TPTP Problem File: SLH0312^1.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/0002_Distributivity/prob_00174_004985__6720088_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2190 ( 288 unt; 911 typ;   0 def)
%            Number of atoms       : 4882 (1001 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 19916 ( 538   ~;  56   |; 380   &;16327   @)
%                                         (   0 <=>;2615  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (  10 avg)
%            Number of types       :  180 ( 179 usr)
%            Number of type conns  : 2379 (2379   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  735 ( 732 usr; 145 con; 0-7 aty)
%            Number of variables   : 4796 (  72   ^;4556   !; 168   ?;4796   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 16:05:02.379
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_I_Eo_M_Eo_J_Mt__Sum____Type__Osum_I_Eo_M_Eo_J_J_J_J,type,
    member952642050024735836um_o_o: produc6322778544207219763um_o_o > set_Pr9196163270038067219um_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_J_J_J,type,
    member6975117218213954584_o_nat: produc1534263882305827951_o_nat > set_Pr6274769524298744783_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member3674313732307032336at_nat: produc6206612473762214375at_nat > set_Pr8921550766394648263at_nat > $o ).

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    member9169340693537962768od_o_o: produc7004997421203748071od_o_o > set_Pr7077720281583819975od_o_o > $o ).

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    member4300370950623927696at_nat: produc7299876168958306407at_nat > set_Pr1604112560894399303at_nat > $o ).

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    member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).

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    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

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    member4387393612980313840od_o_o: set_Pr1932065953672099015od_o_o > set_se9147984907235033255od_o_o > $o ).

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

thf(sy_v_A,type,
    a2: assertion_a_b_c_d ).

thf(sy_v_B,type,
    b2: assertion_a_b_c_d ).

thf(sy_v__092_060Delta_062_H____,type,
    delta: ( d > c ) > set_a ).

thf(sy_v__092_060sigma_062_H____,type,
    sigma: a ).

thf(sy_v__092_060sigma_062____,type,
    sigma2: a ).

thf(sy_v_a____,type,
    a3: a ).

thf(sy_v_aa____,type,
    aa: a ).

thf(sy_v_b____,type,
    b3: a ).

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

thf(sy_v_one,type,
    one: b ).

thf(sy_v_p,type,
    p: b ).

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

thf(sy_v_s____,type,
    s: d > c ).

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 ).

thf(sy_v_x____,type,
    x: a ).

% 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__092_060open_062b_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AB_092_060close_062,axiom,
    sat_a_b_d_c @ plus @ mult @ valid @ b3 @ s @ delta @ b2 ).

% \<open>b, s, \<Delta> \<Turnstile> B\<close>
thf(fact_3_sat_Osimps_I1_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
      = ( ? [A3: a] :
            ( ( Sigma
              = ( mult @ P @ A3 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 ) ) ) ) ).

% sat.simps(1)
thf(fact_4_sat__mult,axiom,
    ! [Sigma: a,P: b,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [A4: a] :
          ( ( Sigma
            = ( mult @ P @ A4 ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ).

% sat_mult
thf(fact_5_unique__inv,axiom,
    ! [A: a,P: b,B: a] :
      ( ( A
        = ( mult @ P @ B ) )
      = ( B
        = ( mult @ ( sinv @ P ) @ A ) ) ) ).

% unique_inv
thf(fact_6_pure__def,axiom,
    ! [A2: assertion_a_b_c_d] :
      ( ( pure_a_b_c_d @ plus @ mult @ valid @ A2 )
      = ( ! [Sigma2: a,Sigma3: a,S2: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta2 @ A2 )
            = ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta3 @ A2 ) ) ) ) ).

% pure_def
thf(fact_7_one__neutral,axiom,
    ! [A: a] :
      ( ( mult @ one @ A )
      = A ) ).

% one_neutral
thf(fact_8__092_060open_062_092_060sigma_062_A_061_Ap_A_092_060odot_062_Ax_092_060close_062,axiom,
    ( sigma2
    = ( mult @ p @ x ) ) ).

% \<open>\<sigma> = p \<odot> x\<close>
thf(fact_9__092_060open_062a_A_061_Ap_A_092_060odot_062_Aaa_092_060close_062,axiom,
    ( a3
    = ( mult @ p @ aa ) ) ).

% \<open>a = p \<odot> aa\<close>
thf(fact_10__092_060open_062aa_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AA_092_060close_062,axiom,
    sat_a_b_d_c @ plus @ mult @ valid @ aa @ s @ delta @ a2 ).

% \<open>aa, s, \<Delta> \<Turnstile> A\<close>
thf(fact_11_equivalentI,axiom,
    ! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S3: d > c] :
          ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ A2 )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ B2 ) )
     => ( ! [Sigma4: a,S3: d > c] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ B2 )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ A2 ) )
       => ( equivalent_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ) ).

% equivalentI
thf(fact_12_sat_Osimps_I10_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ pred_a_b_c_d )
      = ( member_a @ Sigma @ ( Delta @ S ) ) ) ).

% sat.simps(10)
thf(fact_13_DotPos,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
      ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
      = ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ).

% DotPos
thf(fact_14_entailsI,axiom,
    ! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S3: d > c] :
          ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ A2 )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ B2 ) )
     => ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ).

% entailsI
thf(fact_15_entails__def,axiom,
    ! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
      = ( ! [Sigma2: a,S2: d > c] :
            ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ A2 )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ B2 ) ) ) ) ).

% entails_def
thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aa_O_A_092_060lbrakk_062aa_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AA_059_Aa_A_061_Ap_A_092_060odot_062_Aaa_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aa: a] :
        ( ( sat_a_b_d_c @ plus @ mult @ valid @ Aa @ s @ delta @ a2 )
       => ( a3
         != ( mult @ p @ Aa ) ) ) ).

% \<open>\<And>thesis. (\<And>aa. \<lbrakk>aa, s, \<Delta> \<Turnstile> A; a = p \<odot> aa\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_17_not__in__fv__def,axiom,
    ! [A2: assertion_a_b_c_d,S4: set_d] :
      ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ S4 )
      = ( ! [Sigma2: a,S2: d > c,Delta2: ( d > c ) > set_a,S5: d > c] :
            ( ( equal_outside_d_c @ S2 @ S5 @ S4 )
           => ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta2 @ A2 )
              = ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S5 @ Delta2 @ A2 ) ) ) ) ) ).

% not_in_fv_def
thf(fact_18_sat_Osimps_I7_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
        & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).

% sat.simps(7)
thf(fact_19_sat_Osimps_I6_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
        | ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).

% sat.simps(6)
thf(fact_20__092_060open_062_092_060sigma_062_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AMult_Ap_A_IWand_AA_AB_J_092_060close_062,axiom,
    sat_a_b_d_c @ plus @ mult @ valid @ sigma2 @ s @ delta @ ( mult_b_a_c_d @ p @ ( wand_a_b_c_d @ a2 @ b2 ) ) ).

% \<open>\<sigma>, s, \<Delta> \<Turnstile> Mult p (Wand A B)\<close>
thf(fact_21_sat_Osimps_I5_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
      = ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).

% sat.simps(5)
thf(fact_22_sat__imp,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ).

% sat_imp
thf(fact_23__092_060open_062x_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AWand_AA_AB_092_060close_062,axiom,
    sat_a_b_d_c @ plus @ mult @ valid @ x @ s @ delta @ ( wand_a_b_c_d @ a2 @ b2 ) ).

% \<open>x, s, \<Delta> \<Turnstile> Wand A B\<close>
thf(fact_24__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062_092_060sigma_062_A_061_Ap_A_092_060odot_062_Ax_059_Ax_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AWand_AA_AB_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [X: a] :
        ( ( sigma2
          = ( mult @ p @ X ) )
       => ~ ( sat_a_b_d_c @ plus @ mult @ valid @ X @ s @ delta @ ( wand_a_b_c_d @ a2 @ b2 ) ) ) ).

% \<open>\<And>thesis. (\<And>x. \<lbrakk>\<sigma> = p \<odot> x; x, s, \<Delta> \<Turnstile> Wand A B\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_25__092_060open_062Some_Ab_A_061_Ax_A_092_060oplus_062_Aaa_092_060close_062,axiom,
    ( ( some_a @ b3 )
    = ( plus @ x @ aa ) ) ).

% \<open>Some b = x \<oplus> aa\<close>
thf(fact_26__092_060open_062a_M_As_M_A_092_060Delta_062_A_092_060Turnstile_062_AMult_Ap_AA_A_092_060and_062_ASome_A_092_060sigma_062_H_A_061_A_092_060sigma_062_A_092_060oplus_062_Aa_092_060close_062,axiom,
    ( ( sat_a_b_d_c @ plus @ mult @ valid @ a3 @ s @ delta @ ( mult_b_a_c_d @ p @ a2 ) )
    & ( ( some_a @ sigma )
      = ( plus @ sigma2 @ a3 ) ) ) ).

% \<open>a, s, \<Delta> \<Turnstile> Mult p A \<and> Some \<sigma>' = \<sigma> \<oplus> a\<close>
thf(fact_27_DotStar,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% DotStar
thf(fact_28_DotDot,axiom,
    ! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) ) ).

% DotDot
thf(fact_29_dot__star2,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ).

% dot_star2
thf(fact_30_dot__star1,axiom,
    ! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).

% dot_star1
thf(fact_31_WildDot,axiom,
    ! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).

% WildDot
thf(fact_32_DotWild,axiom,
    ! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).

% DotWild
thf(fact_33_dot__mult2,axiom,
    ! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) ) ).

% dot_mult2
thf(fact_34_dot__mult1,axiom,
    ! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) ) ).

% dot_mult1
thf(fact_35_sat__wand,axiom,
    ! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
      ( ! [A4: a,Sigma5: a] :
          ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 )
            & ( ( some_a @ Sigma5 )
              = ( plus @ Sigma @ A4 ) ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B2 ) )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ).

% sat_wand
thf(fact_36_sat_Osimps_I3_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
      = ( ! [A3: a,Sigma3: a] :
            ( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 )
              & ( ( some_a @ Sigma3 )
                = ( plus @ Sigma @ A3 ) ) )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S @ Delta @ B2 ) ) ) ) ).

% sat.simps(3)
thf(fact_37_sat_Osimps_I11_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( bounded_a_b_c_d @ A2 ) )
      = ( ( valid @ Sigma )
       => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 ) ) ) ).

% sat.simps(11)
thf(fact_38_can__factorize,axiom,
    ! [Q: b,P: b] :
    ? [R: b] :
      ( Q
      = ( smult @ R @ P ) ) ).

% can_factorize
thf(fact_39_smult__asso,axiom,
    ! [P: b,Q: b,R2: b] :
      ( ( smult @ ( smult @ P @ Q ) @ R2 )
      = ( smult @ P @ ( smult @ Q @ R2 ) ) ) ).

