TPTP Problem File: SLH0723^1.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    : Separation_Logic_Unbounded/0000_UnboundedLogic/prob_00314_010225__6511282_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2285 ( 276 unt;1006 typ;   0 def)
%            Number of atoms       : 4823 ( 893 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 19443 ( 555   ~;  59   |; 500   &;15828   @)
%                                         (   0 <=>;2501  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (  10 avg)
%            Number of types       :  218 ( 217 usr)
%            Number of type conns  : 2023 (2023   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  792 ( 789 usr; 177 con; 0-7 aty)
%            Number of variables   : 4224 (  66   ^;3968   !; 190   ?;4224   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 16:04:33.700
%------------------------------------------------------------------------------
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Sum____Type__Osum_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__suc_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard_001_Eo,type,
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thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_P,type,
    p: assertion_a_b_d_c ).

thf(sy_v_Q,type,
    q: assertion_a_b_d_c ).

thf(sy_v_R,type,
    r: assertion_a_b_d_c ).

thf(sy_v__092_060Delta_062,type,
    delta: ( c > d ) > set_a ).

thf(sy_v_c,type,
    c2: set_Pr1275464188344874039_a_c_d ).

thf(sy_v_mult,type,
    mult: b > a > a ).

thf(sy_v_one,type,
    one: b ).

thf(sy_v_plus,type,
    plus: a > a > option_a ).

thf(sy_v_sadd,type,
    sadd: b > b > b ).

thf(sy_v_sinv,type,
    sinv: b > b ).

thf(sy_v_smult,type,
    smult: b > b > b ).

thf(sy_v_valid,type,
    valid: a > $o ).

% Relevant facts (1275)
thf(fact_0_commutative,axiom,
    ! [A: a,B: a] :
      ( ( plus @ A @ B )
      = ( plus @ B @ A ) ) ).

% commutative
thf(fact_1_can__divide,axiom,
    ! [P: b,A: a,B: a] :
      ( ( ( mult @ P @ A )
        = ( mult @ P @ B ) )
     => ( A = B ) ) ).

% can_divide
thf(fact_2_assms_I1_J,axiom,
    valid_command_a_c_d @ valid @ c2 ).

% assms(1)
thf(fact_3_assms_I3_J,axiom,
    frame_property_a_c_d @ plus @ valid @ c2 ).

% assms(3)
thf(fact_4_assms_I2_J,axiom,
    safety844553430189520448_a_c_d @ plus @ valid @ c2 ).

% assms(2)
thf(fact_5_assertion_Oinject_I3_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,Y31: assertion_a_b_d_c,Y32: assertion_a_b_d_c] :
      ( ( ( star_a_b_d_c @ X31 @ X32 )
        = ( star_a_b_d_c @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% assertion.inject(3)
thf(fact_6_unique__inv,axiom,
    ! [A: a,P: b,B: a] :
      ( ( A
        = ( mult @ P @ B ) )
      = ( B
        = ( mult @ ( sinv @ P ) @ A ) ) ) ).

% unique_inv
thf(fact_7_assms_I4_J,axiom,
    valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ p @ c2 @ q @ delta ).

% assms(4)
thf(fact_8_one__neutral,axiom,
    ! [A: a] :
      ( ( mult @ one @ A )
      = A ) ).

% one_neutral
thf(fact_9_assms_I5_J,axiom,
    not_in_fv_a_b_d_c @ plus @ mult @ valid @ r @ ( modified_a_c_d @ c2 ) ).

% assms(5)
thf(fact_10_logic_Ovalid__hoare__triple_Ocong,axiom,
    valid_6037315502795721655_b_d_c = valid_6037315502795721655_b_d_c ).

% logic.valid_hoare_triple.cong
thf(fact_11_valid__mono,axiom,
    ! [A: a,B: a] :
      ( ( ( valid @ A )
        & ( pre_larger_a @ plus @ A @ B ) )
     => ( valid @ B ) ) ).

% valid_mono
thf(fact_12_larger__same,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_larger_a @ plus @ A @ B )
      = ( pre_larger_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% larger_same
thf(fact_13_compatible__iff,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% compatible_iff
thf(fact_14_compatible__imp,axiom,
    ! [A: a,B: a,P: b] :
      ( ( pre_compatible_a @ plus @ A @ B )
     => ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).

% compatible_imp
thf(fact_15_compatible__multiples,axiom,
    ! [P: b,A: a,Q: b,B: a] :
      ( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) )
     => ( pre_compatible_a @ plus @ A @ B ) ) ).

% compatible_multiples
thf(fact_16_plus__mult,axiom,
    ! [A: a,B: a,C: a,P: b] :
      ( ( ( some_a @ A )
        = ( plus @ B @ C ) )
     => ( ( some_a @ ( mult @ P @ A ) )
        = ( plus @ ( mult @ P @ B ) @ ( mult @ P @ C ) ) ) ) ).

% plus_mult
thf(fact_17_double__mult,axiom,
    ! [P: b,Q: b,A: a] :
      ( ( mult @ P @ ( mult @ Q @ A ) )
      = ( mult @ ( smult @ P @ Q ) @ A ) ) ).

% double_mult
thf(fact_18_sat_Osimps_I2_J,axiom,
    ! [Sigma: a,S: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
      ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) )
      = ( ? [A3: a,B3: a] :
            ( ( ( some_a @ Sigma )
              = ( plus @ A3 @ B3 ) )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ B3 @ S @ Delta @ B2 ) ) ) ) ).

% sat.simps(2)
thf(fact_19_asso1,axiom,
    ! [A: a,B: a,Ab: a,C: a,Bc: a] :
      ( ( ( ( plus @ A @ B )
          = ( some_a @ Ab ) )
        & ( ( plus @ B @ C )
          = ( some_a @ Bc ) ) )
     => ( ( plus @ Ab @ C )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_20_smult__comm,axiom,
    ! [P: b,Q: b] :
      ( ( smult @ P @ Q )
      = ( smult @ Q @ P ) ) ).

% smult_comm
thf(fact_21_smult__asso,axiom,
    ! [P: b,Q: b,R: b] :
      ( ( smult @ ( smult @ P @ Q ) @ R )
      = ( smult @ P @ ( smult @ Q @ R ) ) ) ).

% smult_asso
thf(fact_22_move__sum,axiom,
    ! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X: a,X1: a,X2: a] :
      ( ( ( some_a @ A )
        = ( plus @ A1 @ A22 ) )
     => ( ( ( some_a @ B )
          = ( plus @ B1 @ B22 ) )
       => ( ( ( some_a @ X )
            = ( plus @ A @ B ) )
         => ( ( ( some_a @ X1 )
              = ( plus @ A1 @ B1 ) )
           => ( ( ( some_a @ X2 )
                = ( plus @ A22 @ B22 ) )
             => ( ( some_a @ X )
                = ( plus @ X1 @ X2 ) ) ) ) ) ) ) ).

% move_sum
thf(fact_23_asso3,axiom,
    ! [A: a,B: a,C: a,Bc: a] :
      ( ~ ( pre_compatible_a @ plus @ A @ B )
     => ( ( ( plus @ B @ C )
          = ( some_a @ Bc ) )
       => ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).

% asso3
thf(fact_24_asso2,axiom,
    ! [A: a,B: a,Ab: a,C: a] :
      ( ( ( ( plus @ A @ B )
          = ( some_a @ Ab ) )
        & ~ ( pre_compatible_a @ plus @ B @ C ) )
     => ~ ( pre_compatible_a @ plus @ Ab @ C ) ) ).

% asso2
thf(fact_25_sum__both__larger,axiom,
    ! [X3: a,A4: a,B4: a,X: a,A: a,B: a] :
      ( ( ( some_a @ X3 )
        = ( plus @ A4 @ B4 ) )
     => ( ( ( some_a @ X )
          = ( plus @ A @ B ) )
       => ( ( pre_larger_a @ plus @ A4 @ A )
         => ( ( pre_larger_a @ plus @ B4 @ B )
           => ( pre_larger_a @ plus @ X3 @ X ) ) ) ) ) ).

% sum_both_larger
thf(fact_26_larger__first__sum,axiom,
    ! [Y: a,A: a,B: a,X: a] :
      ( ( ( some_a @ Y )
        = ( plus @ A @ B ) )
     => ( ( pre_larger_a @ plus @ X @ Y )
       => ? [A5: a] :
            ( ( ( some_a @ X )
              = ( plus @ A5 @ B ) )
            & ( pre_larger_a @ plus @ A5 @ A ) ) ) ) ).

% larger_first_sum
thf(fact_27_larger__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_larger_a @ plus @ A @ B )
      = ( ? [C2: a] :
            ( ( some_a @ A )
            = ( plus @ B @ C2 ) ) ) ) ).

% larger_def
thf(fact_28_larger__implies__compatible,axiom,
    ! [X: a,Y: a] :
      ( ( pre_larger_a @ plus @ X @ Y )
     => ( pre_compatible_a @ plus @ X @ Y ) ) ).

% larger_implies_compatible
thf(fact_29_compatible__smaller,axiom,
    ! [A: a,B: a,X: a] :
      ( ( pre_larger_a @ plus @ A @ B )
     => ( ( pre_compatible_a @ plus @ X @ A )
       => ( pre_compatible_a @ plus @ X @ B ) ) ) ).

% compatible_smaller
thf(fact_30_sone__neutral,axiom,
    ! [P: b] :
      ( ( smult @ one @ P )
      = P ) ).

% sone_neutral
thf(fact_31_sinv__inverse,axiom,
    ! [P: b] :
      ( ( smult @ P @ ( sinv @ P ) )
      = one ) ).

% sinv_inverse
thf(fact_32_not__in__fv__def,axiom,
    ! [A2: assertion_a_b_d_c,S2: set_c] :
      ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A2 @ S2 )
      = ( ! [Sigma2: a,S3: c > d,Delta2: ( c > d ) > set_a,S4: c > d] :
            ( ( equal_outside_c_d @ S3 @ S4 @ S2 )
           => ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta2 @ A2 )
              = ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S4 @ Delta2 @ A2 ) ) ) ) ) ).

% not_in_fv_def
thf(fact_33_pre__logic_Olarger__def,axiom,
    ( pre_larger_a
    = ( ^ [Plus: a > a > option_a,A3: a,B3: a] :
        ? [C2: a] :
          ( ( some_a @ A3 )
          = ( Plus @ B3 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_34_mem__Collect__eq,axiom,
    ! [A: produc2934264451710624999od_o_o,P2: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( collec3302784863738620242od_o_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_35_mem__Collect__eq,axiom,
    ! [A: produc732395585841259969at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( collec1729919521702017196at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_36_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_37_Collect__mem__eq,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o] :
      ( ( collec3302784863738620242od_o_o
        @ ^ [X4: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_38_Collect__mem__eq,axiom,
    ! [A2: set_Pr457366540195662369at_nat] :
      ( ( collec1729919521702017196at_nat
        @ ^ [X4: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_39_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_40_pre__logic_Olarger_Ocong,axiom,
    pre_larger_a = pre_larger_a ).

% pre_logic.larger.cong
thf(fact_41_pre__logic_Ocompatible_Ocong,axiom,
    pre_compatible_a = pre_compatible_a ).

% pre_logic.compatible.cong
thf(fact_42_logic_Onot__in__fv_Ocong,axiom,
    not_in_fv_a_b_d_c = not_in_fv_a_b_d_c ).

% logic.not_in_fv.cong
thf(fact_43_logic_Osafety__monotonicity_Ocong,axiom,
    safety844553430189520448_a_c_d = safety844553430189520448_a_c_d ).

% logic.safety_monotonicity.cong
thf(fact_44_logic_Oframe__property_Ocong,axiom,
    frame_property_a_c_d = frame_property_a_c_d ).

% logic.frame_property.cong
thf(fact_45_logic_Ovalid__command_Ocong,axiom,
    valid_command_a_c_d = valid_command_a_c_d ).

% logic.valid_command.cong
thf(fact_46_distrib__mult,axiom,
    ! [P: b,Q: b,X: a] :
      ( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X ) )
      = ( plus @ ( mult @ P @ X ) @ ( mult @ Q @ X ) ) ) ).

% distrib_mult
thf(fact_47_smult__distrib,axiom,
    ! [P: b,Q: b,R: b] :
      ( ( smult @ P @ ( sadd @ Q @ R ) )
      = ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R ) ) ) ).

% smult_distrib
thf(fact_48_logic__axioms,axiom,
    logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).

% logic_axioms
thf(fact_49_sadd__comm,axiom,
    ! [P: b,Q: b] :
      ( ( sadd @ P @ Q )
      = ( sadd @ Q @ P ) ) ).

% sadd_comm
thf(fact_50_logic_Osmult__distrib,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sadd @ Q @ R ) )
        = ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R ) ) ) ) ).

% logic.smult_distrib
thf(fact_51_logic_Osone__neutral,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ One @ P )
        = P ) ) ).

% logic.sone_neutral
thf(fact_52_logic_Osinv__inverse,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sinv @ P ) )
        = One ) ) ).

% logic.sinv_inverse
thf(fact_53_logic_Oone__neutral,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ One @ A )
        = A ) ) ).

% logic.one_neutral
thf(fact_54_logic_Odouble__mult,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ P @ ( Mult @ Q @ A ) )
        = ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).

% logic.double_mult
thf(fact_55_logic_Ocommutative,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Plus2 @ A @ B )
        = ( Plus2 @ B @ A ) ) ) ).

% logic.commutative
thf(fact_56_logic_Ounique__inv,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,P: b,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( A
          = ( Mult @ P @ B ) )
        = ( B
          = ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).

% logic.unique_inv
thf(fact_57_logic_Osmult__comm,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ Q )
        = ( Smult @ Q @ P ) ) ) ).

% logic.smult_comm
thf(fact_58_logic_Osmult__asso,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ ( Smult @ P @ Q ) @ R )
        = ( Smult @ P @ ( Smult @ Q @ R ) ) ) ) ).

% logic.smult_asso
thf(fact_59_logic_Ocan__divide,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Mult @ P @ A )
          = ( Mult @ P @ B ) )
       => ( A = B ) ) ) ).

% logic.can_divide
thf(fact_60_logic_Osadd__comm,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Sadd @ P @ Q )
        = ( Sadd @ Q @ P ) ) ) ).

% logic.sadd_comm
thf(fact_61_logic_Odistrib__mult,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,X: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X ) )
        = ( Plus2 @ ( Mult @ P @ X ) @ ( Mult @ Q @ X ) ) ) ) ).

% logic.distrib_mult
thf(fact_62_logic_Oplus__mult,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus2 @ B @ C ) )
       => ( ( some_a @ ( Mult @ P @ A ) )
          = ( Plus2 @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).

% logic.plus_mult
thf(fact_63_logic_Omove__sum,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,A1: a,A22: a,B: a,B1: a,B22: a,X: a,X1: a,X2: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus2 @ A1 @ A22 ) )
       => ( ( ( some_a @ B )
            = ( Plus2 @ B1 @ B22 ) )
         => ( ( ( some_a @ X )
              = ( Plus2 @ A @ B ) )
           => ( ( ( some_a @ X1 )
                = ( Plus2 @ A1 @ B1 ) )
             => ( ( ( some_a @ X2 )
                  = ( Plus2 @ A22 @ B22 ) )
               => ( ( some_a @ X )
                  = ( Plus2 @ X1 @ X2 ) ) ) ) ) ) ) ) ).

% logic.move_sum
thf(fact_64_logic_Oasso1,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a,Bc: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus2 @ A @ B )
            = ( some_a @ Ab ) )
          & ( ( Plus2 @ B @ C )
            = ( some_a @ Bc ) ) )
       => ( ( Plus2 @ Ab @ C )
          = ( Plus2 @ A @ Bc ) ) ) ) ).

% logic.asso1
thf(fact_65_logic_Ocompatible__multiples,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,Q: b,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
       => ( pre_compatible_a @ Plus2 @ A @ B ) ) ) ).

% logic.compatible_multiples
thf(fact_66_logic_Ocompatible__imp,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ A @ B )
       => ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_imp
thf(fact_67_logic_Ocompatible__iff,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus2 @ A @ B )
        = ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_iff
thf(fact_68_logic_Olarger__same,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ A @ B )
        = ( pre_larger_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.larger_same
thf(fact_69_logic_Ovalid__mono,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Valid @ A )
          & ( pre_larger_a @ Plus2 @ A @ B ) )
       => ( Valid @ B ) ) ) ).

% logic.valid_mono
thf(fact_70_assertion_Odistinct_I43_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( wand_a_b_d_c @ X41 @ X42 ) ) ).

% assertion.distinct(43)
thf(fact_71_assertion_Odistinct_I23_J,axiom,
    ! [X21: b,X22: assertion_a_b_d_c,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( mult_b_a_d_c @ X21 @ X22 )
     != ( star_a_b_d_c @ X31 @ X32 ) ) ).

% assertion.distinct(23)
thf(fact_72_assertion_Odistinct_I47_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( and_a_b_d_c @ X61 @ X62 ) ) ).

% assertion.distinct(47)
thf(fact_73_assertion_Odistinct_I45_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( or_a_b_d_c @ X51 @ X52 ) ) ).

% assertion.distinct(45)
thf(fact_74_assertion_Odistinct_I59_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( wildcard_a_b_d_c @ X12 ) ) ).

% assertion.distinct(59)
thf(fact_75_assertion_Odistinct_I57_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( bounded_a_b_d_c @ X11 ) ) ).

% assertion.distinct(57)
thf(fact_76_assertion_Odistinct_I55_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != pred_a_b_d_c ) ).

% assertion.distinct(55)
thf(fact_77_logic_Oasso3,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,Bc: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ~ ( pre_compatible_a @ Plus2 @ A @ B )
       => ( ( ( Plus2 @ B @ C )
            = ( some_a @ Bc ) )
         => ~ ( pre_compatible_a @ Plus2 @ A @ Bc ) ) ) ) ).

% logic.asso3
thf(fact_78_logic_Oasso2,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus2 @ A @ B )
            = ( some_a @ Ab ) )
          & ~ ( pre_compatible_a @ Plus2 @ B @ C ) )
       => ~ ( pre_compatible_a @ Plus2 @ Ab @ C ) ) ) ).

% logic.asso2
thf(fact_79_logic_Olarger__first__sum,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Y: a,A: a,B: a,X: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ Y )
          = ( Plus2 @ A @ B ) )
       => ( ( pre_larger_a @ Plus2 @ X @ Y )
         => ? [A5: a] :
              ( ( ( some_a @ X )
                = ( Plus2 @ A5 @ B ) )
              & ( pre_larger_a @ Plus2 @ A5 @ A ) ) ) ) ) ).

% logic.larger_first_sum
thf(fact_80_logic_Osum__both__larger,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X3: a,A4: a,B4: a,X: a,A: a,B: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ X3 )
          = ( Plus2 @ A4 @ B4 ) )
       => ( ( ( some_a @ X )
            = ( Plus2 @ A @ B ) )
         => ( ( pre_larger_a @ Plus2 @ A4 @ A )
           => ( ( pre_larger_a @ Plus2 @ B4 @ B )
             => ( pre_larger_a @ Plus2 @ X3 @ X ) ) ) ) ) ) ).

% logic.sum_both_larger
thf(fact_81_logic_Olarger__implies__compatible,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Y: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ X @ Y )
       => ( pre_compatible_a @ Plus2 @ X @ Y ) ) ) ).

% logic.larger_implies_compatible
thf(fact_82_logic_Ocompatible__smaller,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,X: a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus2 @ A @ B )
       => ( ( pre_compatible_a @ Plus2 @ X @ A )
         => ( pre_compatible_a @ Plus2 @ X @ B ) ) ) ) ).

% logic.compatible_smaller
thf(fact_83_logic_Osat_Osimps_I2_J,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) )
        = ( ? [A3: a,B3: a] :
              ( ( ( some_a @ Sigma )
                = ( Plus2 @ A3 @ B3 ) )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A3 @ S @ Delta @ A2 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B3 @ S @ Delta @ B2 ) ) ) ) ) ).

% logic.sat.simps(2)
thf(fact_84_logic__def,axiom,
    ( logic_a_b
    = ( ^ [Plus: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
          ( ! [A3: a,B3: a] :
              ( ( Plus @ A3 @ B3 )
              = ( Plus @ B3 @ A3 ) )
          & ! [A3: a,B3: a,Ab2: a,C2: a,Bc2: a] :
              ( ( ( ( Plus @ A3 @ B3 )
                  = ( some_a @ Ab2 ) )
                & ( ( Plus @ B3 @ C2 )
                  = ( some_a @ Bc2 ) ) )
             => ( ( Plus @ Ab2 @ C2 )
                = ( Plus @ A3 @ Bc2 ) ) )
          & ! [A3: a,B3: a,Ab2: a,C2: a] :
              ( ( ( ( Plus @ A3 @ B3 )
                  = ( some_a @ Ab2 ) )
                & ~ ( pre_compatible_a @ Plus @ B3 @ C2 ) )
             => ~ ( pre_compatible_a @ Plus @ Ab2 @ C2 ) )
          & ! [P3: b] :
              ( ( Smult2 @ P3 @ ( Sinv2 @ P3 ) )
              = One2 )
          & ! [P3: b] :
              ( ( Smult2 @ One2 @ P3 )
              = P3 )
          & ! [P3: b,Q2: b] :
              ( ( Sadd2 @ P3 @ Q2 )
              = ( Sadd2 @ Q2 @ P3 ) )
          & ! [P3: b,Q2: b] :
              ( ( Smult2 @ P3 @ Q2 )
              = ( Smult2 @ Q2 @ P3 ) )
          & ! [P3: b,Q2: b,R2: b] :
              ( ( Smult2 @ P3 @ ( Sadd2 @ Q2 @ R2 ) )
              = ( Sadd2 @ ( Smult2 @ P3 @ Q2 ) @ ( Smult2 @ P3 @ R2 ) ) )
          & ! [P3: b,Q2: b,R2: b] :
              ( ( Smult2 @ ( Smult2 @ P3 @ Q2 ) @ R2 )
              = ( Smult2 @ P3 @ ( Smult2 @ Q2 @ R2 ) ) )
          & ! [P3: b,Q2: b,A3: a] :
              ( ( Mult2 @ P3 @ ( Mult2 @ Q2 @ A3 ) )
              = ( Mult2 @ ( Smult2 @ P3 @ Q2 ) @ A3 ) )
          & ! [A3: a,B3: a,C2: a,P3: b] :
              ( ( ( some_a @ A3 )
                = ( Plus @ B3 @ C2 ) )
             => ( ( some_a @ ( Mult2 @ P3 @ A3 ) )
                = ( Plus @ ( Mult2 @ P3 @ B3 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
          & ! [P3: b,Q2: b,X4: a] :
              ( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q2 ) @ X4 ) )
              = ( Plus @ ( Mult2 @ P3 @ X4 ) @ ( Mult2 @ Q2 @ X4 ) ) )
          & ! [A3: a] :
              ( ( Mult2 @ One2 @ A3 )
              = A3 )
          & ! [A3: a,B3: a] :
              ( ( ( Valid2 @ A3 )
                & ( pre_larger_a @ Plus @ A3 @ B3 ) )
             => ( Valid2 @ B3 ) ) ) ) ) ).

% logic_def
thf(fact_85_logic_Onot__in__fv__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,S2: set_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ S2 )
        = ( ! [Sigma2: a,S3: c > d,Delta2: ( c > d ) > set_a,S4: c > d] :
              ( ( equal_outside_c_d @ S3 @ S4 @ S2 )
             => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta2 @ A2 )
                = ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S4 @ Delta2 @ A2 ) ) ) ) ) ) ).

% logic.not_in_fv_def
thf(fact_86_option_Oinject,axiom,
    ! [X2: a,Y2: a] :
      ( ( ( some_a @ X2 )
        = ( some_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_87_compatible__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( ( plus @ A @ B )
       != none_a ) ) ).

% compatible_def
thf(fact_88_not__in__fv__mod,axiom,
    ! [A2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S: c > d,Sigma3: a,S5: c > d,X: a,Delta: ( c > d ) > set_a] :
      ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A2 @ ( modified_a_c_d @ C ) )
     => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C )
       => ( ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A2 )
          = ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S5 @ Delta @ A2 ) ) ) ) ).

% not_in_fv_mod
thf(fact_89_safety__monotonicity__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( safety844553430189520448_a_c_d @ plus @ valid @ C )
      = ( ! [Sigma2: a,Sigma4: a,S3: c > d] :
            ( ( ( valid @ Sigma4 )
              & ( pre_larger_a @ plus @ Sigma4 @ Sigma2 )
              & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) ) )
           => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S3 ) ) ) ) ) ).

% safety_monotonicity_def
thf(fact_90_valid__command__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( valid_command_a_c_d @ valid @ C )
      = ( ! [A3: a,B3: a,Sa: c > d,Sb: c > d] :
            ( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A3 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B3 @ Sb ) ) ) @ C )
              & ( valid @ A3 ) )
           => ( valid @ B3 ) ) ) ) ).

% valid_command_def
thf(fact_91_not__None__eq,axiom,
    ! [X: option_a] :
      ( ( X != none_a )
      = ( ? [Y3: a] :
            ( X
            = ( some_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_92_not__Some__eq,axiom,
    ! [X: option_a] :
      ( ( ! [Y3: a] :
            ( X
           != ( some_a @ Y3 ) ) )
      = ( X = none_a ) ) ).

% not_Some_eq
thf(fact_93_frame__property__def,axiom,
    ! [C: set_Pr1275464188344874039_a_c_d] :
      ( ( frame_property_a_c_d @ plus @ valid @ C )
      = ( ! [Sigma2: a,Sigma_0: a,R2: a,Sigma4: a,S3: c > d,S4: c > d] :
            ( ( ( valid @ Sigma2 )
              & ( valid @ Sigma4 )
              & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) )
              & ( ( some_a @ Sigma2 )
                = ( plus @ Sigma_0 @ R2 ) )
              & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S4 ) ) ) @ C ) )
           => ? [Sigma_02: a] :
                ( ( ( some_a @ Sigma4 )
                  = ( plus @ Sigma_02 @ R2 ) )
                & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S4 ) ) ) @ C ) ) ) ) ) ).

% frame_property_def
thf(fact_94_valid__hoare__triple__def,axiom,
    ! [P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q3: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q3 @ Delta )
      = ( ! [Sigma2: a,S3: c > d] :
            ( ( ( valid @ Sigma2 )
              & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta @ P2 ) )
           => ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) )
              & ! [Sigma4: a,S4: c > d] :
                  ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S4 ) ) ) @ C )
                 => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S4 @ Delta @ Q3 ) ) ) ) ) ) ).

% valid_hoare_triple_def
thf(fact_95_valid__hoare__tripleI,axiom,
    ! [Delta: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q3: assertion_a_b_d_c] :
      ( ! [Sigma5: a,S6: c > d] :
          ( ( ( valid @ Sigma5 )
            & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S6 @ Delta @ P2 ) )
         => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S6 ) ) )
     => ( ! [Sigma5: a,S6: c > d,Sigma6: a,S7: c > d] :
            ( ( ( valid @ Sigma5 )
              & ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S6 @ Delta @ P2 ) )
           => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S6 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
             => ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma6 @ S7 @ Delta @ Q3 ) ) )
       => ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q3 @ Delta ) ) ) ).

% valid_hoare_tripleI
thf(fact_96_option_Odistinct_I1_J,axiom,
    ! [X2: a] :
      ( none_a
     != ( some_a @ X2 ) ) ).

% option.distinct(1)
thf(fact_97_option_OdiscI,axiom,
    ! [Option: option_a,X2: a] :
      ( ( Option
        = ( some_a @ X2 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_98_option_Oexhaust,axiom,
    ! [Y: option_a] :
      ( ( Y != none_a )
     => ~ ! [X23: a] :
            ( Y
           != ( some_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_99_split__option__ex,axiom,
    ( ( ^ [P4: option_a > $o] :
        ? [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          | ? [X4: a] : ( P5 @ ( some_a @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_100_split__option__all,axiom,
    ( ( ^ [P4: option_a > $o] :
        ! [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          & ! [X4: a] : ( P5 @ ( some_a @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_101_combine__options__cases,axiom,
    ! [X: option_a,P2: option_a > option_a > $o,Y: option_a] :
      ( ( ( X = none_a )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X @ Y ) )
       => ( ! [A6: a,B5: a] :
              ( ( X
                = ( some_a @ A6 ) )
             => ( ( Y
                  = ( some_a @ B5 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_102_assertion_Odistinct_I3_J,axiom,
    ! [X1: ( c > d ) > a > $o,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
      ( ( sem_c_d_a_b @ X1 )
     != ( star_a_b_d_c @ X31 @ X32 ) ) ).

% assertion.distinct(3)
thf(fact_103_logic_Ovalid__command__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_command_a_c_d @ Valid @ C )
        = ( ! [A3: a,B3: a,Sa: c > d,Sb: c > d] :
              ( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A3 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B3 @ Sb ) ) ) @ C )
                & ( Valid @ A3 ) )
             => ( Valid @ B3 ) ) ) ) ) ).

% logic.valid_command_def
thf(fact_104_pre__logic_Ocompatible__def,axiom,
    ( pre_compatible_a
    = ( ^ [Plus: a > a > option_a,A3: a,B3: a] :
          ( ( Plus @ A3 @ B3 )
         != none_a ) ) ) ).

% pre_logic.compatible_def
thf(fact_105_assertion_Odistinct_I49_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( imp_a_b_d_c @ X71 @ X72 ) ) ).

% assertion.distinct(49)
thf(fact_106_logic_Ovalid__hoare__tripleI,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q3: assertion_a_b_d_c] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma5: a,S6: c > d] :
            ( ( ( Valid @ Sigma5 )
              & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S6 @ Delta @ P2 ) )
           => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S6 ) ) )
       => ( ! [Sigma5: a,S6: c > d,Sigma6: a,S7: c > d] :
              ( ( ( Valid @ Sigma5 )
                & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S6 @ Delta @ P2 ) )
             => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S6 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
               => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma6 @ S7 @ Delta @ Q3 ) ) )
         => ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q3 @ Delta ) ) ) ) ).

