    ( ( produc6121607364029222536t_unit @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_to1996260823553986621t_unit )
    = top_to1835807148980544151t_unit ) ).

% UNIV_Times_UNIV
thf(fact_697_finite__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [A2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ A2 @ A )
           => ( finite2998713641127702882nt_int @ ( B3 @ A2 ) ) )
       => ( finite4752125741994067600nt_int @ ( produc245848125267896712nt_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_698_finite__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [A2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ A2 @ A )
           => ( finite_finite_nat @ ( B3 @ A2 ) ) )
       => ( finite2121020344308190927nt_nat @ ( produc1007865297704999105nt_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_699_finite__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [A2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ A2 @ A )
           => ( finite_finite_int @ ( B3 @ A2 ) ) )
       => ( finite7166541361653770027nt_int @ ( produc1005374827195948829nt_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_700_finite__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite2998713641127702882nt_int @ ( B3 @ A2 ) ) )
       => ( finite7357168692027094261nt_int @ ( produc7854212285064481439nt_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_701_finite__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_nat @ ( B3 @ A2 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_702_finite__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_int] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_int @ ( B3 @ A2 ) ) )
       => ( finite1999359929226648326at_int @ ( produc454536836294682310at_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_703_finite__SigmaI,axiom,
    ! [A: set_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ A )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ( finite2998713641127702882nt_int @ ( B3 @ A2 ) ) )
       => ( finite1171638739636452249nt_int @ ( produc1668682332673839427nt_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_704_finite__SigmaI,axiom,
    ! [A: set_int,B3: int > set_nat] :
      ( ( finite_finite_int @ A )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ( finite_finite_nat @ ( B3 @ A2 ) ) )
       => ( finite7176564660636899590nt_nat @ ( produc1456381018704787142nt_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_705_finite__SigmaI,axiom,
    ! [A: set_int,B3: int > set_int] :
      ( ( finite_finite_int @ A )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ( finite_finite_int @ ( B3 @ A2 ) ) )
       => ( finite2998713641127702882nt_int @ ( produc1453890548195736866nt_int @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_706_UNIV__def,axiom,
    ( top_to4366644338036079209nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X: product_prod_int_int] : $true ) ) ).

% UNIV_def
thf(fact_707_UNIV__def,axiom,
    ( top_to3181862456159035625nt_int
    = ( collec1790188477890212312nt_int
      @ ^ [X: produc2285326912895808259nt_int] : $true ) ) ).

% UNIV_def
thf(fact_708_UNIV__def,axiom,
    ( top_to2069137843433766899nt_int
    = ( collec506566255779805410nt_int
      @ ^ [X: produc7773217078559923341nt_int] : $true ) ) ).

% UNIV_def
thf(fact_709_UNIV__def,axiom,
    ( top_to2321149100101803671nt_int
    = ( collec7447955092554649554nt_int
      @ ^ [X: produc1219242969750017639nt_int] : $true ) ) ).

% UNIV_def
thf(fact_710_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_711_UNIV__def,axiom,
    ( top_top_set_int
    = ( collect_int
      @ ^ [X: int] : $true ) ) ).

% UNIV_def
thf(fact_712_UNIV__def,axiom,
    ( top_to1996260823553986621t_unit
    = ( collect_Product_unit
      @ ^ [X: product_unit] : $true ) ) ).

% UNIV_def
thf(fact_713_UNIV__eq__I,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ! [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A )
     => ( top_to4366644338036079209nt_int = A ) ) ).

% UNIV_eq_I
thf(fact_714_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_715_UNIV__eq__I,axiom,
    ! [A: set_int] :
      ( ! [X3: int] : ( member_int @ X3 @ A )
     => ( top_top_set_int = A ) ) ).

% UNIV_eq_I
thf(fact_716_UNIV__eq__I,axiom,
    ! [A: set_Product_unit] :
      ( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A )
     => ( top_to1996260823553986621t_unit = A ) ) ).

% UNIV_eq_I
thf(fact_717_UNIV__witness,axiom,
    ? [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ top_to4366644338036079209nt_int ) ).

% UNIV_witness
thf(fact_718_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_719_UNIV__witness,axiom,
    ? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).

% UNIV_witness
thf(fact_720_UNIV__witness,axiom,
    ? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).

% UNIV_witness
thf(fact_721_finite__inverse__image,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_722_finite__inverse__image,axiom,
    ! [A: set_int,F2: nat > int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on_nat_int @ F2 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_723_finite__inverse__image,axiom,
    ! [A: set_nat,F2: int > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_int_nat @ F2 @ top_top_set_int )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] : ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_724_finite__inverse__image,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on_int_int @ F2 @ top_top_set_int )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] : ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_725_finite__inverse__image,axiom,
    ! [A: set_nat,F2: product_unit > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on8430439091780834860it_nat @ F2 @ top_to1996260823553986621t_unit )
       => ( finite4290736615968046902t_unit
          @ ( collect_Product_unit
            @ ^ [J: product_unit] : ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_726_finite__inverse__image,axiom,
    ! [A: set_int,F2: product_unit > int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on8427948621271784584it_int @ F2 @ top_to1996260823553986621t_unit )
       => ( finite4290736615968046902t_unit
          @ ( collect_Product_unit
            @ ^ [J: product_unit] : ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_727_finite__inverse__image,axiom,
    ! [A: set_nat,F2: product_prod_int_int > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on4736580515542835928nt_nat @ F2 @ top_to4366644338036079209nt_int )
       => ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [J: product_prod_int_int] : ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_728_finite__inverse__image,axiom,
    ! [A: set_int,F2: product_prod_int_int > int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on4734090045033785652nt_int @ F2 @ top_to4366644338036079209nt_int )
       => ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [J: product_prod_int_int] : ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_729_finite__inverse__image,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: nat > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on2359555466047542454nt_int @ F2 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member5262025264175285858nt_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_730_finite__inverse__image,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on5397397550511676250nt_int @ F2 @ top_top_set_int )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] : ( member5262025264175285858nt_int @ ( F2 @ J ) @ A ) ) ) ) ) ).

% finite_inverse_image
thf(fact_731_finite__UNIV__surj__inj,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
     => ( ( ( image_2653370878348428101nt_int @ F2 @ top_to4366644338036079209nt_int )
          = top_to4366644338036079209nt_int )
       => ( inj_on6454360653267994289nt_int @ F2 @ top_to4366644338036079209nt_int ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_732_finite__UNIV__surj__inj,axiom,
    ! [F2: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
          = top_top_set_nat )
       => ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_733_finite__UNIV__surj__inj,axiom,
    ! [F2: int > int] :
      ( ( finite_finite_int @ top_top_set_int )
     => ( ( ( image_int_int @ F2 @ top_top_set_int )
          = top_top_set_int )
       => ( inj_on_int_int @ F2 @ top_top_set_int ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_734_finite__UNIV__surj__inj,axiom,
    ! [F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
          = top_to1996260823553986621t_unit )
       => ( inj_on8151373323710067377t_unit @ F2 @ top_to1996260823553986621t_unit ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_735_finite__UNIV__inj__surj,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
     => ( ( inj_on6454360653267994289nt_int @ F2 @ top_to4366644338036079209nt_int )
       => ( ( image_2653370878348428101nt_int @ F2 @ top_to4366644338036079209nt_int )
          = top_to4366644338036079209nt_int ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_736_finite__UNIV__inj__surj,axiom,
    ! [F2: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
       => ( ( image_nat_nat @ F2 @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_737_finite__UNIV__inj__surj,axiom,
    ! [F2: int > int] :
      ( ( finite_finite_int @ top_top_set_int )
     => ( ( inj_on_int_int @ F2 @ top_top_set_int )
       => ( ( image_int_int @ F2 @ top_top_set_int )
          = top_top_set_int ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_738_finite__UNIV__inj__surj,axiom,
    ! [F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( ( inj_on8151373323710067377t_unit @ F2 @ top_to1996260823553986621t_unit )
       => ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
          = top_to1996260823553986621t_unit ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_739_SigmaE2,axiom,
    ! [A4: int,B7: nat,A: set_int,B3: int > set_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A4 @ B7 ) @ ( produc1456381018704787142nt_nat @ A @ B3 ) )
     => ~ ( ( member_int @ A4 @ A )
         => ~ ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_740_SigmaE2,axiom,
    ! [A4: nat,B7: int,A: set_nat,B3: nat > set_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A4 @ B7 ) @ ( produc454536836294682310at_int @ A @ B3 ) )
     => ~ ( ( member_nat @ A4 @ A )
         => ~ ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_741_SigmaE2,axiom,
    ! [A4: nat,B7: nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B7 ) @ ( produc457027306803732586at_nat @ A @ B3 ) )
     => ~ ( ( member_nat @ A4 @ A )
         => ~ ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_742_SigmaE2,axiom,
    ! [A4: int,B7: int,A: set_int,B3: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B7 ) @ ( produc1453890548195736866nt_int @ A @ B3 ) )
     => ~ ( ( member_int @ A4 @ A )
         => ~ ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_743_SigmaE2,axiom,
    ! [A4: product_prod_int_int,B7: int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ A4 @ B7 ) @ ( produc1005374827195948829nt_int @ A @ B3 ) )
     => ~ ( ( member5262025264175285858nt_int @ A4 @ A )
         => ~ ( member_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_744_SigmaE2,axiom,
    ! [A4: product_prod_int_int,B7: nat,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( member5907334269321869775nt_nat @ ( produc8909286205500071986nt_nat @ A4 @ B7 ) @ ( produc1007865297704999105nt_nat @ A @ B3 ) )
     => ~ ( ( member5262025264175285858nt_int @ A4 @ A )
         => ~ ( member_nat @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_745_SigmaE2,axiom,
    ! [A4: int,B7: product_prod_int_int,A: set_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ A4 @ B7 ) @ ( produc1668682332673839427nt_int @ A @ B3 ) )
     => ~ ( ( member_int @ A4 @ A )
         => ~ ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_746_SigmaE2,axiom,
    ! [A4: nat,B7: product_prod_int_int,A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( member1920110580185997301nt_int @ ( produc6532261156004778512nt_int @ A4 @ B7 ) @ ( produc7854212285064481439nt_int @ A @ B3 ) )
     => ~ ( ( member_nat @ A4 @ A )
         => ~ ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_747_SigmaE2,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A4 @ B7 ) @ ( produc245848125267896712nt_int @ A @ B3 ) )
     => ~ ( ( member5262025264175285858nt_int @ A4 @ A )
         => ~ ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_748_SigmaE2,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A: set_in3461395444621081367e_term,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B7 ) @ ( produc1318478031590598990nt_int @ A @ B3 ) )
     => ~ ( ( member8845023287901829240e_term @ A4 @ A )
         => ~ ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_749_SigmaD2,axiom,
    ! [A4: int,B7: int,A: set_int,B3: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B7 ) @ ( produc1453890548195736866nt_int @ A @ B3 ) )
     => ( member_int @ B7 @ ( B3 @ A4 ) ) ) ).

% SigmaD2
thf(fact_750_SigmaD2,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A4 @ B7 ) @ ( produc245848125267896712nt_int @ A @ B3 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ).

% SigmaD2
thf(fact_751_SigmaD2,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A: set_in3461395444621081367e_term,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B7 ) @ ( produc1318478031590598990nt_int @ A @ B3 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ).

% SigmaD2
thf(fact_752_SigmaD2,axiom,
    ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int,A: set_Pr3642885161833720865e_term,B3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A4 @ B7 ) @ ( produc5808185664642557892nt_int @ A @ B3 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) ) ) ).

% SigmaD2
thf(fact_753_SigmaD1,axiom,
    ! [A4: int,B7: int,A: set_int,B3: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B7 ) @ ( produc1453890548195736866nt_int @ A @ B3 ) )
     => ( member_int @ A4 @ A ) ) ).

% SigmaD1
thf(fact_754_SigmaD1,axiom,
    ! [A4: product_prod_int_int,B7: product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A4 @ B7 ) @ ( produc245848125267896712nt_int @ A @ B3 ) )
     => ( member5262025264175285858nt_int @ A4 @ A ) ) ).

% SigmaD1
thf(fact_755_SigmaD1,axiom,
    ! [A4: int > option6357759511663192854e_term,B7: product_prod_int_int,A: set_in3461395444621081367e_term,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B7 ) @ ( produc1318478031590598990nt_int @ A @ B3 ) )
     => ( member8845023287901829240e_term @ A4 @ A ) ) ).

% SigmaD1
thf(fact_756_SigmaD1,axiom,
    ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int,A: set_Pr3642885161833720865e_term,B3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A4 @ B7 ) @ ( produc5808185664642557892nt_int @ A @ B3 ) )
     => ( member3222579708246209666e_term @ A4 @ A ) ) ).

% SigmaD1
thf(fact_757_SigmaE,axiom,
    ! [C2: product_prod_int_nat,A: set_int,B3: int > set_nat] :
      ( ( member216504246829706758nt_nat @ C2 @ ( produc1456381018704787142nt_nat @ A @ B3 ) )
     => ~ ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( product_Pair_int_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_758_SigmaE,axiom,
    ! [C2: product_prod_nat_int,A: set_nat,B3: nat > set_int] :
      ( ( member4262671552274231302at_int @ C2 @ ( produc454536836294682310at_int @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: int] :
                ( ( member_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_759_SigmaE,axiom,
    ! [C2: product_prod_nat_nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_760_SigmaE,axiom,
    ! [C2: product_prod_int_int,A: set_int,B3: int > set_int] :
      ( ( member5262025264175285858nt_int @ C2 @ ( produc1453890548195736866nt_int @ A @ B3 ) )
     => ~ ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ! [Y3: int] :
                ( ( member_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( product_Pair_int_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_761_SigmaE,axiom,
    ! [C2: produc6754623152930779156nt_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( member1729483249812673067nt_int @ C2 @ ( produc1005374827195948829nt_int @ A @ B3 ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ! [Y3: int] :
                ( ( member_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc8906795734991021710nt_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_762_SigmaE,axiom,
    ! [C2: produc1709102135585200056nt_nat,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( member5907334269321869775nt_nat @ C2 @ ( produc1007865297704999105nt_nat @ A @ B3 ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc8909286205500071986nt_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_763_SigmaE,axiom,
    ! [C2: produc759720530913461378nt_int,A: set_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( member4957952664650131097nt_int @ C2 @ ( produc1668682332673839427nt_int @ A @ B3 ) )
     => ~ ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc346731203614136500nt_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_764_SigmaE,axiom,
    ! [C2: produc6945250483304103390nt_int,A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( member1920110580185997301nt_int @ C2 @ ( produc7854212285064481439nt_int @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc6532261156004778512nt_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_765_SigmaE,axiom,
    ! [C2: produc1219242969750017639nt_int,A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ C2 @ ( produc245848125267896712nt_int @ A @ B3 ) )
     => ~ ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc3646306378393792727nt_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_766_SigmaE,axiom,
    ! [C2: produc7773217078559923341nt_int,A: set_in3461395444621081367e_term,B3: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ C2 @ ( produc1318478031590598990nt_int @ A @ B3 ) )
     => ~ ! [X3: int > option6357759511663192854e_term] :
            ( ( member8845023287901829240e_term @ X3 @ A )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B3 @ X3 ) )
               => ( C2
                 != ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_767_inj__on__image__Fpow,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A ) ) ) ).

% inj_on_image_Fpow
thf(fact_768_top_Oextremum__uniqueI,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
     => ( A4 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_769_top_Oextremum__uniqueI,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ top_top_set_int @ A4 )
     => ( A4 = top_top_set_int ) ) ).

% top.extremum_uniqueI
thf(fact_770_top_Oextremum__uniqueI,axiom,
    ! [A4: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A4 )
     => ( A4 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_uniqueI
thf(fact_771_top_Oextremum__uniqueI,axiom,
    ! [A4: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ top_to4366644338036079209nt_int @ A4 )
     => ( A4 = top_to4366644338036079209nt_int ) ) ).

% top.extremum_uniqueI
thf(fact_772_top_Oextremum__unique,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
      = ( A4 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_773_top_Oextremum__unique,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ top_top_set_int @ A4 )
      = ( A4 = top_top_set_int ) ) ).

% top.extremum_unique
thf(fact_774_top_Oextremum__unique,axiom,
    ! [A4: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A4 )
      = ( A4 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_unique
thf(fact_775_top_Oextremum__unique,axiom,
    ! [A4: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ top_to4366644338036079209nt_int @ A4 )
      = ( A4 = top_to4366644338036079209nt_int ) ) ).

% top.extremum_unique
thf(fact_776_top__greatest,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).

% top_greatest
thf(fact_777_top__greatest,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ top_top_set_int ) ).

% top_greatest
thf(fact_778_top__greatest,axiom,
    ! [A4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A4 @ top_to1996260823553986621t_unit ) ).

% top_greatest
thf(fact_779_top__greatest,axiom,
    ! [A4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A4 @ top_to4366644338036079209nt_int ) ).