% smult_asso
thf(fact_40_smult__comm,axiom,
    ! [P: b,Q: b] :
      ( ( smult @ P @ Q )
      = ( smult @ Q @ P ) ) ).

% smult_comm
thf(fact_41_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_42_move__sum,axiom,
    ! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X2: a,X1: a,X22: a] :
      ( ( ( some_a @ A )
        = ( plus @ A1 @ A22 ) )
     => ( ( ( some_a @ B )
          = ( plus @ B1 @ B22 ) )
       => ( ( ( some_a @ X2 )
            = ( plus @ A @ B ) )
         => ( ( ( some_a @ X1 )
              = ( plus @ A1 @ B1 ) )
           => ( ( ( some_a @ X22 )
                = ( plus @ A22 @ B22 ) )
             => ( ( some_a @ X2 )
                = ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).

% move_sum
thf(fact_43_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_44_mem__Collect__eq,axiom,
    ! [A: produc732395585841259969at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( collec1729919521702017196at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o] :
      ( ( collec3302784863738620242od_o_o
        @ ^ [X3: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
    ! [A2: set_Pr457366540195662369at_nat] :
      ( ( collec1729919521702017196at_nat
        @ ^ [X3: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_49_double__mult,axiom,
    ! [P: b,Q: b,A: a] :
      ( ( mult @ P @ ( mult @ Q @ A ) )
      = ( mult @ ( smult @ P @ Q ) @ A ) ) ).

% double_mult
thf(fact_50_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_51_sone__neutral,axiom,
    ! [P: b] :
      ( ( smult @ one @ P )
      = P ) ).

% sone_neutral
thf(fact_52_sat_Osimps_I12_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
      = ( ? [A3: a,P3: b] :
            ( ( Sigma
              = ( mult @ P3 @ A3 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 ) ) ) ) ).

% sat.simps(12)
thf(fact_53_sinv__inverse,axiom,
    ! [P: b] :
      ( ( smult @ P @ ( sinv @ P ) )
      = one ) ).

% sinv_inverse
thf(fact_54__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062b_O_ASome_Ab_A_061_Ax_A_092_060oplus_062_Aaa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [B3: a] :
        ( ( some_a @ B3 )
       != ( plus @ x @ aa ) ) ).

% \<open>\<And>thesis. (\<And>b. Some b = x \<oplus> aa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_55_sat_Osimps_I2_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
      = ( ? [A3: a,B4: a] :
            ( ( ( some_a @ Sigma )
              = ( plus @ A3 @ B4 ) )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ B4 @ S @ Delta @ B2 ) ) ) ) ).

% sat.simps(2)
thf(fact_56_distrib__mult,axiom,
    ! [P: b,Q: b,X2: a] :
      ( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X2 ) )
      = ( plus @ ( mult @ P @ X2 ) @ ( mult @ Q @ X2 ) ) ) ).

% distrib_mult
thf(fact_57_smult__distrib,axiom,
    ! [P: b,Q: b,R2: b] :
      ( ( smult @ P @ ( sadd @ Q @ R2 ) )
      = ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R2 ) ) ) ).

% smult_distrib
thf(fact_58_logic__axioms,axiom,
    logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).

% logic_axioms
thf(fact_59_sat_Osimps_I4_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( sem_d_c_a_b @ B ) )
      = ( B @ S @ Sigma ) ) ).

% sat.simps(4)
thf(fact_60_valid__mono,axiom,
    ! [A: a,B: a] :
      ( ( ( valid @ A )
        & ( pre_larger_a @ plus @ A @ B ) )
     => ( valid @ B ) ) ).

% valid_mono
thf(fact_61_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_62_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_63_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_64_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_65_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_66_sadd__comm,axiom,
    ! [P: b,Q: b] :
      ( ( sadd @ P @ Q )
      = ( sadd @ Q @ P ) ) ).

% sadd_comm
thf(fact_67_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_68_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_69_sum__both__larger,axiom,
    ! [X4: a,A5: a,B5: a,X2: a,A: a,B: a] :
      ( ( ( some_a @ X4 )
        = ( plus @ A5 @ B5 ) )
     => ( ( ( some_a @ X2 )
          = ( plus @ A @ B ) )
       => ( ( pre_larger_a @ plus @ A5 @ A )
         => ( ( pre_larger_a @ plus @ B5 @ B )
           => ( pre_larger_a @ plus @ X4 @ X2 ) ) ) ) ) ).

% sum_both_larger
thf(fact_70_larger__first__sum,axiom,
    ! [Y: a,A: a,B: a,X2: a] :
      ( ( ( some_a @ Y )
        = ( plus @ A @ B ) )
     => ( ( pre_larger_a @ plus @ X2 @ Y )
       => ? [A6: a] :
            ( ( ( some_a @ X2 )
              = ( plus @ A6 @ B ) )
            & ( pre_larger_a @ plus @ A6 @ A ) ) ) ) ).

% larger_first_sum
thf(fact_71_larger__implies__compatible,axiom,
    ! [X2: a,Y: a] :
      ( ( pre_larger_a @ plus @ X2 @ Y )
     => ( pre_compatible_a @ plus @ X2 @ Y ) ) ).

% larger_implies_compatible
thf(fact_72_compatible__smaller,axiom,
    ! [A: a,B: a,X2: a] :
      ( ( pre_larger_a @ plus @ A @ B )
     => ( ( pre_compatible_a @ plus @ X2 @ A )
       => ( pre_compatible_a @ plus @ X2 @ B ) ) ) ).

% compatible_smaller
thf(fact_73_intuitionistic__def,axiom,
    ! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S @ Delta @ A2 )
      = ( ! [A3: a,B4: a] :
            ( ( ( pre_larger_a @ plus @ A3 @ B4 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ B4 @ S @ Delta @ A2 ) )
           => ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 ) ) ) ) ).

% intuitionistic_def
thf(fact_74_intuitionisticI,axiom,
    ! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [A4: a,B3: a] :
          ( ( ( pre_larger_a @ plus @ A4 @ B3 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ B3 @ S @ Delta @ A2 ) )
         => ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 ) )
     => ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S @ Delta @ A2 ) ) ).

% intuitionisticI
thf(fact_75_logic_Ocan__factorize,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Q: b,P: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ? [R: b] :
          ( Q
          = ( Smult @ R @ P ) ) ) ).

% logic.can_factorize
thf(fact_76_logic_Odot__mult2,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) ) ) ).

% logic.dot_mult2
thf(fact_77_logic_Odot__mult1,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) ) ) ).

% logic.dot_mult1
thf(fact_78_logic_ODotPos,axiom,
    ! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
        = ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ) ).

% logic.DotPos
thf(fact_79_logic_ODotDot,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) ) ) ).

% logic.DotDot
thf(fact_80_logic_Odot__star1,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.dot_star1
thf(fact_81_logic_Odot__star2,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.dot_star2
thf(fact_82_logic_ODotWild,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).

% logic.DotWild
thf(fact_83_logic_OWildDot,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).

% logic.WildDot
thf(fact_84_logic_ODotStar,axiom,
    ! [Plus: 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,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).

% logic.DotStar
thf(fact_85_logic_Onot__in__fv__def,axiom,
    ! [Plus: 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_c_d,S4: set_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ S4 )
        = ( ! [Sigma2: a,S2: d > c,Delta2: ( d > c ) > set_a,S5: d > c] :
              ( ( equal_outside_d_c @ S2 @ S5 @ S4 )
             => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta2 @ A2 )
                = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S5 @ Delta2 @ A2 ) ) ) ) ) ) ).

% logic.not_in_fv_def
thf(fact_86_logic__def,axiom,
    ( logic_a_b
    = ( ^ [Plus2: 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,B4: a] :
              ( ( Plus2 @ A3 @ B4 )
              = ( Plus2 @ B4 @ A3 ) )
          & ! [A3: a,B4: a,Ab2: a,C2: a,Bc2: a] :
              ( ( ( ( Plus2 @ A3 @ B4 )
                  = ( some_a @ Ab2 ) )
                & ( ( Plus2 @ B4 @ C2 )
                  = ( some_a @ Bc2 ) ) )
             => ( ( Plus2 @ Ab2 @ C2 )
                = ( Plus2 @ A3 @ Bc2 ) ) )
          & ! [A3: a,B4: a,Ab2: a,C2: a] :
              ( ( ( ( Plus2 @ A3 @ B4 )
                  = ( some_a @ Ab2 ) )
                & ~ ( pre_compatible_a @ Plus2 @ B4 @ C2 ) )
             => ~ ( pre_compatible_a @ Plus2 @ 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,R3: b] :
              ( ( Smult2 @ P3 @ ( Sadd2 @ Q2 @ R3 ) )
              = ( Sadd2 @ ( Smult2 @ P3 @ Q2 ) @ ( Smult2 @ P3 @ R3 ) ) )
          & ! [P3: b,Q2: b,R3: b] :
              ( ( Smult2 @ ( Smult2 @ P3 @ Q2 ) @ R3 )
              = ( Smult2 @ P3 @ ( Smult2 @ Q2 @ R3 ) ) )
          & ! [P3: b,Q2: b,A3: a] :
              ( ( Mult2 @ P3 @ ( Mult2 @ Q2 @ A3 ) )
              = ( Mult2 @ ( Smult2 @ P3 @ Q2 ) @ A3 ) )
          & ! [A3: a,B4: a,C2: a,P3: b] :
              ( ( ( some_a @ A3 )
                = ( Plus2 @ B4 @ C2 ) )
             => ( ( some_a @ ( Mult2 @ P3 @ A3 ) )
                = ( Plus2 @ ( Mult2 @ P3 @ B4 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
          & ! [P3: b,Q2: b,X3: a] :
              ( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q2 ) @ X3 ) )
              = ( Plus2 @ ( Mult2 @ P3 @ X3 ) @ ( Mult2 @ Q2 @ X3 ) ) )
          & ! [A3: a] :
              ( ( Mult2 @ One2 @ A3 )
              = A3 )
          & ! [A3: a,B4: a] :
              ( ( ( Valid2 @ A3 )
                & ( pre_larger_a @ Plus2 @ A3 @ B4 ) )
             => ( Valid2 @ B4 ) ) ) ) ) ).

% logic_def
thf(fact_87_logic_Osat_Osimps_I3_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
        = ( ! [A3: a,Sigma3: a] :
              ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 )
                & ( ( some_a @ Sigma3 )
                  = ( Plus @ Sigma @ A3 ) ) )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S @ Delta @ B2 ) ) ) ) ) ).