% logic.valid_hoare_tripleI
thf(fact_107_logic_Ovalid__hoare__triple__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q3: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q3 @ Delta )
        = ( ! [Sigma2: a,S3: c > d] :
              ( ( ( Valid @ Sigma2 )
                & ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta @ P2 ) )
             => ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) )
                & ! [Sigma4: a,S4: c > d] :
                    ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S4 ) ) ) @ C )
                   => ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S4 @ Delta @ Q3 ) ) ) ) ) ) ) ).

% logic.valid_hoare_triple_def
thf(fact_108_assertion_Odistinct_I53_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( forall_c_a_b_d @ X91 @ X92 ) ) ).

% assertion.distinct(53)
thf(fact_109_assertion_Odistinct_I51_J,axiom,
    ! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
      ( ( star_a_b_d_c @ X31 @ X32 )
     != ( exists_c_a_b_d @ X81 @ X82 ) ) ).

% assertion.distinct(51)
thf(fact_110_logic_Oframe__property__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( frame_property_a_c_d @ Plus2 @ Valid @ C )
        = ( ! [Sigma2: a,Sigma_0: a,R2: a,Sigma4: a,S3: c > d,S4: c > d] :
              ( ( ( Valid @ Sigma2 )
                & ( Valid @ Sigma4 )
                & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) )
                & ( ( some_a @ Sigma2 )
                  = ( Plus2 @ Sigma_0 @ R2 ) )
                & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S4 ) ) ) @ C ) )
             => ? [Sigma_02: a] :
                  ( ( ( some_a @ Sigma4 )
                    = ( Plus2 @ Sigma_02 @ R2 ) )
                  & ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S4 ) ) ) @ C ) ) ) ) ) ) ).

% logic.frame_property_def
thf(fact_111_assertion_Oexhaust,axiom,
    ! [Y: assertion_a_b_d_c] :
      ( ! [X13: ( c > d ) > a > $o] :
          ( Y
         != ( sem_c_d_a_b @ X13 ) )
     => ( ! [X212: b,X222: assertion_a_b_d_c] :
            ( Y
           != ( mult_b_a_d_c @ X212 @ X222 ) )
       => ( ! [X312: assertion_a_b_d_c,X322: assertion_a_b_d_c] :
              ( Y
             != ( star_a_b_d_c @ X312 @ X322 ) )
         => ( ! [X412: assertion_a_b_d_c,X422: assertion_a_b_d_c] :
                ( Y
               != ( wand_a_b_d_c @ X412 @ X422 ) )
           => ( ! [X512: assertion_a_b_d_c,X522: assertion_a_b_d_c] :
                  ( Y
                 != ( or_a_b_d_c @ X512 @ X522 ) )
             => ( ! [X612: assertion_a_b_d_c,X622: assertion_a_b_d_c] :
                    ( Y
                   != ( and_a_b_d_c @ X612 @ X622 ) )
               => ( ! [X712: assertion_a_b_d_c,X722: assertion_a_b_d_c] :
                      ( Y
                     != ( imp_a_b_d_c @ X712 @ X722 ) )
                 => ( ! [X812: c,X822: assertion_a_b_d_c] :
                        ( Y
                       != ( exists_c_a_b_d @ X812 @ X822 ) )
                   => ( ! [X912: c,X922: assertion_a_b_d_c] :
                          ( Y
                         != ( forall_c_a_b_d @ X912 @ X922 ) )
                     => ( ( Y != pred_a_b_d_c )
                       => ( ! [X112: assertion_a_b_d_c] :
                              ( Y
                             != ( bounded_a_b_d_c @ X112 ) )
                         => ~ ! [X122: assertion_a_b_d_c] :
                                ( Y
                               != ( wildcard_a_b_d_c @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% assertion.exhaust
thf(fact_112_logic_Onot__in__fv__mod,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S: c > d,Sigma3: a,S5: c > d,X: a,Delta: ( c > d ) > set_a] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ ( modified_a_c_d @ C ) )
       => ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C )
         => ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A2 )
            = ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S5 @ Delta @ A2 ) ) ) ) ) ).

% logic.not_in_fv_mod
thf(fact_113_logic_Osafety__monotonicity__def,axiom,
    ! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
      ( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( safety844553430189520448_a_c_d @ Plus2 @ Valid @ C )
        = ( ! [Sigma2: a,Sigma4: a,S3: c > d] :
              ( ( ( Valid @ Sigma4 )
                & ( pre_larger_a @ Plus2 @ Sigma4 @ Sigma2 )
                & ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) ) )
             => ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma4 @ S3 ) ) ) ) ) ) ).

% logic.safety_monotonicity_def
thf(fact_114_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X2: set_Product_prod_o_o,Y1: set_Product_prod_o_o,Y2: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ X1 @ X2 )
        = ( produc763777882069021527od_o_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_115_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X2: set_Pr1261947904930325089at_nat,Y1: set_Product_prod_o_o,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ X1 @ X2 )
        = ( produc8517790099723286449at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_116_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_117_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_118_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_119_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_120_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_121_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_122_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_123_prod__cases,axiom,
    ! [P2: produc2934264451710624999od_o_o > $o,P: produc2934264451710624999od_o_o] :
      ( ! [A6: set_Product_prod_o_o,B5: set_Product_prod_o_o] : ( P2 @ ( produc763777882069021527od_o_o @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_124_prod__cases,axiom,
    ! [P2: produc732395585841259969at_nat > $o,P: produc732395585841259969at_nat] :
      ( ! [A6: set_Product_prod_o_o,B5: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc8517790099723286449at_nat @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_125_prod__cases,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ! [A6: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_126_surj__pair,axiom,
    ! [P: produc2934264451710624999od_o_o] :
    ? [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
      ( P
      = ( produc763777882069021527od_o_o @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_127_surj__pair,axiom,
    ! [P: produc732395585841259969at_nat] :
    ? [X6: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc8517790099723286449at_nat @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_128_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc2922128104949294807at_nat @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_129_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2934264451710624999od_o_o] :
      ~ ! [A6: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
          ( Y
         != ( produc763777882069021527od_o_o @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_130_old_Oprod_Oexhaust,axiom,
    ! [Y: produc732395585841259969at_nat] :
      ~ ! [A6: set_Product_prod_o_o,B5: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc8517790099723286449at_nat @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_131_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A6: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_132_ssubst__Pair__rhs,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,S5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_133_ssubst__Pair__rhs,axiom,
    ! [R: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat,R3: set_Pr457366540195662369at_nat,S5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_134_ssubst__Pair__rhs,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_135_option_Ocollapse,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( ( some_a @ ( the_a @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_136_curryI,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( F @ ( produc763777882069021527od_o_o @ A @ B ) )
     => ( produc4861507100678042957_o_o_o @ F @ A @ B ) ) ).

% curryI
thf(fact_137_curryI,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ ( produc8517790099723286449at_nat @ A @ B ) )
     => ( produc5440495499350717427_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_138_curryI,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( produc7820828192191229133_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_139_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ~ ( member444158400953824016od_o_o @ K @ ( dom_Pr3509670420972068777_o_o_a @ M ) )
     => ( ( fun_up7252388491999432090tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_140_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ~ ( member4095101504841534314at_nat @ K @ ( dom_Pr8042991038213739471_nat_a @ M ) )
     => ( ( fun_up564627978254483264tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_141_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ~ ( member8757157785044589968at_nat @ K @ ( dom_Pr1957410074072848681_nat_a @ M ) )
     => ( ( fun_up7579534627831735322tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_142_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ M ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) ) ) ).

% in_graphD
thf(fact_143_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% in_graphD
thf(fact_144_in__graphD,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% in_graphD
thf(fact_145_map__add__upd__left,axiom,
    ! [M: produc2934264451710624999od_o_o,E2: produc2934264451710624999od_o_o > option_a,E1: produc2934264451710624999od_o_o > option_a,U1: a] :
      ( ~ ( member444158400953824016od_o_o @ M @ ( dom_Pr3509670420972068777_o_o_a @ E2 ) )
     => ( ( map_ad9054432547132481292_o_o_a @ ( fun_up7252388491999432090tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up7252388491999432090tion_a @ ( map_ad9054432547132481292_o_o_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_146_map__add__upd__left,axiom,
    ! [M: produc732395585841259969at_nat,E2: produc732395585841259969at_nat > option_a,E1: produc732395585841259969at_nat > option_a,U1: a] :
      ( ~ ( member4095101504841534314at_nat @ M @ ( dom_Pr8042991038213739471_nat_a @ E2 ) )
     => ( ( map_ad4947507604491699634_nat_a @ ( fun_up564627978254483264tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up564627978254483264tion_a @ ( map_ad4947507604491699634_nat_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_147_map__add__upd__left,axiom,
    ! [M: produc3843707927480180839at_nat,E2: produc3843707927480180839at_nat > option_a,E1: produc3843707927480180839at_nat > option_a,U1: a] :
      ( ~ ( member8757157785044589968at_nat @ M @ ( dom_Pr1957410074072848681_nat_a @ E2 ) )
     => ( ( map_ad6003791139528408460_nat_a @ ( fun_up7579534627831735322tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
        = ( fun_up7579534627831735322tion_a @ ( map_ad6003791139528408460_nat_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_148_domD,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr3509670420972068777_o_o_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_149_domD,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr8042991038213739471_nat_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_150_domD,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr1957410074072848681_nat_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_151_domI,axiom,
    ! [M: produc2934264451710624999od_o_o > option_a,A: produc2934264451710624999od_o_o,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member444158400953824016od_o_o @ A @ ( dom_Pr3509670420972068777_o_o_a @ M ) ) ) ).

% domI
thf(fact_152_domI,axiom,
    ! [M: produc732395585841259969at_nat > option_a,A: produc732395585841259969at_nat,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member4095101504841534314at_nat @ A @ ( dom_Pr8042991038213739471_nat_a @ M ) ) ) ).

% domI
thf(fact_153_domI,axiom,
    ! [M: produc3843707927480180839at_nat > option_a,A: produc3843707927480180839at_nat,B: a] :
      ( ( ( M @ A )
        = ( some_a @ B ) )
     => ( member8757157785044589968at_nat @ A @ ( dom_Pr1957410074072848681_nat_a @ M ) ) ) ).

% domI
thf(fact_154_domIff,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr3509670420972068777_o_o_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_155_domIff,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr8042991038213739471_nat_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_156_domIff,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr1957410074072848681_nat_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_157_option_Osel,axiom,
    ! [X2: a] :
      ( ( the_a @ ( some_a @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_158_option_Oexpand,axiom,
    ! [Option: option_a,Option2: option_a] :
      ( ( ( Option = none_a )
        = ( Option2 = none_a ) )
     => ( ( ( Option != none_a )
         => ( ( Option2 != none_a )
           => ( ( the_a @ Option )
              = ( the_a @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_159_curryE,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( produc4861507100678042957_o_o_o @ F @ A @ B )
     => ( F @ ( produc763777882069021527od_o_o @ A @ B ) ) ) ).

% curryE
thf(fact_160_curryE,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F @ A @ B )
     => ( F @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_161_curryE,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_162_curryD,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( produc4861507100678042957_o_o_o @ F @ A @ B )
     => ( F @ ( produc763777882069021527od_o_o @ A @ B ) ) ) ).

% curryD
thf(fact_163_curryD,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F @ A @ B )
     => ( F @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_164_curryD,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_165_option_Oexhaust__sel,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( Option
        = ( some_a @ ( the_a @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_166_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V: set_Product_prod_o_o] :
      ( ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ M ) ) ) ).

% in_graphI
thf(fact_167_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ M ) ) ) ).

% in_graphI
thf(fact_168_in__graphI,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ M ) ) ) ).

% in_graphI
thf(fact_169_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V: set_Product_prod_o_o] :
      ( ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ ( some_s1360601300120069868od_o_o @ V ) ) )
      = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ none_s7246968996079588200od_o_o ) ) ) ) ).

% graph_map_upd
thf(fact_170_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat] :
      ( ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V ) ) )
      = ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_171_graph__map__upd,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
      ( ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V ) ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_172_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_173_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_174_graph__restrictD_I2_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_175_restrict__out,axiom,
    ! [X: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ~ ( member444158400953824016od_o_o @ X @ A2 )
     => ( ( restri4233077244345833607_o_o_a @ M @ A2 @ X )
        = none_a ) ) ).

% restrict_out
thf(fact_176_restrict__out,axiom,
    ! [X: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ~ ( member4095101504841534314at_nat @ X @ A2 )
     => ( ( restri896928924502181293_nat_a @ M @ A2 @ X )
        = none_a ) ) ).

% restrict_out
thf(fact_177_restrict__out,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ~ ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( restri4040463761431659527_nat_a @ M @ A2 @ X )
        = none_a ) ) ).

% restrict_out
thf(fact_178_restrict__map__def,axiom,
    ( restri4233077244345833607_o_o_a
    = ( ^ [M2: produc2934264451710624999od_o_o > option_a,A7: set_Pr1932065953672099015od_o_o,X4: produc2934264451710624999od_o_o] : ( if_option_a @ ( member444158400953824016od_o_o @ X4 @ A7 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_179_restrict__map__def,axiom,
    ( restri896928924502181293_nat_a
    = ( ^ [M2: produc732395585841259969at_nat > option_a,A7: set_Pr457366540195662369at_nat,X4: produc732395585841259969at_nat] : ( if_option_a @ ( member4095101504841534314at_nat @ X4 @ A7 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_180_restrict__map__def,axiom,
    ( restri4040463761431659527_nat_a
    = ( ^ [M2: produc3843707927480180839at_nat > option_a,A7: set_Pr4329608150637261639at_nat,X4: produc3843707927480180839at_nat] : ( if_option_a @ ( member8757157785044589968at_nat @ X4 @ A7 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_181_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A2 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_182_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A2 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_183_graph__restrictD_I1_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A2 ) ) )
     => ( member2643936169264416010at_nat @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_184_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B5 ) @ R )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_185_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B5 ) @ R )
           => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_186_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B5 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_187_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B5 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_188_relcompEpair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A @ B5 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_189_relcompEpair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B5 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_190_relcompE,axiom,
    ! [Xz: produc2934264451710624999od_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ Xz @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
            ( ( Xz
              = ( produc763777882069021527od_o_o @ X6 @ Z ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_191_relcompE,axiom,
    ! [Xz: produc2934264451710624999od_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ Xz @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [X6: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat,Z: set_Product_prod_o_o] :
            ( ( Xz
              = ( produc763777882069021527od_o_o @ X6 @ Z ) )
           => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X6 @ Y4 ) @ R )
             => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_192_relcompE,axiom,
    ! [Xz: produc732395585841259969at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ Xz @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc8517790099723286449at_nat @ X6 @ Z ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
             => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_193_relcompE,axiom,
    ! [Xz: produc732395585841259969at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ Xz @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [X6: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc8517790099723286449at_nat @ X6 @ Z ) )
           => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X6 @ Y4 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_194_relcompE,axiom,
    ! [Xz: produc3843707927480180839at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ Xz @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc2922128104949294807at_nat @ X6 @ Z ) )
           => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ X6 @ Y4 ) @ R )
             => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_195_relcompE,axiom,
    ! [Xz: produc3843707927480180839at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ Xz @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc2922128104949294807at_nat @ X6 @ Z ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_196_relcomp_OrelcompI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Product_prod_o_o,R: set_Pr8055703017043224813od_o_o,C: set_Pr1261947904930325089at_nat,S: set_Pr457366540195662369at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom708864787231718074at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_197_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom8634166700253352710od_o_o @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_198_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Pr1261947904930325089at_nat,S: set_Pr457366540195662369at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom526950493951013856at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_199_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,C: set_Product_prod_o_o,S: set_Pr8055703017043224813od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B @ C ) @ S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom3302618809389967788od_o_o @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_200_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,C: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom5236704819349487494at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_201_relcomp_OrelcompI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_202_relcomp_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Product_prod_o_o,C2: set_Product_prod_o_o] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B3 ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_203_relcomp_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat,C2: set_Product_prod_o_o] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B3 ) @ R )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_204_relcomp_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ A22 ) @ ( relcom526950493951013856at_nat @ R @ S ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Product_prod_o_o,C2: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B3 ) @ R )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_205_relcomp_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ A22 ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B3 ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_206_relcomp_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom708864787231718074at_nat @ R @ S ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B3: set_Product_prod_o_o,C2: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A3 @ B3 ) @ R )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_207_relcomp_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_208_relcomp_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ B5 ) @ R )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_209_relcomp_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ B5 ) @ R )
           => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_210_relcomp_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ A22 ) @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ B5 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_211_relcomp_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ A22 ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A1 @ B5 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_212_relcomp_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A1 @ B5 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_213_relcomp_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ B5 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_214_these__insert__Some,axiom,
    ! [X: a,A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A2 ) )
      = ( insert_a @ X @ ( these_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_215_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( range_1318165146934585909od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) ) ) ).

% Range_insert
thf(fact_216_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( range_8412904782085253007at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_8412904782085253007at_nat @ R ) ) ) ).

% Range_insert
thf(fact_217_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_880292732526890933at_nat @ R ) ) ) ).

% Range_insert
thf(fact_218_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( domain8908270702770587084od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) ) ) ).

% Domain_insert
thf(fact_219_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( domain8494901049294654374at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_220_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_221_these__insert__None,axiom,
    ! [A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
      = ( these_a @ A2 ) ) ).

% these_insert_None
thf(fact_222_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_223_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_224_Domain__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_225_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B5 ) @ R ) ) ).

% DomainE
thf(fact_226_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B5 ) @ R ) ) ).

% DomainE
thf(fact_227_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B5 ) @ R ) ) ).

% DomainE
thf(fact_228_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) ) ) ).

% Domain.DomainI
thf(fact_229_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_230_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_231_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
            ( ( A = A3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B3 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_232_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
            ( ( A = A3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_233_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
            ( ( A = A3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_234_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
     => ~ ! [B5: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B5 ) @ R ) ) ).

% Domain.cases
thf(fact_235_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B5 ) @ R ) ) ).

% Domain.cases
thf(fact_236_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B5 ) @ R ) ) ).

% Domain.cases
thf(fact_237_Range__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_238_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y3 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_239_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_240_RangeE,axiom,
    ! [B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) )
     => ~ ! [A6: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A6 @ B ) @ R ) ) ).

% RangeE
thf(fact_241_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R ) )
     => ~ ! [A6: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A6 @ B ) @ R ) ) ).

% RangeE
thf(fact_242_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R ) )
     => ~ ! [A6: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A6 @ B ) @ R ) ) ).

% RangeE
thf(fact_243_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) ) ) ).

% Range.intros
thf(fact_244_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R ) ) ) ).

% Range.intros
thf(fact_245_Range_Ointros,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R ) ) ) ).

% Range.intros
thf(fact_246_Range_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
            ( ( A = B3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B3 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_247_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
      = ( ? [A3: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
            ( ( A = B3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_248_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
            ( ( A = B3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_249_Range_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
     => ~ ! [A6: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A6 @ A ) @ R ) ) ).

% Range.cases
thf(fact_250_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
     => ~ ! [A6: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A6 @ A ) @ R ) ) ).

% Range.cases
thf(fact_251_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
     => ~ ! [A6: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A6 @ A ) @ R ) ) ).

% Range.cases
thf(fact_252_in__these__eq,axiom,
    ! [X: produc2934264451710624999od_o_o,A2: set_op3605315081486705933od_o_o] :
      ( ( member444158400953824016od_o_o @ X @ ( these_2277756779245414180od_o_o @ A2 ) )
      = ( member3623239300863961174od_o_o @ ( some_P7068320699308846284od_o_o @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_253_in__these__eq,axiom,
    ! [X: produc732395585841259969at_nat,A2: set_op7555165197759865831at_nat] :
      ( ( member4095101504841534314at_nat @ X @ ( these_663420305483081086at_nat @ A2 ) )
      = ( member8056002749114821168at_nat @ ( some_P123625206269196070at_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_254_in__these__eq,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_op989438649584373901at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( these_1872268174106123684at_nat @ A2 ) )
      = ( member4288423094740844502at_nat @ ( some_P131194445498448716at_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_255_in__these__eq,axiom,
    ! [X: a,A2: set_option_a] :
      ( ( member_a @ X @ ( these_a @ A2 ) )
      = ( member_option_a @ ( some_a @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_256_fun__upd__None__restrict,axiom,
    ! [X: produc2934264451710624999od_o_o,D: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ( ( member444158400953824016od_o_o @ X @ D )
       => ( ( fun_up7252388491999432090tion_a @ ( restri4233077244345833607_o_o_a @ M @ D ) @ X @ none_a )
          = ( restri4233077244345833607_o_o_a @ M @ ( minus_5704166846132356910od_o_o @ D @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X @ D )
       => ( ( fun_up7252388491999432090tion_a @ ( restri4233077244345833607_o_o_a @ M @ D ) @ X @ none_a )
          = ( restri4233077244345833607_o_o_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_257_fun__upd__None__restrict,axiom,
    ! [X: produc732395585841259969at_nat,D: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ( ( member4095101504841534314at_nat @ X @ D )
       => ( ( fun_up564627978254483264tion_a @ ( restri896928924502181293_nat_a @ M @ D ) @ X @ none_a )
          = ( restri896928924502181293_nat_a @ M @ ( minus_1608234471674494216at_nat @ D @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X @ D )
       => ( ( fun_up564627978254483264tion_a @ ( restri896928924502181293_nat_a @ M @ D ) @ X @ none_a )
          = ( restri896928924502181293_nat_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_258_fun__upd__None__restrict,axiom,
    ! [X: produc3843707927480180839at_nat,D: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ( ( member8757157785044589968at_nat @ X @ D )
       => ( ( fun_up7579534627831735322tion_a @ ( restri4040463761431659527_nat_a @ M @ D ) @ X @ none_a )
          = ( restri4040463761431659527_nat_a @ M @ ( minus_3314409938677909166at_nat @ D @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ D )
       => ( ( fun_up7579534627831735322tion_a @ ( restri4040463761431659527_nat_a @ M @ D ) @ X @ none_a )
          = ( restri4040463761431659527_nat_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_259_these__empty__eq,axiom,
    ! [B2: set_option_a] :
      ( ( ( these_a @ B2 )
        = bot_bot_set_a )
      = ( ( B2 = bot_bot_set_option_a )
        | ( B2
          = ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_empty_eq
thf(fact_260_these__not__empty__eq,axiom,
    ! [B2: set_option_a] :
      ( ( ( these_a @ B2 )
       != bot_bot_set_a )
      = ( ( B2 != bot_bot_set_option_a )
        & ( B2
         != ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_261_refl__on__singleton,axiom,
    ! [X: set_Product_prod_o_o] : ( refl_o935183656833700870od_o_o @ ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X @ X ) @ bot_bo498831808395348787od_o_o ) ) ).

% refl_on_singleton
thf(fact_262_refl__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_263_linear__order__on__singleton,axiom,
    ! [X: set_Product_prod_o_o] : ( order_2625247082195685180od_o_o @ ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X @ X ) @ bot_bo498831808395348787od_o_o ) ) ).

% linear_order_on_singleton
thf(fact_264_linear__order__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_265_image__map__upd,axiom,
    ! [X: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a,Y: a] :
      ( ~ ( member444158400953824016od_o_o @ X @ A2 )
     => ( ( image_1441591353514620194tion_a @ ( fun_up7252388491999432090tion_a @ M @ X @ ( some_a @ Y ) ) @ A2 )
        = ( image_1441591353514620194tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_266_image__map__upd,axiom,
    ! [X: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a,Y: a] :
      ( ~ ( member4095101504841534314at_nat @ X @ A2 )
     => ( ( image_6589722576923083464tion_a @ ( fun_up564627978254483264tion_a @ M @ X @ ( some_a @ Y ) ) @ A2 )
        = ( image_6589722576923083464tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_267_image__map__upd,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a,Y: a] :
      ( ~ ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( image_7626837539197272994tion_a @ ( fun_up7579534627831735322tion_a @ M @ X @ ( some_a @ Y ) ) @ A2 )
        = ( image_7626837539197272994tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_268_these__image__Some__eq,axiom,
    ! [A2: set_a] :
      ( ( these_a @ ( image_a_option_a @ some_a @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_269_refl__on__domain,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
       => ( ( member444158400953824016od_o_o @ A @ A2 )
          & ( member444158400953824016od_o_o @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_270_refl__on__domain,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
       => ( ( member4095101504841534314at_nat @ A @ A2 )
          & ( member4095101504841534314at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_271_refl__on__domain,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
       => ( ( member8757157785044589968at_nat @ A @ A2 )
          & ( member8757157785044589968at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_272_refl__on__domain,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ( member3157015407008874800od_o_o @ A @ A2 )
          & ( member3157015407008874800od_o_o @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_273_refl__on__domain,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( member2643936169264416010at_nat @ A @ A2 )
          & ( member2643936169264416010at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_274_refl__onD2,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R )
       => ( member444158400953824016od_o_o @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_275_refl__onD2,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R )
       => ( member4095101504841534314at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_276_refl__onD2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
       => ( member8757157785044589968at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_277_refl__onD2,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
       => ( member3157015407008874800od_o_o @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_278_refl__onD2,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( member2643936169264416010at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_279_refl__onD1,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R )
       => ( member444158400953824016od_o_o @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_280_refl__onD1,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R )
       => ( member4095101504841534314at_nat @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_281_refl__onD1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
       => ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_282_refl__onD1,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
       => ( member3157015407008874800od_o_o @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_283_refl__onD1,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( member2643936169264416010at_nat @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_284_refl__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ A @ A2 )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_285_refl__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R )
     => ( ( member4095101504841534314at_nat @ A @ A2 )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_286_refl__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_287_refl__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R )
     => ( ( member3157015407008874800od_o_o @ A @ A2 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_288_refl__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_289_None__notin__image__Some,axiom,
    ! [A2: set_a] :
      ~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A2 ) ) ).

% None_notin_image_Some
thf(fact_290_option_Osimps_I15_J,axiom,
    ! [X2: a] :
      ( ( set_option_a2 @ ( some_a @ X2 ) )
      = ( insert_a @ X2 @ bot_bot_set_a ) ) ).

% option.simps(15)
thf(fact_291_set__empty__eq,axiom,
    ! [Xo: option_a] :
      ( ( ( set_option_a2 @ Xo )
        = bot_bot_set_a )
      = ( Xo = none_a ) ) ).

% set_empty_eq
thf(fact_292_elem__set,axiom,
    ! [X: produc2934264451710624999od_o_o,Xo: option8293164206536526765od_o_o] :
      ( ( member444158400953824016od_o_o @ X @ ( set_op3337521775309186165od_o_o @ Xo ) )
      = ( Xo
        = ( some_P7068320699308846284od_o_o @ X ) ) ) ).

% elem_set
thf(fact_293_elem__set,axiom,
    ! [X: produc732395585841259969at_nat,Xo: option7754883786268931591at_nat] :
      ( ( member4095101504841534314at_nat @ X @ ( set_op8338180575893423695at_nat @ Xo ) )
      = ( Xo
        = ( some_P123625206269196070at_nat @ X ) ) ) ).

% elem_set
thf(fact_294_elem__set,axiom,
    ! [X: produc3843707927480180839at_nat,Xo: option8699557107911914541at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_op5866586666149159413at_nat @ Xo ) )
      = ( Xo
        = ( some_P131194445498448716at_nat @ X ) ) ) ).

% elem_set
thf(fact_295_elem__set,axiom,
    ! [X: a,Xo: option_a] :
      ( ( member_a @ X @ ( set_option_a2 @ Xo ) )
      = ( Xo
        = ( some_a @ X ) ) ) ).

% elem_set
thf(fact_296_inj__Some,axiom,
    ! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).

% inj_Some
thf(fact_297_ospec,axiom,
    ! [A2: option_a,P2: a > $o,X: a] :
      ( ! [X6: a] :
          ( ( member_a @ X6 @ ( set_option_a2 @ A2 ) )
         => ( P2 @ X6 ) )
     => ( ( A2
          = ( some_a @ X ) )
       => ( P2 @ X ) ) ) ).

% ospec
thf(fact_298_option_Oset__intros,axiom,
    ! [X2: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X2 @ ( set_op3337521775309186165od_o_o @ ( some_P7068320699308846284od_o_o @ X2 ) ) ) ).

% option.set_intros
thf(fact_299_option_Oset__intros,axiom,
    ! [X2: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X2 @ ( set_op8338180575893423695at_nat @ ( some_P123625206269196070at_nat @ X2 ) ) ) ).

% option.set_intros
thf(fact_300_option_Oset__intros,axiom,
    ! [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ ( set_op5866586666149159413at_nat @ ( some_P131194445498448716at_nat @ X2 ) ) ) ).

% option.set_intros
thf(fact_301_option_Oset__intros,axiom,
    ! [X2: a] : ( member_a @ X2 @ ( set_option_a2 @ ( some_a @ X2 ) ) ) ).

% option.set_intros
thf(fact_302_option_Oset__cases,axiom,
    ! [E: produc2934264451710624999od_o_o,A: option8293164206536526765od_o_o] :
      ( ( member444158400953824016od_o_o @ E @ ( set_op3337521775309186165od_o_o @ A ) )
     => ( A
        = ( some_P7068320699308846284od_o_o @ E ) ) ) ).

% option.set_cases
thf(fact_303_option_Oset__cases,axiom,
    ! [E: produc732395585841259969at_nat,A: option7754883786268931591at_nat] :
      ( ( member4095101504841534314at_nat @ E @ ( set_op8338180575893423695at_nat @ A ) )
     => ( A
        = ( some_P123625206269196070at_nat @ E ) ) ) ).

% option.set_cases
thf(fact_304_option_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A: option8699557107911914541at_nat] :
      ( ( member8757157785044589968at_nat @ E @ ( set_op5866586666149159413at_nat @ A ) )
     => ( A
        = ( some_P131194445498448716at_nat @ E ) ) ) ).

% option.set_cases
thf(fact_305_option_Oset__cases,axiom,
    ! [E: a,A: option_a] :
      ( ( member_a @ E @ ( set_option_a2 @ A ) )
     => ( A
        = ( some_a @ E ) ) ) ).