% top_greatest
thf(fact_780_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_781_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_int @ top_top_set_int )
     => ( finite1999359929226648326at_int @ top_to2179722763343057421at_int ) ) ) ).

% finite_Prod_UNIV
thf(fact_782_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_783_finite__Prod__UNIV,axiom,
    ( ( finite_finite_int @ top_top_set_int )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_784_finite__Prod__UNIV,axiom,
    ( ( finite_finite_int @ top_top_set_int )
   => ( ( finite_finite_int @ top_top_set_int )
     => ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ) ).

% finite_Prod_UNIV
thf(fact_785_finite__Prod__UNIV,axiom,
    ( ( finite_finite_int @ top_top_set_int )
   => ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( finite2229504458203871597t_unit @ top_to876157113852465980t_unit ) ) ) ).

% finite_Prod_UNIV
thf(fact_786_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_787_finite__Prod__UNIV,axiom,
    ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
   => ( ( finite_finite_int @ top_top_set_int )
     => ( finite1009671796988969599it_int @ top_to3484027333181043534it_int ) ) ) ).

% finite_Prod_UNIV
thf(fact_788_finite__Prod__UNIV,axiom,
    ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
   => ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).

% finite_Prod_UNIV
thf(fact_789_finite__Prod__UNIV,axiom,
    ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite2121020344308190927nt_nat @ top_to2571144570446186910nt_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_790_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_791_finite__prod,axiom,
    ( ( finite1999359929226648326at_int @ top_to2179722763343057421at_int )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_prod
thf(fact_792_finite__prod,axiom,
    ( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_793_finite__prod,axiom,
    ( ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_794_finite__prod,axiom,
    ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_prod
thf(fact_795_finite__prod,axiom,
    ( ( finite2229504458203871597t_unit @ top_to876157113852465980t_unit )
    = ( ( finite_finite_int @ top_top_set_int )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_796_finite__prod,axiom,
    ( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_797_finite__prod,axiom,
    ( ( finite1009671796988969599it_int @ top_to3484027333181043534it_int )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite_finite_int @ top_top_set_int ) ) ) ).

% finite_prod
thf(fact_798_finite__prod,axiom,
    ( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
    = ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
      & ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).

% finite_prod
thf(fact_799_finite__prod,axiom,
    ( ( finite2121020344308190927nt_nat @ top_to2571144570446186910nt_nat )
    = ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_800_Finite__Set_Ofinite__set,axiom,
    ( ( finite8744585540193469122nt_int @ top_to6034159466715884489nt_int )
    = ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ).

% Finite_Set.finite_set
thf(fact_801_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_802_Finite__Set_Ofinite__set,axiom,
    ( ( finite6197958912794628473et_int @ top_top_set_set_int )
    = ( finite_finite_int @ top_top_set_int ) ) ).

% Finite_Set.finite_set
thf(fact_803_Finite__Set_Ofinite__set,axiom,
    ( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% Finite_Set.finite_set
thf(fact_804_rangeI,axiom,
    ! [F2: nat > product_prod_int_int,X2: nat] : ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_805_rangeI,axiom,
    ! [F2: nat > int,X2: nat] : ( member_int @ ( F2 @ X2 ) @ ( image_nat_int @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_806_rangeI,axiom,
    ! [F2: nat > nat,X2: nat] : ( member_nat @ ( F2 @ X2 ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_807_rangeI,axiom,
    ! [F2: int > product_prod_int_int,X2: int] : ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) ) ).

% rangeI
thf(fact_808_rangeI,axiom,
    ! [F2: int > int,X2: int] : ( member_int @ ( F2 @ X2 ) @ ( image_int_int @ F2 @ top_top_set_int ) ) ).

% rangeI
thf(fact_809_rangeI,axiom,
    ! [F2: int > nat,X2: int] : ( member_nat @ ( F2 @ X2 ) @ ( image_int_nat @ F2 @ top_top_set_int ) ) ).

% rangeI
thf(fact_810_rangeI,axiom,
    ! [F2: product_unit > product_prod_int_int,X2: product_unit] : ( member5262025264175285858nt_int @ ( F2 @ X2 ) @ ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) ) ).

% rangeI
thf(fact_811_rangeI,axiom,
    ! [F2: product_unit > int,X2: product_unit] : ( member_int @ ( F2 @ X2 ) @ ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) ) ).

% rangeI
thf(fact_812_rangeI,axiom,
    ! [F2: product_unit > nat,X2: product_unit] : ( member_nat @ ( F2 @ X2 ) @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ).

% rangeI
thf(fact_813_range__eqI,axiom,
    ! [B7: product_prod_int_int,F2: nat > product_prod_int_int,X2: nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_814_range__eqI,axiom,
    ! [B7: int,F2: nat > int,X2: nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_int @ B7 @ ( image_nat_int @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_815_range__eqI,axiom,
    ! [B7: nat,F2: nat > nat,X2: nat] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_nat @ B7 @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_816_range__eqI,axiom,
    ! [B7: product_prod_int_int,F2: int > product_prod_int_int,X2: int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) ) ) ).

% range_eqI
thf(fact_817_range__eqI,axiom,
    ! [B7: int,F2: int > int,X2: int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_int @ B7 @ ( image_int_int @ F2 @ top_top_set_int ) ) ) ).

% range_eqI
thf(fact_818_range__eqI,axiom,
    ! [B7: nat,F2: int > nat,X2: int] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_nat @ B7 @ ( image_int_nat @ F2 @ top_top_set_int ) ) ) ).

% range_eqI
thf(fact_819_range__eqI,axiom,
    ! [B7: product_prod_int_int,F2: product_unit > product_prod_int_int,X2: product_unit] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member5262025264175285858nt_int @ B7 @ ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) ) ) ).

% range_eqI
thf(fact_820_range__eqI,axiom,
    ! [B7: int,F2: product_unit > int,X2: product_unit] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_int @ B7 @ ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) ) ) ).

% range_eqI
thf(fact_821_range__eqI,axiom,
    ! [B7: nat,F2: product_unit > nat,X2: product_unit] :
      ( ( B7
        = ( F2 @ X2 ) )
     => ( member_nat @ B7 @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ) ).

% range_eqI
thf(fact_822_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_823_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_char_0
thf(fact_824_ex__new__if__finite,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ~ ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
     => ( ( finite2998713641127702882nt_int @ A )
       => ? [A2: product_prod_int_int] :
            ~ ( member5262025264175285858nt_int @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_825_ex__new__if__finite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A )
       => ? [A2: nat] :
            ~ ( member_nat @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_826_ex__new__if__finite,axiom,
    ! [A: set_int] :
      ( ~ ( finite_finite_int @ top_top_set_int )
     => ( ( finite_finite_int @ A )
       => ? [A2: int] :
            ~ ( member_int @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_827_ex__new__if__finite,axiom,
    ! [A: set_Product_unit] :
      ( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( ( finite4290736615968046902t_unit @ A )
       => ? [A2: product_unit] :
            ~ ( member_Product_unit @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_828_finite__UNIV,axiom,
    finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).

% finite_UNIV
thf(fact_829_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_830_subset__UNIV,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).

% subset_UNIV
thf(fact_831_subset__UNIV,axiom,
    ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).

% subset_UNIV
thf(fact_832_subset__UNIV,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ top_to4366644338036079209nt_int ) ).

% subset_UNIV
thf(fact_833_inj__on__convol__ident,axiom,
    ! [F2: int > int,X5: set_int] :
      ( inj_on5397397550511676250nt_int
      @ ^ [X: int] : ( product_Pair_int_int @ X @ ( F2 @ X ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_834_inj__on__convol__ident,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,X5: set_Pr958786334691620121nt_int] :
      ( inj_on2740507337494710623nt_int
      @ ^ [X: product_prod_int_int] : ( produc3646306378393792727nt_int @ X @ ( F2 @ X ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_835_inj__on__convol__ident,axiom,
    ! [F2: ( int > option6357759511663192854e_term ) > product_prod_int_int,X5: set_in3461395444621081367e_term] :
      ( inj_on5552665323793282529nt_int
      @ ^ [X: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ X @ ( F2 @ X ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_836_inj__on__convol__ident,axiom,
    ! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int,X5: set_Pr3642885161833720865e_term] :
      ( inj_on4793825870093415905nt_int
      @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term] : ( produc5700946648718959541nt_int @ X @ ( F2 @ X ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_837_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F2: nat > nat,D2: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_nat_nat @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_838_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F2: int > nat,D2: set_int] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_int_nat @ F2 @ D2 )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] :
                ( ( member_int @ J @ D2 )
                & ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_839_finite__inverse__image__gen,axiom,
    ! [A: set_int,F2: nat > int,D2: set_nat] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on_nat_int @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_840_finite__inverse__image__gen,axiom,
    ! [A: set_int,F2: int > int,D2: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on_int_int @ F2 @ D2 )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] :
                ( ( member_int @ J @ D2 )
                & ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_841_finite__inverse__image__gen,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: nat > product_prod_int_int,D2: set_nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on2359555466047542454nt_int @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member5262025264175285858nt_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_842_finite__inverse__image__gen,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: int > product_prod_int_int,D2: set_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on5397397550511676250nt_int @ F2 @ D2 )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [J: int] :
                ( ( member_int @ J @ D2 )
                & ( member5262025264175285858nt_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_843_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F2: product_prod_int_int > nat,D2: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on4736580515542835928nt_nat @ F2 @ D2 )
       => ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [J: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ J @ D2 )
                & ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_844_finite__inverse__image__gen,axiom,
    ! [A: set_int,F2: product_prod_int_int > int,D2: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on4734090045033785652nt_int @ F2 @ D2 )
       => ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [J: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ J @ D2 )
                & ( member_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_845_finite__inverse__image__gen,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,D2: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on6454360653267994289nt_int @ F2 @ D2 )
       => ( finite2998713641127702882nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [J: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ J @ D2 )
                & ( member5262025264175285858nt_int @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_846_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F2: produc1219242969750017639nt_int > nat,D2: set_Pr2560585780119916871nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on4424309335961437318nt_nat @ F2 @ D2 )
       => ( finite4752125741994067600nt_int
          @ ( collec7447955092554649554nt_int
            @ ^ [J: produc1219242969750017639nt_int] :
                ( ( member8566619992076573584nt_int @ J @ D2 )
                & ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_847_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_848_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_849_rangeE,axiom,
    ! [B7: product_prod_int_int,F2: nat > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_850_rangeE,axiom,
    ! [B7: int,F2: nat > int] :
      ( ( member_int @ B7 @ ( image_nat_int @ F2 @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_851_rangeE,axiom,
    ! [B7: nat,F2: nat > nat] :
      ( ( member_nat @ B7 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_852_rangeE,axiom,
    ! [B7: product_prod_int_int,F2: int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) )
     => ~ ! [X3: int] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_853_rangeE,axiom,
    ! [B7: int,F2: int > int] :
      ( ( member_int @ B7 @ ( image_int_int @ F2 @ top_top_set_int ) )
     => ~ ! [X3: int] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_854_rangeE,axiom,
    ! [B7: nat,F2: int > nat] :
      ( ( member_nat @ B7 @ ( image_int_nat @ F2 @ top_top_set_int ) )
     => ~ ! [X3: int] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_855_rangeE,axiom,
    ! [B7: product_prod_int_int,F2: product_unit > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ B7 @ ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) )
     => ~ ! [X3: product_unit] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_856_rangeE,axiom,
    ! [B7: int,F2: product_unit > int] :
      ( ( member_int @ B7 @ ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) )
     => ~ ! [X3: product_unit] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_857_rangeE,axiom,
    ! [B7: nat,F2: product_unit > nat] :
      ( ( member_nat @ B7 @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) )
     => ~ ! [X3: product_unit] :
            ( B7
           != ( F2 @ X3 ) ) ) ).

% rangeE
thf(fact_858_finite__imageD,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) )
     => ( ( inj_on6454360653267994289nt_int @ F2 @ A )
       => ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_imageD
thf(fact_859_finite__imageD,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat] :
      ( ( finite2998713641127702882nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) )
     => ( ( inj_on2359555466047542454nt_int @ F2 @ A )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_imageD
thf(fact_860_finite__imageD,axiom,
    ! [F2: int > product_prod_int_int,A: set_int] :
      ( ( finite2998713641127702882nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) )
     => ( ( inj_on5397397550511676250nt_int @ F2 @ A )
       => ( finite_finite_int @ A ) ) ) ).

% finite_imageD
thf(fact_861_finite__imageD,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
     => ( ( inj_on4736580515542835928nt_nat @ F2 @ A )
       => ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_imageD
thf(fact_862_finite__imageD,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
     => ( ( inj_on_nat_nat @ F2 @ A )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_imageD
thf(fact_863_finite__imageD,axiom,
    ! [F2: int > nat,A: set_int] :
      ( ( finite_finite_nat @ ( image_int_nat @ F2 @ A ) )
     => ( ( inj_on_int_nat @ F2 @ A )
       => ( finite_finite_int @ A ) ) ) ).

% finite_imageD
thf(fact_864_finite__imageD,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
     => ( ( inj_on4734090045033785652nt_int @ F2 @ A )
       => ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_imageD
thf(fact_865_finite__imageD,axiom,
    ! [F2: nat > int,A: set_nat] :
      ( ( finite_finite_int @ ( image_nat_int @ F2 @ A ) )
     => ( ( inj_on_nat_int @ F2 @ A )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_imageD
thf(fact_866_finite__imageD,axiom,
    ! [F2: int > int,A: set_int] :
      ( ( finite_finite_int @ ( image_int_int @ F2 @ A ) )
     => ( ( inj_on_int_int @ F2 @ A )
       => ( finite_finite_int @ A ) ) ) ).

% finite_imageD
thf(fact_867_finite__image__iff,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on6454360653267994289nt_int @ F2 @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) )
        = ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_image_iff
thf(fact_868_finite__image__iff,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_869_finite__image__iff,axiom,
    ! [F2: int > product_prod_int_int,A: set_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ A )
     => ( ( finite2998713641127702882nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) )
        = ( finite_finite_int @ A ) ) ) ).

% finite_image_iff
thf(fact_870_finite__image__iff,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4736580515542835928nt_nat @ F2 @ A )
     => ( ( finite_finite_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
        = ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_image_iff
thf(fact_871_finite__image__iff,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_872_finite__image__iff,axiom,
    ! [F2: int > nat,A: set_int] :
      ( ( inj_on_int_nat @ F2 @ A )
     => ( ( finite_finite_nat @ ( image_int_nat @ F2 @ A ) )
        = ( finite_finite_int @ A ) ) ) ).

% finite_image_iff
thf(fact_873_finite__image__iff,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ A )
     => ( ( finite_finite_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
        = ( finite2998713641127702882nt_int @ A ) ) ) ).

% finite_image_iff
thf(fact_874_finite__image__iff,axiom,
    ! [F2: nat > int,A: set_nat] :
      ( ( inj_on_nat_int @ F2 @ A )
     => ( ( finite_finite_int @ ( image_nat_int @ F2 @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_875_finite__image__iff,axiom,
    ! [F2: int > int,A: set_int] :
      ( ( inj_on_int_int @ F2 @ A )
     => ( ( finite_finite_int @ ( image_int_int @ F2 @ A ) )
        = ( finite_finite_int @ A ) ) ) ).

% finite_image_iff
thf(fact_876_card__image,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
        = ( finite_card_nat @ A ) ) ) ).

% card_image
thf(fact_877_card__image,axiom,
    ! [F2: int > int,A: set_int] :
      ( ( inj_on_int_int @ F2 @ A )
     => ( ( finite_card_int @ ( image_int_int @ F2 @ A ) )
        = ( finite_card_int @ A ) ) ) ).

% card_image
thf(fact_878_card__image,axiom,
    ! [F2: product_unit > int,A: set_Product_unit] :
      ( ( inj_on8427948621271784584it_int @ F2 @ A )
     => ( ( finite_card_int @ ( image_873079544045703924it_int @ F2 @ A ) )
        = ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image
thf(fact_879_card__image,axiom,
    ! [F2: int > product_unit,A: set_int] :
      ( ( inj_on4178023183795006766t_unit @ F2 @ A )
     => ( ( finite410649719033368117t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) )
        = ( finite_card_int @ A ) ) ) ).

% card_image
thf(fact_880_card__image,axiom,
    ! [F2: product_unit > product_unit,A: set_Product_unit] :
      ( ( inj_on8151373323710067377t_unit @ F2 @ A )
     => ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) )
        = ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image
thf(fact_881_card__image,axiom,
    ! [F2: int > product_prod_int_int,A: set_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ A )
     => ( ( finite6756421564338198497nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) )
        = ( finite_card_int @ A ) ) ) ).

% card_image
thf(fact_882_card__image,axiom,
    ! [F2: product_unit > product_prod_int_int,A: set_Product_unit] :
      ( ( inj_on6108249006978685533nt_int @ F2 @ A )
     => ( ( finite6756421564338198497nt_int @ ( image_4103462098156874993nt_int @ F2 @ A ) )
        = ( finite410649719033368117t_unit @ A ) ) ) ).