% logic.sat.simps(3)
thf(fact_88_logic_Osat__wand,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a,Sigma5: a] :
            ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 )
              & ( ( some_a @ Sigma5 )
                = ( Plus @ Sigma @ A4 ) ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S @ Delta @ B2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.sat_wand
thf(fact_89_logic_Osat_Osimps_I2_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
        = ( ? [A3: a,B4: a] :
              ( ( ( some_a @ Sigma )
                = ( Plus @ A3 @ B4 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B4 @ S @ Delta @ B2 ) ) ) ) ) ).

% logic.sat.simps(2)
thf(fact_90_logic_Opure__def,axiom,
    ! [Plus: 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_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pure_a_b_c_d @ Plus @ Mult @ Valid @ A2 )
        = ( ! [Sigma2: a,Sigma3: a,S2: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta2 @ A2 )
              = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S2 @ Delta3 @ A2 ) ) ) ) ) ).

% logic.pure_def
thf(fact_91_logic_Ocompatible__smaller,axiom,
    ! [Plus: 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,X2: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ A @ B )
       => ( ( pre_compatible_a @ Plus @ X2 @ A )
         => ( pre_compatible_a @ Plus @ X2 @ B ) ) ) ) ).

% logic.compatible_smaller
thf(fact_92_logic_Olarger__implies__compatible,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: a,Y: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ X2 @ Y )
       => ( pre_compatible_a @ Plus @ X2 @ Y ) ) ) ).

% logic.larger_implies_compatible
thf(fact_93_assertion_Oinject_I2_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,Y21: b,Y22: assertion_a_b_c_d] :
      ( ( ( mult_b_a_c_d @ X21 @ X222 )
        = ( mult_b_a_c_d @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% assertion.inject(2)
thf(fact_94_assertion_Oinject_I4_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,Y41: assertion_a_b_c_d,Y42: assertion_a_b_c_d] :
      ( ( ( wand_a_b_c_d @ X41 @ X42 )
        = ( wand_a_b_c_d @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% assertion.inject(4)
thf(fact_95_logic_Osat_Ocong,axiom,
    sat_a_b_d_c = sat_a_b_d_c ).

% logic.sat.cong
thf(fact_96_logic_Osmult__distrib,axiom,
    ! [Plus: 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,R2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sadd @ Q @ R2 ) )
        = ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R2 ) ) ) ) ).

% logic.smult_distrib
thf(fact_97_logic_Osone__neutral,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ One @ P )
        = P ) ) ).

% logic.sone_neutral
thf(fact_98_logic_Osinv__inverse,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ ( Sinv @ P ) )
        = One ) ) ).

% logic.sinv_inverse
thf(fact_99_logic_Oone__neutral,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ One @ A )
        = A ) ) ).

% logic.one_neutral
thf(fact_100_logic_Odouble__mult,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Mult @ P @ ( Mult @ Q @ A ) )
        = ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).

% logic.double_mult
thf(fact_101_logic_Ocommutative,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Plus @ A @ B )
        = ( Plus @ B @ A ) ) ) ).

% logic.commutative
thf(fact_102_logic_Ounique__inv,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( A
          = ( Mult @ P @ B ) )
        = ( B
          = ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).

% logic.unique_inv
thf(fact_103_logic_Osmult__comm,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ P @ Q )
        = ( Smult @ Q @ P ) ) ) ).

% logic.smult_comm
thf(fact_104_logic_Osmult__asso,axiom,
    ! [Plus: 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,R2: b] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Smult @ ( Smult @ P @ Q ) @ R2 )
        = ( Smult @ P @ ( Smult @ Q @ R2 ) ) ) ) ).

% logic.smult_asso
thf(fact_105_logic_Ocan__divide,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Mult @ P @ A )
          = ( Mult @ P @ B ) )
       => ( A = B ) ) ) ).

% logic.can_divide
thf(fact_106_logic_Osadd__comm,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( Sadd @ P @ Q )
        = ( Sadd @ Q @ P ) ) ) ).

% logic.sadd_comm
thf(fact_107_pre__logic_Ocompatible_Ocong,axiom,
    pre_compatible_a = pre_compatible_a ).

% pre_logic.compatible.cong
thf(fact_108_pre__logic_Olarger_Ocong,axiom,
    pre_larger_a = pre_larger_a ).

% pre_logic.larger.cong
thf(fact_109_logic_OintuitionisticI,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a,B3: a] :
            ( ( ( pre_larger_a @ Plus @ A4 @ B3 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B3 @ S @ Delta @ A2 ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 ) )
       => ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S @ Delta @ A2 ) ) ) ).

% logic.intuitionisticI
thf(fact_110_logic_Ointuitionistic__def,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S @ Delta @ A2 )
        = ( ! [A3: a,B4: a] :
              ( ( ( pre_larger_a @ Plus @ A3 @ B4 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B4 @ S @ Delta @ A2 ) )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 ) ) ) ) ) ).

% logic.intuitionistic_def
thf(fact_111_logic_Opure_Ocong,axiom,
    pure_a_b_c_d = pure_a_b_c_d ).

% logic.pure.cong
thf(fact_112_logic_Odistrib__mult,axiom,
    ! [Plus: 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,X2: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X2 ) )
        = ( Plus @ ( Mult @ P @ X2 ) @ ( Mult @ Q @ X2 ) ) ) ) ).

% logic.distrib_mult
thf(fact_113_logic_Oplus__mult,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus @ B @ C ) )
       => ( ( some_a @ ( Mult @ P @ A ) )
          = ( Plus @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).

% logic.plus_mult
thf(fact_114_logic_Omove__sum,axiom,
    ! [Plus: 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,X2: a,X1: a,X22: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ A )
          = ( Plus @ A1 @ A22 ) )
       => ( ( ( some_a @ B )
            = ( Plus @ B1 @ B22 ) )
         => ( ( ( some_a @ X2 )
              = ( Plus @ A @ B ) )
           => ( ( ( some_a @ X1 )
                = ( Plus @ A1 @ B1 ) )
             => ( ( ( some_a @ X22 )
                  = ( Plus @ A22 @ B22 ) )
               => ( ( some_a @ X2 )
                  = ( Plus @ X1 @ X22 ) ) ) ) ) ) ) ) ).

% logic.move_sum
thf(fact_115_logic_Oasso1,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus @ A @ B )
            = ( some_a @ Ab ) )
          & ( ( Plus @ B @ C )
            = ( some_a @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A @ Bc ) ) ) ) ).

% logic.asso1
thf(fact_116_assertion_Odistinct_I41_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(41)
thf(fact_117_assertion_Odistinct_I23_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( star_a_b_c_d @ X31 @ X32 ) ) ).

% assertion.distinct(23)
thf(fact_118_assertion_Odistinct_I25_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(25)
thf(fact_119_assertion_Odistinct_I39_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(39)
thf(fact_120_assertion_Odistinct_I31_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(31)
thf(fact_121_assertion_Odistinct_I29_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(29)
thf(fact_122_assertion_Odistinct_I27_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(27)
thf(fact_123_assertion_Odistinct_I1_J,axiom,
    ! [X1: ( d > c ) > a > $o,X21: b,X222: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( mult_b_a_c_d @ X21 @ X222 ) ) ).

% assertion.distinct(1)
thf(fact_124_pre__logic_Olarger__def,axiom,
    ( pre_larger_a
    = ( ^ [Plus2: a > a > option_a,A3: a,B4: a] :
        ? [C2: a] :
          ( ( some_a @ A3 )
          = ( Plus2 @ B4 @ C2 ) ) ) ) ).

% pre_logic.larger_def
thf(fact_125_assertion_Odistinct_I75_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( wildcard_a_b_c_d @ X12 ) ) ).

% assertion.distinct(75)
thf(fact_126_assertion_Odistinct_I37_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != pred_a_b_c_d ) ).

% assertion.distinct(37)
thf(fact_127_assertion_Odistinct_I43_J,axiom,
    ! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( star_a_b_c_d @ X31 @ X32 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(43)
thf(fact_128_assertion_Odistinct_I73_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( bounded_a_b_c_d @ X11 ) ) ).

% assertion.distinct(73)
thf(fact_129_assertion_Odistinct_I65_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( imp_a_b_c_d @ X71 @ X72 ) ) ).

% assertion.distinct(65)
thf(fact_130_assertion_Odistinct_I61_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( or_a_b_c_d @ X51 @ X52 ) ) ).

% assertion.distinct(61)
thf(fact_131_assertion_Odistinct_I63_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( and_a_b_c_d @ X61 @ X62 ) ) ).

% assertion.distinct(63)
thf(fact_132_assertion_Odistinct_I5_J,axiom,
    ! [X1: ( d > c ) > a > $o,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( sem_d_c_a_b @ X1 )
     != ( wand_a_b_c_d @ X41 @ X42 ) ) ).

% assertion.distinct(5)
thf(fact_133_assertion_Odistinct_I71_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != pred_a_b_c_d ) ).

% assertion.distinct(71)
thf(fact_134_logic_Ocompatible__multiples,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
       => ( pre_compatible_a @ Plus @ A @ B ) ) ) ).

% logic.compatible_multiples
thf(fact_135_logic_Ocompatible__imp,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ A @ B )
       => ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_imp
thf(fact_136_logic_Ocompatible__iff,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_compatible_a @ Plus @ A @ B )
        = ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.compatible_iff
thf(fact_137_logic_Olarger__same,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( pre_larger_a @ Plus @ A @ B )
        = ( pre_larger_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).

% logic.larger_same
thf(fact_138_logic_Ovalid__mono,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( Valid @ A )
          & ( pre_larger_a @ Plus @ A @ B ) )
       => ( Valid @ B ) ) ) ).

% logic.valid_mono
thf(fact_139_logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
        = ( ? [A3: a] :
              ( ( Sigma
                = ( Mult @ P @ A3 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 ) ) ) ) ) ).

% logic.sat.simps(1)
thf(fact_140_logic_Osat__mult,axiom,
    ! [Plus: 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,P: b,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [A4: a] :
            ( ( Sigma
              = ( Mult @ P @ A4 ) )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ) ).

% logic.sat_mult
thf(fact_141_logic_Osat_Osimps_I12_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
        = ( ? [A3: a,P3: b] :
              ( ( Sigma
                = ( Mult @ P3 @ A3 ) )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 ) ) ) ) ) ).

% logic.sat.simps(12)
thf(fact_142_logic_Osat_Osimps_I11_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( bounded_a_b_c_d @ A2 ) )
        = ( ( Valid @ Sigma )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(11)
thf(fact_143_logic_Osat__imp,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ) ).

% logic.sat_imp
thf(fact_144_logic_Osat_Osimps_I5_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
         => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(5)
thf(fact_145_logic_Osat_Osimps_I6_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
          | ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(6)
thf(fact_146_logic_Osat_Osimps_I7_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
        = ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
          & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).

% logic.sat.simps(7)
thf(fact_147_logic_Osat_Osimps_I4_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( sem_d_c_a_b @ B ) )
        = ( B @ S @ Sigma ) ) ) ).