% option.set_cases
thf(fact_306_inj__img__insertE,axiom,
    ! [F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on7682168121028333361od_o_o @ F @ A2 )
     => ( ~ ( member444158400953824016od_o_o @ X @ B2 )
       => ( ( ( insert3855699742481506999od_o_o @ X @ B2 )
            = ( image_5377701252353234885od_o_o @ F @ A2 ) )
         => ~ ! [X7: produc2934264451710624999od_o_o,A8: set_Pr1932065953672099015od_o_o] :
                ( ~ ( member444158400953824016od_o_o @ X7 @ A8 )
               => ( ( A2
                    = ( insert3855699742481506999od_o_o @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_5377701252353234885od_o_o @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_307_inj__img__insertE,axiom,
    ! [F: produc732395585841259969at_nat > produc2934264451710624999od_o_o,A2: set_Pr457366540195662369at_nat,X: produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on5425124865643856343od_o_o @ F @ A2 )
     => ( ~ ( member444158400953824016od_o_o @ X @ B2 )
       => ( ( ( insert3855699742481506999od_o_o @ X @ B2 )
            = ( image_5018611165457461355od_o_o @ F @ A2 ) )
         => ~ ! [X7: produc732395585841259969at_nat,A8: set_Pr457366540195662369at_nat] :
                ( ~ ( member4095101504841534314at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert8109617542356073361at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_5018611165457461355od_o_o @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_308_inj__img__insertE,axiom,
    ! [F: produc3843707927480180839at_nat > produc2934264451710624999od_o_o,A2: set_Pr4329608150637261639at_nat,X: produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on126908749210227121od_o_o @ F @ A2 )
     => ( ~ ( member444158400953824016od_o_o @ X @ B2 )
       => ( ( ( insert3855699742481506999od_o_o @ X @ B2 )
            = ( image_6049375619684083269od_o_o @ F @ A2 ) )
         => ~ ! [X7: produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
                ( ~ ( member8757157785044589968at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert9069300056098147895at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_6049375619684083269od_o_o @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_309_inj__img__insertE,axiom,
    ! [F: produc2934264451710624999od_o_o > produc732395585841259969at_nat,A2: set_Pr1932065953672099015od_o_o,X: produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on3096523615731442443at_nat @ F @ A2 )
     => ( ~ ( member4095101504841534314at_nat @ X @ B2 )
       => ( ( ( insert8109617542356073361at_nat @ X @ B2 )
            = ( image_2690009915545047455at_nat @ F @ A2 ) )
         => ~ ! [X7: produc2934264451710624999od_o_o,A8: set_Pr1932065953672099015od_o_o] :
                ( ~ ( member444158400953824016od_o_o @ X7 @ A8 )
               => ( ( A2
                    = ( insert3855699742481506999od_o_o @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_2690009915545047455at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_310_inj__img__insertE,axiom,
    ! [F: produc732395585841259969at_nat > produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on4978893938905139889at_nat @ F @ A2 )
     => ( ~ ( member4095101504841534314at_nat @ X @ B2 )
       => ( ( ( insert8109617542356073361at_nat @ X @ B2 )
            = ( image_1677988772524220229at_nat @ F @ A2 ) )
         => ~ ! [X7: produc732395585841259969at_nat,A8: set_Pr457366540195662369at_nat] :
                ( ~ ( member4095101504841534314at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert8109617542356073361at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_1677988772524220229at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_311_inj__img__insertE,axiom,
    ! [F: produc3843707927480180839at_nat > produc732395585841259969at_nat,A2: set_Pr4329608150637261639at_nat,X: produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on7338182502749300107at_nat @ F @ A2 )
     => ( ~ ( member4095101504841534314at_nat @ X @ B2 )
       => ( ( ( insert8109617542356073361at_nat @ X @ B2 )
            = ( image_3981669762529468447at_nat @ F @ A2 ) )
         => ~ ! [X7: produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
                ( ~ ( member8757157785044589968at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert9069300056098147895at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_3981669762529468447at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_312_inj__img__insertE,axiom,
    ! [F: produc2934264451710624999od_o_o > produc3843707927480180839at_nat,A2: set_Pr1932065953672099015od_o_o,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on2506407689725210801at_nat @ F @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B2 )
       => ( ( ( insert9069300056098147895at_nat @ X @ B2 )
            = ( image_8428874560199066949at_nat @ F @ A2 ) )
         => ~ ! [X7: produc2934264451710624999od_o_o,A8: set_Pr1932065953672099015od_o_o] :
                ( ~ ( member444158400953824016od_o_o @ X7 @ A8 )
               => ( ( A2
                    = ( insert3855699742481506999od_o_o @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_8428874560199066949at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_313_inj__img__insertE,axiom,
    ! [F: produc732395585841259969at_nat > produc3843707927480180839at_nat,A2: set_Pr457366540195662369at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6411134530446904663at_nat @ F @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B2 )
       => ( ( ( insert9069300056098147895at_nat @ X @ B2 )
            = ( image_3054621790227073003at_nat @ F @ A2 ) )
         => ~ ! [X7: produc732395585841259969at_nat,A8: set_Pr457366540195662369at_nat] :
                ( ~ ( member4095101504841534314at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert8109617542356073361at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_3054621790227073003at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_314_inj__img__insertE,axiom,
    ! [F: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B2 )
       => ( ( ( insert9069300056098147895at_nat @ X @ B2 )
            = ( image_960689587819713477at_nat @ F @ A2 ) )
         => ~ ! [X7: produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
                ( ~ ( member8757157785044589968at_nat @ X7 @ A8 )
               => ( ( A2
                    = ( insert9069300056098147895at_nat @ X7 @ A8 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B2
                     != ( image_960689587819713477at_nat @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_315_option_Osimps_I14_J,axiom,
    ( ( set_option_a2 @ none_a )
    = bot_bot_set_a ) ).

% option.simps(14)
thf(fact_316_option_Oset__sel,axiom,
    ! [A: option8293164206536526765od_o_o] :
      ( ( A != none_P8254367833036352840od_o_o )
     => ( member444158400953824016od_o_o @ ( the_Pr7254245475272632699od_o_o @ A ) @ ( set_op3337521775309186165od_o_o @ A ) ) ) ).

% option.set_sel
thf(fact_317_option_Oset__sel,axiom,
    ! [A: option7754883786268931591at_nat] :
      ( ( A != none_P5828533823149705634at_nat )
     => ( member4095101504841534314at_nat @ ( the_Pr9097376869753557077at_nat @ A ) @ ( set_op8338180575893423695at_nat @ A ) ) ) ).

% option.set_sel
thf(fact_318_option_Oset__sel,axiom,
    ! [A: option8699557107911914541at_nat] :
      ( ( A != none_P5811896708078067656at_nat )
     => ( member8757157785044589968at_nat @ ( the_Pr6298817542474494203at_nat @ A ) @ ( set_op5866586666149159413at_nat @ A ) ) ) ).

% option.set_sel
thf(fact_319_option_Oset__sel,axiom,
    ! [A: option_a] :
      ( ( A != none_a )
     => ( member_a @ ( the_a @ A ) @ ( set_option_a2 @ A ) ) ) ).

% option.set_sel
thf(fact_320_Field__insert,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) ) ) ).

% Field_insert
thf(fact_321_Field__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).

% Field_insert
thf(fact_322_Field__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( field_6101175518749917524od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( sup_su1477959037604056251od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) ) ) ).

% Field_insert
thf(fact_323_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) ) ) ).

% Field_insert
thf(fact_324_notin__range__Some,axiom,
    ! [X: option_a] :
      ( ( ~ ( member_option_a @ X @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
      = ( X = none_a ) ) ).

% notin_range_Some
thf(fact_325_notin__range__Some,axiom,
    ! [X: option_nat] :
      ( ( ~ ( member_option_nat @ X @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
      = ( X = none_nat ) ) ).

% notin_range_Some
thf(fact_326_FieldI2,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R3 )
     => ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_327_FieldI2,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R3 )
     => ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_328_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R3 )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_329_FieldI2,axiom,
    ! [I: $o,J: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R3 )
     => ( member_o @ J @ ( field_o @ R3 ) ) ) ).

% FieldI2
thf(fact_330_FieldI2,axiom,
    ! [I: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
     => ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_331_FieldI2,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R3 )
     => ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_332_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R3 )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_333_FieldI1,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R3 )
     => ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_334_FieldI1,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R3 )
     => ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_335_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R3 )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_336_FieldI1,axiom,
    ! [I: $o,J: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R3 )
     => ( member_o @ I @ ( field_o @ R3 ) ) ) ).

% FieldI1
thf(fact_337_FieldI1,axiom,
    ! [I: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
     => ( member_nat @ I @ ( field_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_338_FieldI1,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R3 )
     => ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_339_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R3 )
     => ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_340_surj__def,axiom,
    ! [F: nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y3: nat] :
          ? [X4: nat] :
            ( Y3
            = ( F @ X4 ) ) ) ) ).

% surj_def
thf(fact_341_surjI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ! [X6: nat] :
          ( ( G @ ( F @ X6 ) )
          = X6 )
     => ( ( image_nat_nat @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_342_surjE,axiom,
    ! [F: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X6: nat] :
            ( Y
           != ( F @ X6 ) ) ) ).

% surjE
thf(fact_343_surjD,axiom,
    ! [F: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X6: nat] :
          ( Y
          = ( F @ X6 ) ) ) ).

% surjD
thf(fact_344_range__ex1__eq,axiom,
    ! [F: nat > produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( inj_on2663835283014092644od_o_o @ F @ top_top_set_nat )
     => ( ( member444158400953824016od_o_o @ B @ ( image_3802643350302125520od_o_o @ F @ top_top_set_nat ) )
        = ( ? [X4: nat] :
              ( ( B
                = ( F @ X4 ) )
              & ! [Y3: nat] :
                  ( ( B
                    = ( F @ Y3 ) )
                 => ( Y3 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_345_range__ex1__eq,axiom,
    ! [F: nat > produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( inj_on400939309715711422at_nat @ F @ top_top_set_nat )
     => ( ( member4095101504841534314at_nat @ B @ ( image_6796201771609466410at_nat @ F @ top_top_set_nat ) )
        = ( ? [X4: nat] :
              ( ( B
                = ( F @ X4 ) )
              & ! [Y3: nat] :
                  ( ( B
                    = ( F @ Y3 ) )
                 => ( Y3 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_346_range__ex1__eq,axiom,
    ! [F: nat > produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ B @ ( image_2499013659892586576at_nat @ F @ top_top_set_nat ) )
        = ( ? [X4: nat] :
              ( ( B
                = ( F @ X4 ) )
              & ! [Y3: nat] :
                  ( ( B
                    = ( F @ Y3 ) )
                 => ( Y3 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_347_inj__image__mem__iff,axiom,
    ! [F: nat > produc2934264451710624999od_o_o,A: nat,A2: set_nat] :
      ( ( inj_on2663835283014092644od_o_o @ F @ top_top_set_nat )
     => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_3802643350302125520od_o_o @ F @ A2 ) )
        = ( member_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_348_inj__image__mem__iff,axiom,
    ! [F: nat > produc732395585841259969at_nat,A: nat,A2: set_nat] :
      ( ( inj_on400939309715711422at_nat @ F @ top_top_set_nat )
     => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_6796201771609466410at_nat @ F @ A2 ) )
        = ( member_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_349_inj__image__mem__iff,axiom,
    ! [F: nat > produc3843707927480180839at_nat,A: nat,A2: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ ( F @ A ) @ ( image_2499013659892586576at_nat @ F @ A2 ) )
        = ( member_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_350_inj__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on7682168121028333361od_o_o @ F @ top_to271382412937794071od_o_o )
     => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_5377701252353234885od_o_o @ F @ A2 ) )
        = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_351_inj__image__mem__iff,axiom,
    ! [F: produc732395585841259969at_nat > produc2934264451710624999od_o_o,A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( inj_on5425124865643856343od_o_o @ F @ top_to8533465163743739761at_nat )
     => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_5018611165457461355od_o_o @ F @ A2 ) )
        = ( member4095101504841534314at_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_352_inj__image__mem__iff,axiom,
    ! [F: produc3843707927480180839at_nat > produc2934264451710624999od_o_o,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on126908749210227121od_o_o @ F @ top_to6833984726390702231at_nat )
     => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_6049375619684083269od_o_o @ F @ A2 ) )
        = ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_353_inj__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc732395585841259969at_nat,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on3096523615731442443at_nat @ F @ top_to271382412937794071od_o_o )
     => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_2690009915545047455at_nat @ F @ A2 ) )
        = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_354_inj__image__mem__iff,axiom,
    ! [F: produc732395585841259969at_nat > produc732395585841259969at_nat,A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( inj_on4978893938905139889at_nat @ F @ top_to8533465163743739761at_nat )
     => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_1677988772524220229at_nat @ F @ A2 ) )
        = ( member4095101504841534314at_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_355_inj__image__mem__iff,axiom,
    ! [F: produc3843707927480180839at_nat > produc732395585841259969at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7338182502749300107at_nat @ F @ top_to6833984726390702231at_nat )
     => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_3981669762529468447at_nat @ F @ A2 ) )
        = ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_356_inj__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc3843707927480180839at_nat,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on2506407689725210801at_nat @ F @ top_to271382412937794071od_o_o )
     => ( ( member8757157785044589968at_nat @ ( F @ A ) @ ( image_8428874560199066949at_nat @ F @ A2 ) )
        = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_357_reflD,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ).

% reflD
thf(fact_358_reflD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_359_reflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_360_reflI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ X6 ) @ R )
     => ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R ) ) ).

% reflI
thf(fact_361_reflI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ X6 ) @ R )
     => ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% reflI
thf(fact_362_reflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ X6 ) @ R )
     => ( refl_on_nat @ top_top_set_nat @ R ) ) ).

% reflI
thf(fact_363_UNIV__option__conv,axiom,
    ( top_top_set_option_a
    = ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).

% UNIV_option_conv
thf(fact_364_UNIV__option__conv,axiom,
    ( top_to8920198386146353926on_nat
    = ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).

% UNIV_option_conv
thf(fact_365_finite__range__updI,axiom,
    ! [F: nat > option_a,A: nat,B: a] :
      ( ( finite1674126218327898605tion_a @ ( image_nat_option_a @ F @ top_top_set_nat ) )
     => ( finite1674126218327898605tion_a @ ( image_nat_option_a @ ( fun_upd_nat_option_a @ F @ A @ ( some_a @ B ) ) @ top_top_set_nat ) ) ) ).

% finite_range_updI
thf(fact_366_trans__singleton,axiom,
    ! [A: set_Product_prod_o_o] : ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ bot_bo498831808395348787od_o_o ) ) ).

% trans_singleton
thf(fact_367_trans__singleton,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ bot_bo228742789529271731at_nat ) ) ).

% trans_singleton
thf(fact_368_trans__singleton,axiom,
    ! [A: nat] : ( trans_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ bot_bo2099793752762293965at_nat ) ) ).

% trans_singleton
thf(fact_369_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
            = ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ ( minus_5063024033727795502od_o_o @ R @ id_Pro5927853813057311299od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_370_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
            = ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ ( minus_2318593706003940014at_nat @ R @ id_Pro6320881530060322077at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_371_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
            = ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ ( minus_8235382265540049710at_nat @ R @ id_Pro5428700290092274115at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_372_Linear__order__in__diff__Id,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
            = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_373_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
            = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_374_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
            = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_375_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
            = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_376_pair__in__Id__conv,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ id_set2647307015711338339od_o_o )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_377_pair__in__Id__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_378_IdI,axiom,
    ! [A: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ id_set2647307015711338339od_o_o ) ).

% IdI
thf(fact_379_IdI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ id_set1796276052702428605at_nat ) ).

% IdI
thf(fact_380_apsnd__conv,axiom,
    ! [F: set_Product_prod_o_o > set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( produc698598481305428169od_o_o @ F @ ( produc763777882069021527od_o_o @ X @ Y ) )
      = ( produc763777882069021527od_o_o @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_381_apsnd__conv,axiom,
    ! [F: set_Product_prod_o_o > set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( produc215707640099371503od_o_o @ F @ ( produc763777882069021527od_o_o @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_382_apsnd__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc1997673195996962595od_o_o @ F @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc763777882069021527od_o_o @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_383_apsnd__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc4927343412728267593od_o_o @ F @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_384_apsnd__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6468969725218530979at_nat @ F @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_385_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_386_subrelI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ S ) )
     => ( ord_le7586318239878589543od_o_o @ R @ S ) ) ).

% subrelI
thf(fact_387_subrelI,axiom,
    ! [R: set_Pr457366540195662369at_nat,S: set_Pr457366540195662369at_nat] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X6 @ Y4 ) @ R )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X6 @ Y4 ) @ S ) )
     => ( ord_le4841768313324958145at_nat @ R @ S ) ) ).

% subrelI
thf(fact_388_subrelI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R @ S ) ) ).

% subrelI
thf(fact_389_subset__emptyI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: produc2934264451710624999od_o_o] :
          ~ ( member444158400953824016od_o_o @ X6 @ A2 )
     => ( ord_le7586318239878589543od_o_o @ A2 @ bot_bo498831808395348787od_o_o ) ) ).

% subset_emptyI
thf(fact_390_subset__emptyI,axiom,
    ! [A2: set_Pr457366540195662369at_nat] :
      ( ! [X6: produc732395585841259969at_nat] :
          ~ ( member4095101504841534314at_nat @ X6 @ A2 )
     => ( ord_le4841768313324958145at_nat @ A2 @ bot_bo1676454896636820109at_nat ) ) ).

% subset_emptyI
thf(fact_391_subset__emptyI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ! [X6: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X6 @ A2 )
     => ( ord_le1268244103169919719at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_392_insert__subsetI,axiom,
    ! [X: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,X8: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ X @ A2 )
     => ( ( ord_le7586318239878589543od_o_o @ X8 @ A2 )
       => ( ord_le7586318239878589543od_o_o @ ( insert3855699742481506999od_o_o @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_393_insert__subsetI,axiom,
    ! [X: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,X8: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ X @ A2 )
     => ( ( ord_le4841768313324958145at_nat @ X8 @ A2 )
       => ( ord_le4841768313324958145at_nat @ ( insert8109617542356073361at_nat @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_394_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X8: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ X8 @ A2 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_395_trans__on__def,axiom,
    ( trans_7029024365671003047od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
        ! [X4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X4 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ! [Z2: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Z2 @ A7 )
                 => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R2 )
                   => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z2 ) @ R2 )
                     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_396_trans__on__def,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ! [Z2: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Z2 @ A7 )
                 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
                   => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z2 ) @ R2 )
                     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_397_trans__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ! [X6: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o,Z: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X6 @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( member444158400953824016od_o_o @ Z @ A2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X6 @ Y4 ) @ R )
               => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ Z ) @ R )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X6 @ Z ) @ R ) ) ) ) ) )
     => ( trans_2549647969203962503od_o_o @ A2 @ R ) ) ).

% trans_onI
thf(fact_398_trans__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ! [X6: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat,Z: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X6 @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( member4095101504841534314at_nat @ Z @ A2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X6 @ Y4 ) @ R )
               => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ Z ) @ R )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X6 @ Z ) @ R ) ) ) ) ) )
     => ( trans_3798871764278489953at_nat @ A2 @ R ) ) ).

% trans_onI
thf(fact_399_trans__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X6: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( member8757157785044589968at_nat @ Z @ A2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Y4 ) @ R )
               => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ Z ) @ R )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Z ) @ R ) ) ) ) ) )
     => ( trans_4802246114490097159at_nat @ A2 @ R ) ) ).

% trans_onI
thf(fact_400_trans__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X6 @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( member3157015407008874800od_o_o @ Z @ A2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ R )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Z ) @ R ) ) ) ) ) )
     => ( trans_7029024365671003047od_o_o @ A2 @ R ) ) ).

% trans_onI
thf(fact_401_trans__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X6 @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( member2643936169264416010at_nat @ Z @ A2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ R )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Z ) @ R ) ) ) ) ) )
     => ( trans_3732288485672914945at_nat @ A2 @ R ) ) ).

% trans_onI
thf(fact_402_trans__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o,Z3: produc2934264451710624999od_o_o] :
      ( ( trans_2549647969203962503od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ X @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( member444158400953824016od_o_o @ Z3 @ A2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ Z3 ) @ R )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_403_trans__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat,Z3: produc732395585841259969at_nat] :
      ( ( trans_3798871764278489953at_nat @ A2 @ R )
     => ( ( member4095101504841534314at_nat @ X @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( member4095101504841534314at_nat @ Z3 @ A2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ Z3 ) @ R )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_404_trans__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
      ( ( trans_4802246114490097159at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( member8757157785044589968at_nat @ Z3 @ A2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ Z3 ) @ R )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_405_trans__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z3: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ A2 @ R )
     => ( ( member3157015407008874800od_o_o @ X @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( member3157015407008874800od_o_o @ Z3 @ A2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z3 ) @ R )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_406_trans__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ A2 @ R )
     => ( ( member2643936169264416010at_nat @ X @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( member2643936169264416010at_nat @ Z3 @ A2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ R )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_407_IdE,axiom,
    ! [P: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ P @ id_set2647307015711338339od_o_o )
     => ~ ! [X6: set_Product_prod_o_o] :
            ( P
           != ( produc763777882069021527od_o_o @ X6 @ X6 ) ) ) ).

% IdE
thf(fact_408_IdE,axiom,
    ! [P: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ P @ id_set1796276052702428605at_nat )
     => ~ ! [X6: set_Pr1261947904930325089at_nat] :
            ( P
           != ( produc2922128104949294807at_nat @ X6 @ X6 ) ) ) ).

% IdE
thf(fact_409_finite__range__Some,axiom,
    ( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
    = ( finite_finite_a @ top_top_set_a ) ) ).

% finite_range_Some
thf(fact_410_finite__range__Some,axiom,
    ( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_range_Some
thf(fact_411_transI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Z ) @ R ) ) )
     => ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R ) ) ).

% transI
thf(fact_412_transI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Z ) @ R ) ) )
     => ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% transI
thf(fact_413_transI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X6: nat,Y4: nat,Z: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y4 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Z ) @ R ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transI
thf(fact_414_transE,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z3: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z3 ) @ R )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z3 ) @ R ) ) ) ) ).

% transE
thf(fact_415_transE,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ R ) ) ) ) ).

% transE
thf(fact_416_transE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z3: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R ) ) ) ) ).

% transE
thf(fact_417_transD,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z3: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z3 ) @ R )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z3 ) @ R ) ) ) ) ).

% transD
thf(fact_418_transD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ R ) ) ) ) ).

% transD
thf(fact_419_transD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z3: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R ) ) ) ) ).

% transD
thf(fact_420_inj__on__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on7682168121028333361od_o_o @ F @ B2 )
     => ( ( member444158400953824016od_o_o @ A @ B2 )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_5377701252353234885od_o_o @ F @ A2 ) )
            = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_421_inj__on__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc732395585841259969at_nat,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on3096523615731442443at_nat @ F @ B2 )
     => ( ( member444158400953824016od_o_o @ A @ B2 )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_2690009915545047455at_nat @ F @ A2 ) )
            = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_422_inj__on__image__mem__iff,axiom,
    ! [F: produc2934264451710624999od_o_o > produc3843707927480180839at_nat,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on2506407689725210801at_nat @ F @ B2 )
     => ( ( member444158400953824016od_o_o @ A @ B2 )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( ( member8757157785044589968at_nat @ ( F @ A ) @ ( image_8428874560199066949at_nat @ F @ A2 ) )
            = ( member444158400953824016od_o_o @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_423_inj__on__image__mem__iff,axiom,
    ! [F: produc732395585841259969at_nat > produc2934264451710624999od_o_o,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( inj_on5425124865643856343od_o_o @ F @ B2 )
     => ( ( member4095101504841534314at_nat @ A @ B2 )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_5018611165457461355od_o_o @ F @ A2 ) )
            = ( member4095101504841534314at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_424_inj__on__image__mem__iff,axiom,
    ! [F: produc732395585841259969at_nat > produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( inj_on4978893938905139889at_nat @ F @ B2 )
     => ( ( member4095101504841534314at_nat @ A @ B2 )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_1677988772524220229at_nat @ F @ A2 ) )
            = ( member4095101504841534314at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_425_inj__on__image__mem__iff,axiom,
    ! [F: produc732395585841259969at_nat > produc3843707927480180839at_nat,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( inj_on6411134530446904663at_nat @ F @ B2 )
     => ( ( member4095101504841534314at_nat @ A @ B2 )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( ( member8757157785044589968at_nat @ ( F @ A ) @ ( image_3054621790227073003at_nat @ F @ A2 ) )
            = ( member4095101504841534314at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_426_inj__on__image__mem__iff,axiom,
    ! [F: produc3843707927480180839at_nat > produc2934264451710624999od_o_o,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on126908749210227121od_o_o @ F @ B2 )
     => ( ( member8757157785044589968at_nat @ A @ B2 )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( ( member444158400953824016od_o_o @ ( F @ A ) @ ( image_6049375619684083269od_o_o @ F @ A2 ) )
            = ( member8757157785044589968at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_427_inj__on__image__mem__iff,axiom,
    ! [F: produc3843707927480180839at_nat > produc732395585841259969at_nat,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7338182502749300107at_nat @ F @ B2 )
     => ( ( member8757157785044589968at_nat @ A @ B2 )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( ( member4095101504841534314at_nat @ ( F @ A ) @ ( image_3981669762529468447at_nat @ F @ A2 ) )
            = ( member8757157785044589968at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_428_inj__on__image__mem__iff,axiom,
    ! [F: produc3843707927480180839at_nat > produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F @ B2 )
     => ( ( member8757157785044589968at_nat @ A @ B2 )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( ( member8757157785044589968at_nat @ ( F @ A ) @ ( image_960689587819713477at_nat @ F @ A2 ) )
            = ( member8757157785044589968at_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_429_finite__map__freshness,axiom,
    ! [F: nat > option_a] :
      ( ( finite_finite_nat @ ( dom_nat_a @ F ) )
     => ( ~ ( finite_finite_nat @ top_top_set_nat )
       => ? [X6: nat] :
            ( ( F @ X6 )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_430_surj__Compl__image__subset,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_431_the__inv__into__into,axiom,
    ! [F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on7682168121028333361od_o_o @ F @ A2 )
     => ( ( member444158400953824016od_o_o @ X @ ( image_5377701252353234885od_o_o @ F @ A2 ) )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( member444158400953824016od_o_o @ ( the_in8836748171658978163od_o_o @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_432_the__inv__into__into,axiom,
    ! [F: produc732395585841259969at_nat > produc2934264451710624999od_o_o,A2: set_Pr457366540195662369at_nat,X: produc2934264451710624999od_o_o,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on5425124865643856343od_o_o @ F @ A2 )
     => ( ( member444158400953824016od_o_o @ X @ ( image_5018611165457461355od_o_o @ F @ A2 ) )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( member4095101504841534314at_nat @ ( the_in9049430063722188057od_o_o @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_433_the__inv__into__into,axiom,
    ! [F: produc3843707927480180839at_nat > produc2934264451710624999od_o_o,A2: set_Pr4329608150637261639at_nat,X: produc2934264451710624999od_o_o,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on126908749210227121od_o_o @ F @ A2 )
     => ( ( member444158400953824016od_o_o @ X @ ( image_6049375619684083269od_o_o @ F @ A2 ) )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( member8757157785044589968at_nat @ ( the_in4115987936371053555od_o_o @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_434_the__inv__into__into,axiom,
    ! [F: produc2934264451710624999od_o_o > produc732395585841259969at_nat,A2: set_Pr1932065953672099015od_o_o,X: produc732395585841259969at_nat,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on3096523615731442443at_nat @ F @ A2 )
     => ( ( member4095101504841534314at_nat @ X @ ( image_2690009915545047455at_nat @ F @ A2 ) )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( member444158400953824016od_o_o @ ( the_in6720828813809774157at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_435_the__inv__into__into,axiom,
    ! [F: produc732395585841259969at_nat > produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on4978893938905139889at_nat @ F @ A2 )
     => ( ( member4095101504841534314at_nat @ X @ ( image_1677988772524220229at_nat @ F @ A2 ) )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( member4095101504841534314at_nat @ ( the_in8967973126065966323at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_436_the__inv__into__into,axiom,
    ! [F: produc3843707927480180839at_nat > produc732395585841259969at_nat,A2: set_Pr4329608150637261639at_nat,X: produc732395585841259969at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7338182502749300107at_nat @ F @ A2 )
     => ( ( member4095101504841534314at_nat @ X @ ( image_3981669762529468447at_nat @ F @ A2 ) )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( member8757157785044589968at_nat @ ( the_in3101009185138335437at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_437_the__inv__into__into,axiom,
    ! [F: produc2934264451710624999od_o_o > produc3843707927480180839at_nat,A2: set_Pr1932065953672099015od_o_o,X: produc3843707927480180839at_nat,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on2506407689725210801at_nat @ F @ A2 )
     => ( ( member8757157785044589968at_nat @ X @ ( image_8428874560199066949at_nat @ F @ A2 ) )
       => ( ( ord_le7586318239878589543od_o_o @ A2 @ B2 )
         => ( member444158400953824016od_o_o @ ( the_in6495486876886037235at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_438_the__inv__into__into,axiom,
    ! [F: produc732395585841259969at_nat > produc3843707927480180839at_nat,A2: set_Pr457366540195662369at_nat,X: produc3843707927480180839at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on6411134530446904663at_nat @ F @ A2 )
     => ( ( member8757157785044589968at_nat @ X @ ( image_3054621790227073003at_nat @ F @ A2 ) )
       => ( ( ord_le4841768313324958145at_nat @ A2 @ B2 )
         => ( member4095101504841534314at_nat @ ( the_in2173961212835939993at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_439_the__inv__into__into,axiom,
    ! [F: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F @ A2 )
     => ( ( member8757157785044589968at_nat @ X @ ( image_960689587819713477at_nat @ F @ A2 ) )
       => ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
         => ( member8757157785044589968at_nat @ ( the_in8623244896252994419at_nat @ A2 @ F @ X ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_440_IdD,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ id_set2647307015711338339od_o_o )
     => ( A = B ) ) ).

% IdD
thf(fact_441_IdD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
     => ( A = B ) ) ).