% card_image
thf(fact_883_card__image,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ A )
     => ( ( finite_card_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
        = ( finite6756421564338198497nt_int @ A ) ) ) ).

% card_image
thf(fact_884_card__image,axiom,
    ! [F2: product_prod_int_int > product_unit,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4340349633117443333t_unit @ F2 @ A )
     => ( ( finite410649719033368117t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) )
        = ( finite6756421564338198497nt_int @ A ) ) ) ).

% card_image
thf(fact_885_card__image,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on6454360653267994289nt_int @ F2 @ A )
     => ( ( finite6756421564338198497nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) )
        = ( finite6756421564338198497nt_int @ A ) ) ) ).

% card_image
thf(fact_886_Sigma__mono,axiom,
    ! [A: set_int,C: set_int,B3: int > set_int,D2: int > set_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( ord_less_eq_set_int @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le2843351958646193337nt_int @ ( produc1453890548195736866nt_int @ A @ B3 ) @ ( produc1453890548195736866nt_int @ C @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_887_Sigma__mono,axiom,
    ! [A: set_int,C: set_int,B3: int > set_Pr958786334691620121nt_int,D2: int > set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( ord_le2843351958646193337nt_int @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le358265691127597848nt_int @ ( produc1668682332673839427nt_int @ A @ B3 ) @ ( produc1668682332673839427nt_int @ C @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_888_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B3: nat > set_Pr958786334691620121nt_int,D2: nat > set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ord_le2843351958646193337nt_int @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le1455348671771596404nt_int @ ( produc7854212285064481439nt_int @ A @ B3 ) @ ( produc7854212285064481439nt_int @ C @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_889_Sigma__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int,D2: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ C )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ( ord_le2843351958646193337nt_int @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le6090609446090860775nt_int @ ( produc245848125267896712nt_int @ A @ B3 ) @ ( produc245848125267896712nt_int @ C @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_890_range__subsetD,axiom,
    ! [F2: nat > int,B3: set_int,I4: nat] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ top_top_set_nat ) @ B3 )
     => ( member_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_891_range__subsetD,axiom,
    ! [F2: nat > nat,B3: set_nat,I4: nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ B3 )
     => ( member_nat @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_892_range__subsetD,axiom,
    ! [F2: int > int,B3: set_int,I4: int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ top_top_set_int ) @ B3 )
     => ( member_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_893_range__subsetD,axiom,
    ! [F2: int > nat,B3: set_nat,I4: int] :
      ( ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ top_top_set_int ) @ B3 )
     => ( member_nat @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_894_range__subsetD,axiom,
    ! [F2: product_unit > int,B3: set_int,I4: product_unit] :
      ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) @ B3 )
     => ( member_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_895_range__subsetD,axiom,
    ! [F2: product_unit > nat,B3: set_nat,I4: product_unit] :
      ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) @ B3 )
     => ( member_nat @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_896_range__subsetD,axiom,
    ! [F2: nat > product_prod_int_int,B3: set_Pr958786334691620121nt_int,I4: nat] :
      ( ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) @ B3 )
     => ( member5262025264175285858nt_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_897_range__subsetD,axiom,
    ! [F2: int > product_prod_int_int,B3: set_Pr958786334691620121nt_int,I4: int] :
      ( ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) @ B3 )
     => ( member5262025264175285858nt_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_898_range__subsetD,axiom,
    ! [F2: product_unit > product_prod_int_int,B3: set_Pr958786334691620121nt_int,I4: product_unit] :
      ( ( ord_le2843351958646193337nt_int @ ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) @ B3 )
     => ( member5262025264175285858nt_int @ ( F2 @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_899_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_Pr9222295170931077689nt_int] :
      ( ( finite1225166462400338330nt_int @ top_to3181862456159035625nt_int )
     => ( ( ( finite7179617391050183899nt_int @ A )
          = ( finite7179617391050183899nt_int @ top_to3181862456159035625nt_int ) )
       => ( A = top_to3181862456159035625nt_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_900_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_Pr1872883991513573699nt_int] :
      ( ( finite1606273004500963236nt_int @ top_to2069137843433766899nt_int )
     => ( ( ( finite1470244919013085797nt_int @ A )
          = ( finite1470244919013085797nt_int @ top_to2069137843433766899nt_int ) )
       => ( A = top_to2069137843433766899nt_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_901_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_Pr2560585780119916871nt_int] :
      ( ( finite4752125741994067600nt_int @ top_to2321149100101803671nt_int )
     => ( ( ( finite1566866390884520591nt_int @ A )
          = ( finite1566866390884520591nt_int @ top_to2321149100101803671nt_int ) )
       => ( A = top_to2321149100101803671nt_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_902_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
     => ( ( ( finite6756421564338198497nt_int @ A )
          = ( finite6756421564338198497nt_int @ top_to4366644338036079209nt_int ) )
       => ( A = top_to4366644338036079209nt_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_903_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( finite_card_nat @ A )
          = ( finite_card_nat @ top_top_set_nat ) )
       => ( A = top_top_set_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_904_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ top_top_set_int )
     => ( ( ( finite_card_int @ A )
          = ( finite_card_int @ top_top_set_int ) )
       => ( A = top_top_set_int ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_905_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
     => ( ( ( finite410649719033368117t_unit @ A )
          = ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
       => ( A = top_to1996260823553986621t_unit ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_906_Times__subset__cancel2,axiom,
    ! [X2: product_prod_int_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ X2 @ C )
     => ( ( ord_le6090609446090860775nt_int
          @ ( produc245848125267896712nt_int @ A
            @ ^ [Uu: product_prod_int_int] : C )
          @ ( produc245848125267896712nt_int @ B3
            @ ^ [Uu: product_prod_int_int] : C ) )
        = ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_907_Times__subset__cancel2,axiom,
    ! [X2: int,C: set_int,A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( member_int @ X2 @ C )
     => ( ( ord_le4608878813787753898nt_int
          @ ( produc1005374827195948829nt_int @ A
            @ ^ [Uu: product_prod_int_int] : C )
          @ ( produc1005374827195948829nt_int @ B3
            @ ^ [Uu: product_prod_int_int] : C ) )
        = ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_908_Times__subset__cancel2,axiom,
    ! [X2: nat,C: set_nat,A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( member_nat @ X2 @ C )
     => ( ( ord_le7098961958719480654nt_nat
          @ ( produc1007865297704999105nt_nat @ A
            @ ^ [Uu: product_prod_int_int] : C )
          @ ( produc1007865297704999105nt_nat @ B3
            @ ^ [Uu: product_prod_int_int] : C ) )
        = ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_909_Times__subset__cancel2,axiom,
    ! [X2: int,C: set_int,A: set_int,B3: set_int] :
      ( ( member_int @ X2 @ C )
     => ( ( ord_le2843351958646193337nt_int
          @ ( produc1453890548195736866nt_int @ A
            @ ^ [Uu: int] : C )
          @ ( produc1453890548195736866nt_int @ B3
            @ ^ [Uu: int] : C ) )
        = ( ord_less_eq_set_int @ A @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_910_finite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( finite4752125741994067600nt_int
          @ ( produc245848125267896712nt_int @ A
            @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_911_finite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( finite2121020344308190927nt_nat
          @ ( produc1007865297704999105nt_nat @ A
            @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_912_finite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( finite7166541361653770027nt_int
          @ ( produc1005374827195948829nt_int @ A
            @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_913_finite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( finite7357168692027094261nt_int
          @ ( produc7854212285064481439nt_int @ A
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_914_finite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_915_finite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ B3 )
       => ( finite1999359929226648326at_int
          @ ( produc454536836294682310at_int @ A
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_916_finite__cartesian__product,axiom,
    ! [A: set_int,B3: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite2998713641127702882nt_int @ B3 )
       => ( finite1171638739636452249nt_int
          @ ( produc1668682332673839427nt_int @ A
            @ ^ [Uu: int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_917_finite__cartesian__product,axiom,
    ! [A: set_int,B3: set_nat] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( finite7176564660636899590nt_nat
          @ ( produc1456381018704787142nt_nat @ A
            @ ^ [Uu: int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_918_finite__cartesian__product,axiom,
    ! [A: set_int,B3: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_int @ B3 )
       => ( finite2998713641127702882nt_int
          @ ( produc1453890548195736866nt_int @ A
            @ ^ [Uu: int] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_919_infinite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ~ ( finite2998713641127702882nt_int @ B3 )
       => ~ ( finite4752125741994067600nt_int
            @ ( produc245848125267896712nt_int @ A
              @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_920_infinite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_nat] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ~ ( finite_finite_nat @ B3 )
       => ~ ( finite2121020344308190927nt_nat
            @ ( produc1007865297704999105nt_nat @ A
              @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_921_infinite__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int] :
      ( ~ ( finite2998713641127702882nt_int @ A )
     => ( ~ ( finite_finite_int @ B3 )
       => ~ ( finite7166541361653770027nt_int
            @ ( produc1005374827195948829nt_int @ A
              @ ^ [Uu: product_prod_int_int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_922_infinite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_Pr958786334691620121nt_int] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite2998713641127702882nt_int @ B3 )
       => ~ ( finite7357168692027094261nt_int
            @ ( produc7854212285064481439nt_int @ A
              @ ^ [Uu: nat] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_923_infinite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_nat @ B3 )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A
              @ ^ [Uu: nat] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_924_infinite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_int] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_int @ B3 )
       => ~ ( finite1999359929226648326at_int
            @ ( produc454536836294682310at_int @ A
              @ ^ [Uu: nat] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_925_infinite__cartesian__product,axiom,
    ! [A: set_int,B3: set_Pr958786334691620121nt_int] :
      ( ~ ( finite_finite_int @ A )
     => ( ~ ( finite2998713641127702882nt_int @ B3 )
       => ~ ( finite1171638739636452249nt_int
            @ ( produc1668682332673839427nt_int @ A
              @ ^ [Uu: int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_926_infinite__cartesian__product,axiom,
    ! [A: set_int,B3: set_nat] :
      ( ~ ( finite_finite_int @ A )
     => ( ~ ( finite_finite_nat @ B3 )
       => ~ ( finite7176564660636899590nt_nat
            @ ( produc1456381018704787142nt_nat @ A
              @ ^ [Uu: int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_927_infinite__cartesian__product,axiom,
    ! [A: set_int,B3: set_int] :
      ( ~ ( finite_finite_int @ A )
     => ( ~ ( finite_finite_int @ B3 )
       => ~ ( finite2998713641127702882nt_int
            @ ( produc1453890548195736866nt_int @ A
              @ ^ [Uu: int] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_928_finite__range__imageI,axiom,
    ! [G3: nat > nat,F2: nat > nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ G3 @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat
          @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_929_finite__range__imageI,axiom,
    ! [G3: nat > nat,F2: nat > int] :
      ( ( finite_finite_nat @ ( image_nat_nat @ G3 @ top_top_set_nat ) )
     => ( finite_finite_int
        @ ( image_nat_int
          @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_930_finite__range__imageI,axiom,
    ! [G3: nat > int,F2: int > nat] :
      ( ( finite_finite_int @ ( image_nat_int @ G3 @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat
          @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_931_finite__range__imageI,axiom,
    ! [G3: nat > int,F2: int > int] :
      ( ( finite_finite_int @ ( image_nat_int @ G3 @ top_top_set_nat ) )
     => ( finite_finite_int
        @ ( image_nat_int
          @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_932_finite__range__imageI,axiom,
    ! [G3: int > nat,F2: nat > nat] :
      ( ( finite_finite_nat @ ( image_int_nat @ G3 @ top_top_set_int ) )
     => ( finite_finite_nat
        @ ( image_int_nat
          @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_int ) ) ) ).

% finite_range_imageI
thf(fact_933_finite__range__imageI,axiom,
    ! [G3: int > nat,F2: nat > int] :
      ( ( finite_finite_nat @ ( image_int_nat @ G3 @ top_top_set_int ) )
     => ( finite_finite_int
        @ ( image_int_int
          @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_int ) ) ) ).

% finite_range_imageI
thf(fact_934_finite__range__imageI,axiom,
    ! [G3: int > int,F2: int > nat] :
      ( ( finite_finite_int @ ( image_int_int @ G3 @ top_top_set_int ) )
     => ( finite_finite_nat
        @ ( image_int_nat
          @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_int ) ) ) ).

% finite_range_imageI
thf(fact_935_finite__range__imageI,axiom,
    ! [G3: int > int,F2: int > int] :
      ( ( finite_finite_int @ ( image_int_int @ G3 @ top_top_set_int ) )
     => ( finite_finite_int
        @ ( image_int_int
          @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
          @ top_top_set_int ) ) ) ).

% finite_range_imageI
thf(fact_936_finite__range__imageI,axiom,
    ! [G3: product_unit > nat,F2: nat > nat] :
      ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G3 @ top_to1996260823553986621t_unit ) )
     => ( finite_finite_nat
        @ ( image_875570014554754200it_nat
          @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
          @ top_to1996260823553986621t_unit ) ) ) ).

% finite_range_imageI
thf(fact_937_finite__range__imageI,axiom,
    ! [G3: product_unit > nat,F2: nat > int] :
      ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G3 @ top_to1996260823553986621t_unit ) )
     => ( finite_finite_int
        @ ( image_873079544045703924it_int
          @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
          @ top_to1996260823553986621t_unit ) ) ) ).

% finite_range_imageI
thf(fact_938_full__SetCompr__eq,axiom,
    ! [F2: nat > nat] :
      ( ( collect_nat
        @ ^ [U2: nat] :
          ? [X: nat] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_939_full__SetCompr__eq,axiom,
    ! [F2: nat > int] :
      ( ( collect_int
        @ ^ [U2: int] :
          ? [X: nat] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_nat_int @ F2 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_940_full__SetCompr__eq,axiom,
    ! [F2: int > nat] :
      ( ( collect_nat
        @ ^ [U2: nat] :
          ? [X: int] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_int_nat @ F2 @ top_top_set_int ) ) ).

% full_SetCompr_eq
thf(fact_941_full__SetCompr__eq,axiom,
    ! [F2: int > int] :
      ( ( collect_int
        @ ^ [U2: int] :
          ? [X: int] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_int_int @ F2 @ top_top_set_int ) ) ).

% full_SetCompr_eq
thf(fact_942_full__SetCompr__eq,axiom,
    ! [F2: product_unit > nat] :
      ( ( collect_nat
        @ ^ [U2: nat] :
          ? [X: product_unit] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ).

% full_SetCompr_eq
thf(fact_943_full__SetCompr__eq,axiom,
    ! [F2: product_unit > int] :
      ( ( collect_int
        @ ^ [U2: int] :
          ? [X: product_unit] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) ) ).

% full_SetCompr_eq
thf(fact_944_full__SetCompr__eq,axiom,
    ! [F2: nat > product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
          ? [X: nat] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_945_full__SetCompr__eq,axiom,
    ! [F2: int > product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
          ? [X: int] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) ) ).

% full_SetCompr_eq
thf(fact_946_full__SetCompr__eq,axiom,
    ! [F2: product_unit > product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
          ? [X: product_unit] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) ) ).

% full_SetCompr_eq
thf(fact_947_full__SetCompr__eq,axiom,
    ! [F2: nat > produc1219242969750017639nt_int] :
      ( ( collec7447955092554649554nt_int
        @ ^ [U2: produc1219242969750017639nt_int] :
          ? [X: nat] :
            ( U2
            = ( F2 @ X ) ) )
      = ( image_3636395739827918416nt_int @ F2 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_948_finite__surj__inj,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F2 @ A ) )
       => ( inj_on_nat_nat @ F2 @ A ) ) ) ).

% finite_surj_inj
thf(fact_949_finite__surj__inj,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( ord_less_eq_set_int @ A @ ( image_int_int @ F2 @ A ) )
       => ( inj_on_int_int @ F2 @ A ) ) ) ).

% finite_surj_inj
thf(fact_950_finite__surj__inj,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_le2843351958646193337nt_int @ A @ ( image_2653370878348428101nt_int @ F2 @ A ) )
       => ( inj_on6454360653267994289nt_int @ F2 @ A ) ) ) ).

% finite_surj_inj
thf(fact_951_inj__on__finite,axiom,
    ! [F2: product_prod_int_int > nat,A: set_Pr958786334691620121nt_int,B3: set_nat] :
      ( ( inj_on4736580515542835928nt_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( finite2998713641127702882nt_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_952_inj__on__finite,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( finite_finite_nat @ A ) ) ) ) ).

% inj_on_finite
thf(fact_953_inj__on__finite,axiom,
    ! [F2: int > nat,A: set_int,B3: set_nat] :
      ( ( inj_on_int_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( finite_finite_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_954_inj__on__finite,axiom,
    ! [F2: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,B3: set_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( finite2998713641127702882nt_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_955_inj__on__finite,axiom,
    ! [F2: nat > int,A: set_nat,B3: set_int] :
      ( ( inj_on_nat_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( finite_finite_nat @ A ) ) ) ) ).