% logic.sat.simps(4)
thf(fact_148_logic_Osat_Osimps_I10_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ pred_a_b_c_d )
        = ( member_a @ Sigma @ ( Delta @ S ) ) ) ) ).

% logic.sat.simps(10)
thf(fact_149_logic_Oasso2,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( ( Plus @ A @ B )
            = ( some_a @ Ab ) )
          & ~ ( pre_compatible_a @ Plus @ B @ C ) )
       => ~ ( pre_compatible_a @ Plus @ Ab @ C ) ) ) ).

% logic.asso2
thf(fact_150_logic_Oasso3,axiom,
    ! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ~ ( pre_compatible_a @ Plus @ A @ B )
       => ( ( ( Plus @ B @ C )
            = ( some_a @ Bc ) )
         => ~ ( pre_compatible_a @ Plus @ A @ Bc ) ) ) ) ).

% logic.asso3
thf(fact_151_logic_Osum__both__larger,axiom,
    ! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X4: a,A5: a,B5: a,X2: a,A: a,B: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ X4 )
          = ( Plus @ A5 @ B5 ) )
       => ( ( ( some_a @ X2 )
            = ( Plus @ A @ B ) )
         => ( ( pre_larger_a @ Plus @ A5 @ A )
           => ( ( pre_larger_a @ Plus @ B5 @ B )
             => ( pre_larger_a @ Plus @ X4 @ X2 ) ) ) ) ) ) ).

% logic.sum_both_larger
thf(fact_152_logic_Olarger__first__sum,axiom,
    ! [Plus: 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,X2: a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( ( some_a @ Y )
          = ( Plus @ A @ B ) )
       => ( ( pre_larger_a @ Plus @ X2 @ Y )
         => ? [A6: a] :
              ( ( ( some_a @ X2 )
                = ( Plus @ A6 @ B ) )
              & ( pre_larger_a @ Plus @ A6 @ A ) ) ) ) ) ).

% logic.larger_first_sum
thf(fact_153_logic_Oentails__def,axiom,
    ! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
        = ( ! [Sigma2: a,S2: d > c] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ A2 )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ B2 ) ) ) ) ) ).

% logic.entails_def
thf(fact_154_logic_OentailsI,axiom,
    ! [Plus: 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: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S3: d > c] :
            ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ A2 )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ B2 ) )
       => ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ).

% logic.entailsI
thf(fact_155_logic_OequivalentI,axiom,
    ! [Plus: 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: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S3: d > c] :
            ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ A2 )
           => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ B2 ) )
       => ( ! [Sigma4: a,S3: d > c] :
              ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ B2 )
             => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ A2 ) )
         => ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ) ).

% logic.equivalentI
thf(fact_156_option_Oinject,axiom,
    ! [X22: a,Y2: a] :
      ( ( ( some_a @ X22 )
        = ( some_a @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_157_compatible__def,axiom,
    ! [A: a,B: a] :
      ( ( pre_compatible_a @ plus @ A @ B )
      = ( ( plus @ A @ B )
       != none_a ) ) ).

% compatible_def
thf(fact_158_assertion_Oexhaust,axiom,
    ! [Y: assertion_a_b_c_d] :
      ( ! [X13: ( d > c ) > a > $o] :
          ( Y
         != ( sem_d_c_a_b @ X13 ) )
     => ( ! [X212: b,X223: assertion_a_b_c_d] :
            ( Y
           != ( mult_b_a_c_d @ X212 @ X223 ) )
       => ( ! [X312: assertion_a_b_c_d,X322: assertion_a_b_c_d] :
              ( Y
             != ( star_a_b_c_d @ X312 @ X322 ) )
         => ( ! [X412: assertion_a_b_c_d,X422: assertion_a_b_c_d] :
                ( Y
               != ( wand_a_b_c_d @ X412 @ X422 ) )
           => ( ! [X512: assertion_a_b_c_d,X522: assertion_a_b_c_d] :
                  ( Y
                 != ( or_a_b_c_d @ X512 @ X522 ) )
             => ( ! [X612: assertion_a_b_c_d,X622: assertion_a_b_c_d] :
                    ( Y
                   != ( and_a_b_c_d @ X612 @ X622 ) )
               => ( ! [X712: assertion_a_b_c_d,X722: assertion_a_b_c_d] :
                      ( Y
                     != ( imp_a_b_c_d @ X712 @ X722 ) )
                 => ( ! [X81: d,X82: assertion_a_b_c_d] :
                        ( Y
                       != ( exists_d_a_b_c @ X81 @ X82 ) )
                   => ( ! [X91: d,X92: assertion_a_b_c_d] :
                          ( Y
                         != ( forall_d_a_b_c @ X91 @ X92 ) )
                     => ( ( Y != pred_a_b_c_d )
                       => ( ! [X112: assertion_a_b_c_d] :
                              ( Y
                             != ( bounded_a_b_c_d @ X112 ) )
                         => ~ ! [X122: assertion_a_b_c_d] :
                                ( Y
                               != ( wildcard_a_b_c_d @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% assertion.exhaust
thf(fact_159_sat_Osimps_I8_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,X2: d,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( exists_d_a_b_c @ X2 @ A2 ) )
      = ( ? [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V ) @ Delta @ A2 ) ) ) ).

% sat.simps(8)
thf(fact_160_sat_Osimps_I9_J,axiom,
    ! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,X2: d,A2: assertion_a_b_c_d] :
      ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X2 @ A2 ) )
      = ( ! [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V ) @ Delta @ A2 ) ) ) ).

% sat.simps(9)
thf(fact_161_sat__forall,axiom,
    ! [Sigma: a,S: d > c,X2: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ! [V2: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V2 ) @ Delta @ A2 )
     => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X2 @ A2 ) ) ) ).

% sat_forall
thf(fact_162_not__in__fv__mod,axiom,
    ! [A2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S: d > c,Sigma6: a,S6: d > c,X2: a,Delta: ( d > c ) > set_a] :
      ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ ( modified_a_d_c @ C ) )
     => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S6 ) ) ) @ C )
       => ( ( sat_a_b_d_c @ plus @ mult @ valid @ X2 @ S @ Delta @ A2 )
          = ( sat_a_b_d_c @ plus @ mult @ valid @ X2 @ S6 @ Delta @ A2 ) ) ) ) ).

% not_in_fv_mod
thf(fact_163_not__None__eq,axiom,
    ! [X2: option_a] :
      ( ( X2 != none_a )
      = ( ? [Y3: a] :
            ( X2
            = ( some_a @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_164_not__Some__eq,axiom,
    ! [X2: option_a] :
      ( ( ! [Y3: a] :
            ( X2
           != ( some_a @ Y3 ) ) )
      = ( X2 = none_a ) ) ).

% not_Some_eq
thf(fact_165_option_Odistinct_I1_J,axiom,
    ! [X22: a] :
      ( none_a
     != ( some_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_166_option_OdiscI,axiom,
    ! [Option: option_a,X22: a] :
      ( ( Option
        = ( some_a @ X22 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_167_option_Oexhaust,axiom,
    ! [Y: option_a] :
      ( ( Y != none_a )
     => ~ ! [X23: a] :
            ( Y
           != ( some_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_168_split__option__ex,axiom,
    ( ( ^ [P4: option_a > $o] :
        ? [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          | ? [X3: a] : ( P5 @ ( some_a @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_169_split__option__all,axiom,
    ( ( ^ [P4: option_a > $o] :
        ! [X5: option_a] : ( P4 @ X5 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          & ! [X3: a] : ( P5 @ ( some_a @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_170_combine__options__cases,axiom,
    ! [X2: option_a,P2: option_a > option_a > $o,Y: option_a] :
      ( ( ( X2 = none_a )
       => ( P2 @ X2 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P2 @ X2 @ Y ) )
       => ( ! [A4: a,B3: a] :
              ( ( X2
                = ( some_a @ A4 ) )
             => ( ( Y
                  = ( some_a @ B3 ) )
               => ( P2 @ X2 @ Y ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_171_logic_Osat__forall,axiom,
    ! [Plus: 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: d > c,X2: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [V2: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V2 ) @ Delta @ A2 )
       => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X2 @ A2 ) ) ) ) ).

% logic.sat_forall
thf(fact_172_logic_Osat_Osimps_I9_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,X2: d,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X2 @ A2 ) )
        = ( ! [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(9)
thf(fact_173_logic_Osat_Osimps_I8_J,axiom,
    ! [Plus: 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: d > c,Delta: ( d > c ) > set_a,X2: d,A2: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( exists_d_a_b_c @ X2 @ A2 ) )
        = ( ? [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).

% logic.sat.simps(8)
thf(fact_174_assertion_Odistinct_I35_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(35)
thf(fact_175_assertion_Odistinct_I33_J,axiom,
    ! [X21: b,X222: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( mult_b_a_c_d @ X21 @ X222 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(33)
thf(fact_176_pre__logic_Ocompatible__def,axiom,
    ( pre_compatible_a
    = ( ^ [Plus2: a > a > option_a,A3: a,B4: a] :
          ( ( Plus2 @ A3 @ B4 )
         != none_a ) ) ) ).

% pre_logic.compatible_def
thf(fact_177_assertion_Odistinct_I69_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( forall_d_a_b_c @ X912 @ X922 ) ) ).

% assertion.distinct(69)
thf(fact_178_assertion_Odistinct_I67_J,axiom,
    ! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
      ( ( wand_a_b_c_d @ X41 @ X42 )
     != ( exists_d_a_b_c @ X812 @ X822 ) ) ).

% assertion.distinct(67)
thf(fact_179_logic_Onot__in__fv__mod,axiom,
    ! [Plus: 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_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S: d > c,Sigma6: a,S6: d > c,X2: a,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ ( modified_a_d_c @ C ) )
       => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S6 ) ) ) @ C )
         => ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X2 @ S @ Delta @ A2 )
            = ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X2 @ S6 @ Delta @ A2 ) ) ) ) ) ).

% logic.not_in_fv_mod
thf(fact_180_valid__hoare__triple__def,axiom,
    ! [P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q3: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q3 @ Delta )
      = ( ! [Sigma2: a,S2: d > c] :
            ( ( ( valid @ Sigma2 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ P2 ) )
           => ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) )
              & ! [Sigma3: a,S5: d > c] :
                  ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S5 ) ) ) @ C )
                 => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S5 @ Delta @ Q3 ) ) ) ) ) ) ).