% IdD
thf(fact_442_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R @ id_o ) )
        = ( ! [A7: set_o] :
              ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R ) )
             => ( ( A7 != bot_bot_set_o )
               => ? [X4: $o] :
                    ( ( member_o @ X4 @ A7 )
                    & ! [Y3: $o] :
                        ( ( member_o @ Y3 @ A7 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_443_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                    & ! [Y3: nat] :
                        ( ( member_nat @ Y3 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_444_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( wf_set2384217916178216176od_o_o @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) )
        = ( ! [A7: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( A7 != bot_bo8006819982598807891od_o_o )
               => ? [X4: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X4 @ A7 )
                    & ! [Y3: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_445_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( wf_set7675013139414177994at_nat @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X4: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X4 @ A7 )
                    & ! [Y3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_446_Total__subset__Id,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( total_on_o @ ( field_o @ R ) @ R )
     => ( ( ord_le192195778054999463od_o_o @ R @ id_o )
       => ( ( R = bot_bo7073875226086086771od_o_o )
          | ? [A6: $o] :
              ( R
              = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A6 @ A6 ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_447_Total__subset__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ ( field_nat @ R ) @ R )
     => ( ( ord_le3146513528884898305at_nat @ R @ id_nat )
       => ( ( R = bot_bo2099793752762293965at_nat )
          | ? [A6: nat] :
              ( R
              = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A6 @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_448_Total__subset__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( total_5784220159268842891od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( ord_le7586318239878589543od_o_o @ R @ id_set2647307015711338339od_o_o )
       => ( ( R = bot_bo498831808395348787od_o_o )
          | ? [A6: set_Product_prod_o_o] :
              ( R
              = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A6 @ A6 ) @ bot_bo498831808395348787od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_449_Total__subset__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( total_7212417256332265445at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( ord_le1268244103169919719at_nat @ R @ id_set1796276052702428605at_nat )
       => ( ( R = bot_bo228742789529271731at_nat )
          | ? [A6: set_Pr1261947904930325089at_nat] :
              ( R
              = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A6 @ A6 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_450_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o,Phi: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) )
       => ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) )
              | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_451_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
              | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_452_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,Phi: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) )
              | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_453_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) )
              | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_454_apfst__conv,axiom,
    ! [F: set_Product_prod_o_o > set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( produc7567660281052922763od_o_o @ F @ ( produc763777882069021527od_o_o @ X @ Y ) )
      = ( produc763777882069021527od_o_o @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_455_apfst__conv,axiom,
    ! [F: set_Product_prod_o_o > set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc3496622756277034469at_nat @ F @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_456_apfst__conv,axiom,
    ! [F: set_Product_prod_o_o > set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc1719095019421672203at_nat @ F @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_457_apfst__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Product_prod_o_o,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6414627024158678591at_nat @ F @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_458_apfst__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc5380969780118646373at_nat @ F @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_459_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( ( Y5 != X6 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X6 ) @ R ) )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_460_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( ( Y5 != X6 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X6 ) @ R ) )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_461_total__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ! [X6: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X6 @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( X6 != Y4 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X6 @ Y4 ) @ R )
                | ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X6 ) @ R ) ) ) ) )
     => ( total_3055748755304958059od_o_o @ A2 @ R ) ) ).

% total_onI
thf(fact_462_total__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ! [X6: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X6 @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( X6 != Y4 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X6 @ Y4 ) @ R )
                | ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X6 ) @ R ) ) ) ) )
     => ( total_8996415722781341509at_nat @ A2 @ R ) ) ).

% total_onI
thf(fact_463_total__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X6: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( X6 != Y4 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Y4 ) @ R )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X6 ) @ R ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A2 @ R ) ) ).

% total_onI
thf(fact_464_total__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X6 @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( X6 != Y4 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X6 ) @ R ) ) ) ) )
     => ( total_5784220159268842891od_o_o @ A2 @ R ) ) ).

% total_onI
thf(fact_465_total__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X6 @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( X6 != Y4 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X6 ) @ R ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A2 @ R ) ) ).

% total_onI
thf(fact_466_total__on__def,axiom,
    ( total_5784220159268842891od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
        ! [X4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X4 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ( ( X4 != Y3 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R2 )
                  | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_467_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( X4 != Y3 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_468_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o,I: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R ) )
           => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_469_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat,I: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R ) )
           => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_470_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat,I: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R ) )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_471_wo__rel_Oin__notinI,axiom,
    ! [R: set_Product_prod_o_o,J: $o,I: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_o @ I @ ( field_o @ R ) )
         => ( ( member_o @ J @ ( field_o @ R ) )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_472_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,J: nat,I: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_nat @ I @ ( field_nat @ R ) )
         => ( ( member_nat @ J @ ( field_nat @ R ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_473_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o,I: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_474_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_475_wo__rel_OTOTALS,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ! [X9: $o] :
          ( ( member_o @ X9 @ ( field_o @ R ) )
         => ! [Xa: $o] :
              ( ( member_o @ Xa @ ( field_o @ R ) )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X9 @ Xa ) @ R )
                | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa @ X9 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_476_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ! [X9: nat] :
          ( ( member_nat @ X9 @ ( field_nat @ R ) )
         => ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( field_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X9 @ Xa ) @ R )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X9 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_477_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ! [X9: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X9 @ ( field_6101175518749917524od_o_o @ R ) )
         => ! [Xa: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X9 @ Xa ) @ R )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa @ X9 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_478_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ! [X9: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X9 @ ( field_3145993188331922222at_nat @ R ) )
         => ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X9 @ Xa ) @ R )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa @ X9 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_479_well__order__induct__imp,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,P2: produc2934264451710624999od_o_o > $o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ! [X6: produc2934264451710624999od_o_o] :
            ( ! [Y5: produc2934264451710624999od_o_o] :
                ( ( ( Y5 != X6 )
                  & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ X6 ) @ R ) )
               => ( ( member444158400953824016od_o_o @ Y5 @ ( field_5121862139100741428od_o_o @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member444158400953824016od_o_o @ X6 @ ( field_5121862139100741428od_o_o @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_480_well__order__induct__imp,axiom,
    ! [R: set_Pr4427867301352232263at_nat,P2: produc732395585841259969at_nat > $o,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ! [X6: produc732395585841259969at_nat] :
            ( ! [Y5: produc732395585841259969at_nat] :
                ( ( ( Y5 != X6 )
                  & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ X6 ) @ R ) )
               => ( ( member4095101504841534314at_nat @ Y5 @ ( field_7361900449976055182at_nat @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member4095101504841534314at_nat @ X6 @ ( field_7361900449976055182at_nat @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_481_well__order__induct__imp,axiom,
    ! [R: set_Pr5564308138774400199at_nat,P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ! [X6: produc3843707927480180839at_nat] :
            ( ! [Y5: produc3843707927480180839at_nat] :
                ( ( ( Y5 != X6 )
                  & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X6 ) @ R ) )
               => ( ( member8757157785044589968at_nat @ Y5 @ ( field_2862844103579723188at_nat @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member8757157785044589968at_nat @ X6 @ ( field_2862844103579723188at_nat @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_482_well__order__induct__imp,axiom,
    ! [R: set_Product_prod_o_o,P2: $o > $o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ! [X6: $o] :
            ( ! [Y5: $o] :
                ( ( ( Y5 != X6 )
                  & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X6 ) @ R ) )
               => ( ( member_o @ Y5 @ ( field_o @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_o @ X6 @ ( field_o @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member_o @ A @ ( field_o @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_483_well__order__induct__imp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ! [X6: nat] :
            ( ! [Y5: nat] :
                ( ( ( Y5 != X6 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X6 ) @ R ) )
               => ( ( member_nat @ Y5 @ ( field_nat @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_nat @ X6 @ ( field_nat @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_484_well__order__induct__imp,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( ( Y5 != X6 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X6 ) @ R ) )
               => ( ( member3157015407008874800od_o_o @ Y5 @ ( field_6101175518749917524od_o_o @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member3157015407008874800od_o_o @ X6 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_485_well__order__induct__imp,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( ( Y5 != X6 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X6 ) @ R ) )
               => ( ( member2643936169264416010at_nat @ Y5 @ ( field_3145993188331922222at_nat @ R ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member2643936169264416010at_nat @ X6 @ ( field_3145993188331922222at_nat @ R ) )
             => ( P2 @ X6 ) ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_486_totalI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( X6 != Y4 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
            | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X6 ) @ R ) ) )
     => ( total_5784220159268842891od_o_o @ top_to1478448522960722999od_o_o @ R ) ) ).

% totalI
thf(fact_487_totalI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( X6 != Y4 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
            | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X6 ) @ R ) ) )
     => ( total_7212417256332265445at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% totalI
thf(fact_488_totalI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X6: nat,Y4: nat] :
          ( ( X6 != Y4 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y4 ) @ R )
            | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X6 ) @ R ) ) )
     => ( total_on_nat @ top_top_set_nat @ R ) ) ).

% totalI
thf(fact_489_wf__eq__minimal2,axiom,
    ( wf_o
    = ( ^ [R2: set_Product_prod_o_o] :
        ! [A7: set_o] :
          ( ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R2 ) )
            & ( A7 != bot_bot_set_o ) )
         => ? [X4: $o] :
              ( ( member_o @ X4 @ A7 )
              & ! [Y3: $o] :
                  ( ( member_o @ Y3 @ A7 )
                 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_490_wf__eq__minimal2,axiom,
    ( wf_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat] :
        ! [A7: set_nat] :
          ( ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R2 ) )
            & ( A7 != bot_bot_set_nat ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ A7 )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A7 )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_491_wf__eq__minimal2,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R2: set_Pr1932065953672099015od_o_o] :
        ! [A7: set_se491199571785101031od_o_o] :
          ( ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R2 ) )
            & ( A7 != bot_bo8006819982598807891od_o_o ) )
         => ? [X4: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X4 @ A7 )
              & ! [Y3: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                 => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_492_wf__eq__minimal2,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R2: set_Pr4329608150637261639at_nat] :
        ! [A7: set_se7855581050983116737at_nat] :
          ( ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R2 ) )
            & ( A7 != bot_bo3083307316010499117at_nat ) )
         => ? [X4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X4 @ A7 )
              & ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                 => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_493_wo__rel_Ocases__Total,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o,Phi: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) )
       => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_494_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_495_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,Phi: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_496_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_497_wfE__min_H,axiom,
    ! [R3: set_Pr1013884327308616903od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R3 )
     => ( ( Q3 != bot_bo498831808395348787od_o_o )
       => ~ ! [Z: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z @ Q3 )
             => ~ ! [Y5: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ Z ) @ R3 )
                   => ~ ( member444158400953824016od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_498_wfE__min_H,axiom,
    ! [R3: set_Pr4427867301352232263at_nat,Q3: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R3 )
     => ( ( Q3 != bot_bo1676454896636820109at_nat )
       => ~ ! [Z: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z @ Q3 )
             => ~ ! [Y5: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member4095101504841534314at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_499_wfE__min_H,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R3 )
     => ( ( Q3 != bot_bo228742789529271731at_nat )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q3 )
             => ~ ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member8757157785044589968at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_500_wfE__min_H,axiom,
    ! [R3: set_Pr1932065953672099015od_o_o,Q3: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R3 )
     => ( ( Q3 != bot_bo8006819982598807891od_o_o )
       => ~ ! [Z: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z @ Q3 )
             => ~ ! [Y5: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Z ) @ R3 )
                   => ~ ( member3157015407008874800od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_501_wfE__min_H,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,Q3: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ( Q3 != bot_bo3083307316010499117at_nat )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q3 )
             => ~ ! [Y5: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member2643936169264416010at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_502_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ B2 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( bNF_We4719373430050929331od_o_o @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_503_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ B2 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( bNF_We6249471513304245645at_nat @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_504_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ B2 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We7473068394052389427at_nat @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_505_wo__rel_Ominim__least,axiom,
    ! [R: set_Product_prod_o_o,B2: set_o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R ) )
       => ( ( member_o @ B @ B2 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_506_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B2: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ B2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_507_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ B2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We6151898823597705171od_o_o @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_508_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ B2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6832956061294468653at_nat @ R @ B2 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_509_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ A @ B2 )
         => ( ! [B5: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ B5 @ B2 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We4719373430050929331od_o_o @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_510_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ A @ B2 )
         => ( ! [B5: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ B5 @ B2 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We6249471513304245645at_nat @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_511_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ A @ B2 )
         => ( ! [B5: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ B5 @ B2 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We7473068394052389427at_nat @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_512_wo__rel_Oequals__minim,axiom,
    ! [R: set_Product_prod_o_o,B2: set_o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R ) )
       => ( ( member_o @ A @ B2 )
         => ( ! [B5: $o] :
                ( ( member_o @ B5 @ B2 )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We537455116209922916inim_o @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_513_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B2: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R ) )
       => ( ( member_nat @ A @ B2 )
         => ( ! [B5: nat] :
                ( ( member_nat @ B5 @ B2 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_514_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ A @ B2 )
         => ( ! [B5: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ B5 @ B2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We6151898823597705171od_o_o @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_515_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ A @ B2 )
         => ( ! [B5: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ B5 @ B2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B5 ) @ R ) )
           => ( A
              = ( bNF_We6832956061294468653at_nat @ R @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_516_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) @ ( insert3855699742481506999od_o_o @ A @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_517_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member4095101504841534314at_nat @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) @ ( insert8109617542356073361at_nat @ A @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_518_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_519_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member_o @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_520_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_521_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member3157015407008874800od_o_o @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_522_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member2643936169264416010at_nat @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_523_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
         => ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
            = B ) )
        & ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
         => ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_524_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
         => ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
            = B ) )
        & ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
         => ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_525_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_526_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_527_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_528_wo__rel_Omax2__greater,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_529_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_530_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_531_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_532_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R @ A @ B )
              = B )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_533_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R @ A @ B )
              = B )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_534_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R @ A @ B )
              = B )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_535_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R @ A @ B )
              = B )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_536_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = B )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_537_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
              = B )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_538_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
              = B )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_539_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R @ A @ B )
              = A )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_540_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R @ A @ B )
              = A )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_541_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R @ A @ B )
              = A )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_542_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R @ A @ B )
              = A )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_543_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = A )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_544_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
              = A )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_545_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
              = A )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_546_wf__def,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R2: set_Pr1932065953672099015od_o_o] :
        ! [P5: set_Product_prod_o_o > $o] :
          ( ! [X4: set_Product_prod_o_o] :
              ( ! [Y3: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X4 ) @ R2 )
                 => ( P5 @ Y3 ) )
             => ( P5 @ X4 ) )
         => ! [X10: set_Product_prod_o_o] : ( P5 @ X10 ) ) ) ) ).

% wf_def
thf(fact_547_wf__def,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R2: set_Pr4329608150637261639at_nat] :
        ! [P5: set_Pr1261947904930325089at_nat > $o] :
          ( ! [X4: set_Pr1261947904930325089at_nat] :
              ( ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X4 ) @ R2 )
                 => ( P5 @ Y3 ) )
             => ( P5 @ X4 ) )
         => ! [X10: set_Pr1261947904930325089at_nat] : ( P5 @ X10 ) ) ) ) ).

% wf_def
thf(fact_548_wfE__min,axiom,
    ! [R3: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R3 )
     => ( ( member444158400953824016od_o_o @ X @ Q3 )
       => ~ ! [Z: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z @ Q3 )
             => ~ ! [Y5: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ Z ) @ R3 )
                   => ~ ( member444158400953824016od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_549_wfE__min,axiom,
    ! [R3: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Q3: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R3 )
     => ( ( member4095101504841534314at_nat @ X @ Q3 )
       => ~ ! [Z: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z @ Q3 )
             => ~ ! [Y5: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member4095101504841534314at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_550_wfE__min,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ X @ Q3 )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q3 )
             => ~ ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member8757157785044589968at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_551_wfE__min,axiom,
    ! [R3: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Q3: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R3 )
     => ( ( member3157015407008874800od_o_o @ X @ Q3 )
       => ~ ! [Z: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z @ Q3 )
             => ~ ! [Y5: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Z ) @ R3 )
                   => ~ ( member3157015407008874800od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_552_wfE__min,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Q3: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ( member2643936169264416010at_nat @ X @ Q3 )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q3 )
             => ~ ! [Y5: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z ) @ R3 )
                   => ~ ( member2643936169264416010at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_553_wfI__min,axiom,
    ! [R3: set_Pr1013884327308616903od_o_o] :
      ( ! [X6: produc2934264451710624999od_o_o,Q4: set_Pr1932065953672099015od_o_o] :
          ( ( member444158400953824016od_o_o @ X6 @ Q4 )
         => ? [Xa: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Xa @ Q4 )
              & ! [Y4: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ Xa ) @ R3 )
                 => ~ ( member444158400953824016od_o_o @ Y4 @ Q4 ) ) ) )
     => ( wf_Pro3213319136295691984od_o_o @ R3 ) ) ).

% wfI_min
thf(fact_554_wfI__min,axiom,
    ! [R3: set_Pr4427867301352232263at_nat] :
      ( ! [X6: produc732395585841259969at_nat,Q4: set_Pr457366540195662369at_nat] :
          ( ( member4095101504841534314at_nat @ X6 @ Q4 )
         => ? [Xa: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Xa @ Q4 )
              & ! [Y4: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ Xa ) @ R3 )
                 => ~ ( member4095101504841534314at_nat @ Y4 @ Q4 ) ) ) )
     => ( wf_Pro661967360799104810at_nat @ R3 ) ) ).

% wfI_min
thf(fact_555_wfI__min,axiom,
    ! [R3: set_Pr5564308138774400199at_nat] :
      ( ! [X6: produc3843707927480180839at_nat,Q4: set_Pr4329608150637261639at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ Q4 )
         => ? [Xa: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Xa @ Q4 )
              & ! [Y4: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ Xa ) @ R3 )
                 => ~ ( member8757157785044589968at_nat @ Y4 @ Q4 ) ) ) )
     => ( wf_Pro2357547232388600144at_nat @ R3 ) ) ).

% wfI_min
thf(fact_556_wfI__min,axiom,
    ! [R3: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Q4: set_se491199571785101031od_o_o] :
          ( ( member3157015407008874800od_o_o @ X6 @ Q4 )
         => ? [Xa: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa @ Q4 )
              & ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Xa ) @ R3 )
                 => ~ ( member3157015407008874800od_o_o @ Y4 @ Q4 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R3 ) ) ).

% wfI_min
thf(fact_557_wfI__min,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Q4: set_se7855581050983116737at_nat] :
          ( ( member2643936169264416010at_nat @ X6 @ Q4 )
         => ? [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ Q4 )
              & ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa ) @ R3 )
                 => ~ ( member2643936169264416010at_nat @ Y4 @ Q4 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R3 ) ) ).

% wfI_min
thf(fact_558_wfUNIVI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [P6: set_Product_prod_o_o > $o,X6: set_Product_prod_o_o] :
          ( ! [Xa: set_Product_prod_o_o] :
              ( ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Xa ) @ R )
                 => ( P6 @ Y4 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X6 ) )
     => ( wf_set2384217916178216176od_o_o @ R ) ) ).

% wfUNIVI
thf(fact_559_wfUNIVI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [P6: set_Pr1261947904930325089at_nat > $o,X6: set_Pr1261947904930325089at_nat] :
          ( ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa ) @ R )
                 => ( P6 @ Y4 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X6 ) )
     => ( wf_set7675013139414177994at_nat @ R ) ) ).

% wfUNIVI
thf(fact_560_wf__asym,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X ) @ R )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_561_wf__asym,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ R )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_562_wf__induct,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X6 ) @ R )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_563_wf__induct,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X6 ) @ R )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_564_wf__irrefl,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_565_wf__irrefl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_566_wf__not__sym,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X ) @ R )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_567_wf__not__sym,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ R )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_568_wf__not__refl,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_569_wf__not__refl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_570_wf__eq__minimal,axiom,
    ( wf_Pro3213319136295691984od_o_o
    = ( ^ [R2: set_Pr1013884327308616903od_o_o] :
        ! [Q5: set_Pr1932065953672099015od_o_o] :
          ( ? [X4: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X4 @ Q5 )
         => ? [X4: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X4 @ Q5 )
              & ! [Y3: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ X4 ) @ R2 )
                 => ~ ( member444158400953824016od_o_o @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_571_wf__eq__minimal,axiom,
    ( wf_Pro661967360799104810at_nat
    = ( ^ [R2: set_Pr4427867301352232263at_nat] :
        ! [Q5: set_Pr457366540195662369at_nat] :
          ( ? [X4: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X4 @ Q5 )
         => ? [X4: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X4 @ Q5 )
              & ! [Y3: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ X4 ) @ R2 )
                 => ~ ( member4095101504841534314at_nat @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_572_wf__eq__minimal,axiom,
    ( wf_Pro2357547232388600144at_nat
    = ( ^ [R2: set_Pr5564308138774400199at_nat] :
        ! [Q5: set_Pr4329608150637261639at_nat] :
          ( ? [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ Q5 )
         => ? [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ Q5 )
              & ! [Y3: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X4 ) @ R2 )
                 => ~ ( member8757157785044589968at_nat @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_573_wf__eq__minimal,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R2: set_Pr1932065953672099015od_o_o] :
        ! [Q5: set_se491199571785101031od_o_o] :
          ( ? [X4: set_Product_prod_o_o] : ( member3157015407008874800od_o_o @ X4 @ Q5 )
         => ? [X4: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X4 @ Q5 )
              & ! [Y3: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X4 ) @ R2 )
                 => ~ ( member3157015407008874800od_o_o @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_574_wf__eq__minimal,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R2: set_Pr4329608150637261639at_nat] :
        ! [Q5: set_se7855581050983116737at_nat] :
          ( ? [X4: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X4 @ Q5 )
         => ? [X4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X4 @ Q5 )
              & ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X4 ) @ R2 )
                 => ~ ( member2643936169264416010at_nat @ Y3 @ Q5 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_575_wf__induct__rule,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X6 ) @ R )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_576_wf__induct__rule,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X6 ) @ R )
               => ( P2 @ Y5 ) )
           => ( P2 @ X6 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_577_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X6 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_578_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_579_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X6 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_580_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X6 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_581_reduction__pairI,axiom,
    ! [R3: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
      ( ( wf_o @ R3 )
     => ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S2 ) @ R3 )
       => ( fun_reduction_pair_o @ ( produc763777882069021527od_o_o @ R3 @ S2 ) ) ) ) ).

% reduction_pairI
thf(fact_582_reduction__pairI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S2 ) @ R3 )
       => ( fun_re1951171424197210636ir_nat @ ( produc2922128104949294807at_nat @ R3 @ S2 ) ) ) ) ).

% reduction_pairI
thf(fact_583_under__incr,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ord_le2654036014095304327od_o_o @ ( order_8825441150112336748od_o_o @ R @ A ) @ ( order_8825441150112336748od_o_o @ R @ B ) ) ) ) ).

% under_incr
thf(fact_584_under__incr,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ord_le2077887516847798113at_nat @ ( order_8557040927991152966at_nat @ R @ A ) @ ( order_8557040927991152966at_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_585_under__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ord_less_eq_set_nat @ ( order_under_nat @ R @ A ) @ ( order_under_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_586_min__ext__compat,axiom,
    ! [R3: set_Pr7011910401827753159od_o_o,S2: set_Pr7011910401827753159od_o_o] :
      ( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R3 @ S2 ) @ R3 )
     => ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( min_ex2812920868796073424od_o_o @ R3 ) @ ( sup_su4078869789063261851od_o_o @ ( min_ex2812920868796073424od_o_o @ S2 ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( min_ex2812920868796073424od_o_o @ R3 ) ) ) ).

% min_ext_compat
thf(fact_587_min__ext__compat,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) @ R3 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( min_ex6901939911449802026at_nat @ R3 ) @ ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ S2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( min_ex6901939911449802026at_nat @ R3 ) ) ) ).

% min_ext_compat
thf(fact_588_max__ext__compat,axiom,
    ! [R3: set_Pr7011910401827753159od_o_o,S2: set_Pr7011910401827753159od_o_o] :
      ( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R3 @ S2 ) @ R3 )
     => ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( max_ex7572203241849361634od_o_o @ R3 ) @ ( sup_su4078869789063261851od_o_o @ ( max_ex7572203241849361634od_o_o @ S2 ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( max_ex7572203241849361634od_o_o @ R3 ) ) ) ).

% max_ext_compat
thf(fact_589_max__ext__compat,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) @ R3 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( max_ex8135407076693332796at_nat @ R3 ) @ ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ S2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ).

% max_ext_compat
thf(fact_590_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr1932065953672099015od_o_o,Y6: set_Pr1932065953672099015od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( finite820925532130942480od_o_o @ X8 )
     => ( ( finite820925532130942480od_o_o @ Y6 )
       => ( ( Y6 != bot_bo498831808395348787od_o_o )
         => ( ! [X6: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ X6 @ X8 )
               => ? [Xa: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ Xa @ Y6 )
                    & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X6 @ Xa ) @ R3 ) ) )
           => ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ X8 @ Y6 ) @ ( max_ex9141845325704764834od_o_o @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_591_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr457366540195662369at_nat,Y6: set_Pr457366540195662369at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( finite8072506237346428522at_nat @ X8 )
     => ( ( finite8072506237346428522at_nat @ Y6 )
       => ( ( Y6 != bot_bo1676454896636820109at_nat )
         => ( ! [X6: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ X6 @ X8 )
               => ? [Xa: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ Xa @ Y6 )
                    & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X6 @ Xa ) @ R3 ) ) )
           => ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ X8 @ Y6 ) @ ( max_ex1111939896555521276at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_592_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X8 )
     => ( ( finite4343798906461161616at_nat @ Y6 )
       => ( ( Y6 != bot_bo228742789529271731at_nat )
         => ( ! [X6: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X6 @ X8 )
               => ? [Xa: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa @ Y6 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Xa ) @ R3 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X8 @ Y6 ) @ ( max_ex8833738776070432290at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_593_max__ext_Omax__extI,axiom,
    ! [X8: set_Product_prod_o_o,Y6: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o] :
      ( ( finite6120865539452801872od_o_o @ X8 )
     => ( ( finite6120865539452801872od_o_o @ Y6 )
       => ( ( Y6 != bot_bo7073875226086086771od_o_o )
         => ( ! [X6: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ X6 @ X8 )
               => ? [Xa: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ Xa @ Y6 )
                    & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X6 @ Xa ) @ R3 ) ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X8 @ Y6 ) @ ( max_ex7572203241849361634od_o_o @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_594_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X8 )
     => ( ( finite6177210948735845034at_nat @ Y6 )
       => ( ( Y6 != bot_bo2099793752762293965at_nat )
         => ( ! [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ X8 )
               => ? [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ Y6 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Xa ) @ R3 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y6 ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_595_max__ext_Omax__extI,axiom,
    ! [X8: set_se491199571785101031od_o_o,Y6: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( finite9011636195296603184od_o_o @ X8 )
     => ( ( finite9011636195296603184od_o_o @ Y6 )
       => ( ( Y6 != bot_bo8006819982598807891od_o_o )
         => ( ! [X6: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ X6 @ X8 )
               => ? [Xa: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ Xa @ Y6 )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Xa ) @ R3 ) ) )
           => ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ X8 @ Y6 ) @ ( max_ex5401630698724402626od_o_o @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_596_max__ext_Omax__extI,axiom,
    ! [X8: set_se7855581050983116737at_nat,Y6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X8 )
     => ( ( finite9047747110432174090at_nat @ Y6 )
       => ( ( Y6 != bot_bo3083307316010499117at_nat )
         => ( ! [X6: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X6 @ X8 )
               => ? [Xa: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa @ Y6 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Xa ) @ R3 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X8 @ Y6 ) @ ( max_ex4535816423174879388at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_597_max__ext__additive,axiom,
    ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,C3: set_Product_prod_o_o,D: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ ( max_ex7572203241849361634od_o_o @ R3 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ C3 @ D ) @ ( max_ex7572203241849361634od_o_o @ R3 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( sup_su5769328420594410459od_o_o @ A2 @ C3 ) @ ( sup_su5769328420594410459od_o_o @ B2 @ D ) ) @ ( max_ex7572203241849361634od_o_o @ R3 ) ) ) ) ).

% max_ext_additive
thf(fact_598_max__ext__additive,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,C3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C3 @ D ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ C3 ) @ ( sup_su6327502436637775413at_nat @ B2 @ D ) ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ).