% inj_on_finite
thf(fact_956_inj__on__finite,axiom,
    ! [F2: int > int,A: set_int,B3: set_int] :
      ( ( inj_on_int_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 )
       => ( ( finite_finite_int @ B3 )
         => ( finite_finite_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_957_inj__on__finite,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( inj_on6454360653267994289nt_int @ F2 @ A )
     => ( ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) @ B3 )
       => ( ( finite2998713641127702882nt_int @ B3 )
         => ( finite2998713641127702882nt_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_958_inj__on__finite,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat,B3: set_Pr958786334691620121nt_int] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ A )
     => ( ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) @ B3 )
       => ( ( finite2998713641127702882nt_int @ B3 )
         => ( finite_finite_nat @ A ) ) ) ) ).

% inj_on_finite
thf(fact_959_inj__on__finite,axiom,
    ! [F2: int > product_prod_int_int,A: set_int,B3: set_Pr958786334691620121nt_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ A )
     => ( ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) @ B3 )
       => ( ( finite2998713641127702882nt_int @ B3 )
         => ( finite_finite_int @ A ) ) ) ) ).

% inj_on_finite
thf(fact_960_endo__inj__surj,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ A )
       => ( ( inj_on_nat_nat @ F2 @ A )
         => ( ( image_nat_nat @ F2 @ A )
            = A ) ) ) ) ).

% endo_inj_surj
thf(fact_961_endo__inj__surj,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ A )
       => ( ( inj_on_int_int @ F2 @ A )
         => ( ( image_int_int @ F2 @ A )
            = A ) ) ) ) ).

% endo_inj_surj
thf(fact_962_endo__inj__surj,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) @ A )
       => ( ( inj_on6454360653267994289nt_int @ F2 @ A )
         => ( ( image_2653370878348428101nt_int @ F2 @ A )
            = A ) ) ) ) ).

% endo_inj_surj
thf(fact_963_inj__on__iff__eq__card,axiom,
    ! [A: set_Product_unit,F2: product_unit > int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( inj_on8427948621271784584it_int @ F2 @ A )
        = ( ( finite_card_int @ ( image_873079544045703924it_int @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_964_inj__on__iff__eq__card,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( inj_on8151373323710067377t_unit @ F2 @ A )
        = ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_965_inj__on__iff__eq__card,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_nat_nat @ F2 @ A )
        = ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_966_inj__on__iff__eq__card,axiom,
    ! [A: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_nat_int @ F2 @ A )
        = ( ( finite_card_int @ ( image_nat_int @ F2 @ A ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_967_inj__on__iff__eq__card,axiom,
    ! [A: set_nat,F2: nat > product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on7061601236592826506t_unit @ F2 @ A )
        = ( ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F2 @ A ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_968_inj__on__iff__eq__card,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on_int_int @ F2 @ A )
        = ( ( finite_card_int @ ( image_int_int @ F2 @ A ) )
          = ( finite_card_int @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_969_inj__on__iff__eq__card,axiom,
    ! [A: set_int,F2: int > product_unit] :
      ( ( finite_finite_int @ A )
     => ( ( inj_on4178023183795006766t_unit @ F2 @ A )
        = ( ( finite410649719033368117t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) )
          = ( finite_card_int @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_970_inj__on__iff__eq__card,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_prod_int_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( inj_on6108249006978685533nt_int @ F2 @ A )
        = ( ( finite6756421564338198497nt_int @ ( image_4103462098156874993nt_int @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_971_inj__on__iff__eq__card,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on4734090045033785652nt_int @ F2 @ A )
        = ( ( finite_card_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
          = ( finite6756421564338198497nt_int @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_972_inj__on__iff__eq__card,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( inj_on4340349633117443333t_unit @ F2 @ A )
        = ( ( finite410649719033368117t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) )
          = ( finite6756421564338198497nt_int @ A ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_973_eq__card__imp__inj__on,axiom,
    ! [A: set_Product_unit,F2: product_unit > int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ( finite_card_int @ ( image_873079544045703924it_int @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) )
       => ( inj_on8427948621271784584it_int @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_974_eq__card__imp__inj__on,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) )
       => ( inj_on8151373323710067377t_unit @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_975_eq__card__imp__inj__on,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
          = ( finite_card_nat @ A ) )
       => ( inj_on_nat_nat @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_976_eq__card__imp__inj__on,axiom,
    ! [A: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_int @ ( image_nat_int @ F2 @ A ) )
          = ( finite_card_nat @ A ) )
       => ( inj_on_nat_int @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_977_eq__card__imp__inj__on,axiom,
    ! [A: set_nat,F2: nat > product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F2 @ A ) )
          = ( finite_card_nat @ A ) )
       => ( inj_on7061601236592826506t_unit @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_978_eq__card__imp__inj__on,axiom,
    ! [A: set_int,F2: int > int] :
      ( ( finite_finite_int @ A )
     => ( ( ( finite_card_int @ ( image_int_int @ F2 @ A ) )
          = ( finite_card_int @ A ) )
       => ( inj_on_int_int @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_979_eq__card__imp__inj__on,axiom,
    ! [A: set_int,F2: int > product_unit] :
      ( ( finite_finite_int @ A )
     => ( ( ( finite410649719033368117t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) )
          = ( finite_card_int @ A ) )
       => ( inj_on4178023183795006766t_unit @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_980_eq__card__imp__inj__on,axiom,
    ! [A: set_Product_unit,F2: product_unit > product_prod_int_int] :
      ( ( finite4290736615968046902t_unit @ A )
     => ( ( ( finite6756421564338198497nt_int @ ( image_4103462098156874993nt_int @ F2 @ A ) )
          = ( finite410649719033368117t_unit @ A ) )
       => ( inj_on6108249006978685533nt_int @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_981_eq__card__imp__inj__on,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ( finite_card_int @ ( image_5042161079198086560nt_int @ F2 @ A ) )
          = ( finite6756421564338198497nt_int @ A ) )
       => ( inj_on4734090045033785652nt_int @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_982_eq__card__imp__inj__on,axiom,
    ! [A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( ( finite410649719033368117t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) )
          = ( finite6756421564338198497nt_int @ A ) )
       => ( inj_on4340349633117443333t_unit @ F2 @ A ) ) ) ).

% eq_card_imp_inj_on
thf(fact_983_surjective__iff__injective__gen,axiom,
    ! [S: set_Product_unit,T2: set_Product_unit,F2: product_unit > product_unit] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ( ( finite410649719033368117t_unit @ S )
            = ( finite410649719033368117t_unit @ T2 ) )
         => ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F2 @ S ) @ T2 )
           => ( ( ! [X: product_unit] :
                    ( ( member_Product_unit @ X @ T2 )
                   => ? [Y: product_unit] :
                        ( ( member_Product_unit @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on8151373323710067377t_unit @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_984_surjective__iff__injective__gen,axiom,
    ! [S: set_Product_unit,T2: set_nat,F2: product_unit > nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ( ( finite410649719033368117t_unit @ S )
            = ( finite_card_nat @ T2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F2 @ S ) @ T2 )
           => ( ( ! [X: nat] :
                    ( ( member_nat @ X @ T2 )
                   => ? [Y: product_unit] :
                        ( ( member_Product_unit @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on8430439091780834860it_nat @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_985_surjective__iff__injective__gen,axiom,
    ! [S: set_Product_unit,T2: set_int,F2: product_unit > int] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ( ( finite410649719033368117t_unit @ S )
            = ( finite_card_int @ T2 ) )
         => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F2 @ S ) @ T2 )
           => ( ( ! [X: int] :
                    ( ( member_int @ X @ T2 )
                   => ? [Y: product_unit] :
                        ( ( member_Product_unit @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on8427948621271784584it_int @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_986_surjective__iff__injective__gen,axiom,
    ! [S: set_nat,T2: set_Product_unit,F2: nat > product_unit] :
      ( ( finite_finite_nat @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ( ( finite_card_nat @ S )
            = ( finite410649719033368117t_unit @ T2 ) )
         => ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F2 @ S ) @ T2 )
           => ( ( ! [X: product_unit] :
                    ( ( member_Product_unit @ X @ T2 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on7061601236592826506t_unit @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_987_surjective__iff__injective__gen,axiom,
    ! [S: set_nat,T2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ( ( finite_card_nat @ S )
            = ( finite_card_nat @ T2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ T2 )
           => ( ( ! [X: nat] :
                    ( ( member_nat @ X @ T2 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_nat_nat @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_988_surjective__iff__injective__gen,axiom,
    ! [S: set_nat,T2: set_int,F2: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ( ( finite_card_nat @ S )
            = ( finite_card_int @ T2 ) )
         => ( ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ S ) @ T2 )
           => ( ( ! [X: int] :
                    ( ( member_int @ X @ T2 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_nat_int @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_989_surjective__iff__injective__gen,axiom,
    ! [S: set_int,T2: set_Product_unit,F2: int > product_unit] :
      ( ( finite_finite_int @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ( ( finite_card_int @ S )
            = ( finite410649719033368117t_unit @ T2 ) )
         => ( ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F2 @ S ) @ T2 )
           => ( ( ! [X: product_unit] :
                    ( ( member_Product_unit @ X @ T2 )
                   => ? [Y: int] :
                        ( ( member_int @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on4178023183795006766t_unit @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_990_surjective__iff__injective__gen,axiom,
    ! [S: set_int,T2: set_nat,F2: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ( ( finite_card_int @ S )
            = ( finite_card_nat @ T2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ S ) @ T2 )
           => ( ( ! [X: nat] :
                    ( ( member_nat @ X @ T2 )
                   => ? [Y: int] :
                        ( ( member_int @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_int_nat @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_991_surjective__iff__injective__gen,axiom,
    ! [S: set_int,T2: set_int,F2: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ( ( finite_card_int @ S )
            = ( finite_card_int @ T2 ) )
         => ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ S ) @ T2 )
           => ( ( ! [X: int] :
                    ( ( member_int @ X @ T2 )
                   => ? [Y: int] :
                        ( ( member_int @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_int_int @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_992_surjective__iff__injective__gen,axiom,
    ! [S: set_Pr958786334691620121nt_int,T2: set_Product_unit,F2: product_prod_int_int > product_unit] :
      ( ( finite2998713641127702882nt_int @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ( ( finite6756421564338198497nt_int @ S )
            = ( finite410649719033368117t_unit @ T2 ) )
         => ( ( ord_le3507040750410214029t_unit @ ( image_2335562724295632793t_unit @ F2 @ S ) @ T2 )
           => ( ( ! [X: product_unit] :
                    ( ( member_Product_unit @ X @ T2 )
                   => ? [Y: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ Y @ S )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on4340349633117443333t_unit @ F2 @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_993_card__bij__eq,axiom,
    ! [F2: product_unit > product_unit,A: set_Product_unit,B3: set_Product_unit,G3: product_unit > product_unit] :
      ( ( inj_on8151373323710067377t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) @ B3 )
       => ( ( inj_on8151373323710067377t_unit @ G3 @ B3 )
         => ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ G3 @ B3 ) @ A )
           => ( ( finite4290736615968046902t_unit @ A )
             => ( ( finite4290736615968046902t_unit @ B3 )
               => ( ( finite410649719033368117t_unit @ A )
                  = ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_994_card__bij__eq,axiom,
    ! [F2: product_unit > nat,A: set_Product_unit,B3: set_nat,G3: nat > product_unit] :
      ( ( inj_on8430439091780834860it_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F2 @ A ) @ B3 )
       => ( ( inj_on7061601236592826506t_unit @ G3 @ B3 )
         => ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ G3 @ B3 ) @ A )
           => ( ( finite4290736615968046902t_unit @ A )
             => ( ( finite_finite_nat @ B3 )
               => ( ( finite410649719033368117t_unit @ A )
                  = ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_995_card__bij__eq,axiom,
    ! [F2: product_unit > int,A: set_Product_unit,B3: set_int,G3: int > product_unit] :
      ( ( inj_on8427948621271784584it_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F2 @ A ) @ B3 )
       => ( ( inj_on4178023183795006766t_unit @ G3 @ B3 )
         => ( ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ G3 @ B3 ) @ A )
           => ( ( finite4290736615968046902t_unit @ A )
             => ( ( finite_finite_int @ B3 )
               => ( ( finite410649719033368117t_unit @ A )
                  = ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_996_card__bij__eq,axiom,
    ! [F2: nat > product_unit,A: set_nat,B3: set_Product_unit,G3: product_unit > nat] :
      ( ( inj_on7061601236592826506t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F2 @ A ) @ B3 )
       => ( ( inj_on8430439091780834860it_nat @ G3 @ B3 )
         => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ G3 @ B3 ) @ A )
           => ( ( finite_finite_nat @ A )
             => ( ( finite4290736615968046902t_unit @ B3 )
               => ( ( finite_card_nat @ A )
                  = ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_997_card__bij__eq,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat,G3: nat > nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B3 )
       => ( ( inj_on_nat_nat @ G3 @ B3 )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G3 @ B3 ) @ A )
           => ( ( finite_finite_nat @ A )
             => ( ( finite_finite_nat @ B3 )
               => ( ( finite_card_nat @ A )
                  = ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_998_card__bij__eq,axiom,
    ! [F2: nat > int,A: set_nat,B3: set_int,G3: int > nat] :
      ( ( inj_on_nat_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A ) @ B3 )
       => ( ( inj_on_int_nat @ G3 @ B3 )
         => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G3 @ B3 ) @ A )
           => ( ( finite_finite_nat @ A )
             => ( ( finite_finite_int @ B3 )
               => ( ( finite_card_nat @ A )
                  = ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_999_card__bij__eq,axiom,
    ! [F2: int > product_unit,A: set_int,B3: set_Product_unit,G3: product_unit > int] :
      ( ( inj_on4178023183795006766t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F2 @ A ) @ B3 )
       => ( ( inj_on8427948621271784584it_int @ G3 @ B3 )
         => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ G3 @ B3 ) @ A )
           => ( ( finite_finite_int @ A )
             => ( ( finite4290736615968046902t_unit @ B3 )
               => ( ( finite_card_int @ A )
                  = ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1000_card__bij__eq,axiom,
    ! [F2: int > nat,A: set_int,B3: set_nat,G3: nat > int] :
      ( ( inj_on_int_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B3 )
       => ( ( inj_on_nat_int @ G3 @ B3 )
         => ( ( ord_less_eq_set_int @ ( image_nat_int @ G3 @ B3 ) @ A )
           => ( ( finite_finite_int @ A )
             => ( ( finite_finite_nat @ B3 )
               => ( ( finite_card_int @ A )
                  = ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1001_card__bij__eq,axiom,
    ! [F2: int > int,A: set_int,B3: set_int,G3: int > int] :
      ( ( inj_on_int_int @ F2 @ A )
     => ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 )
       => ( ( inj_on_int_int @ G3 @ B3 )
         => ( ( ord_less_eq_set_int @ ( image_int_int @ G3 @ B3 ) @ A )
           => ( ( finite_finite_int @ A )
             => ( ( finite_finite_int @ B3 )
               => ( ( finite_card_int @ A )
                  = ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1002_card__bij__eq,axiom,
    ! [F2: product_prod_int_int > product_unit,A: set_Pr958786334691620121nt_int,B3: set_Product_unit,G3: product_unit > product_prod_int_int] :
      ( ( inj_on4340349633117443333t_unit @ F2 @ A )
     => ( ( ord_le3507040750410214029t_unit @ ( image_2335562724295632793t_unit @ F2 @ A ) @ B3 )
       => ( ( inj_on6108249006978685533nt_int @ G3 @ B3 )
         => ( ( ord_le2843351958646193337nt_int @ ( image_4103462098156874993nt_int @ G3 @ B3 ) @ A )
           => ( ( finite2998713641127702882nt_int @ A )
             => ( ( finite4290736615968046902t_unit @ B3 )
               => ( ( finite6756421564338198497nt_int @ A )
                  = ( finite410649719033368117t_unit @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1003_full__exhaustive__int_H_Ocases,axiom,
    ! [X2: produc2285326912895808259nt_int] :
      ~ ! [F4: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I2: int] :
          ( X2
         != ( produc5700946648718959541nt_int @ F4 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_1004_finite__option__UNIV,axiom,
    ( ( finite6429242118585374504nt_int @ top_to1656296329633014575nt_int )
    = ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ).

% finite_option_UNIV
thf(fact_1005_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_1006_finite__option__UNIV,axiom,
    ( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
    = ( finite_finite_int @ top_top_set_int ) ) ).