% valid_hoare_triple_def
thf(fact_181_valid__hoare__tripleI,axiom,
    ! [Delta: ( d > c ) > set_a,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q3: assertion_a_b_c_d] :
      ( ! [Sigma4: a,S3: d > c] :
          ( ( ( valid @ Sigma4 )
            & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ P2 ) )
         => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S3 ) ) )
     => ( ! [Sigma4: a,S3: d > c,Sigma5: a,S7: d > c] :
            ( ( ( valid @ Sigma4 )
              & ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S3 @ Delta @ P2 ) )
           => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S3 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S7 ) ) ) @ C )
             => ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S7 @ Delta @ Q3 ) ) )
       => ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P2 @ C @ Q3 @ Delta ) ) ) ).

% valid_hoare_tripleI
thf(fact_182_logic_Ovalid__hoare__tripleI,axiom,
    ! [Plus: 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: ( d > c ) > set_a,P2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q3: assertion_a_b_c_d] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ! [Sigma4: a,S3: d > c] :
            ( ( ( Valid @ Sigma4 )
              & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ P2 ) )
           => ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S3 ) ) )
       => ( ! [Sigma4: a,S3: d > c,Sigma5: a,S7: d > c] :
              ( ( ( Valid @ Sigma4 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S3 @ Delta @ P2 ) )
             => ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S3 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S7 ) ) ) @ C )
               => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S7 @ Delta @ Q3 ) ) )
         => ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q3 @ Delta ) ) ) ) ).

% logic.valid_hoare_tripleI
thf(fact_183_logic_Ovalid__hoare__triple__def,axiom,
    ! [Plus: 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_c_d,C: set_Pr7868159745199425715_a_d_c,Q3: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
      ( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
     => ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P2 @ C @ Q3 @ Delta )
        = ( ! [Sigma2: a,S2: d > c] :
              ( ( ( Valid @ Sigma2 )
                & ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ P2 ) )
             => ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) )
                & ! [Sigma3: a,S5: d > c] :
                    ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S5 ) ) ) @ C )
                   => ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S5 @ Delta @ Q3 ) ) ) ) ) ) ) ).

% logic.valid_hoare_triple_def
thf(fact_184_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A5: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_185_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A5: set_Product_prod_o_o,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_186_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_187_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Product_prod_o_o,Y1: set_Product_prod_o_o,Y2: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ X1 @ X22 )
        = ( produc763777882069021527od_o_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_188_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat,Y1: set_Product_prod_o_o,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ X1 @ X22 )
        = ( produc8517790099723286449at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_189_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X22 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_190_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2934264451710624999od_o_o] :
      ~ ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
          ( Y
         != ( produc763777882069021527od_o_o @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_191_old_Oprod_Oexhaust,axiom,
    ! [Y: produc732395585841259969at_nat] :
      ~ ! [A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc8517790099723286449at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_192_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_193_surj__pair,axiom,
    ! [P: produc2934264451710624999od_o_o] :
    ? [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
      ( P
      = ( produc763777882069021527od_o_o @ X @ Y4 ) ) ).

% surj_pair
thf(fact_194_surj__pair,axiom,
    ! [P: produc732395585841259969at_nat] :
    ? [X: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc8517790099723286449at_nat @ X @ Y4 ) ) ).

% surj_pair
thf(fact_195_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc2922128104949294807at_nat @ X @ Y4 ) ) ).

% surj_pair
thf(fact_196_prod__cases,axiom,
    ! [P2: produc2934264451710624999od_o_o > $o,P: produc2934264451710624999od_o_o] :
      ( ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] : ( P2 @ ( produc763777882069021527od_o_o @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_197_prod__cases,axiom,
    ! [P2: produc732395585841259969at_nat > $o,P: produc732395585841259969at_nat] :
      ( ! [A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc8517790099723286449at_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_198_prod__cases,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_199_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A5: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_200_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A5: set_Product_prod_o_o,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_201_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_202_ssubst__Pair__rhs,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o,S6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ S ) @ R4 )
     => ( ( S6 = S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ S6 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_203_ssubst__Pair__rhs,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat,R4: set_Pr457366540195662369at_nat,S6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ S ) @ R4 )
     => ( ( S6 = S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ S6 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_204_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat,S6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ R4 )
     => ( ( S6 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S6 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_205_option_Ocollapse,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( ( some_a @ ( the_a @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_domD,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr3509670420972068777_o_o_a @ M ) )
     => ? [B3: a] :
          ( ( M @ A )
          = ( some_a @ B3 ) ) ) ).

% domD
thf(fact_216_domD,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr8042991038213739471_nat_a @ M ) )
     => ? [B3: a] :
          ( ( M @ A )
          = ( some_a @ B3 ) ) ) ).

% domD
thf(fact_217_domD,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr1957410074072848681_nat_a @ M ) )
     => ? [B3: a] :
          ( ( M @ A )
          = ( some_a @ B3 ) ) ) ).

% domD
thf(fact_218_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_219_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_220_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_221_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_222_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_223_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_224_option_Osel,axiom,
    ! [X22: a] :
      ( ( the_a @ ( some_a @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_option_Oexhaust__sel,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( Option
        = ( some_a @ ( the_a @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_233_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_234_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_235_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_236_option_Oset__sel,axiom,
    ! [A: option_a] :
      ( ( A != none_a )
     => ( member_a @ ( the_a @ A ) @ ( set_option_a2 @ A ) ) ) ).

% option.set_sel
thf(fact_237_bind__eq__None__conv,axiom,
    ! [A: option_a,F: a > option_a] :
      ( ( ( bind_a_a @ A @ F )
        = none_a )
      = ( ( A = none_a )
        | ( ( F @ ( the_a @ A ) )
          = none_a ) ) ) ).

% bind_eq_None_conv
thf(fact_238_elem__set,axiom,
    ! [X2: produc2934264451710624999od_o_o,Xo: option8293164206536526765od_o_o] :
      ( ( member444158400953824016od_o_o @ X2 @ ( set_op3337521775309186165od_o_o @ Xo ) )
      = ( Xo
        = ( some_P7068320699308846284od_o_o @ X2 ) ) ) ).

% elem_set
thf(fact_239_elem__set,axiom,
    ! [X2: produc732395585841259969at_nat,Xo: option7754883786268931591at_nat] :
      ( ( member4095101504841534314at_nat @ X2 @ ( set_op8338180575893423695at_nat @ Xo ) )
      = ( Xo
        = ( some_P123625206269196070at_nat @ X2 ) ) ) ).

% elem_set
thf(fact_240_elem__set,axiom,
    ! [X2: produc3843707927480180839at_nat,Xo: option8699557107911914541at_nat] :
      ( ( member8757157785044589968at_nat @ X2 @ ( set_op5866586666149159413at_nat @ Xo ) )
      = ( Xo
        = ( some_P131194445498448716at_nat @ X2 ) ) ) ).

% elem_set
thf(fact_241_elem__set,axiom,
    ! [X2: a,Xo: option_a] :
      ( ( member_a @ X2 @ ( set_option_a2 @ Xo ) )
      = ( Xo
        = ( some_a @ X2 ) ) ) ).

% elem_set
thf(fact_242_bind__runit,axiom,
    ! [X2: option_a] :
      ( ( bind_a_a @ X2 @ some_a )
      = X2 ) ).

% bind_runit
thf(fact_243_set__empty__eq,axiom,
    ! [Xo: option_a] :
      ( ( ( set_option_a2 @ Xo )
        = bot_bot_set_a )
      = ( Xo = none_a ) ) ).

% set_empty_eq
thf(fact_244_option_Osimps_I14_J,axiom,
    ( ( set_option_a2 @ none_a )
    = bot_bot_set_a ) ).

% option.simps(14)
thf(fact_245_bind__eq__Some__conv,axiom,
    ! [F: option_a,G: a > option_a,X2: a] :
      ( ( ( bind_a_a @ F @ G )
        = ( some_a @ X2 ) )
      = ( ? [Y3: a] :
            ( ( F
              = ( some_a @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_a @ X2 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_246_bind_Obind__lzero,axiom,
    ! [F: a > option_a] :
      ( ( bind_a_a @ none_a @ F )
      = none_a ) ).

% bind.bind_lzero
thf(fact_247_ospec,axiom,
    ! [A2: option_a,P2: a > $o,X2: a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_option_a2 @ A2 ) )
         => ( P2 @ X ) )
     => ( ( A2
          = ( some_a @ X2 ) )
       => ( P2 @ X2 ) ) ) ).

% ospec
thf(fact_248_option_Oset__intros,axiom,
    ! [X22: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X22 @ ( set_op3337521775309186165od_o_o @ ( some_P7068320699308846284od_o_o @ X22 ) ) ) ).

% option.set_intros
thf(fact_249_option_Oset__intros,axiom,
    ! [X22: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X22 @ ( set_op8338180575893423695at_nat @ ( some_P123625206269196070at_nat @ X22 ) ) ) ).

% option.set_intros
thf(fact_250_option_Oset__intros,axiom,
    ! [X22: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X22 @ ( set_op5866586666149159413at_nat @ ( some_P131194445498448716at_nat @ X22 ) ) ) ).

% option.set_intros
thf(fact_251_option_Oset__intros,axiom,
    ! [X22: a] : ( member_a @ X22 @ ( set_option_a2 @ ( some_a @ X22 ) ) ) ).

% option.set_intros
thf(fact_252_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_253_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_254_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_255_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_256_bind__split__asm,axiom,
    ! [P2: option_a > $o,M: option_a,F: a > option_a] :
      ( ( P2 @ ( bind_a_a @ M @ F ) )
      = ( ~ ( ( ( M = none_a )
              & ~ ( P2 @ none_a ) )
            | ? [X3: a] :
                ( ( M
                  = ( some_a @ X3 ) )
                & ~ ( P2 @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_257_bind__split,axiom,
    ! [P2: option_a > $o,M: option_a,F: a > option_a] :
      ( ( P2 @ ( bind_a_a @ M @ F ) )
      = ( ( ( M = none_a )
         => ( P2 @ none_a ) )
        & ! [V: a] :
            ( ( M
              = ( some_a @ V ) )
           => ( P2 @ ( F @ V ) ) ) ) ) ).

% bind_split
thf(fact_258_option_Osimps_I15_J,axiom,
    ! [X22: a] :
      ( ( set_option_a2 @ ( some_a @ X22 ) )
      = ( insert_a @ X22 @ bot_bot_set_a ) ) ).

% option.simps(15)
thf(fact_259_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V3: set_Product_prod_o_o] :
      ( ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V3 ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V3 ) @ ( graph_5719763352399009470od_o_o @ M ) ) ) ).

% in_graphI
thf(fact_260_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V3: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V3 ) @ ( graph_2163970799110208408at_nat @ M ) ) ) ).

% in_graphI
thf(fact_261_in__graphI,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V3 ) @ ( graph_2380648924068268862at_nat @ M ) ) ) ).

% in_graphI
thf(fact_262_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V3: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V3 ) @ ( graph_5719763352399009470od_o_o @ M ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V3 ) ) ) ).