% max_ext_additive
thf(fact_599_max__ext_Ocases,axiom,
    ! [A1: set_Pr1932065953672099015od_o_o,A22: set_Pr1932065953672099015od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A1 @ A22 ) @ ( max_ex9141845325704764834od_o_o @ R3 ) )
     => ~ ( ( finite820925532130942480od_o_o @ A1 )
         => ( ( finite820925532130942480od_o_o @ A22 )
           => ( ( A22 != bot_bo498831808395348787od_o_o )
             => ~ ! [X9: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ X9 @ A1 )
                   => ? [Xa2: produc2934264451710624999od_o_o] :
                        ( ( member444158400953824016od_o_o @ Xa2 @ A22 )
                        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_600_max__ext_Ocases,axiom,
    ! [A1: set_Pr457366540195662369at_nat,A22: set_Pr457366540195662369at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A1 @ A22 ) @ ( max_ex1111939896555521276at_nat @ R3 ) )
     => ~ ( ( finite8072506237346428522at_nat @ A1 )
         => ( ( finite8072506237346428522at_nat @ A22 )
           => ( ( A22 != bot_bo1676454896636820109at_nat )
             => ~ ! [X9: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ X9 @ A1 )
                   => ? [Xa2: produc732395585841259969at_nat] :
                        ( ( member4095101504841534314at_nat @ Xa2 @ A22 )
                        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_601_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A22 )
           => ( ( A22 != bot_bo228742789529271731at_nat )
             => ~ ! [X9: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X9 @ A1 )
                   => ? [Xa2: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa2 @ A22 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_602_max__ext_Ocases,axiom,
    ! [A1: set_se491199571785101031od_o_o,A22: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A1 @ A22 ) @ ( max_ex5401630698724402626od_o_o @ R3 ) )
     => ~ ( ( finite9011636195296603184od_o_o @ A1 )
         => ( ( finite9011636195296603184od_o_o @ A22 )
           => ( ( A22 != bot_bo8006819982598807891od_o_o )
             => ~ ! [X9: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X9 @ A1 )
                   => ? [Xa2: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Xa2 @ A22 )
                        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_603_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R3 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A22 )
           => ( ( A22 != bot_bo3083307316010499117at_nat )
             => ~ ! [X9: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X9 @ A1 )
                   => ? [Xa2: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa2 @ A22 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_604_max__ext_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R3 ) )
     => ~ ( ( finite6120865539452801872od_o_o @ A1 )
         => ( ( finite6120865539452801872od_o_o @ A22 )
           => ( ( A22 != bot_bo7073875226086086771od_o_o )
             => ~ ! [X9: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X9 @ A1 )
                   => ? [Xa2: product_prod_o_o] :
                        ( ( member7466972457876170832od_o_o @ Xa2 @ A22 )
                        & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_605_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X9: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X9 @ A1 )
                   => ? [Xa2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa2 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X9 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_606_max__ext_Osimps,axiom,
    ! [A1: set_Pr1932065953672099015od_o_o,A22: set_Pr1932065953672099015od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A1 @ A22 ) @ ( max_ex9141845325704764834od_o_o @ R3 ) )
      = ( ( finite820925532130942480od_o_o @ A1 )
        & ( finite820925532130942480od_o_o @ A22 )
        & ( A22 != bot_bo498831808395348787od_o_o )
        & ! [X4: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X4 @ A1 )
           => ? [Y3: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ Y3 @ A22 )
                & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_607_max__ext_Osimps,axiom,
    ! [A1: set_Pr457366540195662369at_nat,A22: set_Pr457366540195662369at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A1 @ A22 ) @ ( max_ex1111939896555521276at_nat @ R3 ) )
      = ( ( finite8072506237346428522at_nat @ A1 )
        & ( finite8072506237346428522at_nat @ A22 )
        & ( A22 != bot_bo1676454896636820109at_nat )
        & ! [X4: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X4 @ A1 )
           => ? [Y3: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ Y3 @ A22 )
                & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_608_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A22 )
        & ( A22 != bot_bo228742789529271731at_nat )
        & ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A1 )
           => ? [Y3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y3 @ A22 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_609_max__ext_Osimps,axiom,
    ! [A1: set_se491199571785101031od_o_o,A22: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A1 @ A22 ) @ ( max_ex5401630698724402626od_o_o @ R3 ) )
      = ( ( finite9011636195296603184od_o_o @ A1 )
        & ( finite9011636195296603184od_o_o @ A22 )
        & ( A22 != bot_bo8006819982598807891od_o_o )
        & ! [X4: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X4 @ A1 )
           => ? [Y3: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Y3 @ A22 )
                & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_610_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R3 ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A22 )
        & ( A22 != bot_bo3083307316010499117at_nat )
        & ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ A1 )
           => ? [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ A22 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_611_max__ext_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R3: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R3 ) )
      = ( ( finite6120865539452801872od_o_o @ A1 )
        & ( finite6120865539452801872od_o_o @ A22 )
        & ( A22 != bot_bo7073875226086086771od_o_o )
        & ! [X4: product_prod_o_o] :
            ( ( member7466972457876170832od_o_o @ X4 @ A1 )
           => ? [Y3: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ Y3 @ A22 )
                & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_612_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A1 )
           => ? [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_613_Linear__order__Well__order__iff,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        = ( ! [A7: set_o] :
              ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R ) )
             => ( ( A7 != bot_bot_set_o )
               => ? [X4: $o] :
                    ( ( member_o @ X4 @ A7 )
                    & ! [Y3: $o] :
                        ( ( member_o @ Y3 @ A7 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_614_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                    & ! [Y3: nat] :
                        ( ( member_nat @ Y3 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_615_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
        = ( ! [A7: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( A7 != bot_bo8006819982598807891od_o_o )
               => ? [X4: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X4 @ A7 )
                    & ! [Y3: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_616_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X4: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X4 @ A7 )
                    & ! [Y3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_617_underS__incl__iff,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( order_1929855754104155309od_o_o @ R @ A ) @ ( order_1929855754104155309od_o_o @ R @ B ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_618_underS__incl__iff,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ord_le4841768313324958145at_nat @ ( order_6745690092771398279at_nat @ R @ A ) @ ( order_6745690092771398279at_nat @ R @ B ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_619_underS__incl__iff,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ord_le1268244103169919719at_nat @ ( order_3736018788886245933at_nat @ R @ A ) @ ( order_3736018788886245933at_nat @ R @ B ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_620_underS__incl__iff,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ord_less_eq_set_o @ ( order_underS_o @ R @ A ) @ ( order_underS_o @ R @ B ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_621_underS__incl__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_622_underS__incl__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R @ A ) @ ( order_3915675016455977933od_o_o @ R @ B ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_623_underS__incl__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R @ A ) @ ( order_2644262826035875111at_nat @ R @ B ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_624_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( bNF_We6886001019485585053od_o_o @ R @ A2 @ B )
        = ( ( member444158400953824016od_o_o @ B @ A2 )
          & ! [X4: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X4 @ A2 )
             => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_625_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( bNF_We6170040095471856759at_nat @ R @ A2 @ B )
        = ( ( member4095101504841534314at_nat @ B @ A2 )
          & ! [X4: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X4 @ A2 )
             => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_626_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( bNF_We2939539520719410205at_nat @ R @ A2 @ B )
        = ( ( member8757157785044589968at_nat @ B @ A2 )
          & ! [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ A2 )
             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_627_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( bNF_We794377122088811965od_o_o @ R @ A2 @ B )
        = ( ( member3157015407008874800od_o_o @ B @ A2 )
          & ! [X4: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X4 @ A2 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_628_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( bNF_We943883530777384727at_nat @ R @ A2 @ B )
        = ( ( member2643936169264416010at_nat @ B @ A2 )
          & ! [X4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X4 @ A2 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_629_underS__I,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( I != J )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R3 )
       => ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_630_underS__I,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( I != J )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R3 )
       => ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_631_underS__I,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( I != J )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R3 )
       => ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_632_underS__I,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( I != J )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R3 )
       => ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_633_underS__I,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( I != J )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R3 )
       => ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_634_underS__E,axiom,
    ! [I: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R3 @ J ) )
     => ( ( I != J )
        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_635_underS__E,axiom,
    ! [I: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R3 @ J ) )
     => ( ( I != J )
        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_636_underS__E,axiom,
    ! [I: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R3 @ J ) )
     => ( ( I != J )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_637_underS__E,axiom,
    ! [I: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R3 @ J ) )
     => ( ( I != J )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_638_underS__E,axiom,
    ! [I: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R3 @ J ) )
     => ( ( I != J )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_639_well__order__on__domain,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_2783320356396525957od_o_o @ A2 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
       => ( ( member444158400953824016od_o_o @ A @ A2 )
          & ( member444158400953824016od_o_o @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_640_well__order__on__domain,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_1951249980326167391at_nat @ A2 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
       => ( ( member4095101504841534314at_nat @ A @ A2 )
          & ( member4095101504841534314at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_641_well__order__on__domain,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ A2 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
       => ( ( member8757157785044589968at_nat @ A @ A2 )
          & ( member8757157785044589968at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_642_well__order__on__domain,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ( member3157015407008874800od_o_o @ A @ A2 )
          & ( member3157015407008874800od_o_o @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_643_well__order__on__domain,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( member2643936169264416010at_nat @ A @ A2 )
          & ( member2643936169264416010at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_644_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R3 @ J ) )
     => ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_645_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R3 @ J ) )
     => ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_646_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R3 @ J ) )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_647_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: $o,R3: set_Product_prod_o_o,J: $o] :
      ( ( member_o @ I @ ( order_underS_o @ R3 @ J ) )
     => ( member_o @ I @ ( field_o @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_648_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
      ( ( member_nat @ I @ ( order_underS_nat @ R3 @ J ) )
     => ( member_nat @ I @ ( field_nat @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_649_in__finite__psubset,axiom,
    ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ finite1053609180499918954od_o_o )
      = ( ( ord_le3253702748690112691od_o_o @ A2 @ B2 )
        & ( finite6120865539452801872od_o_o @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_650_in__finite__psubset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
        & ( finite6177210948735845034at_nat @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_651_underS__incr,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
         => ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R @ A ) @ ( order_3915675016455977933od_o_o @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_652_underS__incr,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
         => ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R @ A ) @ ( order_2644262826035875111at_nat @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_653_underS__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( antisym_on_nat @ top_top_set_nat @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
         => ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_654_bsqr__max2,axiom,
    ! [R: set_Product_prod_o_o,A1: $o,A22: $o,B1: $o,B22: $o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( product_Pair_o_o @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr_o @ R ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We3104086342435940326max2_o @ R @ A1 @ A22 ) @ ( bNF_We3104086342435940326max2_o @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_655_bsqr__max2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_656_bsqr__max2,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,B1: set_Product_prod_o_o,B22: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( produc763777882069021527od_o_o @ B1 @ B22 ) ) @ ( bNF_We3935152771354962724od_o_o @ R ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We9141258785624724309od_o_o @ R @ A1 @ A22 ) @ ( bNF_We9141258785624724309od_o_o @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_657_bsqr__max2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,B1: set_Pr1261947904930325089at_nat,B22: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( produc2922128104949294807at_nat @ B1 @ B22 ) ) @ ( bNF_We6127972852007642366at_nat @ R ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6094461251688579247at_nat @ R @ A1 @ A22 ) @ ( bNF_We6094461251688579247at_nat @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_658_wf__bounded__measure,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,Ub: set_Product_prod_o_o > nat,F: set_Product_prod_o_o > nat] :
      ( ! [A6: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B5 @ A6 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_eq_nat @ ( F @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_nat @ ( F @ A6 ) @ ( F @ B5 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R ) ) ).

% wf_bounded_measure
thf(fact_659_wf__bounded__measure,axiom,
    ! [R: set_Pr4329608150637261639at_nat,Ub: set_Pr1261947904930325089at_nat > nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ! [A6: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ A6 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_eq_nat @ ( F @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_nat @ ( F @ A6 ) @ ( F @ B5 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R ) ) ).

% wf_bounded_measure
thf(fact_660_antisym__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( antisy2259881126919591626od_o_o @ A2 @ R )
     => ( ( member444158400953824016od_o_o @ X @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X ) @ R )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_661_antisym__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( antisy1857962754316978724at_nat @ A2 @ R )
     => ( ( member4095101504841534314at_nat @ X @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X ) @ R )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_662_antisym__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( antisy6856624793204979530at_nat @ A2 @ R )
     => ( ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_663_antisym__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_o_o @ A2 @ R )
     => ( ( member3157015407008874800od_o_o @ X @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_664_antisym__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ A2 @ R )
     => ( ( member2643936169264416010at_nat @ X @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_665_antisym__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ! [X6: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X6 @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X6 @ Y4 ) @ R )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X6 ) @ R )
               => ( X6 = Y4 ) ) ) ) )
     => ( antisy2259881126919591626od_o_o @ A2 @ R ) ) ).

% antisym_onI
thf(fact_666_antisym__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ! [X6: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X6 @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X6 @ Y4 ) @ R )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X6 ) @ R )
               => ( X6 = Y4 ) ) ) ) )
     => ( antisy1857962754316978724at_nat @ A2 @ R ) ) ).

% antisym_onI
thf(fact_667_antisym__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X6: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Y4 ) @ R )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X6 ) @ R )
               => ( X6 = Y4 ) ) ) ) )
     => ( antisy6856624793204979530at_nat @ A2 @ R ) ) ).

% antisym_onI
thf(fact_668_antisym__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X6 @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X6 ) @ R )
               => ( X6 = Y4 ) ) ) ) )
     => ( antisy777991898044409578od_o_o @ A2 @ R ) ) ).

% antisym_onI
thf(fact_669_antisym__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X6 @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X6 ) @ R )
               => ( X6 = Y4 ) ) ) ) )
     => ( antisy1130263102092445124at_nat @ A2 @ R ) ) ).

% antisym_onI
thf(fact_670_antisym__on__def,axiom,
    ( antisy777991898044409578od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
        ! [X4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X4 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R2 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X4 ) @ R2 )
                 => ( X4 = Y3 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_671_antisym__on__def,axiom,
    ( antisy1130263102092445124at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X4 ) @ R2 )
                 => ( X4 = Y3 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_672_antisymI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [X6: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Y4 ) @ R )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X6 ) @ R )
           => ( X6 = Y4 ) ) )
     => ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R ) ) ).

% antisymI
thf(fact_673_antisymI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X6: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y4 ) @ R )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X6 ) @ R )
           => ( X6 = Y4 ) ) )
     => ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% antisymI
thf(fact_674_antisymI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X6: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y4 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X6 ) @ R )
           => ( X6 = Y4 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R ) ) ).

% antisymI
thf(fact_675_antisymD,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_676_antisymD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_677_antisymD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( antisym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_678_cofinal__def,axiom,
    ( bNF_Ca4363140458462388713inal_o
    = ( ^ [A7: set_o,R2: set_Product_prod_o_o] :
        ! [X4: $o] :
          ( ( member_o @ X4 @ ( field_o @ R2 ) )
         => ? [Y3: $o] :
              ( ( member_o @ Y3 @ A7 )
              & ( X4 = (~ Y3) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R2 ) ) ) ) ) ).

% cofinal_def
thf(fact_679_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A7: set_nat,R2: set_Pr1261947904930325089at_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ ( field_nat @ R2 ) )
         => ? [Y3: nat] :
              ( ( member_nat @ Y3 @ A7 )
              & ( X4 != Y3 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 ) ) ) ) ) ).

% cofinal_def
thf(fact_680_cofinal__def,axiom,
    ( bNF_Ca3407573850950408152od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
        ! [X4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X4 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ? [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
              & ( X4 != Y3 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X4 @ Y3 ) @ R2 ) ) ) ) ) ).

% cofinal_def
thf(fact_681_cofinal__def,axiom,
    ( bNF_Ca8448034394003515314at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ? [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
              & ( X4 != Y3 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 ) ) ) ) ) ).

% cofinal_def
thf(fact_682_in__measure,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,F: set_Product_prod_o_o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( measur5705974739304014543od_o_o @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_683_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_684_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Product_prod_o_o,A4: set_Pr1261947904930325089at_nat,B4: set_Product_prod_o_o,R: set_Pr4329608150637261639at_nat,S: set_Pr1932065953672099015od_o_o] :
      ( ( member625886800346054288od_o_o @ ( produc4770972565181134295od_o_o @ ( produc2070086378307464573od_o_o @ A @ B ) @ ( produc2070086378307464573od_o_o @ A4 @ B4 ) ) @ ( lex_pr3343786553352869269od_o_o @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B4 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_685_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B4: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc763777882069021527od_o_o @ A4 @ B4 ) ) @ ( lex_pr2209867684007004015od_o_o @ R @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B4 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_686_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr4329608150637261639at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc8517790099723286449at_nat @ A4 @ B4 ) ) @ ( lex_pr568118237913915337at_nat @ R @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B4 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_687_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A4 @ B4 ) ) @ ( lex_pr8000699672184251119at_nat @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B4 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_688_same__fstI,axiom,
    ! [P2: set_Product_prod_o_o > $o,X: set_Product_prod_o_o,Y7: set_Product_prod_o_o,Y: set_Product_prod_o_o,R3: set_Product_prod_o_o > set_Pr1932065953672099015od_o_o] :
      ( ( P2 @ X )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ X @ Y7 ) @ ( produc763777882069021527od_o_o @ X @ Y ) ) @ ( same_f884114232143252012od_o_o @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_689_same__fstI,axiom,
    ! [P2: set_Product_prod_o_o > $o,X: set_Product_prod_o_o,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Product_prod_o_o > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ X @ Y7 ) @ ( produc8517790099723286449at_nat @ X @ Y ) ) @ ( same_f6937299433074352134at_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_690_same__fstI,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X @ Y7 ) @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( same_f1460074399991292076at_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_691_mlex__leq,axiom,
    ! [F: set_Product_prod_o_o > nat,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_692_mlex__leq,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_693_mlex__less,axiom,
    ! [F: set_Product_prod_o_o > nat,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F @ R3 ) ) ) ).

% mlex_less
thf(fact_694_mlex__less,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R3 ) ) ) ).

% mlex_less
thf(fact_695_mlex__iff,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,F: set_Product_prod_o_o > nat,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F @ R3 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_696_mlex__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R3 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_697_finite__ordLess__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( finite_finite_nat @ ( field_nat @ R ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R4 ) )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_698_finite__ordLess__infinite,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( finite_finite_o @ ( field_o @ R ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R4 ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_699_finite__ordLess__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( finite_finite_nat @ ( field_nat @ R ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R4 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_700_finite__ordLess__infinite,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( finite_finite_o @ ( field_o @ R ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R4 ) )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_701_ordLess__irreflexive,axiom,
    ! [R: set_Product_prod_o_o] :
      ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We2568191115274586186ss_o_o ) ).

% ordLess_irreflexive
thf(fact_702_ordLess__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We3448473429330749420at_nat ) ).

% ordLess_irreflexive
thf(fact_703_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_704_ordLess__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_705_ordLess__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_706_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_707_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_708_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_709_card__of__ordLess,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ~ ? [F2: $o > $o] :
              ( ( inj_on_o_o @ F2 @ A2 )
              & ( ord_less_eq_set_o @ ( image_o_o @ F2 @ A2 ) @ B2 ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% card_of_ordLess
thf(fact_710_card__of__ordLess,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ~ ? [F2: nat > nat] :
              ( ( inj_on_nat_nat @ F2 @ A2 )
              & ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3448473429330749420at_nat ) ) ).

% card_of_ordLess
thf(fact_711_card__of__ordLess,axiom,
    ! [A2: set_nat,B2: set_o] :
      ( ( ~ ? [F2: nat > $o] :
              ( ( inj_on_nat_o @ F2 @ A2 )
              & ( ord_less_eq_set_o @ ( image_nat_o @ F2 @ A2 ) @ B2 ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We8182288985678559134_o_nat ) ) ).

% card_of_ordLess
thf(fact_712_ordLess__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
      = ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
        & ~ ? [X10: $o > nat] : ( bNF_We4981740601763737620_o_nat @ R4 @ R @ X10 ) ) ) ).

% ordLess_iff
thf(fact_713_ordLess__iff,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
      = ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
        & ~ ? [X10: $o > $o] : ( bNF_We7100138119184918100ed_o_o @ R4 @ R @ X10 ) ) ) ).

% ordLess_iff
thf(fact_714_ordLess__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
      = ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
        & ~ ? [X10: nat > nat] : ( bNF_We650101353357441270at_nat @ R4 @ R @ X10 ) ) ) ).

% ordLess_iff
thf(fact_715_ordLess__iff,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
      = ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
        & ~ ? [X10: nat > $o] : ( bNF_We8084411962871922930_nat_o @ R4 @ R @ X10 ) ) ) ).

% ordLess_iff
thf(fact_716_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We2061588309931968636_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_717_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We2568191115274586186ss_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_718_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We3448473429330749420at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_719_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We8182288985678559134_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_720_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_721_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_722_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_723_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We6538615181293537951_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_724_card__of__least,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ A2 @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_least
thf(fact_725_card__of__least,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ A2 @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_least
thf(fact_726_card__of__Field__ordLess,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_Field_ordLess
thf(fact_727_card__of__Field__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_Field_ordLess
thf(fact_728_infinite__iff__card__of__nat,axiom,
    ! [A2: set_nat] :
      ( ( ~ ( finite_finite_nat @ A2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% infinite_iff_card_of_nat
thf(fact_729_card__of__ordLeq__finite,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( finite_finite_o @ B2 )
       => ( finite_finite_o @ A2 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_730_card__of__ordLeq__finite,axiom,
    ! [A2: set_o,B2: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_o @ A2 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_731_card__of__ordLeq__finite,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_732_card__of__ordLeq__infinite,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ~ ( finite_finite_o @ A2 )
       => ~ ( finite_finite_o @ B2 ) ) ) ).

% card_of_ordLeq_infinite
thf(fact_733_card__of__ordLeq__infinite,axiom,
    ! [A2: set_o,B2: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ~ ( finite_finite_o @ A2 )
       => ~ ( finite_finite_nat @ B2 ) ) ) ).

% card_of_ordLeq_infinite
thf(fact_734_card__of__ordLeq__infinite,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ~ ( finite_finite_nat @ A2 )
       => ~ ( finite_finite_nat @ B2 ) ) ) ).

% card_of_ordLeq_infinite
thf(fact_735_card__of__mono2,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R4 ) ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_mono2
thf(fact_736_card__of__mono2,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R4 ) ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_mono2
thf(fact_737_card__of__mono2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R4 ) ) ) @ bNF_We6538615181293537951_nat_o ) ) ).

% card_of_mono2
thf(fact_738_card__of__mono2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R4 ) ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_mono2
thf(fact_739_card__of__singl__ordLeq,axiom,
    ! [A2: set_o,B: $o] :
      ( ( A2 != bot_bot_set_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_singl_ordLeq
thf(fact_740_card__of__singl__ordLeq,axiom,
    ! [A2: set_nat,B: $o] :
      ( ( A2 != bot_bot_set_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_singl_ordLeq
thf(fact_741_card__of__singl__ordLeq,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_singl_ordLeq
thf(fact_742_card__of__empty,axiom,
    ! [A2: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o ) ).

% card_of_empty
thf(fact_743_card__of__empty,axiom,
    ! [A2: set_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat ) ).

% card_of_empty
thf(fact_744_card__of__empty,axiom,
    ! [A2: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ).

% card_of_empty
thf(fact_745_card__of__empty3,axiom,
    ! [A2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( A2 = bot_bot_set_o ) ) ).

% card_of_empty3
thf(fact_746_card__of__empty3,axiom,
    ! [A2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We3435943820185352641_o_nat )
     => ( A2 = bot_bot_set_o ) ) ).

% card_of_empty3
thf(fact_747_card__of__empty3,axiom,
    ! [A2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We2031159114419172489at_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% card_of_empty3
thf(fact_748_ordLeq__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_749_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_750_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_751_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_752_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_753_ordLeq__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_754_card__of__image,axiom,
    ! [F: $o > $o,A2: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( image_o_o @ F @ A2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o ) ).

% card_of_image
thf(fact_755_card__of__image,axiom,
    ! [F: nat > $o,A2: set_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( image_nat_o @ F @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat ) ).

% card_of_image
thf(fact_756_card__of__image,axiom,
    ! [F: nat > nat,A2: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ).

% card_of_image
thf(fact_757_card__of__ordLeq2,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( ? [G2: $o > $o] :
              ( ( image_o_o @ G2 @ B2 )
              = A2 ) )
        = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% card_of_ordLeq2
thf(fact_758_card__of__ordLeq2,axiom,
    ! [A2: set_o,B2: set_nat] :
      ( ( A2 != bot_bot_set_o )
     => ( ( ? [G2: nat > $o] :
              ( ( image_nat_o @ G2 @ B2 )
              = A2 ) )
        = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% card_of_ordLeq2
thf(fact_759_card__of__ordLeq2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ( ? [G2: nat > nat] :
              ( ( image_nat_nat @ G2 @ B2 )
              = A2 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% card_of_ordLeq2
thf(fact_760_card__of__ordLeqI,axiom,
    ! [F: $o > $o,A2: set_o,B2: set_o] :
      ( ( inj_on_o_o @ F @ A2 )
     => ( ! [A6: $o] :
            ( ( member_o @ A6 @ A2 )
           => ( member_o @ ( F @ A6 ) @ B2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% card_of_ordLeqI
thf(fact_761_card__of__ordLeqI,axiom,
    ! [F: $o > nat,A2: set_o,B2: set_nat] :
      ( ( inj_on_o_nat @ F @ A2 )
     => ( ! [A6: $o] :
            ( ( member_o @ A6 @ A2 )
           => ( member_nat @ ( F @ A6 ) @ B2 ) )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_762_card__of__ordLeqI,axiom,
    ! [F: nat > nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( member_nat @ ( F @ A6 ) @ B2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_763_card__of__ordLeqI,axiom,
    ! [F: produc2934264451710624999od_o_o > nat,A2: set_Pr1932065953672099015od_o_o,B2: set_nat] :
      ( ( inj_on9163735697967710470_o_nat @ F @ A2 )
     => ( ! [A6: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ A6 @ A2 )
           => ( member_nat @ ( F @ A6 ) @ B2 ) )
       => ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We7173072801150825232_o_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_764_card__of__ordLeqI,axiom,
    ! [F: produc732395585841259969at_nat > nat,A2: set_Pr457366540195662369at_nat,B2: set_nat] :
      ( ( inj_on9206849020120836832at_nat @ F @ A2 )
     => ( ! [A6: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ A6 @ A2 )
           => ( member_nat @ ( F @ A6 ) @ B2 ) )
       => ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We8470213457150283754at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_765_card__of__ordLeqI,axiom,
    ! [F: produc3843707927480180839at_nat > nat,A2: set_Pr4329608150637261639at_nat,B2: set_nat] :
      ( ( inj_on7963584518437281414at_nat @ F @ A2 )
     => ( ! [A6: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ A6 @ A2 )
           => ( member_nat @ ( F @ A6 ) @ B2 ) )
       => ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We8384597595354199696at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_766_card__of__ordLeqI,axiom,
    ! [F: nat > produc2934264451710624999od_o_o,A2: set_nat,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on2663835283014092644od_o_o @ F @ A2 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( member444158400953824016od_o_o @ ( F @ A6 ) @ B2 ) )
       => ( member1944519956519572214od_o_o @ ( produc3027304428310723133od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca6121500631330824515od_o_o @ B2 ) ) @ bNF_We673172386197207406od_o_o ) ) ) ).

% card_of_ordLeqI
thf(fact_767_card__of__ordLeqI,axiom,
    ! [F: nat > produc732395585841259969at_nat,A2: set_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on400939309715711422at_nat @ F @ A2 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( member4095101504841534314at_nat @ ( F @ A6 ) @ B2 ) )
       => ( member156447988302929782at_nat @ ( produc5313346920894465213at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca8664924986033767453at_nat @ B2 ) ) @ bNF_We8887675783599934152at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_768_card__of__ordLeqI,axiom,
    ! [F: nat > produc3843707927480180839at_nat,A2: set_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F @ A2 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( member8757157785044589968at_nat @ ( F @ A6 ) @ B2 ) )
       => ( member8293877710313108214at_nat @ ( produc5916770540428940861at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca9194600574093320387at_nat @ B2 ) ) @ bNF_We7685085753668031982at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_769_card__of__ordLeqI,axiom,
    ! [F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on7682168121028333361od_o_o @ F @ A2 )
     => ( ! [A6: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ A6 @ A2 )
           => ( member444158400953824016od_o_o @ ( F @ A6 ) @ B2 ) )
       => ( member2182030262393914128od_o_o @ ( produc23025780167396183od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ ( bNF_Ca6121500631330824515od_o_o @ B2 ) ) @ bNF_We8627497140504843687od_o_o ) ) ) ).

% card_of_ordLeqI
thf(fact_770_surj__imp__ordLeq,axiom,
    ! [B2: set_o,F: $o > $o,A2: set_o] :
      ( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% surj_imp_ordLeq
thf(fact_771_surj__imp__ordLeq,axiom,
    ! [B2: set_o,F: nat > $o,A2: set_nat] :
      ( ( ord_less_eq_set_o @ B2 @ ( image_nat_o @ F @ A2 ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% surj_imp_ordLeq
thf(fact_772_surj__imp__ordLeq,axiom,
    ! [B2: set_nat,F: nat > nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% surj_imp_ordLeq
thf(fact_773_card__of__mono1,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_mono1
thf(fact_774_card__of__mono1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_mono1
thf(fact_775_card__of__ordLeq,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ? [F2: $o > $o] :
            ( ( inj_on_o_o @ F2 @ A2 )
            & ( ord_less_eq_set_o @ ( image_o_o @ F2 @ A2 ) @ B2 ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_ordLeq
thf(fact_776_card__of__ordLeq,axiom,
    ! [A2: set_o,B2: set_nat] :
      ( ( ? [F2: $o > nat] :
            ( ( inj_on_o_nat @ F2 @ A2 )
            & ( ord_less_eq_set_nat @ ( image_o_nat @ F2 @ A2 ) @ B2 ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_ordLeq
thf(fact_777_card__of__ordLeq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ? [F2: nat > nat] :
            ( ( inj_on_nat_nat @ F2 @ A2 )
            & ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_ordLeq
thf(fact_778_not__ordLeq__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% not_ordLeq_ordLess
thf(fact_779_not__ordLeq__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We2061588309931968636_nat_o ) ) ).

% not_ordLeq_ordLess
thf(fact_780_not__ordLeq__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We3448473429330749420at_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_781_not__ordLeq__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We8182288985678559134_o_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_782_not__ordLess__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_783_not__ordLess__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% not_ordLess_ordLeq
thf(fact_784_not__ordLess__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_785_not__ordLess__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ).