% finite_option_UNIV
thf(fact_1007_finite__option__UNIV,axiom,
    ( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% finite_option_UNIV
thf(fact_1008_card__cartesian__product,axiom,
    ! [A: set_int,B3: set_Product_unit] :
      ( ( finite5262300457862123438t_unit
        @ ( produc2622961094310184599t_unit @ A
          @ ^ [Uu: int] : B3 ) )
      = ( times_times_nat @ ( finite_card_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1009_card__cartesian__product,axiom,
    ! [A: set_Product_unit,B3: set_int] :
      ( ( finite4042467796647221440it_int
        @ ( produc6872886531786962417it_int @ A
          @ ^ [Uu: product_unit] : B3 ) )
      = ( times_times_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1010_card__cartesian__product,axiom,
    ! [A: set_Product_unit,B3: set_Product_unit] :
      ( ( finite5501798543469036431t_unit
        @ ( produc6121607364029222536t_unit @ A
          @ ^ [Uu: product_unit] : B3 ) )
      = ( times_times_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1011_card__cartesian__product,axiom,
    ! [A: set_int,B3: set_int] :
      ( ( finite6756421564338198497nt_int
        @ ( produc1453890548195736866nt_int @ A
          @ ^ [Uu: int] : B3 ) )
      = ( times_times_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1012_card__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_int] :
      ( ( finite3787662430427006444nt_int
        @ ( produc1005374827195948829nt_int @ A
          @ ^ [Uu: product_prod_int_int] : B3 ) )
      = ( times_times_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite_card_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1013_card__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Product_unit] :
      ( ( finite4484788426335282531t_unit
        @ ( produc8181708692561065052t_unit @ A
          @ ^ [Uu: product_prod_int_int] : B3 ) )
      = ( times_times_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite410649719033368117t_unit @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1014_card__cartesian__product,axiom,
    ! [A: set_int,B3: set_Pr958786334691620121nt_int] :
      ( ( finite7016131845264464474nt_int
        @ ( produc1668682332673839427nt_int @ A
          @ ^ [Uu: int] : B3 ) )
      = ( times_times_nat @ ( finite_card_int @ A ) @ ( finite6756421564338198497nt_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1015_card__cartesian__product,axiom,
    ! [A: set_Product_unit,B3: set_Pr958786334691620121nt_int] :
      ( ( finite3348091610864412347nt_int
        @ ( produc726236029567531444nt_int @ A
          @ ^ [Uu: product_unit] : B3 ) )
      = ( times_times_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite6756421564338198497nt_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1016_card__cartesian__product,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( finite1566866390884520591nt_int
        @ ( produc245848125267896712nt_int @ A
          @ ^ [Uu: product_prod_int_int] : B3 ) )
      = ( times_times_nat @ ( finite6756421564338198497nt_int @ A ) @ ( finite6756421564338198497nt_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1017_card__cartesian__product,axiom,
    ! [A: set_int,B3: set_Pr2560585780119916871nt_int] :
      ( ( finite8313665907029225736nt_int
        @ ( produc8936478010656452849nt_int @ A
          @ ^ [Uu: int] : B3 ) )
      = ( times_times_nat @ ( finite_card_int @ A ) @ ( finite1566866390884520591nt_int @ B3 ) ) ) ).

% card_cartesian_product
thf(fact_1018_inj__image__subset__iff,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ ( image_nat_nat @ F2 @ B3 ) )
        = ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_1019_inj__image__subset__iff,axiom,
    ! [F2: nat > product_prod_int_int,A: set_nat,B3: set_nat] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ top_top_set_nat )
     => ( ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F2 @ A ) @ ( image_2667626500211843362nt_int @ F2 @ B3 ) )
        = ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_1020_inj__image__subset__iff,axiom,
    ! [F2: int > product_prod_int_int,A: set_int,B3: set_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ top_top_set_int )
     => ( ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F2 @ A ) @ ( image_5705468584675977158nt_int @ F2 @ B3 ) )
        = ( ord_less_eq_set_int @ A @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_1021_inj__image__subset__iff,axiom,
    ! [F2: product_unit > product_prod_int_int,A: set_Product_unit,B3: set_Product_unit] :
      ( ( inj_on6108249006978685533nt_int @ F2 @ top_to1996260823553986621t_unit )
     => ( ( ord_le2843351958646193337nt_int @ ( image_4103462098156874993nt_int @ F2 @ A ) @ ( image_4103462098156874993nt_int @ F2 @ B3 ) )
        = ( ord_le3507040750410214029t_unit @ A @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_1022_inj__image__subset__iff,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
      ( ( inj_on6454360653267994289nt_int @ F2 @ top_to4366644338036079209nt_int )
     => ( ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F2 @ A ) @ ( image_2653370878348428101nt_int @ F2 @ B3 ) )
        = ( ord_le2843351958646193337nt_int @ A @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_1023_subset__image__inj,axiom,
    ! [S: set_nat,F2: nat > nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F2 @ T2 ) )
      = ( ? [U3: set_nat] :
            ( ( ord_less_eq_set_nat @ U3 @ T2 )
            & ( inj_on_nat_nat @ F2 @ U3 )
            & ( S
              = ( image_nat_nat @ F2 @ U3 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1024_subset__image__inj,axiom,
    ! [S: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int,T2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ S @ ( image_2653370878348428101nt_int @ F2 @ T2 ) )
      = ( ? [U3: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ U3 @ T2 )
            & ( inj_on6454360653267994289nt_int @ F2 @ U3 )
            & ( S
              = ( image_2653370878348428101nt_int @ F2 @ U3 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1025_inj__on__image__mem__iff,axiom,
    ! [F2: int > product_prod_int_int,B3: set_int,A4: int,A: set_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ B3 )
     => ( ( member_int @ A4 @ B3 )
       => ( ( ord_less_eq_set_int @ A @ B3 )
         => ( ( member5262025264175285858nt_int @ ( F2 @ A4 ) @ ( image_5705468584675977158nt_int @ F2 @ A ) )
            = ( member_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1026_inj__on__image__mem__iff,axiom,
    ! [F2: int > int,B3: set_int,A4: int,A: set_int] :
      ( ( inj_on_int_int @ F2 @ B3 )
     => ( ( member_int @ A4 @ B3 )
       => ( ( ord_less_eq_set_int @ A @ B3 )
         => ( ( member_int @ ( F2 @ A4 ) @ ( image_int_int @ F2 @ A ) )
            = ( member_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1027_inj__on__image__mem__iff,axiom,
    ! [F2: int > nat,B3: set_int,A4: int,A: set_int] :
      ( ( inj_on_int_nat @ F2 @ B3 )
     => ( ( member_int @ A4 @ B3 )
       => ( ( ord_less_eq_set_int @ A @ B3 )
         => ( ( member_nat @ ( F2 @ A4 ) @ ( image_int_nat @ F2 @ A ) )
            = ( member_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1028_inj__on__image__mem__iff,axiom,
    ! [F2: nat > product_prod_int_int,B3: set_nat,A4: nat,A: set_nat] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ B3 )
     => ( ( member_nat @ A4 @ B3 )
       => ( ( ord_less_eq_set_nat @ A @ B3 )
         => ( ( member5262025264175285858nt_int @ ( F2 @ A4 ) @ ( image_2667626500211843362nt_int @ F2 @ A ) )
            = ( member_nat @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1029_inj__on__image__mem__iff,axiom,
    ! [F2: nat > int,B3: set_nat,A4: nat,A: set_nat] :
      ( ( inj_on_nat_int @ F2 @ B3 )
     => ( ( member_nat @ A4 @ B3 )
       => ( ( ord_less_eq_set_nat @ A @ B3 )
         => ( ( member_int @ ( F2 @ A4 ) @ ( image_nat_int @ F2 @ A ) )
            = ( member_nat @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1030_inj__on__image__mem__iff,axiom,
    ! [F2: nat > nat,B3: set_nat,A4: nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ B3 )
     => ( ( member_nat @ A4 @ B3 )
       => ( ( ord_less_eq_set_nat @ A @ B3 )
         => ( ( member_nat @ ( F2 @ A4 ) @ ( image_nat_nat @ F2 @ A ) )
            = ( member_nat @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1031_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,B3: set_Pr958786334691620121nt_int,A4: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on6454360653267994289nt_int @ F2 @ B3 )
     => ( ( member5262025264175285858nt_int @ A4 @ B3 )
       => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
         => ( ( member5262025264175285858nt_int @ ( F2 @ A4 ) @ ( image_2653370878348428101nt_int @ F2 @ A ) )
            = ( member5262025264175285858nt_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1032_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_int_int > int,B3: set_Pr958786334691620121nt_int,A4: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ B3 )
     => ( ( member5262025264175285858nt_int @ A4 @ B3 )
       => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
         => ( ( member_int @ ( F2 @ A4 ) @ ( image_5042161079198086560nt_int @ F2 @ A ) )
            = ( member5262025264175285858nt_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1033_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_int_int > nat,B3: set_Pr958786334691620121nt_int,A4: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4736580515542835928nt_nat @ F2 @ B3 )
     => ( ( member5262025264175285858nt_int @ A4 @ B3 )
       => ( ( ord_le2843351958646193337nt_int @ A @ B3 )
         => ( ( member_nat @ ( F2 @ A4 ) @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
            = ( member5262025264175285858nt_int @ A4 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1034_inj__on__image__eq__iff,axiom,
    ! [F2: nat > nat,C: set_nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ C )
     => ( ( ord_less_eq_set_nat @ A @ C )
       => ( ( ord_less_eq_set_nat @ B3 @ C )
         => ( ( ( image_nat_nat @ F2 @ A )
              = ( image_nat_nat @ F2 @ B3 ) )
            = ( A = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1035_range__ex1__eq,axiom,
    ! [F2: nat > product_prod_int_int,B7: product_prod_int_int] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ top_top_set_nat )
     => ( ( member5262025264175285858nt_int @ B7 @ ( image_2667626500211843362nt_int @ F2 @ top_top_set_nat ) )
        = ( ? [X: nat] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: nat] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1036_range__ex1__eq,axiom,
    ! [F2: nat > int,B7: int] :
      ( ( inj_on_nat_int @ F2 @ top_top_set_nat )
     => ( ( member_int @ B7 @ ( image_nat_int @ F2 @ top_top_set_nat ) )
        = ( ? [X: nat] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: nat] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1037_range__ex1__eq,axiom,
    ! [F2: nat > nat,B7: nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( member_nat @ B7 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
        = ( ? [X: nat] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: nat] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1038_range__ex1__eq,axiom,
    ! [F2: int > product_prod_int_int,B7: product_prod_int_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ top_top_set_int )
     => ( ( member5262025264175285858nt_int @ B7 @ ( image_5705468584675977158nt_int @ F2 @ top_top_set_int ) )
        = ( ? [X: int] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: int] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1039_range__ex1__eq,axiom,
    ! [F2: int > int,B7: int] :
      ( ( inj_on_int_int @ F2 @ top_top_set_int )
     => ( ( member_int @ B7 @ ( image_int_int @ F2 @ top_top_set_int ) )
        = ( ? [X: int] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: int] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1040_range__ex1__eq,axiom,
    ! [F2: int > nat,B7: nat] :
      ( ( inj_on_int_nat @ F2 @ top_top_set_int )
     => ( ( member_nat @ B7 @ ( image_int_nat @ F2 @ top_top_set_int ) )
        = ( ? [X: int] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: int] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1041_range__ex1__eq,axiom,
    ! [F2: product_unit > product_prod_int_int,B7: product_prod_int_int] :
      ( ( inj_on6108249006978685533nt_int @ F2 @ top_to1996260823553986621t_unit )
     => ( ( member5262025264175285858nt_int @ B7 @ ( image_4103462098156874993nt_int @ F2 @ top_to1996260823553986621t_unit ) )
        = ( ? [X: product_unit] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: product_unit] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1042_range__ex1__eq,axiom,
    ! [F2: product_unit > int,B7: int] :
      ( ( inj_on8427948621271784584it_int @ F2 @ top_to1996260823553986621t_unit )
     => ( ( member_int @ B7 @ ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit ) )
        = ( ? [X: product_unit] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: product_unit] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1043_range__ex1__eq,axiom,
    ! [F2: product_unit > nat,B7: nat] :
      ( ( inj_on8430439091780834860it_nat @ F2 @ top_to1996260823553986621t_unit )
     => ( ( member_nat @ B7 @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) )
        = ( ? [X: product_unit] :
              ( ( B7
                = ( F2 @ X ) )
              & ! [Y: product_unit] :
                  ( ( B7
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1044_inj__image__eq__iff,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( ( image_nat_nat @ F2 @ A )
          = ( image_nat_nat @ F2 @ B3 ) )
        = ( A = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1045_top__empty__eq,axiom,
    ( top_to1578927101902068148_int_o
    = ( ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ top_to4366644338036079209nt_int ) ) ) ).

% top_empty_eq
thf(fact_1046_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_1047_top__empty__eq,axiom,
    ( top_top_int_o
    = ( ^ [X: int] : ( member_int @ X @ top_top_set_int ) ) ) ).

% top_empty_eq
thf(fact_1048_top__empty__eq,axiom,
    ( top_to2465898995584390880unit_o
    = ( ^ [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ) ) ).

% top_empty_eq
thf(fact_1049_top__set__def,axiom,
    ( top_to4366644338036079209nt_int
    = ( collec213857154873943460nt_int @ top_to1578927101902068148_int_o ) ) ).

% top_set_def
thf(fact_1050_top__set__def,axiom,
    ( top_to3181862456159035625nt_int
    = ( collec1790188477890212312nt_int @ top_to3619306743234492124_int_o ) ) ).

% top_set_def
thf(fact_1051_top__set__def,axiom,
    ( top_to2069137843433766899nt_int
    = ( collec506566255779805410nt_int @ top_to8953086782060878418_int_o ) ) ).

% top_set_def
thf(fact_1052_top__set__def,axiom,
    ( top_to2321149100101803671nt_int
    = ( collec7447955092554649554nt_int @ top_to425394652897361158_int_o ) ) ).

% top_set_def
thf(fact_1053_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_1054_top__set__def,axiom,
    ( top_top_set_int
    = ( collect_int @ top_top_int_o ) ) ).

% top_set_def
thf(fact_1055_top__set__def,axiom,
    ( top_to1996260823553986621t_unit
    = ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).

% top_set_def
thf(fact_1056_top__empty__eq2,axiom,
    ( top_top_int_int_o
    = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ top_to4366644338036079209nt_int ) ) ) ).

% top_empty_eq2
thf(fact_1057_top__empty__eq2,axiom,
    ( top_to7876356747670055550_int_o
    = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ top_to2321149100101803671nt_int ) ) ) ).

% top_empty_eq2
thf(fact_1058_top__empty__eq2,axiom,
    ( top_to6513043852502318900_int_o
    = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ top_to2069137843433766899nt_int ) ) ) ).

% top_empty_eq2
thf(fact_1059_top__empty__eq2,axiom,
    ( top_to6261655714344447806_int_o
    = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ top_to3181862456159035625nt_int ) ) ) ).

% top_empty_eq2
thf(fact_1060_inj__on__id2,axiom,
    ! [A: set_nat] :
      ( inj_on_nat_nat
      @ ^ [X: nat] : X
      @ A ) ).

% inj_on_id2
thf(fact_1061_surj__def,axiom,
    ! [F2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: nat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1062_surj__def,axiom,
    ! [F2: nat > int] :
      ( ( ( image_nat_int @ F2 @ top_top_set_nat )
        = top_top_set_int )
      = ( ! [Y: int] :
          ? [X: nat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1063_surj__def,axiom,
    ! [F2: nat > product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
      = ( ! [Y: product_unit] :
          ? [X: nat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1064_surj__def,axiom,
    ! [F2: int > nat] :
      ( ( ( image_int_nat @ F2 @ top_top_set_int )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: int] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1065_surj__def,axiom,
    ! [F2: int > int] :
      ( ( ( image_int_int @ F2 @ top_top_set_int )
        = top_top_set_int )
      = ( ! [Y: int] :
          ? [X: int] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1066_surj__def,axiom,
    ! [F2: int > product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F2 @ top_top_set_int )
        = top_to1996260823553986621t_unit )
      = ( ! [Y: product_unit] :
          ? [X: int] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1067_surj__def,axiom,
    ! [F2: product_unit > nat] :
      ( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: product_unit] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1068_surj__def,axiom,
    ! [F2: product_unit > int] :
      ( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_int )
      = ( ! [Y: int] :
          ? [X: product_unit] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1069_surj__def,axiom,
    ! [F2: product_unit > product_unit] :
      ( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
      = ( ! [Y: product_unit] :
          ? [X: product_unit] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1070_surjI,axiom,
    ! [G3: nat > nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_nat_nat @ G3 @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_1071_surjI,axiom,
    ! [G3: nat > int,F2: int > nat] :
      ( ! [X3: int] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_nat_int @ G3 @ top_top_set_nat )
        = top_top_set_int ) ) ).