% in_graphD
thf(fact_263_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V3: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V3 ) @ ( graph_2163970799110208408at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) ) ) ).

% in_graphD
thf(fact_264_in__graphD,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V3 ) @ ( graph_2380648924068268862at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) ) ) ).

% in_graphD
thf(fact_265_restrict__out,axiom,
    ! [X2: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ~ ( member444158400953824016od_o_o @ X2 @ A2 )
     => ( ( restri4233077244345833607_o_o_a @ M @ A2 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_266_restrict__out,axiom,
    ! [X2: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ~ ( member4095101504841534314at_nat @ X2 @ A2 )
     => ( ( restri896928924502181293_nat_a @ M @ A2 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_267_restrict__out,axiom,
    ! [X2: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ~ ( member8757157785044589968at_nat @ X2 @ A2 )
     => ( ( restri4040463761431659527_nat_a @ M @ A2 @ X2 )
        = none_a ) ) ).

% restrict_out
thf(fact_268_is__none__code_I2_J,axiom,
    ! [X2: a] :
      ~ ( is_none_a @ ( some_a @ X2 ) ) ).

% is_none_code(2)
thf(fact_269_is__none__code_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_code(1)
thf(fact_270_these__insert__None,axiom,
    ! [A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
      = ( these_a @ A2 ) ) ).

% these_insert_None
thf(fact_271_these__insert__Some,axiom,
    ! [X2: a,A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ ( some_a @ X2 ) @ A2 ) )
      = ( insert_a @ X2 @ ( these_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_272_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V3: set_Product_prod_o_o] :
      ( ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ ( some_s1360601300120069868od_o_o @ V3 ) ) )
      = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ K @ V3 ) @ ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ none_s7246968996079588200od_o_o ) ) ) ) ).

% graph_map_upd
thf(fact_273_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V3: set_Pr1261947904930325089at_nat] :
      ( ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V3 ) ) )
      = ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ K @ V3 ) @ ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_274_graph__map__upd,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat] :
      ( ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V3 ) ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ K @ V3 ) @ ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_275_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V3: 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 @ V3 ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A2 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_276_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V3: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V3 ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A2 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_277_graph__restrictD_I1_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V3 ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A2 ) ) )
     => ( member2643936169264416010at_nat @ K @ A2 ) ) ).

% graph_restrictD(1)
thf(fact_278_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V3: 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 @ V3 ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_279_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V3: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V3 ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_280_graph__restrictD_I2_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V3 ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A2 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V3 ) ) ) ).

% graph_restrictD(2)
thf(fact_281_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_282_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_283_restrict__map__def,axiom,
    ( restri4233077244345833607_o_o_a
    = ( ^ [M2: produc2934264451710624999od_o_o > option_a,A7: set_Pr1932065953672099015od_o_o,X3: produc2934264451710624999od_o_o] : ( if_option_a @ ( member444158400953824016od_o_o @ X3 @ A7 ) @ ( M2 @ X3 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_284_restrict__map__def,axiom,
    ( restri896928924502181293_nat_a
    = ( ^ [M2: produc732395585841259969at_nat > option_a,A7: set_Pr457366540195662369at_nat,X3: produc732395585841259969at_nat] : ( if_option_a @ ( member4095101504841534314at_nat @ X3 @ A7 ) @ ( M2 @ X3 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_285_restrict__map__def,axiom,
    ( restri4040463761431659527_nat_a
    = ( ^ [M2: produc3843707927480180839at_nat > option_a,A7: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] : ( if_option_a @ ( member8757157785044589968at_nat @ X3 @ A7 ) @ ( M2 @ X3 ) @ none_a ) ) ) ).

% restrict_map_def
thf(fact_286_is__none__simps_I2_J,axiom,
    ! [X2: a] :
      ~ ( is_none_a @ ( some_a @ X2 ) ) ).

% is_none_simps(2)
thf(fact_287_is__none__simps_I1_J,axiom,
    is_none_a @ none_a ).

% is_none_simps(1)
thf(fact_288_Option_Ois__none__def,axiom,
    ( is_none_a
    = ( ^ [X3: option_a] : ( X3 = none_a ) ) ) ).

% Option.is_none_def
thf(fact_289_in__these__eq,axiom,
    ! [X2: produc2934264451710624999od_o_o,A2: set_op3605315081486705933od_o_o] :
      ( ( member444158400953824016od_o_o @ X2 @ ( these_2277756779245414180od_o_o @ A2 ) )
      = ( member3623239300863961174od_o_o @ ( some_P7068320699308846284od_o_o @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_290_in__these__eq,axiom,
    ! [X2: produc732395585841259969at_nat,A2: set_op7555165197759865831at_nat] :
      ( ( member4095101504841534314at_nat @ X2 @ ( these_663420305483081086at_nat @ A2 ) )
      = ( member8056002749114821168at_nat @ ( some_P123625206269196070at_nat @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_291_in__these__eq,axiom,
    ! [X2: produc3843707927480180839at_nat,A2: set_op989438649584373901at_nat] :
      ( ( member8757157785044589968at_nat @ X2 @ ( these_1872268174106123684at_nat @ A2 ) )
      = ( member4288423094740844502at_nat @ ( some_P131194445498448716at_nat @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_292_in__these__eq,axiom,
    ! [X2: a,A2: set_option_a] :
      ( ( member_a @ X2 @ ( these_a @ A2 ) )
      = ( member_option_a @ ( some_a @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_293_fun__upd__None__restrict,axiom,
    ! [X2: produc2934264451710624999od_o_o,D: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a] :
      ( ( ( member444158400953824016od_o_o @ X2 @ D )
       => ( ( fun_up7252388491999432090tion_a @ ( restri4233077244345833607_o_o_a @ M @ D ) @ X2 @ none_a )
          = ( restri4233077244345833607_o_o_a @ M @ ( minus_5704166846132356910od_o_o @ D @ ( insert3855699742481506999od_o_o @ X2 @ bot_bo498831808395348787od_o_o ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X2 @ D )
       => ( ( fun_up7252388491999432090tion_a @ ( restri4233077244345833607_o_o_a @ M @ D ) @ X2 @ none_a )
          = ( restri4233077244345833607_o_o_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_294_fun__upd__None__restrict,axiom,
    ! [X2: produc732395585841259969at_nat,D: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a] :
      ( ( ( member4095101504841534314at_nat @ X2 @ D )
       => ( ( fun_up564627978254483264tion_a @ ( restri896928924502181293_nat_a @ M @ D ) @ X2 @ none_a )
          = ( restri896928924502181293_nat_a @ M @ ( minus_1608234471674494216at_nat @ D @ ( insert8109617542356073361at_nat @ X2 @ bot_bo1676454896636820109at_nat ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X2 @ D )
       => ( ( fun_up564627978254483264tion_a @ ( restri896928924502181293_nat_a @ M @ D ) @ X2 @ none_a )
          = ( restri896928924502181293_nat_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_295_fun__upd__None__restrict,axiom,
    ! [X2: produc3843707927480180839at_nat,D: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a] :
      ( ( ( member8757157785044589968at_nat @ X2 @ D )
       => ( ( fun_up7579534627831735322tion_a @ ( restri4040463761431659527_nat_a @ M @ D ) @ X2 @ none_a )
          = ( restri4040463761431659527_nat_a @ M @ ( minus_3314409938677909166at_nat @ D @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X2 @ D )
       => ( ( fun_up7579534627831735322tion_a @ ( restri4040463761431659527_nat_a @ M @ D ) @ X2 @ none_a )
          = ( restri4040463761431659527_nat_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_296_linear__order__on__singleton,axiom,
    ! [X2: set_Product_prod_o_o] : ( order_2625247082195685180od_o_o @ ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X2 ) @ bot_bo498831808395348787od_o_o ) ) ).

% linear_order_on_singleton
thf(fact_297_linear__order__on__singleton,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_298_image__map__upd,axiom,
    ! [X2: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_a,Y: a] :
      ( ~ ( member444158400953824016od_o_o @ X2 @ A2 )
     => ( ( image_1441591353514620194tion_a @ ( fun_up7252388491999432090tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
        = ( image_1441591353514620194tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_299_image__map__upd,axiom,
    ! [X2: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_a,Y: a] :
      ( ~ ( member4095101504841534314at_nat @ X2 @ A2 )
     => ( ( image_6589722576923083464tion_a @ ( fun_up564627978254483264tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
        = ( image_6589722576923083464tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_300_image__map__upd,axiom,
    ! [X2: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_a,Y: a] :
      ( ~ ( member8757157785044589968at_nat @ X2 @ A2 )
     => ( ( image_7626837539197272994tion_a @ ( fun_up7579534627831735322tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
        = ( image_7626837539197272994tion_a @ M @ A2 ) ) ) ).

% image_map_upd
thf(fact_301_these__image__Some__eq,axiom,
    ! [A2: set_a] :
      ( ( these_a @ ( image_a_option_a @ some_a @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_302_inj__Some,axiom,
    ! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).

% inj_Some
thf(fact_303_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_304_Field__insert,axiom,
    ! [A: $o,B: $o,R2: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).

% Field_insert
thf(fact_305_Field__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_306_Field__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( field_6101175518749917524od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) )
      = ( sup_su1477959037604056251od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R2 ) ) ) ).

% Field_insert
thf(fact_307_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_308_refl__on__singleton,axiom,
    ! [X2: set_Product_prod_o_o] : ( refl_o935183656833700870od_o_o @ ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X2 ) @ bot_bo498831808395348787od_o_o ) ) ).

% refl_on_singleton
thf(fact_309_refl__on__singleton,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_310_prod__set__simps_I1_J,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( basic_3307925540671692871od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) )
      = ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(1)
thf(fact_311_prod__set__simps_I1_J,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3533134110227475617at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y ) )
      = ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(1)
thf(fact_312_prod__set__simps_I1_J,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) )
      = ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(1)
thf(fact_313_prod__set__simps_I2_J,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( basic_2174982398181630341od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) )
      = ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(2)
thf(fact_314_prod__set__simps_I2_J,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_7607750056555722463at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_315_prod__set__simps_I2_J,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_316_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( range_1318165146934585909od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) )
      = ( insert3817420688681518039od_o_o @ B @ ( range_1318165146934585909od_o_o @ R2 ) ) ) ).

% Range_insert
thf(fact_317_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( range_8412904782085253007at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_318_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_319_refl__onD2,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
       => ( member444158400953824016od_o_o @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_320_refl__onD2,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
       => ( member4095101504841534314at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_321_refl__onD2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
       => ( member8757157785044589968at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_322_refl__onD2,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( member3157015407008874800od_o_o @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_323_refl__onD2,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( member2643936169264416010at_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_324_refl__onD1,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
       => ( member444158400953824016od_o_o @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_325_refl__onD1,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
       => ( member4095101504841534314at_nat @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_326_refl__onD1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
       => ( member8757157785044589968at_nat @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_327_refl__onD1,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( member3157015407008874800od_o_o @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_328_refl__onD1,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( member2643936169264416010at_nat @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_329_Range__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_330_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_331_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_332_refl__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ A2 )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_333_refl__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ A2 )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_334_refl__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_335_refl__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A2 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_336_refl__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_337_FieldI2,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R4 )
     => ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R4 ) ) ) ).