% not_ordLess_ordLeq
thf(fact_786_ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLess_imp_ordLeq
thf(fact_787_ordLess__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_788_ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_789_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_790_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_791_ordLeq__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_792_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_793_ordLeq__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_794_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_795_ordLess__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_796_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_797_ordLess__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_798_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_799_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_800_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_801_ordLess__not__embed,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ? [X_1: $o > $o] : ( bNF_We7100138119184918100ed_o_o @ R4 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_802_ordLess__not__embed,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ~ ? [X_1: nat > nat] : ( bNF_We650101353357441270at_nat @ R4 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_803_ordLess__not__embed,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ? [X_1: nat > $o] : ( bNF_We8084411962871922930_nat_o @ R4 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_804_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ R4 ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_805_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ R4 ) @ bNF_We3448473429330749420at_nat ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_806_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ R4 ) @ bNF_We8182288985678559134_o_nat ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_807_exists__minim__Well__order,axiom,
    ! [R3: set_se491199571785101031od_o_o] :
      ( ( R3 != bot_bo8006819982598807891od_o_o )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X6 @ R3 )
           => ( order_2615775362358406454r_on_o @ ( field_o @ X6 ) @ X6 ) )
       => ? [X6: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X6 @ R3 )
            & ! [Xa: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Xa @ R3 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Xa ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_808_exists__minim__Well__order,axiom,
    ! [R3: set_se7855581050983116737at_nat] :
      ( ( R3 != bot_bo3083307316010499117at_nat )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X6 @ R3 )
           => ( order_2888998067076097458on_nat @ ( field_nat @ X6 ) @ X6 ) )
       => ? [X6: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X6 @ R3 )
            & ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R3 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Xa ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_809_ordLeq__total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLeq_total
thf(fact_810_ordLeq__total,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLeq_total
thf(fact_811_ordLeq__total,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLeq_total
thf(fact_812_ordLeq__total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLeq_total
thf(fact_813_ordLeq__reflexive,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLeq_reflexive
thf(fact_814_ordLeq__reflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_reflexive
thf(fact_815_ordLeq__Well__order__simp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_816_ordLeq__Well__order__simp,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_817_ordLeq__Well__order__simp,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_818_ordLeq__Well__order__simp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_819_card__of__ordLess2,axiom,
    ! [B2: set_o,A2: set_o] :
      ( ( B2 != bot_bot_set_o )
     => ( ( ~ ? [F2: $o > $o] :
                ( ( image_o_o @ F2 @ A2 )
                = B2 ) )
        = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% card_of_ordLess2
thf(fact_820_card__of__ordLess2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( B2 != bot_bot_set_nat )
     => ( ( ~ ? [F2: nat > nat] :
                ( ( image_nat_nat @ F2 @ A2 )
                = B2 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% card_of_ordLess2
thf(fact_821_card__of__ordLess2,axiom,
    ! [B2: set_nat,A2: set_o] :
      ( ( B2 != bot_bot_set_nat )
     => ( ( ~ ? [F2: $o > nat] :
                ( ( image_o_nat @ F2 @ A2 )
                = B2 ) )
        = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% card_of_ordLess2
thf(fact_822_card__of__Un__ordLess__infinite,axiom,
    ! [C3: set_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ C3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_823_card__of__Un__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_824_card__of__Un__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_825_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o,F: nat > $o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( bNF_We8084411962871922930_nat_o @ R1 @ R22 @ F )
       => ( ( image_nat_o @ F @ ( field_nat @ R1 ) )
         != ( field_o @ R22 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_826_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o,F: $o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( bNF_We7100138119184918100ed_o_o @ R1 @ R22 @ F )
       => ( ( image_o_o @ F @ ( field_o @ R1 ) )
         != ( field_o @ R22 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_827_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We3448473429330749420at_nat )
     => ( ( bNF_We650101353357441270at_nat @ R1 @ R22 @ F )
       => ( ( image_nat_nat @ F @ ( field_nat @ R1 ) )
         != ( field_nat @ R22 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_828_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat,F: $o > nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( bNF_We4981740601763737620_o_nat @ R1 @ R22 @ F )
       => ( ( image_o_nat @ F @ ( field_o @ R1 ) )
         != ( field_nat @ R22 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_829_ordLess__or__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_830_ordLess__or__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_831_ordLess__or__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_832_ordLess__or__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_833_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o,R32: set_Product_prod_o_o,F13: nat > $o,F23: $o > $o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R22 @ R32 ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( bNF_We8084411962871922930_nat_o @ R1 @ R32 @ F13 )
         => ( ( bNF_We7100138119184918100ed_o_o @ R22 @ R32 @ F23 )
           => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ ( image_nat_o @ F13 @ ( field_nat @ R1 ) ) @ ( image_o_o @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We2873360996177905586Incl_o @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_834_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o,R32: set_Product_prod_o_o,F13: $o > $o,F23: $o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R22 @ R32 ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( bNF_We7100138119184918100ed_o_o @ R1 @ R32 @ F13 )
         => ( ( bNF_We7100138119184918100ed_o_o @ R22 @ R32 @ F23 )
           => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ ( image_o_o @ F13 @ ( field_o @ R1 ) ) @ ( image_o_o @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We2873360996177905586Incl_o @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_835_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat,R32: set_Pr1261947904930325089at_nat,F13: nat > nat,F23: nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R22 @ R32 ) @ bNF_We3448473429330749420at_nat )
       => ( ( bNF_We650101353357441270at_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We650101353357441270at_nat @ R22 @ R32 @ F23 )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( image_nat_nat @ F13 @ ( field_nat @ R1 ) ) @ ( image_nat_nat @ F23 @ ( field_nat @ R22 ) ) ) @ ( bNF_We720474143800091190cl_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_836_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o,R32: set_Pr1261947904930325089at_nat,F13: nat > nat,F23: $o > nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R22 @ R32 ) @ bNF_We8182288985678559134_o_nat )
       => ( ( bNF_We650101353357441270at_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We4981740601763737620_o_nat @ R22 @ R32 @ F23 )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( image_nat_nat @ F13 @ ( field_nat @ R1 ) ) @ ( image_o_nat @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We720474143800091190cl_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_837_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o,R32: set_Pr1261947904930325089at_nat,F13: $o > nat,F23: $o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R22 @ R32 ) @ bNF_We8182288985678559134_o_nat )
       => ( ( bNF_We4981740601763737620_o_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We4981740601763737620_o_nat @ R22 @ R32 @ F23 )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( image_o_nat @ F13 @ ( field_o @ R1 ) ) @ ( image_o_nat @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We720474143800091190cl_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_838_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat,R32: set_Pr1261947904930325089at_nat,F13: $o > nat,F23: nat > nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R22 @ R32 ) @ bNF_We3448473429330749420at_nat )
       => ( ( bNF_We4981740601763737620_o_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We650101353357441270at_nat @ R22 @ R32 @ F23 )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( image_o_nat @ F13 @ ( field_o @ R1 ) ) @ ( image_nat_nat @ F23 @ ( field_nat @ R22 ) ) ) @ ( bNF_We720474143800091190cl_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_839_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o,R32: set_Pr7011910401827753159od_o_o,F13: nat > product_prod_o_o,F23: $o > product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member3902862663760325584od_o_o @ ( produc324748338109790743od_o_o @ R22 @ R32 ) @ bNF_We247743418333764825od_o_o )
       => ( ( bNF_We8360211665025712513od_o_o @ R1 @ R32 @ F13 )
         => ( ( bNF_We2791365878847735267od_o_o @ R22 @ R32 @ F23 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( image_102748095387869968od_o_o @ F13 @ ( field_nat @ R1 ) ) @ ( image_4057150146340385428od_o_o @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We4431449992149936193od_o_o @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_840_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o,R32: set_Pr8693737435421807431at_nat,F13: nat > product_prod_nat_nat,F23: $o > product_prod_nat_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member2350708248842576976at_nat @ ( produc6344041504541683863at_nat @ R22 @ R32 ) @ bNF_We2311539218253673395at_nat )
       => ( ( bNF_We8202045019506091099at_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We7878404351426547645at_nat @ R22 @ R32 @ F23 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( image_5846123807819985514at_nat @ F13 @ ( field_nat @ R1 ) ) @ ( image_3855930084881510382at_nat @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We1681723038778501915at_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_841_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o,R32: set_Pr7011910401827753159od_o_o,F13: $o > product_prod_o_o,F23: $o > product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member3902862663760325584od_o_o @ ( produc324748338109790743od_o_o @ R22 @ R32 ) @ bNF_We247743418333764825od_o_o )
       => ( ( bNF_We2791365878847735267od_o_o @ R1 @ R32 @ F13 )
         => ( ( bNF_We2791365878847735267od_o_o @ R22 @ R32 @ F23 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( image_4057150146340385428od_o_o @ F13 @ ( field_o @ R1 ) ) @ ( image_4057150146340385428od_o_o @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We4431449992149936193od_o_o @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_842_embed__ordLess__ofilterIncl,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o,R32: set_Pr8693737435421807431at_nat,F13: $o > product_prod_nat_nat,F23: $o > product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member2350708248842576976at_nat @ ( produc6344041504541683863at_nat @ R22 @ R32 ) @ bNF_We2311539218253673395at_nat )
       => ( ( bNF_We7878404351426547645at_nat @ R1 @ R32 @ F13 )
         => ( ( bNF_We7878404351426547645at_nat @ R22 @ R32 @ F23 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( image_3855930084881510382at_nat @ F13 @ ( field_o @ R1 ) ) @ ( image_3855930084881510382at_nat @ F23 @ ( field_o @ R22 ) ) ) @ ( bNF_We1681723038778501915at_nat @ R32 ) ) ) ) ) ) ).

% embed_ordLess_ofilterIncl
thf(fact_843_card__of__empty1,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        | ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_empty1
thf(fact_844_card__of__empty1,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        | ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_empty1
thf(fact_845_card__of__empty1,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        | ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_empty1
thf(fact_846_card__of__empty1,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        | ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We6538615181293537951_nat_o ) ) ).

% card_of_empty1
thf(fact_847_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We3448473429330749420at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We3448473429330749420at_nat )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_848_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_849_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_850_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_851_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_852_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_853_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_854_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_855_Card__order__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z3: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( X != Y )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
         => ( ( Y != Z3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ R )
             => ( ( X != Z3 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_856_Card__order__trans,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z3: set_Product_prod_o_o] :
      ( ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( X != Y )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R )
         => ( ( Y != Z3 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z3 ) @ R )
             => ( ( X != Z3 )
                & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_857_Card__order__trans,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( X != Y )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
         => ( ( Y != Z3 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ R )
             => ( ( X != Z3 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_858_Card__order__trans,axiom,
    ! [R: set_Product_prod_o_o,X: $o,Y: $o,Z3: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( X != Y )
       => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
         => ( ( Y != Z3 )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z3 ) @ R )
             => ( ( X != Z3 )
                & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Z3 ) @ R ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_859_Card__order__wo__rel,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( bNF_We3818239936649020644el_nat @ R ) ) ).

% Card_order_wo_rel
thf(fact_860_Card__order__wo__rel,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( bNF_We6437023980930612612_rel_o @ R ) ) ).

% Card_order_wo_rel
thf(fact_861_infinite__Card__order__limit,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ~ ( finite820925532130942480od_o_o @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
         => ? [X6: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X6 @ ( field_5121862139100741428od_o_o @ R ) )
              & ( A != X6 )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_862_infinite__Card__order__limit,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ~ ( finite8072506237346428522at_nat @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
         => ? [X6: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X6 @ ( field_7361900449976055182at_nat @ R ) )
              & ( A != X6 )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_863_infinite__Card__order__limit,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ~ ( finite4343798906461161616at_nat @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
         => ? [X6: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X6 @ ( field_2862844103579723188at_nat @ R ) )
              & ( A != X6 )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_864_infinite__Card__order__limit,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
       => ( ( member_nat @ A @ ( field_nat @ R ) )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ ( field_nat @ R ) )
              & ( A != X6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_865_infinite__Card__order__limit,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ~ ( finite9011636195296603184od_o_o @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
         => ? [X6: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X6 @ ( field_6101175518749917524od_o_o @ R ) )
              & ( A != X6 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_866_infinite__Card__order__limit,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ~ ( finite9047747110432174090at_nat @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
         => ? [X6: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X6 @ ( field_3145993188331922222at_nat @ R ) )
              & ( A != X6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_867_infinite__Card__order__limit,axiom,
    ! [R: set_Product_prod_o_o,A: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ~ ( finite_finite_o @ ( field_o @ R ) )
       => ( ( member_o @ A @ ( field_o @ R ) )
         => ? [X6: $o] :
              ( ( member_o @ X6 @ ( field_o @ R ) )
              & ( A = (~ X6) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ X6 ) @ R ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_868_exists__minim__Card__order,axiom,
    ! [R3: set_se7855581050983116737at_nat] :
      ( ( R3 != bot_bo3083307316010499117at_nat )
     => ( ! [X6: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X6 @ R3 )
           => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ X6 ) @ X6 ) )
       => ? [X6: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X6 @ R3 )
            & ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R3 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Xa ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Card_order
thf(fact_869_exists__minim__Card__order,axiom,
    ! [R3: set_se491199571785101031od_o_o] :
      ( ( R3 != bot_bo8006819982598807891od_o_o )
     => ( ! [X6: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X6 @ R3 )
           => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ X6 ) @ X6 ) )
       => ? [X6: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X6 @ R3 )
            & ! [Xa: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Xa @ R3 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X6 @ Xa ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% exists_minim_Card_order
thf(fact_870_card__order__on__def,axiom,
    ( bNF_Ca1281551314933786834on_nat
    = ( ^ [A7: set_nat,R2: set_Pr1261947904930325089at_nat] :
          ( ( order_2888998067076097458on_nat @ A7 @ R2 )
          & ! [R6: set_Pr1261947904930325089at_nat] :
              ( ( order_2888998067076097458on_nat @ A7 @ R6 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% card_order_on_def
thf(fact_871_card__order__on__def,axiom,
    ( bNF_Ca8331644756375544342r_on_o
    = ( ^ [A7: set_o,R2: set_Product_prod_o_o] :
          ( ( order_2615775362358406454r_on_o @ A7 @ R2 )
          & ! [R6: set_Product_prod_o_o] :
              ( ( order_2615775362358406454r_on_o @ A7 @ R6 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% card_order_on_def
thf(fact_872_Card__order__iff__ordLeq__card__of,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% Card_order_iff_ordLeq_card_of
thf(fact_873_Card__order__iff__ordLeq__card__of,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% Card_order_iff_ordLeq_card_of
thf(fact_874_Card__order__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We3435943820185352641_o_nat ) ) ).

% Card_order_empty
thf(fact_875_Card__order__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% Card_order_empty
thf(fact_876_Card__order__empty,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% Card_order_empty
thf(fact_877_Card__order__empty,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We6538615181293537951_nat_o ) ) ).

% Card_order_empty
thf(fact_878_card__of__underS,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( member2182030262393914128od_o_o @ ( produc23025780167396183od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ ( order_1929855754104155309od_o_o @ R @ A ) ) @ R ) @ bNF_We9142809693765176714od_o_o ) ) ) ).

% card_of_underS
thf(fact_879_card__of__underS,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( member4079835277323035408at_nat @ ( produc8612946581105484631at_nat @ ( bNF_Ca8664924986033767453at_nat @ ( order_6745690092771398279at_nat @ R @ A ) ) @ R ) @ bNF_We1318246335796365322at_nat ) ) ) ).

% card_of_underS
thf(fact_880_card__of__underS,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( member2845199937591886608at_nat @ ( produc7545183385222921047at_nat @ ( bNF_Ca9194600574093320387at_nat @ ( order_3736018788886245933at_nat @ R @ A ) ) @ R ) @ bNF_We2594139806602030986at_nat ) ) ) ).

% card_of_underS
thf(fact_881_card__of__underS,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( order_underS_nat @ R @ A ) ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% card_of_underS
thf(fact_882_card__of__underS,axiom,
    ! [R: set_Product_prod_o_o,A: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( order_underS_o @ R @ A ) ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% card_of_underS
thf(fact_883_Card__order__singl__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B: $o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( ( field_nat @ R )
         != bot_bot_set_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_884_Card__order__singl__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( ( field_nat @ R )
         != bot_bot_set_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_885_Card__order__singl__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,B: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( ( field_o @ R )
         != bot_bot_set_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_886_Card__order__singl__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,B: nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( ( field_o @ R )
         != bot_bot_set_o )
       => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_887_ordLeq__refl,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_refl
thf(fact_888_ordLeq__refl,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLeq_refl
thf(fact_889_ex__toCard__pred,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ? [X_12: $o > nat] : ( bNF_Gr988296774780055034_o_nat @ A2 @ R @ X_12 ) ) ) ).

% ex_toCard_pred
thf(fact_890_ex__toCard__pred,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ? [X_12: nat > nat] : ( bNF_Gr4367463400984379536at_nat @ A2 @ R @ X_12 ) ) ) ).

% ex_toCard_pred
thf(fact_891_ex__toCard__pred,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ? [X_12: $o > $o] : ( bNF_Gr3194703283472089838ed_o_o @ A2 @ R @ X_12 ) ) ) ).

% ex_toCard_pred
thf(fact_892_ex__toCard__pred,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ? [X_12: nat > $o] : ( bNF_Gr4090968135888240344_nat_o @ A2 @ R @ X_12 ) ) ) ).

% ex_toCard_pred
thf(fact_893_toCard__inj,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1261947904930325089at_nat,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R ) @ bNF_We7173072801150825232_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y @ A2 )
           => ( ( ( bNF_Gr2687907518133588746_o_nat @ A2 @ R @ X )
                = ( bNF_Gr2687907518133588746_o_nat @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_894_toCard__inj,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr1261947904930325089at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R ) @ bNF_We8470213457150283754at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr859799933468256996at_nat @ A2 @ R @ X )
                = ( bNF_Gr859799933468256996at_nat @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_895_toCard__inj,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr1261947904930325089at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R ) @ bNF_We8384597595354199696at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr8319706499545975946at_nat @ A2 @ R @ X )
                = ( bNF_Gr8319706499545975946at_nat @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_896_toCard__inj,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat,X: $o,Y: $o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member_o @ X @ A2 )
         => ( ( member_o @ Y @ A2 )
           => ( ( ( bNF_Gr1235702096034787515_o_nat @ A2 @ R @ X )
                = ( bNF_Gr1235702096034787515_o_nat @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_897_toCard__inj,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member_nat @ X @ A2 )
         => ( ( member_nat @ Y @ A2 )
           => ( ( ( bNF_Gr5321390424033619983at_nat @ A2 @ R @ X )
                = ( bNF_Gr5321390424033619983at_nat @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_898_toCard__inj,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Product_prod_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( member5998968164356017744od_o_o @ ( produc8369371535328480407od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R ) @ bNF_We5212295730361461848_o_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y @ A2 )
           => ( ( ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R @ X )
                = ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_899_toCard__inj,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Product_prod_o_o,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( member1980351565637979088od_o_o @ ( produc8102740133251482647od_o_o @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R ) @ bNF_We6477715483879687422_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr7020601193219610500_nat_o @ A2 @ R @ X )
                = ( bNF_Gr7020601193219610500_nat_o @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_900_toCard__inj,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Product_prod_o_o,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( member8532720315100152400od_o_o @ ( produc3565432567573771415od_o_o @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R ) @ bNF_We3095294111201691864_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr9013816254238203998_nat_o @ A2 @ R @ X )
                = ( bNF_Gr9013816254238203998_nat_o @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_901_toCard__inj,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,X: $o,Y: $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member_o @ X @ A2 )
         => ( ( member_o @ Y @ A2 )
           => ( ( ( bNF_Gr748490023993654765rd_o_o @ A2 @ R @ X )
                = ( bNF_Gr748490023993654765rd_o_o @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_902_toCard__inj,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o,X: nat,Y: nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member_nat @ X @ A2 )
         => ( ( member_nat @ Y @ A2 )
           => ( ( ( bNF_Gr4338373457142972825_nat_o @ A2 @ R @ X )
                = ( bNF_Gr4338373457142972825_nat_o @ A2 @ R @ Y ) )
              = ( X = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_903_toCard__pred__toCard,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( bNF_Gr988296774780055034_o_nat @ A2 @ R @ ( bNF_Gr1235702096034787515_o_nat @ A2 @ R ) ) ) ) ).

% toCard_pred_toCard
thf(fact_904_toCard__pred__toCard,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( bNF_Gr4367463400984379536at_nat @ A2 @ R @ ( bNF_Gr5321390424033619983at_nat @ A2 @ R ) ) ) ) ).

% toCard_pred_toCard
thf(fact_905_toCard__pred__toCard,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( bNF_Gr3194703283472089838ed_o_o @ A2 @ R @ ( bNF_Gr748490023993654765rd_o_o @ A2 @ R ) ) ) ) ).

% toCard_pred_toCard
thf(fact_906_toCard__pred__toCard,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( bNF_Gr4090968135888240344_nat_o @ A2 @ R @ ( bNF_Gr4338373457142972825_nat_o @ A2 @ R ) ) ) ) ).

% toCard_pred_toCard
thf(fact_907_fromCard__toCard,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Pr1261947904930325089at_nat,B: produc2934264451710624999od_o_o] :
      ( ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R ) @ bNF_We7173072801150825232_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( ( bNF_Gr6179234852794809723_o_nat @ A2 @ R @ ( bNF_Gr2687907518133588746_o_nat @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_908_fromCard__toCard,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Pr1261947904930325089at_nat,B: produc732395585841259969at_nat] :
      ( ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R ) @ bNF_We8470213457150283754at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( ( bNF_Gr6550272361737169109at_nat @ A2 @ R @ ( bNF_Gr859799933468256996at_nat @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_909_fromCard__toCard,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat] :
      ( ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R ) @ bNF_We8384597595354199696at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( ( bNF_Gr2810406598121987579at_nat @ A2 @ R @ ( bNF_Gr8319706499545975946at_nat @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_910_fromCard__toCard,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat,B: $o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member_o @ B @ A2 )
         => ( ( bNF_Gr6736587135095141548_o_nat @ A2 @ R @ ( bNF_Gr1235702096034787515_o_nat @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_911_fromCard__toCard,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member_nat @ B @ A2 )
         => ( ( bNF_Gr6953169559552526942at_nat @ A2 @ R @ ( bNF_Gr5321390424033619983at_nat @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_912_fromCard__toCard,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R: set_Product_prod_o_o,B: produc2934264451710624999od_o_o] :
      ( ( member5998968164356017744od_o_o @ ( produc8369371535328480407od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R ) @ bNF_We5212295730361461848_o_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( ( bNF_Gr3541370040355373357_o_o_o @ A2 @ R @ ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_913_fromCard__toCard,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R: set_Product_prod_o_o,B: produc732395585841259969at_nat] :
      ( ( member1980351565637979088od_o_o @ ( produc8102740133251482647od_o_o @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R ) @ bNF_We6477715483879687422_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( ( bNF_Gr2632823273528503123_nat_o @ A2 @ R @ ( bNF_Gr7020601193219610500_nat_o @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_914_fromCard__toCard,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R: set_Product_prod_o_o,B: produc3843707927480180839at_nat] :
      ( ( member8532720315100152400od_o_o @ ( produc3565432567573771415od_o_o @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R ) @ bNF_We3095294111201691864_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( ( bNF_Gr5397592485871042733_nat_o @ A2 @ R @ ( bNF_Gr9013816254238203998_nat_o @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_915_fromCard__toCard,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,B: $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member_o @ B @ A2 )
         => ( ( bNF_Gr5731841836078664636rd_o_o @ A2 @ R @ ( bNF_Gr748490023993654765rd_o_o @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_916_fromCard__toCard,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o,B: nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member_nat @ B @ A2 )
         => ( ( bNF_Gr615886459348551050_nat_o @ A2 @ R @ ( bNF_Gr4338373457142972825_nat_o @ A2 @ R @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_917_card__suc__least,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ R )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ bNF_We3448473429330749420at_nat )
         => ( member5767526579600973610at_nat @ ( produc317013393018064241at_nat @ ( bNF_Ca2802100949066778438uc_nat @ R ) @ S ) @ bNF_We3002834159454949218at_nat ) ) ) ) ).

% card_suc_least
thf(fact_918_card__suc__least,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ R )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ S ) @ bNF_We2061588309931968636_nat_o )
         => ( member4096496664939243216od_o_o @ ( produc4704006580933601559od_o_o @ ( bNF_Ca2802100949066778438uc_nat @ R ) @ S ) @ bNF_We9012773109738928006_nat_o ) ) ) ) ).

% card_suc_least
thf(fact_919_card__suc__least,axiom,
    ! [R: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ top_top_set_o @ R )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S ) @ bNF_We8182288985678559134_o_nat )
         => ( member8460311871040685094at_nat @ ( produc4234488011337487725at_nat @ ( bNF_Ca274681290319134370_suc_o @ R ) @ S ) @ bNF_We4725572391957805758_o_nat ) ) ) ) ).

% card_suc_least
thf(fact_920_card__suc__least,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ top_top_set_o @ R )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ bNF_We2568191115274586186ss_o_o )
         => ( member8462173001240246732od_o_o @ ( produc5711127959335640851od_o_o @ ( bNF_Ca274681290319134370_suc_o @ R ) @ S ) @ bNF_We998535745932950826uc_o_o ) ) ) ) ).

% card_suc_least
thf(fact_921_card__suc__ordLess__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ R4 )
         => ( ( member2760641190231654006uc_nat @ ( produc2392970816633672125uc_nat @ R @ ( bNF_Ca2802100949066778438uc_nat @ R4 ) ) @ bNF_We1638477839083004957uc_nat )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% card_suc_ordLess_imp_ordLeq
thf(fact_922_card__suc__ordLess__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( bNF_Ca8331644756375544342r_on_o @ top_top_set_o @ R4 )
         => ( ( member8317308649106419066_suc_o @ ( produc1256764871752459201_suc_o @ R @ ( bNF_Ca274681290319134370_suc_o @ R4 ) ) @ bNF_We6106446344456452641_suc_o )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ) ).

% card_suc_ordLess_imp_ordLeq
thf(fact_923_card__suc__ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 )
       => ( ( bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ R4 )
         => ( ( member8784710198487953232uc_nat @ ( produc418357114914907543uc_nat @ R @ ( bNF_Ca2802100949066778438uc_nat @ R4 ) ) @ bNF_We4310403136949278955uc_nat )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ) ).

% card_suc_ordLess_imp_ordLeq
thf(fact_924_card__suc__ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 )
       => ( ( bNF_Ca8331644756375544342r_on_o @ top_top_set_o @ R4 )
         => ( ( member1523031322834883412_suc_o @ ( produc327304624436735131_suc_o @ R @ ( bNF_Ca274681290319134370_suc_o @ R4 ) ) @ bNF_We4836139883986425299_suc_o )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% card_suc_ordLess_imp_ordLeq
thf(fact_925_card__suc__greater,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ R )
     => ( member2760641190231654006uc_nat @ ( produc2392970816633672125uc_nat @ R @ ( bNF_Ca2802100949066778438uc_nat @ R ) ) @ bNF_We1638477839083004957uc_nat ) ) ).

% card_suc_greater
thf(fact_926_card__suc__greater,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ top_top_set_o @ R )
     => ( member1523031322834883412_suc_o @ ( produc327304624436735131_suc_o @ R @ ( bNF_Ca274681290319134370_suc_o @ R ) ) @ bNF_We4836139883986425299_suc_o ) ) ).

% card_suc_greater
thf(fact_927_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We3448473429330749420at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We3448473429330749420at_nat )
           => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R ) @ bNF_We7041958966232365723at_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_928_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We3448473429330749420at_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
           => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R ) @ bNF_We4935586570105487173_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_929_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We3448473429330749420at_nat )
           => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R ) @ bNF_We1194315142955236767at_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_930_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
           => ( member989342599902505446at_nat @ ( produc4966347635449911085at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R ) @ bNF_We7066948511106741057_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_931_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
           => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R ) @ bNF_We7847786823715778023_o_o_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_932_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
           => ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R ) @ bNF_We2130777192033087497_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_933_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
           => ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R ) @ bNF_We1503459268740929763at_o_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_934_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
           => ( member3285728679383846360od_o_o @ ( produc9174727453167716639od_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R ) @ bNF_We9222787287886609421_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_935_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We3448473429330749420at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_936_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat,B2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_937_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_938_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o,B2: set_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_939_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member989342599902505446at_nat @ ( produc4966347635449911085at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R ) @ bNF_We3900823710022635230_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_940_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R ) @ bNF_We7395407223230661442at_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_941_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R ) @ bNF_We1913306613526136040_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_942_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
           => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R ) @ bNF_We5086521355153908664at_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_943_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R ) @ bNF_We3513324756044439818_o_o_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_944_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R ) @ bNF_We8188024427803757478_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_945_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R ) @ bNF_We7560706504511599744at_o_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_946_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
           => ( member3285728679383846360od_o_o @ ( produc9174727453167716639od_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R ) @ bNF_We6200507331307258288_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_947_cinfinite__mono,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca7082918126634695480nite_o @ R1 )
       => ( bNF_Ca7082918126634695480nite_o @ R22 ) ) ) ).

% cinfinite_mono
thf(fact_948_cinfinite__mono,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca7082918126634695480nite_o @ R1 )
       => ( bNF_Ca7751713578713111600te_nat @ R22 ) ) ) ).

% cinfinite_mono
thf(fact_949_cinfinite__mono,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca7751713578713111600te_nat @ R1 )
       => ( bNF_Ca7751713578713111600te_nat @ R22 ) ) ) ).

% cinfinite_mono
thf(fact_950_card__of__Un__Plus__ordLeq,axiom,
    ! [A2: set_nat,B2: set_nat] : ( member7606898804734335598at_nat @ ( produc7668796723872103605at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) ) @ bNF_We4444846861891280440at_nat ) ).