% surjI
thf(fact_1072_surjI,axiom,
    ! [G3: nat > product_unit,F2: product_unit > nat] :
      ( ! [X3: product_unit] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_8730104196221521654t_unit @ G3 @ top_top_set_nat )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_1073_surjI,axiom,
    ! [G3: int > nat,F2: nat > int] :
      ( ! [X3: nat] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_int_nat @ G3 @ top_top_set_int )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_1074_surjI,axiom,
    ! [G3: int > int,F2: int > int] :
      ( ! [X3: int] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_int_int @ G3 @ top_top_set_int )
        = top_top_set_int ) ) ).

% surjI
thf(fact_1075_surjI,axiom,
    ! [G3: int > product_unit,F2: product_unit > int] :
      ( ! [X3: product_unit] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_5846526143423701914t_unit @ G3 @ top_top_set_int )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_1076_surjI,axiom,
    ! [G3: product_unit > nat,F2: nat > product_unit] :
      ( ! [X3: nat] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_875570014554754200it_nat @ G3 @ top_to1996260823553986621t_unit )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_1077_surjI,axiom,
    ! [G3: product_unit > int,F2: int > product_unit] :
      ( ! [X3: int] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_873079544045703924it_int @ G3 @ top_to1996260823553986621t_unit )
        = top_top_set_int ) ) ).

% surjI
thf(fact_1078_surjI,axiom,
    ! [G3: product_unit > product_unit,F2: product_unit > product_unit] :
      ( ! [X3: product_unit] :
          ( ( G3 @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_405062704495631173t_unit @ G3 @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_1079_surjE,axiom,
    ! [F2: nat > nat,Y4: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: nat] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1080_surjE,axiom,
    ! [F2: nat > int,Y4: int] :
      ( ( ( image_nat_int @ F2 @ top_top_set_nat )
        = top_top_set_int )
     => ~ ! [X3: nat] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1081_surjE,axiom,
    ! [F2: nat > product_unit,Y4: product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ~ ! [X3: nat] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1082_surjE,axiom,
    ! [F2: int > nat,Y4: nat] :
      ( ( ( image_int_nat @ F2 @ top_top_set_int )
        = top_top_set_nat )
     => ~ ! [X3: int] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1083_surjE,axiom,
    ! [F2: int > int,Y4: int] :
      ( ( ( image_int_int @ F2 @ top_top_set_int )
        = top_top_set_int )
     => ~ ! [X3: int] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1084_surjE,axiom,
    ! [F2: int > product_unit,Y4: product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F2 @ top_top_set_int )
        = top_to1996260823553986621t_unit )
     => ~ ! [X3: int] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1085_surjE,axiom,
    ! [F2: product_unit > nat,Y4: nat] :
      ( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
     => ~ ! [X3: product_unit] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1086_surjE,axiom,
    ! [F2: product_unit > int,Y4: int] :
      ( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_int )
     => ~ ! [X3: product_unit] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1087_surjE,axiom,
    ! [F2: product_unit > product_unit,Y4: product_unit] :
      ( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
     => ~ ! [X3: product_unit] :
            ( Y4
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_1088_surjD,axiom,
    ! [F2: nat > nat,Y4: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X3: nat] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1089_surjD,axiom,
    ! [F2: nat > int,Y4: int] :
      ( ( ( image_nat_int @ F2 @ top_top_set_nat )
        = top_top_set_int )
     => ? [X3: nat] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1090_surjD,axiom,
    ! [F2: nat > product_unit,Y4: product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ? [X3: nat] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1091_surjD,axiom,
    ! [F2: int > nat,Y4: nat] :
      ( ( ( image_int_nat @ F2 @ top_top_set_int )
        = top_top_set_nat )
     => ? [X3: int] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1092_surjD,axiom,
    ! [F2: int > int,Y4: int] :
      ( ( ( image_int_int @ F2 @ top_top_set_int )
        = top_top_set_int )
     => ? [X3: int] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1093_surjD,axiom,
    ! [F2: int > product_unit,Y4: product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F2 @ top_top_set_int )
        = top_to1996260823553986621t_unit )
     => ? [X3: int] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1094_surjD,axiom,
    ! [F2: product_unit > nat,Y4: nat] :
      ( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
     => ? [X3: product_unit] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1095_surjD,axiom,
    ! [F2: product_unit > int,Y4: int] :
      ( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
        = top_top_set_int )
     => ? [X3: product_unit] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1096_surjD,axiom,
    ! [F2: product_unit > product_unit,Y4: product_unit] :
      ( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
     => ? [X3: product_unit] :
          ( Y4
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_1097_infinite__iff__countable__subset,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ? [F3: nat > nat] :
            ( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
            & ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1098_infinite__iff__countable__subset,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ? [F3: nat > int] :
            ( ( inj_on_nat_int @ F3 @ top_top_set_nat )
            & ( ord_less_eq_set_int @ ( image_nat_int @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1099_infinite__iff__countable__subset,axiom,
    ! [S: set_Pr958786334691620121nt_int] :
      ( ( ~ ( finite2998713641127702882nt_int @ S ) )
      = ( ? [F3: nat > product_prod_int_int] :
            ( ( inj_on2359555466047542454nt_int @ F3 @ top_top_set_nat )
            & ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1100_infinite__countable__subset,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ? [F4: nat > nat] :
          ( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
          & ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_1101_infinite__countable__subset,axiom,
    ! [S: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ? [F4: nat > int] :
          ( ( inj_on_nat_int @ F4 @ top_top_set_nat )
          & ( ord_less_eq_set_int @ ( image_nat_int @ F4 @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_1102_infinite__countable__subset,axiom,
    ! [S: set_Pr958786334691620121nt_int] :
      ( ~ ( finite2998713641127702882nt_int @ S )
     => ? [F4: nat > product_prod_int_int] :
          ( ( inj_on2359555466047542454nt_int @ F4 @ top_top_set_nat )
          & ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F4 @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_1103_inj__on__image__iff,axiom,
    ! [A: set_nat,G3: nat > nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ! [Xa3: nat] :
              ( ( member_nat @ Xa3 @ A )
             => ( ( ( G3 @ ( F2 @ X3 ) )
                  = ( G3 @ ( F2 @ Xa3 ) ) )
                = ( ( G3 @ X3 )
                  = ( G3 @ Xa3 ) ) ) ) )
     => ( ( inj_on_nat_nat @ F2 @ A )
       => ( ( inj_on_nat_nat @ G3 @ ( image_nat_nat @ F2 @ A ) )
          = ( inj_on_nat_nat @ G3 @ A ) ) ) ) ).

% inj_on_image_iff
thf(fact_1104_inj__on__subset,axiom,
    ! [F2: nat > nat,A: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A )
     => ( ( ord_less_eq_set_nat @ B3 @ A )
       => ( inj_on_nat_nat @ F2 @ B3 ) ) ) ).

% inj_on_subset
thf(fact_1105_subset__inj__on,axiom,
    ! [F2: nat > nat,B3: set_nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( inj_on_nat_nat @ F2 @ A ) ) ) ).

% subset_inj_on
thf(fact_1106_inj__image__mem__iff,axiom,
    ! [F2: nat > int,A4: nat,A: set_nat] :
      ( ( inj_on_nat_int @ F2 @ top_top_set_nat )
     => ( ( member_int @ ( F2 @ A4 ) @ ( image_nat_int @ F2 @ A ) )
        = ( member_nat @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1107_inj__image__mem__iff,axiom,
    ! [F2: nat > nat,A4: nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( member_nat @ ( F2 @ A4 ) @ ( image_nat_nat @ F2 @ A ) )
        = ( member_nat @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1108_inj__image__mem__iff,axiom,
    ! [F2: int > int,A4: int,A: set_int] :
      ( ( inj_on_int_int @ F2 @ top_top_set_int )
     => ( ( member_int @ ( F2 @ A4 ) @ ( image_int_int @ F2 @ A ) )
        = ( member_int @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1109_inj__image__mem__iff,axiom,
    ! [F2: int > nat,A4: int,A: set_int] :
      ( ( inj_on_int_nat @ F2 @ top_top_set_int )
     => ( ( member_nat @ ( F2 @ A4 ) @ ( image_int_nat @ F2 @ A ) )
        = ( member_int @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1110_inj__image__mem__iff,axiom,
    ! [F2: product_unit > int,A4: product_unit,A: set_Product_unit] :
      ( ( inj_on8427948621271784584it_int @ F2 @ top_to1996260823553986621t_unit )
     => ( ( member_int @ ( F2 @ A4 ) @ ( image_873079544045703924it_int @ F2 @ A ) )
        = ( member_Product_unit @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1111_inj__image__mem__iff,axiom,
    ! [F2: product_unit > nat,A4: product_unit,A: set_Product_unit] :
      ( ( inj_on8430439091780834860it_nat @ F2 @ top_to1996260823553986621t_unit )
     => ( ( member_nat @ ( F2 @ A4 ) @ ( image_875570014554754200it_nat @ F2 @ A ) )
        = ( member_Product_unit @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1112_inj__image__mem__iff,axiom,
    ! [F2: product_prod_int_int > int,A4: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4734090045033785652nt_int @ F2 @ top_to4366644338036079209nt_int )
     => ( ( member_int @ ( F2 @ A4 ) @ ( image_5042161079198086560nt_int @ F2 @ A ) )
        = ( member5262025264175285858nt_int @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1113_inj__image__mem__iff,axiom,
    ! [F2: product_prod_int_int > nat,A4: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( inj_on4736580515542835928nt_nat @ F2 @ top_to4366644338036079209nt_int )
     => ( ( member_nat @ ( F2 @ A4 ) @ ( image_5044651549707136836nt_nat @ F2 @ A ) )
        = ( member5262025264175285858nt_int @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1114_inj__image__mem__iff,axiom,
    ! [F2: nat > product_prod_int_int,A4: nat,A: set_nat] :
      ( ( inj_on2359555466047542454nt_int @ F2 @ top_top_set_nat )
     => ( ( member5262025264175285858nt_int @ ( F2 @ A4 ) @ ( image_2667626500211843362nt_int @ F2 @ A ) )
        = ( member_nat @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1115_inj__image__mem__iff,axiom,
    ! [F2: int > product_prod_int_int,A4: int,A: set_int] :
      ( ( inj_on5397397550511676250nt_int @ F2 @ top_top_set_int )
     => ( ( member5262025264175285858nt_int @ ( F2 @ A4 ) @ ( image_5705468584675977158nt_int @ F2 @ A ) )
        = ( member_int @ A4 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1116_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_nat_nat
    @ ^ [X: nat] : X
    @ top_top_set_nat ) ).

% sorted_list_of_set.inj_on
thf(fact_1117_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_int_int
    @ ^ [X: int] : X
    @ top_top_set_int ) ).

% sorted_list_of_set.inj_on
thf(fact_1118_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on8151373323710067377t_unit
    @ ^ [X: product_unit] : X
    @ top_to1996260823553986621t_unit ) ).

% sorted_list_of_set.inj_on
thf(fact_1119_Gr__incl,axiom,
    ! [A: set_int,F2: int > int,B3: set_int] :
      ( ( ord_le2843351958646193337nt_int @ ( bNF_Gr_int_int @ A @ F2 )
        @ ( produc1453890548195736866nt_int @ A
          @ ^ [Uu: int] : B3 ) )
      = ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 ) ) ).

% Gr_incl
thf(fact_1120_inj__on__apsnd,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc3094765310956027504at_nat @ F2 )
        @ ( produc457027306803732586at_nat @ top_top_set_nat
          @ ^ [Uu: nat] : A ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apsnd
thf(fact_1121_inj__on__apsnd,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on2505922616566413745nt_nat @ ( produc3092274840446977228at_int @ F2 )
        @ ( produc1456381018704787142nt_nat @ top_top_set_int
          @ ^ [Uu: int] : A ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apsnd
thf(fact_1122_inj__on__apsnd,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on8401517491650455179it_nat @ ( produc8838893277175244653t_unit @ F2 )
        @ ( produc6875377002296012693it_nat @ top_to1996260823553986621t_unit
          @ ^ [Uu: product_unit] : A ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apsnd
thf(fact_1123_inj__on__apfst,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc986720760941809198at_nat @ F2 )
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apfst
thf(fact_1124_inj__on__apfst,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on3694970277613828529at_int @ ( produc984230290432758922at_int @ F2 )
        @ ( produc454536836294682310at_int @ A
          @ ^ [Uu: nat] : top_top_set_int ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apfst
thf(fact_1125_inj__on__apfst,axiom,
    ! [F2: nat > nat,A: set_nat] :
      ( ( inj_on6643003698537017303t_unit @ ( produc8285911118168523055t_unit @ F2 )
        @ ( produc5506539147108004339t_unit @ A
          @ ^ [Uu: nat] : top_to1996260823553986621t_unit ) )
      = ( inj_on_nat_nat @ F2 @ A ) ) ).

% inj_on_apfst
thf(fact_1126_card__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_Product_unit] :
      ( ( finite_finite_nat @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( finite4290736615968046902t_unit @ ( B3 @ X3 ) ) )
       => ( ( finite8145878510659943178t_unit @ ( produc5506539147108004339t_unit @ A @ B3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [A3: nat] : ( finite410649719033368117t_unit @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1127_card__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( finite_finite_nat @ ( B3 @ X3 ) ) )
       => ( ( finite711546835091564841at_nat @ ( produc457027306803732586at_nat @ A @ B3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [A3: nat] : ( finite_card_nat @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1128_card__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_int] :
      ( ( finite_finite_nat @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( finite_finite_int @ ( B3 @ X3 ) ) )
       => ( ( finite5757067852437143941at_int @ ( produc454536836294682310at_int @ A @ B3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [A3: nat] : ( finite_card_int @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1129_card__SigmaI,axiom,
    ! [A: set_int,B3: int > set_Product_unit] :
      ( ( finite_finite_int @ A )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( finite4290736615968046902t_unit @ ( B3 @ X3 ) ) )
       => ( ( finite5262300457862123438t_unit @ ( produc2622961094310184599t_unit @ A @ B3 ) )
          = ( groups4541462559716669496nt_nat
            @ ^ [A3: int] : ( finite410649719033368117t_unit @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1130_card__SigmaI,axiom,
    ! [A: set_int,B3: int > set_nat] :
      ( ( finite_finite_int @ A )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( finite_finite_nat @ ( B3 @ X3 ) ) )
       => ( ( finite1710900546992619397nt_nat @ ( produc1456381018704787142nt_nat @ A @ B3 ) )
          = ( groups4541462559716669496nt_nat
            @ ^ [A3: int] : ( finite_card_nat @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1131_card__SigmaI,axiom,
    ! [A: set_int,B3: int > set_int] :
      ( ( finite_finite_int @ A )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( finite_finite_int @ ( B3 @ X3 ) ) )
       => ( ( finite6756421564338198497nt_int @ ( produc1453890548195736866nt_int @ A @ B3 ) )
          = ( groups4541462559716669496nt_nat
            @ ^ [A3: int] : ( finite_card_int @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1132_card__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Product_unit] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ( finite4290736615968046902t_unit @ ( B3 @ X3 ) ) )
       => ( ( finite4484788426335282531t_unit @ ( produc8181708692561065052t_unit @ A @ B3 ) )
          = ( groups3536494975961350351nt_nat
            @ ^ [A3: product_prod_int_int] : ( finite410649719033368117t_unit @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1133_card__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ( finite_finite_nat @ ( B3 @ X3 ) ) )
       => ( ( finite7965513449936203152nt_nat @ ( produc1007865297704999105nt_nat @ A @ B3 ) )
          = ( groups3536494975961350351nt_nat
            @ ^ [A3: product_prod_int_int] : ( finite_card_nat @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1134_card__SigmaI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ( finite_finite_int @ ( B3 @ X3 ) ) )
       => ( ( finite3787662430427006444nt_int @ ( produc1005374827195948829nt_int @ A @ B3 ) )
          = ( groups3536494975961350351nt_nat
            @ ^ [A3: product_prod_int_int] : ( finite_card_int @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1135_card__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( finite2998713641127702882nt_int @ ( B3 @ X3 ) ) )
       => ( ( finite3978289760800330678nt_int @ ( produc7854212285064481439nt_int @ A @ B3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [A3: nat] : ( finite6756421564338198497nt_int @ ( B3 @ A3 ) )
            @ A ) ) ) ) ).