% FieldI2
thf(fact_338_FieldI2,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R4 )
     => ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_339_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_340_FieldI2,axiom,
    ! [I: $o,J: $o,R4: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R4 )
     => ( member_o @ J @ ( field_o @ R4 ) ) ) ).

% FieldI2
thf(fact_341_FieldI2,axiom,
    ! [I: nat,J: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R4 )
     => ( member_nat @ J @ ( field_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_342_FieldI2,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R4 )
     => ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R4 ) ) ) ).

% FieldI2
thf(fact_343_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_344_FieldI1,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R4 )
     => ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R4 ) ) ) ).

% FieldI1
thf(fact_345_FieldI1,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R4 )
     => ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_346_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_347_FieldI1,axiom,
    ! [I: $o,J: $o,R4: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R4 )
     => ( member_o @ I @ ( field_o @ R4 ) ) ) ).

% FieldI1
thf(fact_348_FieldI1,axiom,
    ! [I: nat,J: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R4 )
     => ( member_nat @ I @ ( field_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_349_FieldI1,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R4 )
     => ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R4 ) ) ) ).

% FieldI1
thf(fact_350_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 )
     => ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_351_RangeE,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R2 ) )
     => ~ ! [A4: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A4 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_352_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) )
     => ~ ! [A4: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A4 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_353_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) )
     => ~ ! [A4: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_354_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R2 ) ) ) ).

% Range.intros
thf(fact_355_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_356_Range_Ointros,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_357_Range_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
      = ( ? [A3: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
            ( ( A = B4 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_358_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
      = ( ? [A3: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = B4 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_359_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = B4 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_360_Range_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
     => ~ ! [A4: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A4 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_361_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
     => ~ ! [A4: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A4 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_362_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
     => ~ ! [A4: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_363_refl__on__domain,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A2 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( ( member444158400953824016od_o_o @ A @ A2 )
          & ( member444158400953824016od_o_o @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_364_refl__on__domain,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A2 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( ( member4095101504841534314at_nat @ A @ A2 )
          & ( member4095101504841534314at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_365_refl__on__domain,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A2 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ( member8757157785044589968at_nat @ A @ A2 )
          & ( member8757157785044589968at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_366_refl__on__domain,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( member3157015407008874800od_o_o @ A @ A2 )
          & ( member3157015407008874800od_o_o @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_367_refl__on__domain,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( member2643936169264416010at_nat @ A @ A2 )
          & ( member2643936169264416010at_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_368_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( domain8908270702770587084od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) ) ) ).

% Domain_insert
thf(fact_369_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( domain8494901049294654374at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_370_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_371_notin__range__Some,axiom,
    ! [X2: option_a] :
      ( ( ~ ( member_option_a @ X2 @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
      = ( X2 = none_a ) ) ).

% notin_range_Some
thf(fact_372_notin__range__Some,axiom,
    ! [X2: option_nat] :
      ( ( ~ ( member_option_nat @ X2 @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
      = ( X2 = none_nat ) ) ).

% notin_range_Some
thf(fact_373_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
            = ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ ( minus_5063024033727795502od_o_o @ R2 @ id_Pro5927853813057311299od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_374_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
            = ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ ( minus_2318593706003940014at_nat @ R2 @ id_Pro6320881530060322077at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_375_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
            = ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ ( minus_8235382265540049710at_nat @ R2 @ id_Pro5428700290092274115at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_376_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 )
            = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_377_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
            = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_378_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
            = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R2 @ id_set2647307015711338339od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_379_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
            = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R2 @ id_set1796276052702428605at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_380_cofinal__def,axiom,
    ( bNF_Ca4363140458462388713inal_o
    = ( ^ [A7: set_o,R3: set_Product_prod_o_o] :
        ! [X3: $o] :
          ( ( member_o @ X3 @ ( field_o @ R3 ) )
         => ? [Y3: $o] :
              ( ( member_o @ Y3 @ A7 )
              & ( X3 = (~ Y3) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_381_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A7: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ ( field_nat @ R3 ) )
         => ? [Y3: nat] :
              ( ( member_nat @ Y3 @ A7 )
              & ( X3 != Y3 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_382_cofinal__def,axiom,
    ( bNF_Ca3407573850950408152od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ ( field_6101175518749917524od_o_o @ R3 ) )
         => ? [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
              & ( X3 != Y3 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_383_cofinal__def,axiom,
    ( bNF_Ca8448034394003515314at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ? [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
              & ( X3 != Y3 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_384_Image__singleton__iff,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_385_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A: set_Product_prod_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_386_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_387_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A2 )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_388_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A2 )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_389_ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_390_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ A2 )
       => ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_391_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc732395585841259969at_nat,R2: set_Pr6579314390632234657at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ A @ B ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ A2 )
       => ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_392_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc3843707927480180839at_nat,R2: set_Pr2596487125543962183at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member7900332092847368336at_nat @ ( produc2313366179012357847at_nat @ A @ B ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ A2 )
       => ( member8757157785044589968at_nat @ B @ ( image_8716920961790427923at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_393_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc2934264451710624999od_o_o,R2: set_Pr7090123087317550701od_o_o,A2: set_Pr457366540195662369at_nat] :
      ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ A @ B ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ A2 )
       => ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_394_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ A2 )
       => ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_395_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc3843707927480180839at_nat,R2: set_Pr6877924626643063277at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( member3999834080665787702at_nat @ ( produc1020012403072254077at_nat @ A @ B ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ A2 )
       => ( member8757157785044589968at_nat @ B @ ( image_8447118000439789241at_nat @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_396_ImageI,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc2934264451710624999od_o_o,R2: set_Pr4084129990900558663od_o_o,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8203642551620930960od_o_o @ ( produc9157239275352149975od_o_o @ A @ B ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( member444158400953824016od_o_o @ B @ ( image_6337422021275444243od_o_o @ R2 @ A2 ) ) ) ) ).

% ImageI
thf(fact_397_IdI,axiom,
    ! [A: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ id_set2647307015711338339od_o_o ) ).

% IdI
thf(fact_398_IdI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ id_set1796276052702428605at_nat ) ).

% IdI
thf(fact_399_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_400_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_401_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ A2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_402_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A2: set_se491199571785101031od_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ A2 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_403_rev__ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_404_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ A @ A2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_405_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc732395585841259969at_nat,R2: set_Pr6579314390632234657at_nat] :
      ( ( member444158400953824016od_o_o @ A @ A2 )
     => ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ A @ B ) @ R2 )
       => ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_406_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc3843707927480180839at_nat,R2: set_Pr2596487125543962183at_nat] :
      ( ( member444158400953824016od_o_o @ A @ A2 )
     => ( ( member7900332092847368336at_nat @ ( produc2313366179012357847at_nat @ A @ B ) @ R2 )
       => ( member8757157785044589968at_nat @ B @ ( image_8716920961790427923at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_407_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,B: produc2934264451710624999od_o_o,R2: set_Pr7090123087317550701od_o_o] :
      ( ( member4095101504841534314at_nat @ A @ A2 )
     => ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ A @ B ) @ R2 )
       => ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_408_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_409_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat,B: produc3843707927480180839at_nat,R2: set_Pr6877924626643063277at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A2 )
     => ( ( member3999834080665787702at_nat @ ( produc1020012403072254077at_nat @ A @ B ) @ R2 )
       => ( member8757157785044589968at_nat @ B @ ( image_8447118000439789241at_nat @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_410_rev__ImageI,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc2934264451710624999od_o_o,R2: set_Pr4084129990900558663od_o_o] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( ( member8203642551620930960od_o_o @ ( produc9157239275352149975od_o_o @ A @ B ) @ R2 )
       => ( member444158400953824016od_o_o @ B @ ( image_6337422021275444243od_o_o @ R2 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_411_Image__iff,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A2 ) )
      = ( ? [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_412_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A2 ) )
      = ( ? [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A2 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_413_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A2 ) )
      = ( ? [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_414_ImageE,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A2 ) )
     => ~ ! [X: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ B ) @ R2 )
           => ~ ( member3157015407008874800od_o_o @ X @ A2 ) ) ) ).

% ImageE
thf(fact_415_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A2: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A2 ) )
     => ~ ! [X: set_Product_prod_o_o] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ B ) @ R2 )
           => ~ ( member3157015407008874800od_o_o @ X @ A2 ) ) ) ).

% ImageE
thf(fact_416_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A2 ) )
     => ~ ! [X: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ B ) @ R2 )
           => ~ ( member2643936169264416010at_nat @ X @ A2 ) ) ) ).

% ImageE
thf(fact_417_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R2 @ A2 ) )
     => ~ ! [X: produc2934264451710624999od_o_o] :
            ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ B ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ X @ A2 ) ) ) ).

% ImageE
thf(fact_418_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R2: set_Pr7090123087317550701od_o_o,A2: set_Pr457366540195662369at_nat] :
      ( ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R2 @ A2 ) )
     => ~ ! [X: produc732395585841259969at_nat] :
            ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ X @ B ) @ R2 )
           => ~ ( member4095101504841534314at_nat @ X @ A2 ) ) ) ).

% ImageE
thf(fact_419_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R2: set_Pr4084129990900558663od_o_o,A2: set_Pr4329608150637261639at_nat] :
      ( ( member444158400953824016od_o_o @ B @ ( image_6337422021275444243od_o_o @ R2 @ A2 ) )
     => ~ ! [X: produc3843707927480180839at_nat] :
            ( ( member8203642551620930960od_o_o @ ( produc9157239275352149975od_o_o @ X @ B ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).

% ImageE
thf(fact_420_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R2: set_Pr6579314390632234657at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R2 @ A2 ) )
     => ~ ! [X: produc2934264451710624999od_o_o] :
            ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ X @ B ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ X @ A2 ) ) ) ).

% ImageE
thf(fact_421_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R2 @ A2 ) )
     => ~ ! [X: produc732395585841259969at_nat] :
            ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ B ) @ R2 )
           => ~ ( member4095101504841534314at_nat @ X @ A2 ) ) ) ).

% ImageE
thf(fact_422_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R2: set_Pr4797530719494571297at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( image_150793935887408877at_nat @ R2 @ A2 ) )
     => ~ ! [X: produc3843707927480180839at_nat] :
            ( ( member3985572510901480042at_nat @ ( produc1947060375374649521at_nat @ X @ B ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).