% card_of_Un_Plus_ordLeq
thf(fact_951_Cinfinite__limit2,axiom,
    ! [X1: produc2934264451710624999od_o_o,R: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ X1 @ ( field_5121862139100741428od_o_o @ R ) )
     => ( ( member444158400953824016od_o_o @ X2 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( ( bNF_Ca9205897942416659591od_o_o @ R )
            & ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R ) )
         => ? [X6: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X6 @ ( field_5121862139100741428od_o_o @ R ) )
              & ( X1 != X6 )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_952_Cinfinite__limit2,axiom,
    ! [X1: produc732395585841259969at_nat,R: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ X1 @ ( field_7361900449976055182at_nat @ R ) )
     => ( ( member4095101504841534314at_nat @ X2 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( ( bNF_Ca6670480423259178337at_nat @ R )
            & ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R ) @ R ) )
         => ? [X6: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X6 @ ( field_7361900449976055182at_nat @ R ) )
              & ( X1 != X6 )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_953_Cinfinite__limit2,axiom,
    ! [X1: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ X1 @ ( field_2862844103579723188at_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ X2 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( ( bNF_Ca8759257893432750087at_nat @ R )
            & ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R ) @ R ) )
         => ? [X6: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X6 @ ( field_2862844103579723188at_nat @ R ) )
              & ( X1 != X6 )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_954_Cinfinite__limit2,axiom,
    ! [X1: nat,R: set_Pr1261947904930325089at_nat,X2: nat] :
      ( ( member_nat @ X1 @ ( field_nat @ R ) )
     => ( ( member_nat @ X2 @ ( field_nat @ R ) )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ ( field_nat @ R ) )
              & ( X1 != X6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_955_Cinfinite__limit2,axiom,
    ! [X1: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ X1 @ ( field_6101175518749917524od_o_o @ R ) )
     => ( ( member3157015407008874800od_o_o @ X2 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( bNF_Ca140147938854208935od_o_o @ R )
            & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R ) )
         => ? [X6: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X6 @ ( field_6101175518749917524od_o_o @ R ) )
              & ( X1 != X6 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_956_Cinfinite__limit2,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ X1 @ ( field_3145993188331922222at_nat @ R ) )
     => ( ( member2643936169264416010at_nat @ X2 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( bNF_Ca4089899758288986625at_nat @ R )
            & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R ) @ R ) )
         => ? [X6: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X6 @ ( field_3145993188331922222at_nat @ R ) )
              & ( X1 != X6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X1 @ X6 ) @ R )
              & ( X2 != X6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_957_Cinfinite__limit2,axiom,
    ! [X1: $o,R: set_Product_prod_o_o,X2: $o] :
      ( ( member_o @ X1 @ ( field_o @ R ) )
     => ( ( member_o @ X2 @ ( field_o @ R ) )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ? [X6: $o] :
              ( ( member_o @ X6 @ ( field_o @ R ) )
              & ( X1 = (~ X6) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X1 @ X6 ) @ R )
              & ( X2 = (~ X6) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X6 ) @ R ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_958_Cinfinite__limit,axiom,
    ! [X: produc2934264451710624999od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ X @ ( field_5121862139100741428od_o_o @ R ) )
     => ( ( ( bNF_Ca9205897942416659591od_o_o @ R )
          & ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R ) )
       => ? [X6: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X6 @ ( field_5121862139100741428od_o_o @ R ) )
            & ( X != X6 )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_959_Cinfinite__limit,axiom,
    ! [X: produc732395585841259969at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ X @ ( field_7361900449976055182at_nat @ R ) )
     => ( ( ( bNF_Ca6670480423259178337at_nat @ R )
          & ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R ) @ R ) )
       => ? [X6: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X6 @ ( field_7361900449976055182at_nat @ R ) )
            & ( X != X6 )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_960_Cinfinite__limit,axiom,
    ! [X: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( field_2862844103579723188at_nat @ R ) )
     => ( ( ( bNF_Ca8759257893432750087at_nat @ R )
          & ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R ) @ R ) )
       => ? [X6: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X6 @ ( field_2862844103579723188at_nat @ R ) )
            & ( X != X6 )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_961_Cinfinite__limit,axiom,
    ! [X: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ X @ ( field_nat @ R ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ ( field_nat @ R ) )
            & ( X != X6 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_962_Cinfinite__limit,axiom,
    ! [X: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R ) )
     => ( ( ( bNF_Ca140147938854208935od_o_o @ R )
          & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R ) )
       => ? [X6: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X6 @ ( field_6101175518749917524od_o_o @ R ) )
            & ( X != X6 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_963_Cinfinite__limit,axiom,
    ! [X: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R ) )
     => ( ( ( bNF_Ca4089899758288986625at_nat @ R )
          & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R ) @ R ) )
       => ? [X6: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X6 @ ( field_3145993188331922222at_nat @ R ) )
            & ( X != X6 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_964_Cinfinite__limit,axiom,
    ! [X: $o,R: set_Product_prod_o_o] :
      ( ( member_o @ X @ ( field_o @ R ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
       => ? [X6: $o] :
            ( ( member_o @ X6 @ ( field_o @ R ) )
            & ( X = (~ X6) )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X6 ) @ R ) ) ) ) ).

% Cinfinite_limit
thf(fact_965_ordLeq__Plus__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We6538615181293537951_nat_o )
       => ( member952642050024735836um_o_o @ ( produc3837620720005901475um_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We7217180388603359757um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_966_ordLeq__Plus__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ P7 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member8262324088718329556um_o_o @ ( produc4585484280506290715um_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We1079009337437905725um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_967_ordLeq__Plus__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ P7 ) @ bNF_We3435943820185352641_o_nat )
       => ( member6655200516074465936_o_nat @ ( produc4410791399313715927_o_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We2094493454282013185_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_968_ordLeq__Plus__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ P7 ) @ bNF_We2031159114419172489at_nat )
       => ( member6975117218213954584_o_nat @ ( produc2864506647045777759_o_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7239028013353664305_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_969_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We6538615181293537951_nat_o )
       => ( member8826396524424052564um_o_o @ ( produc5149556716212013723um_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8976728017661452183um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_970_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ P7 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member3483138326940501008um_o_o @ ( produc1030987845608174167um_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8987810345020088115um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_971_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ P7 ) @ bNF_We3435943820185352641_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We3854041083340105611_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_972_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ P7 ) @ bNF_We2031159114419172489at_nat )
       => ( member2972727061379769104_o_nat @ ( produc728317944619019095_o_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We8273322497166968871_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_973_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We6538615181293537951_nat_o )
       => ( member7051533951421365136_nat_o @ ( produc5765749630380524503_nat_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We2558484984367686093_nat_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_974_ordLeq__Plus__mono,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ P7 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We7362575607395598641_nat_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_975_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_o,C3: set_o,D: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( member3483138326940501008um_o_o @ ( produc1030987845608174167um_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ B2 @ D ) ) ) @ bNF_We8987810345020088115um_o_o ) ) ) ).

% card_of_Plus_mono
thf(fact_976_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_o,C3: set_o,D: set_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We3435943820185352641_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ D ) ) ) @ bNF_We3854041083340105611_o_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_977_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_o,C3: set_nat,D: set_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We2031159114419172489at_nat )
       => ( member2972727061379769104_o_nat @ ( produc728317944619019095_o_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ C3 ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ D ) ) ) @ bNF_We8273322497166968871_o_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_978_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_o,D: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ B2 @ D ) ) ) @ bNF_We7362575607395598641_nat_o ) ) ) ).

% card_of_Plus_mono
thf(fact_979_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_o,D: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We3435943820185352641_o_nat )
       => ( member9006821182645044420at_nat @ ( produc6767169964153735947at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We1329699954732187917at_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_980_card__of__Plus__mono,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_nat,D: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We2031159114419172489at_nat )
       => ( member234890756356643848at_nat @ ( produc5352410120312684367at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We7431878391334027121at_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_981_card__of__Plus__mono,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_o,D: set_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( member1510635369261286160_nat_o @ ( produc224851048220445527_nat_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ C3 ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ B2 @ D ) ) ) @ bNF_We5603027978337506215_nat_o ) ) ) ).

% card_of_Plus_mono
thf(fact_982_card__of__Plus__mono,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_o,D: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We3435943820185352641_o_nat )
       => ( member7654646017578591112at_nat @ ( produc3548793344679855823at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We295405470918883351at_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_983_card__of__Plus__mono,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat,D: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We2031159114419172489at_nat )
       => ( member3674313732307032336at_nat @ ( produc2112825387572549975at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We5082115074515246823at_nat ) ) ) ).

% card_of_Plus_mono
thf(fact_984_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ C3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
         => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We7847786823715778023_o_o_o ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_985_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_o,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_o @ C3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2061588309931968636_nat_o )
         => ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2130777192033087497_nat_o ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_986_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_o,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_o @ C3 )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2061588309931968636_nat_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We2568191115274586186ss_o_o )
         => ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We1503459268740929763at_o_o ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_987_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
         => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We7041958966232365723at_nat ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_988_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
         => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We4935586570105487173_o_nat ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_989_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3448473429330749420at_nat )
         => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We1194315142955236767at_nat ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_990_card__of__Plus__ordLess__infinite,axiom,
    ! [C3: set_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ C3 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We8182288985678559134_o_nat )
         => ( member989342599902505446at_nat @ ( produc4966347635449911085at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We7066948511106741057_o_nat ) ) ) ) ).

% card_of_Plus_ordLess_infinite
thf(fact_991_Cinfinite__limit__finite,axiom,
    ! [X8: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X8 )
     => ( ( ord_less_eq_set_nat @ X8 @ ( field_nat @ R ) )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ ( field_nat @ R ) )
              & ! [Xa: nat] :
                  ( ( member_nat @ Xa @ X8 )
                 => ( ( Xa != X6 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X6 ) @ R ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_992_Cinfinite__limit__finite,axiom,
    ! [X8: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( finite9011636195296603184od_o_o @ X8 )
     => ( ( ord_le2654036014095304327od_o_o @ X8 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( bNF_Ca140147938854208935od_o_o @ R )
            & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R ) )
         => ? [X6: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X6 @ ( field_6101175518749917524od_o_o @ R ) )
              & ! [Xa: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Xa @ X8 )
                 => ( ( Xa != X6 )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa @ X6 ) @ R ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_993_Cinfinite__limit__finite,axiom,
    ! [X8: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X8 )
     => ( ( ord_le2077887516847798113at_nat @ X8 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( bNF_Ca4089899758288986625at_nat @ R )
            & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R ) @ R ) )
         => ? [X6: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X6 @ ( field_3145993188331922222at_nat @ R ) )
              & ! [Xa: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Xa @ X8 )
                 => ( ( Xa != X6 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa @ X6 ) @ R ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_994_Cinfinite__limit__finite,axiom,
    ! [X8: set_o,R: set_Product_prod_o_o] :
      ( ( finite_finite_o @ X8 )
     => ( ( ord_less_eq_set_o @ X8 @ ( field_o @ R ) )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ? [X6: $o] :
              ( ( member_o @ X6 @ ( field_o @ R ) )
              & ! [Xa: $o] :
                  ( ( member_o @ Xa @ X8 )
                 => ( ( Xa = (~ X6) )
                    & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa @ X6 ) @ R ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_995_Un__Cinfinite__bound,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat,B2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_996_Un__Cinfinite__bound,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_997_Un__Cinfinite__bound,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_998_Un__Cinfinite__bound,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o,B2: set_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
         => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_999_Cinfinite__gt__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ).

% Cinfinite_gt_empty
thf(fact_1000_Cinfinite__gt__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We8182288985678559134_o_nat ) ) ).

% Cinfinite_gt_empty
thf(fact_1001_Cinfinite__gt__empty,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% Cinfinite_gt_empty
thf(fact_1002_Cinfinite__gt__empty,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R ) @ bNF_We2061588309931968636_nat_o ) ) ).

% Cinfinite_gt_empty
thf(fact_1003_Cfinite__ordLess__Cinfinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1004_Cfinite__ordLess__Cinfinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ S ) @ bNF_We2061588309931968636_nat_o ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1005_Cfinite__ordLess__Cinfinite,axiom,
    ! [R: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1006_Cfinite__ordLess__Cinfinite,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1007_Card__order__Plus__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ R ) @ bNF_We3435943820185352641_o_nat )
         => ( ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R ) @ ( field_o @ P ) ) ) @ R ) @ bNF_We5202279374879649339_o_nat )
            & ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ P ) @ ( field_nat @ R ) ) ) @ R ) @ bNF_We1461007947729398933at_nat ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1008_Card__order__Plus__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ R ) @ bNF_We2031159114419172489at_nat )
         => ( ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R ) @ ( field_nat @ P ) ) ) @ R ) @ bNF_We4717344193235055909at_nat )
            & ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ P ) @ ( field_nat @ R ) ) ) @ R ) @ bNF_We4717344193235055909at_nat ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1009_Card__order__Plus__infinite,axiom,
    ! [R: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ R ) @ bNF_We6538615181293537951_nat_o )
         => ( ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ R ) @ bNF_We8455499808419870611_nat_o )
            & ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ P ) @ ( field_o @ R ) ) ) @ R ) @ bNF_We7828181885127712877at_o_o ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1010_Card__order__Plus__infinite,axiom,
    ! [R: set_Product_prod_o_o,P: set_Product_prod_o_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ R ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ R ) @ bNF_We5455623833347018717_o_o_o )
            & ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ P ) @ ( field_o @ R ) ) ) @ R ) @ bNF_We5455623833347018717_o_o_o ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1011_ordIso__symmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We4116350175183677332_o_nat ) ) ).

% ordIso_symmetric
thf(fact_1012_ordIso__symmetric,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We7219021536291862642_nat_o ) ) ).

% ordIso_symmetric
thf(fact_1013_ordIso__symmetric,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We2654380646378065620so_o_o ) ) ).

% ordIso_symmetric
thf(fact_1014_ordIso__symmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_symmetric
thf(fact_1015_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1016_ordIso__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1017_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1018_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1019_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1020_ordIso__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1021_ordIso__Plus__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We7219021536291862642_nat_o )
       => ( member952642050024735836um_o_o @ ( produc3837620720005901475um_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We3692099834942423674um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1022_ordIso__Plus__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ P7 ) @ bNF_We4116350175183677332_o_nat )
       => ( member6655200516074465936_o_nat @ ( produc4410791399313715927_o_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7792784937475852910_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1023_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We7219021536291862642_nat_o )
       => ( member7051533951421365136_nat_o @ ( produc5765749630380524503_nat_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8256776467561525818_nat_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1024_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ P7 ) @ bNF_We4116350175183677332_o_nat )
       => ( member9006821182645044420at_nat @ ( produc6767169964153735947at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7027991437926027642at_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1025_ordIso__Plus__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ P7 ) @ bNF_We2654380646378065620so_o_o )
       => ( member8262324088718329556um_o_o @ ( produc4585484280506290715um_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We5813203082145254800um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1026_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ P7 ) @ bNF_We2654380646378065620so_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We2873397315248171908_nat_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1027_ordIso__Plus__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ P7 ) @ bNF_We5258908940166488438at_nat )
       => ( member6975117218213954584_o_nat @ ( produc2864506647045777759_o_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We6180393435421844484_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1028_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ P7 ) @ bNF_We5258908940166488438at_nat )
       => ( member234890756356643848at_nat @ ( produc5352410120312684367at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We6373243813402207300at_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1029_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ P7 ) @ bNF_We7219021536291862642_nat_o )
       => ( member8826396524424052564um_o_o @ ( produc5149556716212013723um_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R4 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We4487549725514025450um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1030_ordIso__Plus__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ P7 ) @ bNF_We4116350175183677332_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R4 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We8588234828047454686_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1031_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_o,C3: set_o,D: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We2654380646378065620so_o_o )
       => ( member3483138326940501008um_o_o @ ( produc1030987845608174167um_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ B2 @ D ) ) ) @ bNF_We2736418005100055072um_o_o ) ) ) ).

% card_of_Plus_cong
thf(fact_1032_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_o,C3: set_o,D: set_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We4116350175183677332_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ D ) ) ) @ bNF_We8588234828047454686_o_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1033_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_o,C3: set_nat,D: set_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We5258908940166488438at_nat )
       => ( member2972727061379769104_o_nat @ ( produc728317944619019095_o_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ C3 ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ D ) ) ) @ bNF_We4748241943506032788_o_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1034_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_o,D: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We2654380646378065620so_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ B2 @ D ) ) ) @ bNF_We2873397315248171908_nat_o ) ) ) ).

% card_of_Plus_cong
thf(fact_1035_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_o,D: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We4116350175183677332_o_nat )
       => ( member9006821182645044420at_nat @ ( produc6767169964153735947at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We7027991437926027642at_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1036_card__of__Plus__cong,axiom,
    ! [A2: set_o,B2: set_nat,C3: set_nat,D: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We5258908940166488438at_nat )
       => ( member234890756356643848at_nat @ ( produc5352410120312684367at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We6373243813402207300at_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1037_card__of__Plus__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_o,D: set_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We5258908940166488438at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3172636760949973492d_of_o @ D ) ) @ bNF_We2654380646378065620so_o_o )
       => ( member1510635369261286160_nat_o @ ( produc224851048220445527_nat_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ C3 ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ B2 @ D ) ) ) @ bNF_We2077947424676570132_nat_o ) ) ) ).

% card_of_Plus_cong
thf(fact_1038_card__of__Plus__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_o,D: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We5258908940166488438at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We4116350175183677332_o_nat )
       => ( member7654646017578591112at_nat @ ( produc3548793344679855823at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We8460142929841839338at_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1039_card__of__Plus__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat,D: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ C3 ) @ ( bNF_Ca3793111618940312692of_nat @ D ) ) @ bNF_We5258908940166488438at_nat )
       => ( member3674313732307032336at_nat @ ( produc2112825387572549975at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ C3 ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ D ) ) ) @ bNF_We3455965832816432340at_nat ) ) ) ).

% card_of_Plus_cong
thf(fact_1040_card__of__refl,axiom,
    ! [A2: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We2654380646378065620so_o_o ) ).

% card_of_refl
thf(fact_1041_card__of__refl,axiom,
    ! [A2: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5258908940166488438at_nat ) ).

% card_of_refl
thf(fact_1042_card__of__ordIso__subst,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( A2 = B2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_ordIso_subst
thf(fact_1043_card__of__ordIso__subst,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We5258908940166488438at_nat ) ) ).

% card_of_ordIso_subst
thf(fact_1044_card__order__on__ordIso,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ A2 @ R )
     => ( ( bNF_Ca1281551314933786834on_nat @ A2 @ R4 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% card_order_on_ordIso
thf(fact_1045_card__order__on__ordIso,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ A2 @ R )
     => ( ( bNF_Ca8331644756375544342r_on_o @ A2 @ R4 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% card_order_on_ordIso
thf(fact_1046_ordLeq__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1047_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1048_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1049_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1050_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1051_ordLeq__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1052_ordIso__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1053_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1054_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1055_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1056_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1057_ordIso__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1058_ordIso__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1059_ordIso__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordIso_imp_ordLeq
thf(fact_1060_ordIso__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1061_ordIso__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
      = ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We6538615181293537951_nat_o )
        & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1062_ordIso__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
      = ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
        & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1063_ordIso__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1064_ordIso__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1065_ordLess__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1066_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1067_ordLess__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1068_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1069_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1070_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1071_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R5 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1072_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1073_ordIso__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1074_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R5 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1075_ordIso__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1076_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R5 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1077_not__ordLess__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o ) ) ).

% not_ordLess_ordIso
thf(fact_1078_not__ordLess__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1079_not__ordLess__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1080_card__of__empty__ordIso,axiom,
    member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We2654380646378065620so_o_o ).

% card_of_empty_ordIso
thf(fact_1081_card__of__empty__ordIso,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We4116350175183677332_o_nat ).

% card_of_empty_ordIso
thf(fact_1082_card__of__empty__ordIso,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We5258908940166488438at_nat ).

% card_of_empty_ordIso
thf(fact_1083_card__of__empty2,axiom,
    ! [A2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We2654380646378065620so_o_o )
     => ( A2 = bot_bot_set_o ) ) ).

% card_of_empty2
thf(fact_1084_card__of__empty2,axiom,
    ! [A2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We4116350175183677332_o_nat )
     => ( A2 = bot_bot_set_o ) ) ).

% card_of_empty2
thf(fact_1085_card__of__empty2,axiom,
    ! [A2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We5258908940166488438at_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% card_of_empty2
thf(fact_1086_card__of__ordIso__finite,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( ( finite_finite_o @ A2 )
        = ( finite_finite_o @ B2 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1087_card__of__ordIso__finite,axiom,
    ! [A2: set_o,B2: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( ( finite_finite_o @ A2 )
        = ( finite_finite_nat @ B2 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1088_card__of__ordIso__finite,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We5258908940166488438at_nat )
     => ( ( finite_finite_nat @ A2 )
        = ( finite_finite_nat @ B2 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1089_card__of__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R4 ) ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_cong
thf(fact_1090_card__of__cong,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R4 ) ) ) @ bNF_We4116350175183677332_o_nat ) ) ).

% card_of_cong
thf(fact_1091_card__of__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R4 ) ) ) @ bNF_We7219021536291862642_nat_o ) ) ).

% card_of_cong
thf(fact_1092_card__of__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R4 ) ) ) @ bNF_We5258908940166488438at_nat ) ) ).

% card_of_cong
thf(fact_1093_Card__order__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We5258908940166488438at_nat )
       => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso
thf(fact_1094_Card__order__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We4116350175183677332_o_nat )
       => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso
thf(fact_1095_Card__order__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R4 @ R ) @ bNF_We7219021536291862642_nat_o )
       => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso
thf(fact_1096_Card__order__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We2654380646378065620so_o_o )
       => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso
thf(fact_1097_Card__order__ordIso2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat )
       => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso2
thf(fact_1098_Card__order__ordIso2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R4 ) @ bNF_We7219021536291862642_nat_o )
       => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso2
thf(fact_1099_Card__order__ordIso2,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat )
       => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso2
thf(fact_1100_Card__order__ordIso2,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o )
       => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R4 ) @ R4 ) ) ) ).

% Card_order_ordIso2
thf(fact_1101_ordIso__refl,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_refl
thf(fact_1102_ordIso__refl,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We2654380646378065620so_o_o ) ) ).

% ordIso_refl
thf(fact_1103_finite__well__order__on__ordIso,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( order_2615775362358406454r_on_o @ A2 @ R )
       => ( ( order_2615775362358406454r_on_o @ A2 @ R4 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o ) ) ) ) ).

% finite_well_order_on_ordIso
thf(fact_1104_finite__well__order__on__ordIso,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( order_2888998067076097458on_nat @ A2 @ R )
       => ( ( order_2888998067076097458on_nat @ A2 @ R4 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat ) ) ) ) ).

% finite_well_order_on_ordIso
thf(fact_1105_card__of__unique,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ A2 @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5258908940166488438at_nat ) ) ).

% card_of_unique
thf(fact_1106_card__of__unique,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ A2 @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_unique
thf(fact_1107_ordIso__reflexive,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We2654380646378065620so_o_o ) ) ).

% ordIso_reflexive
thf(fact_1108_ordIso__reflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_reflexive
thf(fact_1109_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We8115293805659342695eq_o_o )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2568191115274586186ss_o_o )
        | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R4 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
thf(fact_1110_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We2031159114419172489at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We3448473429330749420at_nat )
        | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R4 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
thf(fact_1111_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R4: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We3435943820185352641_o_nat )
      = ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We8182288985678559134_o_nat )
        | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R4 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
thf(fact_1112_internalize__ordLeq,axiom,
    ! [R4: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [P3: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_eq_set_nat @ ( field_nat @ P3 ) @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ P3 ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P3 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1113_internalize__ordLeq,axiom,
    ! [R4: set_Product_prod_o_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [P3: set_Product_prod_o_o] :
            ( ( ord_less_eq_set_o @ ( field_o @ P3 ) @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ P3 ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P3 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_ordLeq
thf(fact_1114_internalize__ordLeq,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We2031159114419172489at_nat )
      = ( ? [P3: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_eq_set_nat @ ( field_nat @ P3 ) @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ P3 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P3 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1115_internalize__card__of__ordLeq2,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ C3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1116_internalize__card__of__ordLeq2,axiom,
    ! [A2: set_o,C3: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ C3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1117_internalize__card__of__ordLeq2,axiom,
    ! [A2: set_nat,C3: set_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We6538615181293537951_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ C3 )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ ( bNF_Ca3172636760949973492d_of_o @ C3 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1118_internalize__card__of__ordLeq2,axiom,
    ! [A2: set_nat,C3: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We2031159114419172489at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ C3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ ( bNF_Ca3793111618940312692of_nat @ C3 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1119_ordIso__card__of__imp__Card__order,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5258908940166488438at_nat )
     => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) ) ).

% ordIso_card_of_imp_Card_order
thf(fact_1120_ordIso__card__of__imp__Card__order,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) ) ).

% ordIso_card_of_imp_Card_order
thf(fact_1121_ordIso__card__of__imp__Card__order,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) ) ).

% ordIso_card_of_imp_Card_order
thf(fact_1122_Card__order__iff__ordIso__card__of,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) ) @ bNF_We5258908940166488438at_nat ) ) ).

% Card_order_iff_ordIso_card_of
thf(fact_1123_Card__order__iff__ordIso__card__of,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% Card_order_iff_ordIso_card_of
thf(fact_1124_card__of__Field__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ R ) @ bNF_We5258908940166488438at_nat ) ) ).

% card_of_Field_ordIso
thf(fact_1125_card__of__Field__ordIso,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ R ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_Field_ordIso
thf(fact_1126_internalize__ordLess,axiom,
    ! [R4: set_Product_prod_o_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ R ) @ bNF_We2568191115274586186ss_o_o )
      = ( ? [P3: set_Product_prod_o_o] :
            ( ( ord_less_set_o @ ( field_o @ P3 ) @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R4 @ P3 ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P3 @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_ordLess
thf(fact_1127_internalize__ordLess,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ R ) @ bNF_We3448473429330749420at_nat )
      = ( ? [P3: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_set_nat @ ( field_nat @ P3 ) @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R4 @ P3 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P3 @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1128_internalize__ordLess,axiom,
    ! [R4: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ R ) @ bNF_We8182288985678559134_o_nat )
      = ( ? [P3: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_set_nat @ ( field_nat @ P3 ) @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R4 @ P3 ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P3 @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1129_Cinfinite__cong,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We5258908940166488438at_nat )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R1 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
       => ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) ) ) ) ).

% Cinfinite_cong
thf(fact_1130_Cinfinite__cong,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R1 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
       => ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) ) ) ) ).

% Cinfinite_cong
thf(fact_1131_Cinfinite__cong,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
       => ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) ) ) ) ).

% Cinfinite_cong
thf(fact_1132_Cinfinite__cong,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
       => ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) ) ) ) ).

% Cinfinite_cong
thf(fact_1133_card__of__ordIso__finite__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: set_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5258908940166488438at_nat )
       => ( ( finite_finite_nat @ ( field_nat @ R ) )
          = ( finite_finite_nat @ A2 ) ) ) ) ).

% card_of_ordIso_finite_Field
thf(fact_1134_card__of__ordIso__finite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We2654380646378065620so_o_o )
       => ( ( finite_finite_o @ ( field_o @ R ) )
          = ( finite_finite_o @ A2 ) ) ) ) ).

% card_of_ordIso_finite_Field
thf(fact_1135_card__of__ordIso__finite__Field,axiom,
    ! [R: set_Product_prod_o_o,A2: set_nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4116350175183677332_o_nat )
       => ( ( finite_finite_o @ ( field_o @ R ) )
          = ( finite_finite_nat @ A2 ) ) ) ) ).

% card_of_ordIso_finite_Field
thf(fact_1136_internalize__card__of__ordLeq,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1137_internalize__card__of__ordLeq,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1138_internalize__card__of__ordLeq,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We6538615181293537951_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ ( field_o @ R ) )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1139_internalize__card__of__ordLeq,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2031159114419172489at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1140_internalize__card__of__ordLess,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_set_o @ B6 @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1141_internalize__card__of__ordLess,axiom,
    ! [A2: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We2061588309931968636_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_set_o @ B6 @ ( field_o @ R ) )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1142_internalize__card__of__ordLess,axiom,
    ! [A2: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R ) @ bNF_We3448473429330749420at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1143_internalize__card__of__ordLess,axiom,
    ! [A2: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R ) @ bNF_We8182288985678559134_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1144_card__of__Plus__infinite2,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ B2 @ A2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We5455623833347018717_o_o_o ) ) ) ).

% card_of_Plus_infinite2
thf(fact_1145_card__of__Plus__infinite2,axiom,
    ! [A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat )
       => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We1461007947729398933at_nat ) ) ) ).

% card_of_Plus_infinite2
thf(fact_1146_card__of__Plus__infinite2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat )
       => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4717344193235055909at_nat ) ) ) ).

% card_of_Plus_infinite2
thf(fact_1147_card__of__Plus__infinite1,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We5455623833347018717_o_o_o ) ) ) ).

% card_of_Plus_infinite1
thf(fact_1148_card__of__Plus__infinite1,axiom,
    ! [A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat )
       => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5202279374879649339_o_nat ) ) ) ).

% card_of_Plus_infinite1
thf(fact_1149_card__of__Plus__infinite1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat )
       => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4717344193235055909at_nat ) ) ) ).

% card_of_Plus_infinite1
thf(fact_1150_card__of__Plus__infinite,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We5455623833347018717_o_o_o )
          & ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ B2 @ A2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We5455623833347018717_o_o_o ) ) ) ) ).

% card_of_Plus_infinite
thf(fact_1151_card__of__Plus__infinite,axiom,
    ! [A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We5202279374879649339_o_nat )
          & ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ B2 @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We1461007947729398933at_nat ) ) ) ) ).

% card_of_Plus_infinite
thf(fact_1152_card__of__Plus__infinite,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat )
       => ( ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4717344193235055909at_nat )
          & ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ B2 @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We4717344193235055909at_nat ) ) ) ) ).

% card_of_Plus_infinite
thf(fact_1153_regularCard__def,axiom,
    ( bNF_Ca1459541525017835783Card_o
    = ( ^ [R2: set_Product_prod_o_o] :
        ! [K2: set_o] :
          ( ( ( ord_less_eq_set_o @ K2 @ ( field_o @ R2 ) )
            & ( bNF_Ca4363140458462388713inal_o @ K2 @ R2 ) )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ K2 ) @ R2 ) @ bNF_We2654380646378065620so_o_o ) ) ) ) ).

% regularCard_def
thf(fact_1154_regularCard__def,axiom,
    ( bNF_Ca6818034935745638433rd_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat] :
        ! [K2: set_nat] :
          ( ( ( ord_less_eq_set_nat @ K2 @ ( field_nat @ R2 ) )
            & ( bNF_Ca5159621140400407487al_nat @ K2 @ R2 ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ K2 ) @ R2 ) @ bNF_We5258908940166488438at_nat ) ) ) ) ).

% regularCard_def
thf(fact_1155_regularCard__ordIso,axiom,
    ! [K: set_Pr1261947904930325089at_nat,K3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ K3 ) @ bNF_We5258908940166488438at_nat )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ K )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ K ) @ K ) )
       => ( ( bNF_Ca6818034935745638433rd_nat @ K )
         => ( bNF_Ca6818034935745638433rd_nat @ K3 ) ) ) ) ).

% regularCard_ordIso
thf(fact_1156_regularCard__ordIso,axiom,
    ! [K: set_Product_prod_o_o,K3: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ K3 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ K )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ K ) @ K ) )
       => ( ( bNF_Ca1459541525017835783Card_o @ K )
         => ( bNF_Ca6818034935745638433rd_nat @ K3 ) ) ) ) ).

% regularCard_ordIso
thf(fact_1157_regularCard__ordIso,axiom,
    ! [K: set_Product_prod_o_o,K3: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ K3 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ K )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ K ) @ K ) )
       => ( ( bNF_Ca1459541525017835783Card_o @ K )
         => ( bNF_Ca1459541525017835783Card_o @ K3 ) ) ) ) ).

% regularCard_ordIso
thf(fact_1158_Cnotzero__UNIV,axiom,
    ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
    & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) ) @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) ) ) ).

% Cnotzero_UNIV
thf(fact_1159_Cnotzero__UNIV,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) ) @ ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) ) ) ).

% Cnotzero_UNIV
thf(fact_1160_czero__ordIso,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_czero_o @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We4116350175183677332_o_nat ).

% czero_ordIso
thf(fact_1161_czero__ordIso,axiom,
    member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_czero_o @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o ).