% card_SigmaI
thf(fact_1136_apfst__conv,axiom,
    ! [F2: int > int,X2: int,Y4: int] :
      ( ( produc2355237562570658114nt_int @ F2 @ ( product_Pair_int_int @ X2 @ Y4 ) )
      = ( product_Pair_int_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1137_apfst__conv,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,X2: product_prod_int_int,Y4: product_prod_int_int] :
      ( ( produc3605865892030788029nt_int @ F2 @ ( produc3646306378393792727nt_int @ X2 @ Y4 ) )
      = ( produc3646306378393792727nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1138_apfst__conv,axiom,
    ! [F2: product_prod_int_int > int > option6357759511663192854e_term,X2: product_prod_int_int,Y4: product_prod_int_int] :
      ( ( produc1363703303993119577nt_int @ F2 @ ( produc3646306378393792727nt_int @ X2 @ Y4 ) )
      = ( produc4305682042979456191nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1139_apfst__conv,axiom,
    ! [F2: product_prod_int_int > produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y4: product_prod_int_int] :
      ( ( produc222693570364858191nt_int @ F2 @ ( produc3646306378393792727nt_int @ X2 @ Y4 ) )
      = ( produc5700946648718959541nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1140_apfst__conv,axiom,
    ! [F2: ( int > option6357759511663192854e_term ) > product_prod_int_int,X2: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc4373787890801669085nt_int @ F2 @ ( produc4305682042979456191nt_int @ X2 @ Y4 ) )
      = ( produc3646306378393792727nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1141_apfst__conv,axiom,
    ! [F2: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X2: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc7131893314732144953nt_int @ F2 @ ( produc4305682042979456191nt_int @ X2 @ Y4 ) )
      = ( produc4305682042979456191nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1142_apfst__conv,axiom,
    ! [F2: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X2: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc6067946493426971951nt_int @ F2 @ ( produc4305682042979456191nt_int @ X2 @ Y4 ) )
      = ( produc5700946648718959541nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1143_apfst__conv,axiom,
    ! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int,X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc9015291761996157799nt_int @ F2 @ ( produc5700946648718959541nt_int @ X2 @ Y4 ) )
      = ( produc3646306378393792727nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1144_apfst__conv,axiom,
    ! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc6919319362177967663nt_int @ F2 @ ( produc5700946648718959541nt_int @ X2 @ Y4 ) )
      = ( produc4305682042979456191nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1145_apfst__conv,axiom,
    ! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc7277921532047535397nt_int @ F2 @ ( produc5700946648718959541nt_int @ X2 @ Y4 ) )
      = ( produc5700946648718959541nt_int @ ( F2 @ X2 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1146_apsnd__conv,axiom,
    ! [F2: int > int,X2: int,Y4: int] :
      ( ( produc4463282112584876420nt_int @ F2 @ ( product_Pair_int_int @ X2 @ Y4 ) )
      = ( product_Pair_int_int @ X2 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1147_apsnd__conv,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,X2: product_prod_int_int,Y4: product_prod_int_int] :
      ( ( produc312515743181951483nt_int @ F2 @ ( produc3646306378393792727nt_int @ X2 @ Y4 ) )
      = ( produc3646306378393792727nt_int @ X2 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1148_apsnd__conv,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,X2: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc7011684061226097695e_term @ F2 @ ( produc4305682042979456191nt_int @ X2 @ Y4 ) )
      = ( produc4305682042979456191nt_int @ X2 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1149_apsnd__conv,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( ( produc4973430039190721449e_term @ F2 @ ( produc5700946648718959541nt_int @ X2 @ Y4 ) )
      = ( produc5700946648718959541nt_int @ X2 @ ( F2 @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1150_sum__mono,axiom,
    ! [K2: set_Pr958786334691620121nt_int,F2: product_prod_int_int > nat,G3: product_prod_int_int > nat] :
      ( ! [I2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ I2 @ K2 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups3536494975961350351nt_nat @ F2 @ K2 ) @ ( groups3536494975961350351nt_nat @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1151_sum__mono,axiom,
    ! [K2: set_int,F2: int > nat,G3: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K2 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ K2 ) @ ( groups4541462559716669496nt_nat @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1152_sum__mono,axiom,
    ! [K2: set_nat,F2: nat > nat,G3: nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K2 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ K2 ) @ ( groups3542108847815614940at_nat @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1153_sum__mono,axiom,
    ! [K2: set_Pr958786334691620121nt_int,F2: product_prod_int_int > int,G3: product_prod_int_int > int] :
      ( ! [I2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ I2 @ K2 )
         => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3534004505452300075nt_int @ F2 @ K2 ) @ ( groups3534004505452300075nt_int @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1154_sum__mono,axiom,
    ! [K2: set_int,F2: int > int,G3: int > int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K2 )
         => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ K2 ) @ ( groups4538972089207619220nt_int @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1155_sum__mono,axiom,
    ! [K2: set_nat,F2: nat > int,G3: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K2 )
         => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ K2 ) @ ( groups3539618377306564664at_int @ G3 @ K2 ) ) ) ).

% sum_mono
thf(fact_1156_GrD2,axiom,
    ! [X2: int,Fx: int,A: set_int,F2: int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Fx ) @ ( bNF_Gr_int_int @ A @ F2 ) )
     => ( ( F2 @ X2 )
        = Fx ) ) ).

% GrD2
thf(fact_1157_GrD2,axiom,
    ! [X2: product_prod_int_int,Fx: product_prod_int_int,A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Fx ) @ ( bNF_Gr1573321696747050978nt_int @ A @ F2 ) )
     => ( ( F2 @ X2 )
        = Fx ) ) ).

% GrD2
thf(fact_1158_GrD2,axiom,
    ! [X2: int > option6357759511663192854e_term,Fx: product_prod_int_int,A: set_in3461395444621081367e_term,F2: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Fx ) @ ( bNF_Gr5793145711989525364nt_int @ A @ F2 ) )
     => ( ( F2 @ X2 )
        = Fx ) ) ).

% GrD2
thf(fact_1159_GrD2,axiom,
    ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Fx: product_prod_int_int,A: set_Pr3642885161833720865e_term,F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Fx ) @ ( bNF_Gr3160678035669724394nt_int @ A @ F2 ) )
     => ( ( F2 @ X2 )
        = Fx ) ) ).

% GrD2
thf(fact_1160_GrD1,axiom,
    ! [X2: int,Fx: int,A: set_int,F2: int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Fx ) @ ( bNF_Gr_int_int @ A @ F2 ) )
     => ( member_int @ X2 @ A ) ) ).

% GrD1
thf(fact_1161_GrD1,axiom,
    ! [X2: product_prod_int_int,Fx: product_prod_int_int,A: set_Pr958786334691620121nt_int,F2: product_prod_int_int > product_prod_int_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Fx ) @ ( bNF_Gr1573321696747050978nt_int @ A @ F2 ) )
     => ( member5262025264175285858nt_int @ X2 @ A ) ) ).

% GrD1
thf(fact_1162_GrD1,axiom,
    ! [X2: int > option6357759511663192854e_term,Fx: product_prod_int_int,A: set_in3461395444621081367e_term,F2: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Fx ) @ ( bNF_Gr5793145711989525364nt_int @ A @ F2 ) )
     => ( member8845023287901829240e_term @ X2 @ A ) ) ).

% GrD1
thf(fact_1163_GrD1,axiom,
    ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Fx: product_prod_int_int,A: set_Pr3642885161833720865e_term,F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Fx ) @ ( bNF_Gr3160678035669724394nt_int @ A @ F2 ) )
     => ( member3222579708246209666e_term @ X2 @ A ) ) ).

% GrD1
thf(fact_1164_sum__mono__inv,axiom,
    ! [F2: product_prod_int_int > nat,I5: set_Pr958786334691620121nt_int,G3: product_prod_int_int > nat,I4: product_prod_int_int] :
      ( ( ( groups3536494975961350351nt_nat @ F2 @ I5 )
        = ( groups3536494975961350351nt_nat @ G3 @ I5 ) )
     => ( ! [I2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member5262025264175285858nt_int @ I4 @ I5 )
         => ( ( finite2998713641127702882nt_int @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1165_sum__mono__inv,axiom,
    ! [F2: nat > nat,I5: set_nat,G3: nat > nat,I4: nat] :
      ( ( ( groups3542108847815614940at_nat @ F2 @ I5 )
        = ( groups3542108847815614940at_nat @ G3 @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_nat @ I4 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1166_sum__mono__inv,axiom,
    ! [F2: int > nat,I5: set_int,G3: int > nat,I4: int] :
      ( ( ( groups4541462559716669496nt_nat @ F2 @ I5 )
        = ( groups4541462559716669496nt_nat @ G3 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_int @ I4 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1167_sum__mono__inv,axiom,
    ! [F2: product_prod_int_int > int,I5: set_Pr958786334691620121nt_int,G3: product_prod_int_int > int,I4: product_prod_int_int] :
      ( ( ( groups3534004505452300075nt_int @ F2 @ I5 )
        = ( groups3534004505452300075nt_int @ G3 @ I5 ) )
     => ( ! [I2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member5262025264175285858nt_int @ I4 @ I5 )
         => ( ( finite2998713641127702882nt_int @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1168_sum__mono__inv,axiom,
    ! [F2: nat > int,I5: set_nat,G3: nat > int,I4: nat] :
      ( ( ( groups3539618377306564664at_int @ F2 @ I5 )
        = ( groups3539618377306564664at_int @ G3 @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_nat @ I4 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1169_sum__mono__inv,axiom,
    ! [F2: int > int,I5: set_int,G3: int > int,I4: int] :
      ( ( ( groups4538972089207619220nt_int @ F2 @ I5 )
        = ( groups4538972089207619220nt_int @ G3 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_int @ I4 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F2 @ I4 )
              = ( G3 @ I4 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_1170_sum_Oimage__eq,axiom,
    ! [G3: nat > nat,A: set_nat] :
      ( ( inj_on_nat_nat @ G3 @ A )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( image_nat_nat @ G3 @ A ) )
        = ( groups3542108847815614940at_nat @ G3 @ A ) ) ) ).

% sum.image_eq
thf(fact_1171_sum__multicount__gen,axiom,
    ! [S2: set_Product_unit,T4: set_Product_unit,R2: product_unit > product_unit > $o,K: product_unit > nat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( finite4290736615968046902t_unit @ T4 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T4 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8766797177574957219it_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1172_sum__multicount__gen,axiom,
    ! [S2: set_Product_unit,T4: set_nat,R2: product_unit > nat > $o,K: nat > nat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( finite_finite_nat @ T4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T4 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1173_sum__multicount__gen,axiom,
    ! [S2: set_Product_unit,T4: set_int,R2: product_unit > int > $o,K: int > nat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T4 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1174_sum__multicount__gen,axiom,
    ! [S2: set_nat,T4: set_Product_unit,R2: nat > product_unit > $o,K: product_unit > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite4290736615968046902t_unit @ T4 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T4 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8766797177574957219it_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1175_sum__multicount__gen,axiom,
    ! [S2: set_nat,T4: set_nat,R2: nat > nat > $o,K: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T4 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1176_sum__multicount__gen,axiom,
    ! [S2: set_nat,T4: set_int,R2: nat > int > $o,K: int > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T4 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1177_sum__multicount__gen,axiom,
    ! [S2: set_int,T4: set_Product_unit,R2: int > product_unit > $o,K: product_unit > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite4290736615968046902t_unit @ T4 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T4 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8766797177574957219it_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1178_sum__multicount__gen,axiom,
    ! [S2: set_int,T4: set_nat,R2: int > nat > $o,K: nat > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_nat @ T4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T4 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1179_sum__multicount__gen,axiom,
    ! [S2: set_int,T4: set_int,R2: int > int > $o,K: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T4 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1180_sum__multicount__gen,axiom,
    ! [S2: set_Product_unit,T4: set_Pr958786334691620121nt_int,R2: product_unit > product_prod_int_int > $o,K: product_prod_int_int > nat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( finite2998713641127702882nt_int @ T4 )
       => ( ! [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ T4 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S2 )
                        & ( R2 @ I @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite6756421564338198497nt_int
                  @ ( collec213857154873943460nt_int
                    @ ^ [J: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ J @ T4 )
                        & ( R2 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3536494975961350351nt_nat @ K @ T4 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1181_sum__multicount,axiom,
    ! [S: set_Product_unit,T2: set_Product_unit,R2: product_unit > product_unit > $o,K: nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T2 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite410649719033368117t_unit @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1182_sum__multicount,axiom,
    ! [S: set_Product_unit,T2: set_nat,R2: product_unit > nat > $o,K: nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T2 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1183_sum__multicount,axiom,
    ! [S: set_Product_unit,T2: set_int,R2: product_unit > int > $o,K: nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1184_sum__multicount,axiom,
    ! [S: set_nat,T2: set_Product_unit,R2: nat > product_unit > $o,K: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T2 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite410649719033368117t_unit @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1185_sum__multicount,axiom,
    ! [S: set_nat,T2: set_nat,R2: nat > nat > $o,K: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T2 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1186_sum__multicount,axiom,
    ! [S: set_nat,T2: set_int,R2: nat > int > $o,K: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1187_sum__multicount,axiom,
    ! [S: set_int,T2: set_Product_unit,R2: int > product_unit > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite4290736615968046902t_unit @ T2 )
       => ( ! [X3: product_unit] :
              ( ( member_Product_unit @ X3 @ T2 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [J: product_unit] :
                        ( ( member_Product_unit @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite410649719033368117t_unit @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1188_sum__multicount,axiom,
    ! [S: set_int,T2: set_nat,R2: int > nat > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T2 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1189_sum__multicount,axiom,
    ! [S: set_int,T2: set_int,R2: int > int > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1190_sum__multicount,axiom,
    ! [S: set_Product_unit,T2: set_Pr958786334691620121nt_int,R2: product_unit > product_prod_int_int > $o,K: nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite2998713641127702882nt_int @ T2 )
       => ( ! [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ T2 )
             => ( ( finite410649719033368117t_unit
                  @ ( collect_Product_unit
                    @ ^ [I: product_unit] :
                        ( ( member_Product_unit @ I @ S )
                        & ( R2 @ I @ X3 ) ) ) )
                = K ) )
         => ( ( groups8766797177574957219it_nat
              @ ^ [I: product_unit] :
                  ( finite6756421564338198497nt_int
                  @ ( collec213857154873943460nt_int
                    @ ^ [J: product_prod_int_int] :
                        ( ( member5262025264175285858nt_int @ J @ T2 )
                        & ( R2 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite6756421564338198497nt_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_1191_Gr__def,axiom,
    ( bNF_Gr_int_int
    = ( ^ [A6: set_int,F3: int > int] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [A3: int] :
              ( ( Uu
                = ( product_Pair_int_int @ A3 @ ( F3 @ A3 ) ) )
              & ( member_int @ A3 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_1192_Gr__def,axiom,
    ( bNF_Gr1573321696747050978nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,F3: product_prod_int_int > product_prod_int_int] :
          ( collec7447955092554649554nt_int
          @ ^ [Uu: produc1219242969750017639nt_int] :
            ? [A3: product_prod_int_int] :
              ( ( Uu
                = ( produc3646306378393792727nt_int @ A3 @ ( F3 @ A3 ) ) )
              & ( member5262025264175285858nt_int @ A3 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_1193_Gr__def,axiom,
    ( bNF_Gr5793145711989525364nt_int
    = ( ^ [A6: set_in3461395444621081367e_term,F3: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
          ( collec506566255779805410nt_int
          @ ^ [Uu: produc7773217078559923341nt_int] :
            ? [A3: int > option6357759511663192854e_term] :
              ( ( Uu
                = ( produc4305682042979456191nt_int @ A3 @ ( F3 @ A3 ) ) )
              & ( member8845023287901829240e_term @ A3 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_1194_Gr__def,axiom,
    ( bNF_Gr3160678035669724394nt_int
    = ( ^ [A6: set_Pr3642885161833720865e_term,F3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
          ( collec1790188477890212312nt_int
          @ ^ [Uu: produc2285326912895808259nt_int] :
            ? [A3: produc8551481072490612790e_term > option6357759511663192854e_term] :
              ( ( Uu
                = ( produc5700946648718959541nt_int @ A3 @ ( F3 @ A3 ) ) )
              & ( member3222579708246209666e_term @ A3 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_1195_sum__fun__comp,axiom,
    ! [S: set_Product_unit,R2: set_nat,G3: product_unit > nat,F2: nat > nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ G3 @ S ) @ R2 )
         => ( ( groups8766797177574957219it_nat
              @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups3542108847815614940at_nat
              @ ^ [Y: nat] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite410649719033368117t_unit
                      @ ( collect_Product_unit
                        @ ^ [X: product_unit] :
                            ( ( member_Product_unit @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1196_sum__fun__comp,axiom,
    ! [S: set_Product_unit,R2: set_int,G3: product_unit > int,F2: int > nat] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_int @ R2 )
       => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ G3 @ S ) @ R2 )
         => ( ( groups8766797177574957219it_nat
              @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups4541462559716669496nt_nat
              @ ^ [Y: int] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite410649719033368117t_unit
                      @ ( collect_Product_unit
                        @ ^ [X: product_unit] :
                            ( ( member_Product_unit @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1197_sum__fun__comp,axiom,
    ! [S: set_nat,R2: set_nat,G3: nat > nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G3 @ S ) @ R2 )
         => ( ( groups3542108847815614940at_nat
              @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups3542108847815614940at_nat
              @ ^ [Y: nat] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite_card_nat
                      @ ( collect_nat
                        @ ^ [X: nat] :
                            ( ( member_nat @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1198_sum__fun__comp,axiom,
    ! [S: set_nat,R2: set_int,G3: nat > int,F2: int > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ R2 )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G3 @ S ) @ R2 )
         => ( ( groups3542108847815614940at_nat
              @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups4541462559716669496nt_nat
              @ ^ [Y: int] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite_card_nat
                      @ ( collect_nat
                        @ ^ [X: nat] :
                            ( ( member_nat @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1199_sum__fun__comp,axiom,
    ! [S: set_int,R2: set_nat,G3: int > nat,F2: nat > nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G3 @ S ) @ R2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups3542108847815614940at_nat
              @ ^ [Y: nat] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite_card_int
                      @ ( collect_int
                        @ ^ [X: int] :
                            ( ( member_int @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1200_sum__fun__comp,axiom,
    ! [S: set_int,R2: set_int,G3: int > int,F2: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ R2 )
       => ( ( ord_less_eq_set_int @ ( image_int_int @ G3 @ S ) @ R2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [X: int] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups4541462559716669496nt_nat
              @ ^ [Y: int] :
                  ( times_times_nat
                  @ ( semiri1316708129612266289at_nat
                    @ ( finite_card_int
                      @ ( collect_int
                        @ ^ [X: int] :
                            ( ( member_int @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1201_sum__fun__comp,axiom,
    ! [S: set_Product_unit,R2: set_nat,G3: product_unit > nat,F2: nat > int] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ G3 @ S ) @ R2 )
         => ( ( groups8764306707065906943it_int
              @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups3539618377306564664at_int
              @ ^ [Y: nat] :
                  ( times_times_int
                  @ ( semiri1314217659103216013at_int
                    @ ( finite410649719033368117t_unit
                      @ ( collect_Product_unit
                        @ ^ [X: product_unit] :
                            ( ( member_Product_unit @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1202_sum__fun__comp,axiom,
    ! [S: set_Product_unit,R2: set_int,G3: product_unit > int,F2: int > int] :
      ( ( finite4290736615968046902t_unit @ S )
     => ( ( finite_finite_int @ R2 )
       => ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ G3 @ S ) @ R2 )
         => ( ( groups8764306707065906943it_int
              @ ^ [X: product_unit] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups4538972089207619220nt_int
              @ ^ [Y: int] :
                  ( times_times_int
                  @ ( semiri1314217659103216013at_int
                    @ ( finite410649719033368117t_unit
                      @ ( collect_Product_unit
                        @ ^ [X: product_unit] :
                            ( ( member_Product_unit @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1203_sum__fun__comp,axiom,
    ! [S: set_nat,R2: set_nat,G3: nat > nat,F2: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ R2 )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G3 @ S ) @ R2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups3539618377306564664at_int
              @ ^ [Y: nat] :
                  ( times_times_int
                  @ ( semiri1314217659103216013at_int
                    @ ( finite_card_nat
                      @ ( collect_nat
                        @ ^ [X: nat] :
                            ( ( member_nat @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1204_sum__fun__comp,axiom,
    ! [S: set_nat,R2: set_int,G3: nat > int,F2: int > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ R2 )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G3 @ S ) @ R2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [X: nat] : ( F2 @ ( G3 @ X ) )
              @ S )
            = ( groups4538972089207619220nt_int
              @ ^ [Y: int] :
                  ( times_times_int
                  @ ( semiri1314217659103216013at_int
                    @ ( finite_card_nat
                      @ ( collect_nat
                        @ ^ [X: nat] :
                            ( ( member_nat @ X @ S )
                            & ( ( G3 @ X )
                              = Y ) ) ) ) )
                  @ ( F2 @ Y ) )
              @ R2 ) ) ) ) ) ).

% sum_fun_comp
thf(fact_1205_card__UN__le,axiom,
    ! [I5: set_nat,A: nat > set_int] :
      ( ( finite_finite_nat @ I5 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ A @ I5 ) ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( finite_card_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1206_card__UN__le,axiom,
    ! [I5: set_nat,A: nat > set_Product_unit] :
      ( ( finite_finite_nat @ I5 )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ A @ I5 ) ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( finite410649719033368117t_unit @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1207_card__UN__le,axiom,
    ! [I5: set_int,A: int > set_int] :
      ( ( finite_finite_int @ I5 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ A @ I5 ) ) )
        @ ( groups4541462559716669496nt_nat
          @ ^ [I: int] : ( finite_card_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1208_card__UN__le,axiom,
    ! [I5: set_int,A: int > set_Product_unit] :
      ( ( finite_finite_int @ I5 )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ A @ I5 ) ) )
        @ ( groups4541462559716669496nt_nat
          @ ^ [I: int] : ( finite410649719033368117t_unit @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1209_card__UN__le,axiom,
    ! [I5: set_Pr958786334691620121nt_int,A: product_prod_int_int > set_int] :
      ( ( finite2998713641127702882nt_int @ I5 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( comple3221217463730067765et_int @ ( image_8760337445960573782et_int @ A @ I5 ) ) )
        @ ( groups3536494975961350351nt_nat
          @ ^ [I: product_prod_int_int] : ( finite_card_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1210_card__UN__le,axiom,
    ! [I5: set_Pr958786334691620121nt_int,A: product_prod_int_int > set_Product_unit] :
      ( ( finite2998713641127702882nt_int @ I5 )
     => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ ( image_6821120353060102265t_unit @ A @ I5 ) ) )
        @ ( groups3536494975961350351nt_nat
          @ ^ [I: product_prod_int_int] : ( finite410649719033368117t_unit @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1211_card__UN__le,axiom,
    ! [I5: set_nat,A: nat > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ I5 )
     => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ ( comple5382143125604098054nt_int @ ( image_6875168181789618818nt_int @ A @ I5 ) ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( finite6756421564338198497nt_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1212_card__UN__le,axiom,
    ! [I5: set_int,A: int > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ I5 )
     => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ ( comple5382143125604098054nt_int @ ( image_8635204845542730022nt_int @ A @ I5 ) ) )
        @ ( groups4541462559716669496nt_nat
          @ ^ [I: int] : ( finite6756421564338198497nt_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1213_card__UN__le,axiom,
    ! [I5: set_Pr958786334691620121nt_int,A: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ I5 )
     => ( ord_less_eq_nat @ ( finite6756421564338198497nt_int @ ( comple5382143125604098054nt_int @ ( image_9050163826262591653nt_int @ A @ I5 ) ) )
        @ ( groups3536494975961350351nt_nat
          @ ^ [I: product_prod_int_int] : ( finite6756421564338198497nt_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1214_card__UN__le,axiom,
    ! [I5: set_nat,A: nat > set_Pr2560585780119916871nt_int] :
      ( ( finite_finite_nat @ I5 )
     => ( ord_less_eq_nat @ ( finite1566866390884520591nt_int @ ( comple5838755015762642100nt_int @ ( image_7683095892744924464nt_int @ A @ I5 ) ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( finite1566866390884520591nt_int @ ( A @ I ) )
          @ I5 ) ) ) ).

% card_UN_le
thf(fact_1215_member__product,axiom,
    ! [X2: product_prod_int_int,A: set_int,B3: set_int] :
      ( ( member5262025264175285858nt_int @ X2 @ ( produc1926197756957559084nt_int @ A @ B3 ) )
      = ( member5262025264175285858nt_int @ X2
        @ ( produc1453890548195736866nt_int @ A
          @ ^ [Uu: int] : B3 ) ) ) ).

% member_product
thf(fact_1216_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: int > set_int,I5: set_int] :
    ? [F4: int > product_prod_int_int] :
      ( ( inj_on5397397550511676250nt_int @ F4 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ A @ I5 ) ) )
      & ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F4 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ A @ I5 ) ) ) @ ( produc1453890548195736866nt_int @ I5 @ A ) ) ) ).

% Ex_inj_on_UNION_Sigma
thf(fact_1217_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_1218_UN__I,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: product_prod_int_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( comple5382143125604098054nt_int @ ( image_9050163826262591653nt_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1219_UN__I,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: int,B3: product_prod_int_int > set_int] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member_int @ B7 @ ( comple3221217463730067765et_int @ ( image_8760337445960573782et_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1220_UN__I,axiom,
    ! [A4: product_prod_int_int,A: set_Pr958786334691620121nt_int,B7: nat,B3: product_prod_int_int > set_nat] :
      ( ( member5262025264175285858nt_int @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member_nat @ B7 @ ( comple7399068483239264473et_nat @ ( image_3714816428614994682et_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1221_UN__I,axiom,
    ! [A4: int,A: set_int,B7: product_prod_int_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( member_int @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( comple5382143125604098054nt_int @ ( image_8635204845542730022nt_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1222_UN__I,axiom,
    ! [A4: int,A: set_int,B7: int,B3: int > set_int] :
      ( ( member_int @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member_int @ B7 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1223_UN__I,axiom,
    ! [A4: int,A: set_int,B7: nat,B3: int > set_nat] :
      ( ( member_int @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member_nat @ B7 @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1224_UN__I,axiom,
    ! [A4: nat,A: set_nat,B7: product_prod_int_int,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( member_nat @ A4 @ A )
     => ( ( member5262025264175285858nt_int @ B7 @ ( B3 @ A4 ) )
       => ( member5262025264175285858nt_int @ B7 @ ( comple5382143125604098054nt_int @ ( image_6875168181789618818nt_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1225_UN__I,axiom,
    ! [A4: nat,A: set_nat,B7: int,B3: nat > set_int] :
      ( ( member_nat @ A4 @ A )
     => ( ( member_int @ B7 @ ( B3 @ A4 ) )
       => ( member_int @ B7 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1226_UN__I,axiom,
    ! [A4: nat,A: set_nat,B7: nat,B3: nat > set_nat] :
      ( ( member_nat @ A4 @ A )
     => ( ( member_nat @ B7 @ ( B3 @ A4 ) )
       => ( member_nat @ B7 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_1227_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1228_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1229_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_1230_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_1231_finite__UN,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite2998713641127702882nt_int @ ( comple5382143125604098054nt_int @ ( image_9050163826262591653nt_int @ B3 @ A ) ) )
        = ( ! [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
             => ( finite2998713641127702882nt_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1232_finite__UN,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_nat] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_3714816428614994682et_nat @ B3 @ A ) ) )
        = ( ! [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
             => ( finite_finite_nat @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1233_finite__UN,axiom,
    ! [A: set_Pr958786334691620121nt_int,B3: product_prod_int_int > set_int] :
      ( ( finite2998713641127702882nt_int @ A )
     => ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_8760337445960573782et_int @ B3 @ A ) ) )
        = ( ! [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A )
             => ( finite_finite_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1234_finite__UN,axiom,
    ! [A: set_nat,B3: nat > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite2998713641127702882nt_int @ ( comple5382143125604098054nt_int @ ( image_6875168181789618818nt_int @ B3 @ A ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ( finite2998713641127702882nt_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1235_finite__UN,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ( finite_finite_nat @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1236_finite__UN,axiom,
    ! [A: set_nat,B3: nat > set_int] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ( finite_finite_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1237_finite__UN,axiom,
    ! [A: set_int,B3: int > set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite2998713641127702882nt_int @ ( comple5382143125604098054nt_int @ ( image_8635204845542730022nt_int @ B3 @ A ) ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ A )
             => ( finite2998713641127702882nt_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1238_finite__UN,axiom,
    ! [A: set_int,B3: int > set_nat] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A ) ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ A )
             => ( finite_finite_nat @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1239_finite__UN,axiom,
    ! [A: set_int,B3: int > set_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ A )
             => ( finite_finite_int @ ( B3 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1240_finite__Union,axiom,
    ! [A: set_se6260736226359567993nt_int] :
      ( ( finite8744585540193469122nt_int @ A )
     => ( ! [M5: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ M5 @ A )
           => ( finite2998713641127702882nt_int @ M5 ) )
       => ( finite2998713641127702882nt_int @ ( comple5382143125604098054nt_int @ A ) ) ) ) ).

% finite_Union
thf(fact_1241_finite__Union,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ! [M5: set_nat] :
            ( ( member_set_nat @ M5 @ A )
           => ( finite_finite_nat @ M5 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).

% finite_Union
thf(fact_1242_finite__Union,axiom,
    ! [A: set_set_int] :
      ( ( finite6197958912794628473et_int @ A )
     => ( ! [M5: set_int] :
            ( ( member_set_int @ M5 @ A )
           => ( finite_finite_int @ M5 ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ A ) ) ) ) ).

% finite_Union
thf(fact_1243_finite__UN__I,axiom,
    ! [A: set_int,B3: int > set_int] :
      ( ( finite_finite_int @ A )
     => ( ! [A2: int] :
            ( ( member_int @ A2 @ A )
           => ( finite_finite_int @ ( B3 @ A2 ) ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) ) ) ) ).

% finite_UN_I
thf(fact_1244_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_1245_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% nat_leq_as_int
thf(fact_1246_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_1247_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_1248_int__ops_I7_J,axiom,
    ! [A4: nat,B7: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A4 @ B7 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ).

% int_ops(7)
thf(fact_1249_verit__la__generic,axiom,
    ! [A4: int,X2: int] :
      ( ( ord_less_eq_int @ A4 @ X2 )
      | ( A4 = X2 )
      | ( ord_less_eq_int @ X2 @ A4 ) ) ).

% verit_la_generic
thf(fact_1250_board__def,axiom,
    ( board
    = ( ^ [N2: nat,M3: nat] :
          ( collec213857154873943460nt_int
          @ ^ [Uu: product_prod_int_int] :
            ? [I: int,J: int] :
              ( ( Uu
                = ( product_Pair_int_int @ I @ J ) )
              & ( ord_less_eq_int @ one_one_int @ I )
              & ( ord_less_eq_int @ I @ ( semiri1314217659103216013at_int @ N2 ) )
              & ( ord_less_eq_int @ one_one_int @ J )
              & ( ord_less_eq_int @ J @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ) ) ).

% board_def
thf(fact_1251_board__exec__leq,axiom,
    ! [I4: int,J4: int,N: nat,M: nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J4 ) @ ( board_exec @ N @ M ) )
      = ( ( ord_less_eq_int @ one_one_int @ I4 )
        & ( ord_less_eq_int @ I4 @ ( semiri1314217659103216013at_int @ N ) )
        & ( ord_less_eq_int @ one_one_int @ J4 )
        & ( ord_less_eq_int @ J4 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% board_exec_leq
thf(fact_1252_row__exec__leq,axiom,
    ! [J4: int,M: nat] :
      ( ( member_int @ J4 @ ( row_exec @ M ) )
      = ( ( ord_less_eq_int @ one_one_int @ J4 )
        & ( ord_less_eq_int @ J4 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% row_exec_leq
thf(fact_1253_board__exec__aux__leq__mem,axiom,
    ! [I4: int,J4: int,K: nat,M2: set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I4 @ J4 ) @ ( board_exec_aux @ K @ M2 ) )
      = ( ( ord_less_eq_int @ one_one_int @ I4 )
        & ( ord_less_eq_int @ I4 @ ( semiri1314217659103216013at_int @ K ) )
        & ( member_int @ J4 @ M2 ) ) ) ).

% board_exec_aux_leq_mem
thf(fact_1254_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_1255_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_1256_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_1257_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1258_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1259_knights__path__board__non__empty,axiom,
    ! [B7: set_Pr958786334691620121nt_int,Ps: list_P5707943133018811711nt_int] :
      ( ( knights_path @ B7 @ Ps )
     => ( B7 != bot_bo1796632182523588997nt_int ) ) ).

% knights_path_board_non_empty
thf(fact_1260_diff__diff__cancel,axiom,
    ! [I4: nat,N: nat] :
      ( ( ord_less_eq_nat @ I4 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
        = I4 ) ) ).

% diff_diff_cancel
thf(fact_1261_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N4: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% int_diff_cases
thf(fact_1262_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_1263_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_1264_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_1265_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1266_diff__commute,axiom,
    ! [I4: nat,J4: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J4 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J4 ) ) ).

% diff_commute
thf(fact_1267_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_1268_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_1269_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_1270_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_1271_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1272_le__diff__iff_H,axiom,
    ! [A4: nat,C2: nat,B7: nat] :
      ( ( ord_less_eq_nat @ A4 @ C2 )
     => ( ( ord_less_eq_nat @ B7 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A4 ) @ ( minus_minus_nat @ C2 @ B7 ) )
          = ( ord_less_eq_nat @ B7 @ A4 ) ) ) ) ).

% le_diff_iff'
thf(fact_1273_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_1274_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N5 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_1275_int__le__induct,axiom,
    ! [I4: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I4 @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I4 ) ) ) ) ).

% int_le_induct

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( transpose_board @ ( board @ n @ m ) )
    = ( board @ m @ n ) ) ).

%------------------------------------------------------------------------------