% czero_ordIso
thf(fact_1162_czero__ordIso,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bNF_Ca8004134252417476708ro_nat @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat ).

% czero_ordIso
thf(fact_1163_card__of__bool,axiom,
    ! [A1: $o,A22: $o] :
      ( ( A1 != A22 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ A1 @ ( insert_o @ A22 @ bot_bot_set_o ) ) ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_bool
thf(fact_1164_card__of__bool,axiom,
    ! [A1: nat,A22: nat] :
      ( ( A1 != A22 )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ A1 @ ( insert_nat @ A22 @ bot_bot_set_nat ) ) ) ) @ bNF_We4116350175183677332_o_nat ) ) ).

% card_of_bool
thf(fact_1165_cinfinite__not__czero,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca7082918126634695480nite_o @ R )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We4116350175183677332_o_nat ) ) ).

% cinfinite_not_czero
thf(fact_1166_cinfinite__not__czero,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca7082918126634695480nite_o @ R )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o ) ) ).

% cinfinite_not_czero
thf(fact_1167_cinfinite__not__czero,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca7751713578713111600te_nat @ R )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat ) ) ).

% cinfinite_not_czero
thf(fact_1168_czeroE,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We4116350175183677332_o_nat )
     => ( ( field_o @ R )
        = bot_bot_set_o ) ) ).

% czeroE
thf(fact_1169_czeroE,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We7219021536291862642_nat_o )
     => ( ( field_nat @ R )
        = bot_bot_set_nat ) ) ).

% czeroE
thf(fact_1170_czeroE,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
     => ( ( field_o @ R )
        = bot_bot_set_o ) ) ).

% czeroE
thf(fact_1171_czeroE,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
     => ( ( field_nat @ R )
        = bot_bot_set_nat ) ) ).

% czeroE
thf(fact_1172_card__of__ordIso__czero__iff__empty,axiom,
    ! [A2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We4116350175183677332_o_nat )
      = ( A2 = bot_bot_set_o ) ) ).

% card_of_ordIso_czero_iff_empty
thf(fact_1173_card__of__ordIso__czero__iff__empty,axiom,
    ! [A2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
      = ( A2 = bot_bot_set_o ) ) ).

% card_of_ordIso_czero_iff_empty
thf(fact_1174_card__of__ordIso__czero__iff__empty,axiom,
    ! [A2: set_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ bNF_Cardinal_czero_o ) @ bNF_We7219021536291862642_nat_o )
      = ( A2 = bot_bot_set_nat ) ) ).

% card_of_ordIso_czero_iff_empty
thf(fact_1175_card__of__ordIso__czero__iff__empty,axiom,
    ! [A2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% card_of_ordIso_czero_iff_empty
thf(fact_1176_czeroI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( ( field_nat @ R )
          = bot_bot_set_nat )
       => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We7219021536291862642_nat_o ) ) ) ).

% czeroI
thf(fact_1177_czeroI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R )
     => ( ( ( field_nat @ R )
          = bot_bot_set_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% czeroI
thf(fact_1178_czeroI,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( ( field_o @ R )
          = bot_bot_set_o )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% czeroI
thf(fact_1179_czeroI,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R )
     => ( ( ( field_o @ R )
          = bot_bot_set_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% czeroI
thf(fact_1180_Cnotzero__imp__not__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( field_nat @ R )
       != bot_bot_set_nat ) ) ).

% Cnotzero_imp_not_empty
thf(fact_1181_Cnotzero__imp__not__empty,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( field_o @ R )
       != bot_bot_set_o ) ) ).

% Cnotzero_imp_not_empty
thf(fact_1182_Cnotzero__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ Q ) @ bNF_We2031159114419172489at_nat )
         => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Q @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q ) ) ) ) ) ).

% Cnotzero_mono
thf(fact_1183_Cnotzero__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Q: set_Product_prod_o_o] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ Q ) @ bNF_We6538615181293537951_nat_o )
         => ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Q @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q ) ) ) ) ) ).

% Cnotzero_mono
thf(fact_1184_Cnotzero__mono,axiom,
    ! [R: set_Product_prod_o_o,Q: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ Q ) @ bNF_We3435943820185352641_o_nat )
         => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Q @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q ) ) ) ) ) ).

% Cnotzero_mono
thf(fact_1185_Cnotzero__mono,axiom,
    ! [R: set_Product_prod_o_o,Q: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ Q ) @ bNF_We8115293805659342695eq_o_o )
         => ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Q @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q ) ) ) ) ) ).

% Cnotzero_mono
thf(fact_1186_Cinfinite__Cnotzero,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) ) ) ).

% Cinfinite_Cnotzero
thf(fact_1187_Cinfinite__Cnotzero,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) ) ) ).

% Cinfinite_Cnotzero
thf(fact_1188_cone__ordLeq__Cnotzero,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member5908366503767856682at_nat @ ( produc8927032219841499249at_nat @ bNF_Cardinal_cone @ R ) @ bNF_We6582606825125044278it_nat ) ) ).

% cone_ordLeq_Cnotzero
thf(fact_1189_cone__ordLeq__Cnotzero,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member1093438524718783952od_o_o @ ( produc7098341156072696855od_o_o @ bNF_Cardinal_cone @ R ) @ bNF_We5767726630029298098unit_o ) ) ).

% cone_ordLeq_Cnotzero
thf(fact_1190_cprod__infinite1_H,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P ) )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ R ) @ bNF_We2031159114419172489at_nat )
         => ( member1413612341684349482at_nat @ ( produc8587262944202267249at_nat @ ( bNF_Ca5918273672090100260at_nat @ R @ P ) @ R ) @ bNF_We2750463662153845245at_nat ) ) ) ) ).

% cprod_infinite1'
thf(fact_1191_cprod__infinite1_H,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P ) )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ R ) @ bNF_We3435943820185352641_o_nat )
         => ( member5535817393624548534at_nat @ ( produc2339070416962590717at_nat @ ( bNF_Ca3771748454224040004_nat_o @ R @ P ) @ R ) @ bNF_We5935404193577612131_o_nat ) ) ) ) ).

% cprod_infinite1'
thf(fact_1192_cprod__infinite1_H,axiom,
    ! [R: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P ) )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ R ) @ bNF_We6538615181293537951_nat_o )
         => ( member980006822203139752od_o_o @ ( produc7741497687623863791od_o_o @ ( bNF_Ca669077093115854694_o_nat @ R @ P ) @ R ) @ bNF_We2588849498452662379_nat_o ) ) ) ) ).

% cprod_infinite1'
thf(fact_1193_cprod__infinite1_H,axiom,
    ! [R: set_Product_prod_o_o,P: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P ) )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ R ) @ bNF_We8115293805659342695eq_o_o )
         => ( member506192354440972880od_o_o @ ( produc3444145290598225047od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R @ P ) @ R ) @ bNF_We2355029620655671557_o_o_o ) ) ) ) ).

% cprod_infinite1'
thf(fact_1194_cprod__infinite2_H,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We2031159114419172489at_nat )
         => ( member1413612341684349482at_nat @ ( produc8587262944202267249at_nat @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) @ R22 ) @ bNF_We2750463662153845245at_nat ) ) ) ) ).

% cprod_infinite2'
thf(fact_1195_cprod__infinite2_H,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R1 @ R22 ) @ bNF_We6538615181293537951_nat_o )
         => ( member602287860883522140od_o_o @ ( produc7363778726304246179od_o_o @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) @ R22 ) @ bNF_We1961531575160504645at_o_o ) ) ) ) ).

% cprod_infinite2'
thf(fact_1196_cprod__infinite2_H,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We3435943820185352641_o_nat )
         => ( member13643011425145602at_nat @ ( produc6040268071617963593at_nat @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) @ R22 ) @ bNF_We2194132766427361725at_nat ) ) ) ) ).

% cprod_infinite2'
thf(fact_1197_cprod__infinite2_H,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
         => ( member506192354440972880od_o_o @ ( produc3444145290598225047od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) @ R22 ) @ bNF_We2355029620655671557_o_o_o ) ) ) ) ).

% cprod_infinite2'
thf(fact_1198_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member7930742895388902160od_o_o @ ( produc189021232855538519od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) ) @ bNF_We5263191010062085543od_o_o ) ) ) ).

% cprod_mono
thf(fact_1199_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member1643650677906951352_o_nat @ ( produc8026554252588472319_o_nat @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) @ bNF_We5526594851599572503_o_nat ) ) ) ).

% cprod_mono
thf(fact_1200_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member6130509595031242000_o_nat @ ( produc5910034646884183383_o_nat @ ( bNF_Ca669077093115854694_o_nat @ P1 @ P22 ) @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) @ bNF_We5517640669336891927_o_nat ) ) ) ).

% cprod_mono
thf(fact_1201_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member8824421929673164292_nat_o @ ( produc3218210373639305035_nat_o @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) @ bNF_We9035129375655065533_nat_o ) ) ) ).

% cprod_mono
thf(fact_1202_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8717407822839053200at_nat @ ( produc6996284652319478231at_nat @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We7027591586743403905at_nat ) ) ) ).

% cprod_mono
thf(fact_1203_cprod__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member2414491960730527720at_nat @ ( produc6295173111627586351at_nat @ ( bNF_Ca669077093115854694_o_nat @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We1398974386449396097at_nat ) ) ) ).

% cprod_mono
thf(fact_1204_cprod__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member6208704584088763408_nat_o @ ( produc3222486542081100375_nat_o @ ( bNF_Ca3771748454224040004_nat_o @ P1 @ P22 ) @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) @ bNF_We5283883798296455779_nat_o ) ) ) ).

% cprod_mono
thf(fact_1205_cprod__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member1892715368018688924at_nat @ ( produc5773396518915747555at_nat @ ( bNF_Ca3771748454224040004_nat_o @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We4116928198778210395at_nat ) ) ) ).

% cprod_mono
thf(fact_1206_cprod__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ ( bNF_Ca5918273672090100260at_nat @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We145940432564263207at_nat ) ) ) ).

% cprod_mono
thf(fact_1207_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8717407822839053200at_nat @ ( produc6996284652319478231at_nat @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We4425979316902024558at_nat ) ) ) ).

% cprod_cong
thf(fact_1208_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member8824421929673164292_nat_o @ ( produc3218210373639305035_nat_o @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) @ bNF_We6436225757993239184_nat_o ) ) ) ).

% cprod_cong
thf(fact_1209_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member2414491960730527720at_nat @ ( produc6295173111627586351at_nat @ ( bNF_Ca669077093115854694_o_nat @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We5155846448255121108at_nat ) ) ) ).

% cprod_cong
thf(fact_1210_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member1643650677906951352_o_nat @ ( produc8026554252588472319_o_nat @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) @ bNF_We2927691233937746154_o_nat ) ) ) ).

% cprod_cong
thf(fact_1211_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member7930742895388902160od_o_o @ ( produc189021232855538519od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) ) @ bNF_We3478797130297825300od_o_o ) ) ) ).

% cprod_cong
thf(fact_1212_cprod__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member6130509595031242000_o_nat @ ( produc5910034646884183383_o_nat @ ( bNF_Ca669077093115854694_o_nat @ P1 @ P22 ) @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) @ bNF_We2916028399495512580_o_nat ) ) ) ).

% cprod_cong
thf(fact_1213_cprod__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member1892715368018688924at_nat @ ( produc5773396518915747555at_nat @ ( bNF_Ca3771748454224040004_nat_o @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We7873800260583935406at_nat ) ) ) ).

% cprod_cong
thf(fact_1214_cprod__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member6208704584088763408_nat_o @ ( produc3222486542081100375_nat_o @ ( bNF_Ca3771748454224040004_nat_o @ P1 @ P22 ) @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) @ bNF_We2682271528455076432_nat_o ) ) ) ).

% cprod_cong
thf(fact_1215_cprod__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ ( bNF_Ca5918273672090100260at_nat @ P1 @ P22 ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ bNF_We4319090096570846868at_nat ) ) ) ).

% cprod_cong
thf(fact_1216_cone__not__czero,axiom,
    ~ ( member1093438524718783952od_o_o @ ( produc7098341156072696855od_o_o @ bNF_Cardinal_cone @ bNF_Cardinal_czero_o ) @ bNF_We8819814353795312031unit_o ) ).

% cone_not_czero
thf(fact_1217_cprod__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( member1413612341684349482at_nat @ ( produc8587262944202267249at_nat @ ( bNF_Ca5918273672090100260at_nat @ R @ R ) @ R ) @ bNF_We2750463662153845245at_nat ) ) ).

% cprod_infinite
thf(fact_1218_cprod__infinite,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( member506192354440972880od_o_o @ ( produc3444145290598225047od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R @ R ) @ R ) @ bNF_We2355029620655671557_o_o_o ) ) ).

% cprod_infinite
thf(fact_1219_Cfinite__cprod__Cinfinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member1413612341684349482at_nat @ ( produc8587262944202267249at_nat @ ( bNF_Ca5918273672090100260at_nat @ R @ S ) @ S ) @ bNF_We1581642880855175978at_nat ) ) ) ).

% Cfinite_cprod_Cinfinite
thf(fact_1220_Cfinite__cprod__Cinfinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member602287860883522140od_o_o @ ( produc7363778726304246179od_o_o @ ( bNF_Ca3771748454224040004_nat_o @ R @ S ) @ S ) @ bNF_We5569858094202414322at_o_o ) ) ) ).

% Cfinite_cprod_Cinfinite
thf(fact_1221_Cfinite__cprod__Cinfinite,axiom,
    ! [R: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member13643011425145602at_nat @ ( produc6040268071617963593at_nat @ ( bNF_Ca669077093115854694_o_nat @ R @ S ) @ S ) @ bNF_We383578267767362384at_nat ) ) ) ).

% Cfinite_cprod_Cinfinite
thf(fact_1222_Cfinite__cprod__Cinfinite,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member506192354440972880od_o_o @ ( produc3444145290598225047od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R @ S ) @ S ) @ bNF_We3336912373155303448_o_o_o ) ) ) ).

% Cfinite_cprod_Cinfinite
thf(fact_1223_cprod__cinfinite__bound,axiom,
    ! [P: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Q @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
           => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
                & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
             => ( member1413612341684349482at_nat @ ( produc8587262944202267249at_nat @ ( bNF_Ca5918273672090100260at_nat @ P @ Q ) @ R ) @ bNF_We1581642880855175978at_nat ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1224_cprod__cinfinite__bound,axiom,
    ! [P: set_Pr1261947904930325089at_nat,R: set_Product_prod_o_o,Q: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ Q @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
           => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
                & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
             => ( member4285047842233315792od_o_o @ ( produc5921634641268913687od_o_o @ ( bNF_Ca5918273672090100260at_nat @ P @ Q ) @ R ) @ bNF_We8412050412698735038_nat_o ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1225_cprod__cinfinite__bound,axiom,
    ! [P: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,Q: set_Product_prod_o_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ R ) @ bNF_We2031159114419172489at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Q @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
           => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
                & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
             => ( member5535817393624548534at_nat @ ( produc2339070416962590717at_nat @ ( bNF_Ca3771748454224040004_nat_o @ P @ Q ) @ R ) @ bNF_We4124849694917612790_o_nat ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1226_cprod__cinfinite__bound,axiom,
    ! [P: set_Pr1261947904930325089at_nat,R: set_Product_prod_o_o,Q: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ R ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Q @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P ) @ P )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
           => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
                & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
             => ( member602287860883522140od_o_o @ ( produc7363778726304246179od_o_o @ ( bNF_Ca3771748454224040004_nat_o @ P @ Q ) @ R ) @ bNF_We5569858094202414322at_o_o ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1227_cprod__cinfinite__bound,axiom,
    ! [P: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Q @ R ) @ bNF_We2031159114419172489at_nat )
       => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
           => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
                & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
             => ( member13643011425145602at_nat @ ( produc6040268071617963593at_nat @ ( bNF_Ca669077093115854694_o_nat @ P @ Q ) @ R ) @ bNF_We383578267767362384at_nat ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1228_cprod__cinfinite__bound,axiom,
    ! [P: set_Product_prod_o_o,R: set_Product_prod_o_o,Q: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ Q @ R ) @ bNF_We6538615181293537951_nat_o )
       => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ Q ) @ Q )
           => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
                & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
             => ( member980006822203139752od_o_o @ ( produc7741497687623863791od_o_o @ ( bNF_Ca669077093115854694_o_nat @ P @ Q ) @ R ) @ bNF_We6197176017494572056_nat_o ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1229_cprod__cinfinite__bound,axiom,
    ! [P: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat,Q: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ R ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Q @ R ) @ bNF_We3435943820185352641_o_nat )
       => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
           => ( ( ( bNF_Ca7751713578713111600te_nat @ R )
                & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R ) @ R ) )
             => ( member14561935767372458at_nat @ ( produc1942719025490540785at_nat @ ( bNF_Ca8659498892590988930od_o_o @ P @ Q ) @ R ) @ bNF_We1909975299713449808_o_nat ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1230_cprod__cinfinite__bound,axiom,
    ! [P: set_Product_prod_o_o,R: set_Product_prod_o_o,Q: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ R ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Q @ R ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P ) @ P )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ Q ) @ Q )
           => ( ( ( bNF_Ca7082918126634695480nite_o @ R )
                & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R ) @ R ) )
             => ( member506192354440972880od_o_o @ ( produc3444145290598225047od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ P @ Q ) @ R ) @ bNF_We3336912373155303448_o_o_o ) ) ) ) ) ) ).

% cprod_cinfinite_bound
thf(fact_1231_ordLeq__cprod2,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P1 ) @ P1 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P22 ) @ P22 )
       => ( member6798280822823052662at_nat @ ( produc2715267583992483517at_nat @ P22 @ ( bNF_Ca5918273672090100260at_nat @ P1 @ P22 ) ) @ bNF_We4941690273897891080at_nat ) ) ) ).

% ordLeq_cprod2
thf(fact_1232_ordLeq__cprod2,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P1 ) @ P1 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P22 ) @ P22 )
       => ( member6196234723914714820_nat_o @ ( produc4826005137656083979_nat_o @ P22 @ ( bNF_Ca3771748454224040004_nat_o @ P1 @ P22 ) ) @ bNF_We8004284360722039214_nat_o ) ) ) ).

% ordLeq_cprod2
thf(fact_1233_ordLeq__cprod2,axiom,
    ! [P1: set_Product_prod_o_o,P22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P1 ) @ P1 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ P22 ) @ P22 )
       => ( member621746330869129630_o_nat @ ( produc6115727808940752869_o_nat @ P22 @ ( bNF_Ca669077093115854694_o_nat @ P1 @ P22 ) ) @ bNF_We2400814313052847824_o_nat ) ) ) ).

% ordLeq_cprod2
thf(fact_1234_ordLeq__cprod2,axiom,
    ! [P1: set_Product_prod_o_o,P22: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P1 ) @ P1 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ P22 ) @ P22 )
       => ( member3902862663760325584od_o_o @ ( produc324748338109790743od_o_o @ P22 @ ( bNF_Ca8659498892590988930od_o_o @ P1 @ P22 ) ) @ bNF_We8720988762226505078od_o_o ) ) ) ).

% ordLeq_cprod2
thf(fact_1235_Cinfinite__cprod2,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( ( bNF_Ca1166716789977911457at_nat @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) )
          & ( bNF_Ca7505599351223300479at_nat @ ( field_4706345496283612622at_nat @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) ) ) ) ).

% Cinfinite_cprod2
thf(fact_1236_Cinfinite__cprod2,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( ( bNF_Ca5820406274200315037_nat_o @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) )
          & ( bNF_Ca4568735273429938239_nat_o @ ( field_8110745030544226864_nat_o @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) ) ) ) ).

% Cinfinite_cprod2
thf(fact_1237_Cinfinite__cprod2,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( ( bNF_Ca2311871750144822007_o_nat @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) )
          & ( bNF_Ca1060200749374445209_o_nat @ ( field_4602210506488733834_o_nat @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) ) ) ) ).

% Cinfinite_cprod2
thf(fact_1238_Cinfinite__cprod2,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( ( bNF_Ca753335641802049735od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) )
          & ( bNF_Ca6022239954640842917od_o_o @ ( field_7351064555842910324od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) ) @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) ) ) ) ) ).

% Cinfinite_cprod2
thf(fact_1239_cinfinite__cprod2,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( bNF_Ca1166716789977911457at_nat @ ( bNF_Ca5918273672090100260at_nat @ R1 @ R22 ) ) ) ) ).

% cinfinite_cprod2
thf(fact_1240_cinfinite__cprod2,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Product_prod_o_o] :
      ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ bNF_Ca8004134252417476708ro_nat ) @ bNF_We5258908940166488438at_nat )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( bNF_Ca5820406274200315037_nat_o @ ( bNF_Ca3771748454224040004_nat_o @ R1 @ R22 ) ) ) ) ).

% cinfinite_cprod2
thf(fact_1241_cinfinite__cprod2,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
       => ( bNF_Ca2311871750144822007_o_nat @ ( bNF_Ca669077093115854694_o_nat @ R1 @ R22 ) ) ) ) ).

% cinfinite_cprod2
thf(fact_1242_cinfinite__cprod2,axiom,
    ! [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R1 ) @ R1 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
       => ( bNF_Ca753335641802049735od_o_o @ ( bNF_Ca8659498892590988930od_o_o @ R1 @ R22 ) ) ) ) ).

% cinfinite_cprod2
thf(fact_1243_Un__csum,axiom,
    ! [A2: set_nat,B2: set_nat] : ( member7606898804734335598at_nat @ ( produc7668796723872103605at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( bNF_Ca591209840194543554at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) ) @ bNF_We4444846861891280440at_nat ) ).

% Un_csum
thf(fact_1244_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member9006821182645044420at_nat @ ( produc6767169964153735947at_nat @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We7027991437926027642at_nat ) ) ) ).

% csum_cong
thf(fact_1245_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca8004039653520920358_nat_o @ R1 @ R22 ) ) @ bNF_We2873397315248171908_nat_o ) ) ) ).

% csum_cong
thf(fact_1246_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member234890756356643848at_nat @ ( produc5352410120312684367at_nat @ ( bNF_Ca4901368292412735048_o_nat @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We6373243813402207300at_nat ) ) ) ).

% csum_cong
thf(fact_1247_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca4901368292412735048_o_nat @ R1 @ R22 ) ) @ bNF_We8588234828047454686_o_nat ) ) ) ).

% csum_cong
thf(fact_1248_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member3483138326940501008um_o_o @ ( produc1030987845608174167um_o_o @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca3327395247694603040um_o_o @ R1 @ R22 ) ) @ bNF_We2736418005100055072um_o_o ) ) ) ).

% csum_cong
thf(fact_1249_csum__cong,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member2972727061379769104_o_nat @ ( produc728317944619019095_o_nat @ ( bNF_Ca4901368292412735048_o_nat @ P1 @ P22 ) @ ( bNF_Ca4901368292412735048_o_nat @ R1 @ R22 ) ) @ bNF_We4748241943506032788_o_nat ) ) ) ).

% csum_cong
thf(fact_1250_csum__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We4116350175183677332_o_nat )
       => ( member7654646017578591112at_nat @ ( produc3548793344679855823at_nat @ ( bNF_Ca8004039653520920358_nat_o @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We8460142929841839338at_nat ) ) ) ).

% csum_cong
thf(fact_1251_csum__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We2654380646378065620so_o_o )
       => ( member1510635369261286160_nat_o @ ( produc224851048220445527_nat_o @ ( bNF_Ca8004039653520920358_nat_o @ P1 @ P22 ) @ ( bNF_Ca8004039653520920358_nat_o @ R1 @ R22 ) ) @ bNF_We2077947424676570132_nat_o ) ) ) ).

% csum_cong
thf(fact_1252_csum__cong,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We5258908940166488438at_nat )
       => ( member3674313732307032336at_nat @ ( produc2112825387572549975at_nat @ ( bNF_Ca591209840194543554at_nat @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We3455965832816432340at_nat ) ) ) ).

% csum_cong
thf(fact_1253_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member3483138326940501008um_o_o @ ( produc1030987845608174167um_o_o @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca3327395247694603040um_o_o @ R1 @ R22 ) ) @ bNF_We8987810345020088115um_o_o ) ) ) ).

% csum_mono
thf(fact_1254_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member3365704633057040844_o_nat @ ( produc3782849074957862163_o_nat @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca4901368292412735048_o_nat @ R1 @ R22 ) ) @ bNF_We3854041083340105611_o_nat ) ) ) ).

% csum_mono
thf(fact_1255_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Product_prod_o_o,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P1 @ R1 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member2972727061379769104_o_nat @ ( produc728317944619019095_o_nat @ ( bNF_Ca4901368292412735048_o_nat @ P1 @ P22 ) @ ( bNF_Ca4901368292412735048_o_nat @ R1 @ R22 ) ) @ bNF_We8273322497166968871_o_nat ) ) ) ).

% csum_mono
thf(fact_1256_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member7444511523098636876_nat_o @ ( produc8820280760719367571_nat_o @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca8004039653520920358_nat_o @ R1 @ R22 ) ) @ bNF_We7362575607395598641_nat_o ) ) ) ).

% csum_mono
thf(fact_1257_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member9006821182645044420at_nat @ ( produc6767169964153735947at_nat @ ( bNF_Ca3327395247694603040um_o_o @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We1329699954732187917at_nat ) ) ) ).

% csum_mono
thf(fact_1258_csum__mono,axiom,
    ! [P1: set_Product_prod_o_o,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P1 @ R1 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member234890756356643848at_nat @ ( produc5352410120312684367at_nat @ ( bNF_Ca4901368292412735048_o_nat @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We7431878391334027121at_nat ) ) ) ).

% csum_mono
thf(fact_1259_csum__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P22 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member1510635369261286160_nat_o @ ( produc224851048220445527_nat_o @ ( bNF_Ca8004039653520920358_nat_o @ P1 @ P22 ) @ ( bNF_Ca8004039653520920358_nat_o @ R1 @ R22 ) ) @ bNF_We5603027978337506215_nat_o ) ) ) ).

% csum_mono
thf(fact_1260_csum__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Product_prod_o_o,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P22 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( member7654646017578591112at_nat @ ( produc3548793344679855823at_nat @ ( bNF_Ca8004039653520920358_nat_o @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We295405470918883351at_nat ) ) ) ).

% csum_mono
thf(fact_1261_csum__mono,axiom,
    ! [P1: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat,P22: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P1 @ R1 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P22 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member3674313732307032336at_nat @ ( produc2112825387572549975at_nat @ ( bNF_Ca591209840194543554at_nat @ P1 @ P22 ) @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) ) @ bNF_We5082115074515246823at_nat ) ) ) ).

% csum_mono
thf(fact_1262_Plus__csum,axiom,
    ! [A2: set_nat,B2: set_nat] : ( member3674313732307032336at_nat @ ( produc2112825387572549975at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ ( bNF_Ca591209840194543554at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) ) @ bNF_We3455965832816432340at_nat ) ).

% Plus_csum
thf(fact_1263_csum__absorb2_H,axiom,
    ! [R22: set_Pr1261947904930325089at_nat,R1: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
            | ( bNF_Ca7751713578713111600te_nat @ R22 ) )
         => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca4901368292412735048_o_nat @ R1 @ R22 ) @ R22 ) @ bNF_We1461007947729398933at_nat ) ) ) ) ).

% csum_absorb2'
thf(fact_1264_csum__absorb2_H,axiom,
    ! [R22: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R1 )
            | ( bNF_Ca7751713578713111600te_nat @ R22 ) )
         => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) @ R22 ) @ bNF_We4717344193235055909at_nat ) ) ) ) ).

% csum_absorb2'
thf(fact_1265_csum__absorb2_H,axiom,
    ! [R22: set_Product_prod_o_o,R1: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
            | ( bNF_Ca7082918126634695480nite_o @ R22 ) )
         => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca3327395247694603040um_o_o @ R1 @ R22 ) @ R22 ) @ bNF_We5455623833347018717_o_o_o ) ) ) ) ).

% csum_absorb2'
thf(fact_1266_csum__absorb1_H,axiom,
    ! [R22: set_Pr1261947904930325089at_nat,R1: set_Product_prod_o_o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R1 @ R22 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
            | ( bNF_Ca7751713578713111600te_nat @ R22 ) )
         => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca8004039653520920358_nat_o @ R22 @ R1 ) @ R22 ) @ bNF_We5202279374879649339_o_nat ) ) ) ) ).

% csum_absorb1'
thf(fact_1267_csum__absorb1_H,axiom,
    ! [R22: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R1 )
            | ( bNF_Ca7751713578713111600te_nat @ R22 ) )
         => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca591209840194543554at_nat @ R22 @ R1 ) @ R22 ) @ bNF_We4717344193235055909at_nat ) ) ) ) ).

% csum_absorb1'
thf(fact_1268_csum__absorb1_H,axiom,
    ! [R22: set_Product_prod_o_o,R1: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R1 )
            | ( bNF_Ca7082918126634695480nite_o @ R22 ) )
         => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca3327395247694603040um_o_o @ R22 @ R1 ) @ R22 ) @ bNF_We5455623833347018717_o_o_o ) ) ) ) ).

% csum_absorb1'
thf(fact_1269_csum__absorb2,axiom,
    ! [R22: set_Pr1261947904930325089at_nat,R1: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R22 )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R22 ) @ R22 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R1 @ R22 ) @ bNF_We2031159114419172489at_nat )
       => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca591209840194543554at_nat @ R1 @ R22 ) @ R22 ) @ bNF_We4717344193235055909at_nat ) ) ) ).

% csum_absorb2
thf(fact_1270_csum__absorb2,axiom,
    ! [R22: set_Product_prod_o_o,R1: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R22 )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R22 ) @ R22 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R1 @ R22 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca3327395247694603040um_o_o @ R1 @ R22 ) @ R22 ) @ bNF_We5455623833347018717_o_o_o ) ) ) ).

% csum_absorb2
thf(fact_1271_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_1272_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_1273_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_1274_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat

% Helper facts (3)
thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X: option_a,Y: option_a] :
      ( ( if_option_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X: option_a,Y: option_a] :
      ( ( if_option_a @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ p @ r ) @ c2 @ ( star_a_b_d_c @ q @ r ) @ delta ).

%------------------------------------------------------------------------------