% ImageE
thf(fact_423_ImageE,axiom,
    ! [B: produc3843707927480180839at_nat,R2: set_Pr2596487125543962183at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( member8757157785044589968at_nat @ B @ ( image_8716920961790427923at_nat @ R2 @ A2 ) )
     => ~ ! [X: produc2934264451710624999od_o_o] :
            ( ( member7900332092847368336at_nat @ ( produc2313366179012357847at_nat @ X @ B ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ X @ A2 ) ) ) ).

% ImageE
thf(fact_424_IdE,axiom,
    ! [P: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ P @ id_set2647307015711338339od_o_o )
     => ~ ! [X: set_Product_prod_o_o] :
            ( P
           != ( produc763777882069021527od_o_o @ X @ X ) ) ) ).

% IdE
thf(fact_425_IdE,axiom,
    ! [P: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ P @ id_set1796276052702428605at_nat )
     => ~ ! [X: set_Pr1261947904930325089at_nat] :
            ( P
           != ( produc2922128104949294807at_nat @ X @ X ) ) ) ).

% IdE
thf(fact_426_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
     => ~ ! [B3: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B3 ) @ R2 ) ) ).

% Domain.cases
thf(fact_427_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B3 ) @ R2 ) ) ).

% Domain.cases
thf(fact_428_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ R2 ) ) ).

% Domain.cases
thf(fact_429_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
      = ( ? [A3: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
            ( ( A = A3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_430_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
      = ( ? [A3: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = A3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_431_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = A3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B4 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_432_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_433_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_434_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_435_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
     => ~ ! [B3: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B3 ) @ R2 ) ) ).

% DomainE
thf(fact_436_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B3 ) @ R2 ) ) ).

% DomainE
thf(fact_437_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ R2 ) ) ).

% DomainE
thf(fact_438_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
      = ( ? [Y3: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_439_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_440_Domain__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
      = ( ? [Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_441_reflD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_442_reflD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_443_reflD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_444_reflI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ X ) @ R2 )
     => ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% reflI
thf(fact_445_reflI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ R2 )
     => ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% reflI
thf(fact_446_reflI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
     => ( refl_on_nat @ top_top_set_nat @ R2 ) ) ).

% reflI
thf(fact_447_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_448_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_449_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_450_IdD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
     => ( A = B ) ) ).

% IdD
thf(fact_451_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_452_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_453_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_454_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_455_asymI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
         => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 ) )
     => ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% asymI
thf(fact_456_asymI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
         => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 ) )
     => ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% asymI
thf(fact_457_asymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R2 ) )
     => ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymI
thf(fact_458_finite__map__freshness,axiom,
    ! [F: nat > option_a] :
      ( ( finite_finite_nat @ ( dom_nat_a @ F ) )
     => ( ~ ( finite_finite_nat @ top_top_set_nat )
       => ? [X: nat] :
            ( ( F @ X )
            = none_a ) ) ) ).

% finite_map_freshness
thf(fact_459_asym__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y4 ) @ R2 )
             => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X ) @ R2 ) ) ) )
     => ( asym_o7797749731265203737od_o_o @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_460_asym__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y4 ) @ R2 )
             => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X ) @ R2 ) ) ) )
     => ( asym_o194845430079802867at_nat @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_461_asym__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y4 ) @ R2 )
             => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X ) @ R2 ) ) ) )
     => ( asym_o9101232109321078681at_nat @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_462_asym__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 ) ) ) )
     => ( asym_o8381145231034218809od_o_o @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_463_asym__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 ) ) ) )
     => ( asym_o5033339280856310931at_nat @ A2 @ R2 ) ) ).

% asym_onI
thf(fact_464_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_465_trans__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o,Z: produc2934264451710624999od_o_o] :
      ( ( trans_2549647969203962503od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( member444158400953824016od_o_o @ Z @ A2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ Z ) @ R2 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_466_trans__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat,Z: produc732395585841259969at_nat] :
      ( ( trans_3798871764278489953at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( member4095101504841534314at_nat @ Z @ A2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ Z ) @ R2 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_467_trans__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] :
      ( ( trans_4802246114490097159at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( member8757157785044589968at_nat @ Z @ A2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ Z ) @ R2 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_468_trans__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( member3157015407008874800od_o_o @ Z @ A2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_469_trans__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( member2643936169264416010at_nat @ Z @ A2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_470_trans__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o,Z2: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( member444158400953824016od_o_o @ Z2 @ A2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y4 ) @ R2 )
               => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ Z2 ) @ R2 )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_2549647969203962503od_o_o @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_471_trans__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat,Z2: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( member4095101504841534314at_nat @ Z2 @ A2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y4 ) @ R2 )
               => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ Z2 ) @ R2 )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_3798871764278489953at_nat @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_472_trans__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( member8757157785044589968at_nat @ Z2 @ A2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y4 ) @ R2 )
               => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ Z2 ) @ R2 )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_4802246114490097159at_nat @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_473_trans__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( member3157015407008874800od_o_o @ Z2 @ A2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z2 ) @ R2 )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_7029024365671003047od_o_o @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_474_trans__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( member2643936169264416010at_nat @ Z2 @ A2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z2 ) @ R2 )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_3732288485672914945at_nat @ A2 @ R2 ) ) ).

% trans_onI
thf(fact_475_trans__on__def,axiom,
    ( trans_7029024365671003047od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ! [Z3: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Z3 @ A7 )
                 => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 )
                   => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z3 ) @ R3 )
                     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z3 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_476_trans__on__def,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ! [Z3: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Z3 @ A7 )
                 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
                   => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z3 ) @ R3 )
                     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_477_asym__on__def,axiom,
    ( asym_o8381145231034218809od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 )
               => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_478_asym__on__def,axiom,
    ( asym_o5033339280856310931at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
               => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_479_asym__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( asym_o7797749731265203737od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
           => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_480_asym__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( asym_o194845430079802867at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
           => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_481_asym__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( asym_o9101232109321078681at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
           => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_482_asym__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_483_asym__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_484_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_485_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_486_transD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_487_transD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_488_transD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_489_transE,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_490_transE,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_491_transE,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_492_transI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z2 ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 ) ) )
     => ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% transI
thf(fact_493_transI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z2 ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 ) ) )
     => ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% transI
thf(fact_494_transI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y4: nat,Z2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R2 ) ) ).

% transI
thf(fact_495_asymD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 ) ) ) ).

% asymD
thf(fact_496_asymD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 ) ) ) ).

% asymD
thf(fact_497_asymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X2 ) @ R2 ) ) ) ).

% asymD
thf(fact_498_asym__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 )
      = ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_499_asym__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 )
      = ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_500_asym__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_501_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc2934264451710624999od_o_o > set_Product_prod_o_o,X2: produc2934264451710624999od_o_o,C: set_Product_prod_o_o,G: produc2934264451710624999od_o_o > set_Product_prod_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member444158400953824016od_o_o @ X2 @ A2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( bNF_Gr1939940824387609677od_o_o @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_502_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc732395585841259969at_nat > set_Product_prod_o_o,X2: produc732395585841259969at_nat,C: set_Product_prod_o_o,G: produc732395585841259969at_nat > set_Product_prod_o_o,A2: set_Pr457366540195662369at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member4095101504841534314at_nat @ X2 @ A2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( bNF_Gr4234126200300082471od_o_o @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_503_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc3843707927480180839at_nat > set_Product_prod_o_o,X2: produc3843707927480180839at_nat,C: set_Product_prod_o_o,G: produc3843707927480180839at_nat > set_Product_prod_o_o,A2: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member8757157785044589968at_nat @ X2 @ A2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( bNF_Gr325834636694211021od_o_o @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_504_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc2934264451710624999od_o_o > set_Product_prod_o_o,X2: produc2934264451710624999od_o_o,C: set_Pr1261947904930325089at_nat,G: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member444158400953824016od_o_o @ X2 @ A2 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ ( bNF_Gr9251090155517351at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_505_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc732395585841259969at_nat > set_Product_prod_o_o,X2: produc732395585841259969at_nat,C: set_Pr1261947904930325089at_nat,G: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member4095101504841534314at_nat @ X2 @ A2 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ ( bNF_Gr3107587960415891841at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_506_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F: produc3843707927480180839at_nat > set_Product_prod_o_o,X2: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat,G: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member8757157785044589968at_nat @ X2 @ A2 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ ( bNF_Gr6396972118715063079at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_507_image2__eqI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,F: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat,X2: produc2934264451710624999od_o_o,C: set_Pr1261947904930325089at_nat,G: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat,A2: set_Pr1932065953672099015od_o_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member444158400953824016od_o_o @ X2 @ A2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( bNF_Gr3579140021203399117at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_508_image2__eqI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,F: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat,X2: produc732395585841259969at_nat,C: set_Pr1261947904930325089at_nat,G: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member4095101504841534314at_nat @ X2 @ A2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( bNF_Gr1969212064759494823at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_509_image2__eqI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,F: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,X2: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat,G: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member8757157785044589968at_nat @ X2 @ A2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( bNF_Gr1057879175231919949at_nat @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_510_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_3482285145583873534od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_511_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_8058285401901086552at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_512_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_4077073789452270206at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_513_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_514_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_515_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_6208692358724846110od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_516_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_517_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R2 @ id_o ) )
        = ( ! [A7: set_o] :
              ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R2 ) )
             => ( ( A7 != bot_bot_set_o )
               => ? [X3: $o] :
                    ( ( member_o @ X3 @ A7 )
                    & ! [Y3: $o] :
                        ( ( member_o @ Y3 @ A7 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_518_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R2 ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                    & ! [Y3: nat] :
                        ( ( member_nat @ Y3 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_519_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( wf_set2384217916178216176od_o_o @ ( minus_5704166846132356910od_o_o @ R2 @ id_set2647307015711338339od_o_o ) )
        = ( ! [A7: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( A7 != bot_bo8006819982598807891od_o_o )
               => ? [X3: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X3 @ A7 )
                    & ! [Y3: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_520_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( wf_set7675013139414177994at_nat @ ( minus_3314409938677909166at_nat @ R2 @ id_set1796276052702428605at_nat ) )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X3: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X3 @ A7 )
                    & ! [Y3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_521_equiv__class__eq__iff,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
        = ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ X2 @ bot_bo498831808395348787od_o_o ) )
            = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) ) )
          & ( member444158400953824016od_o_o @ X2 @ A2 )
          & ( member444158400953824016od_o_o @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_522_equiv__class__eq__iff,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
        = ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ X2 @ bot_bo1676454896636820109at_nat ) )
            = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) ) )
          & ( member4095101504841534314at_nat @ X2 @ A2 )
          & ( member4095101504841534314at_nat @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_523_equiv__class__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
        = ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
            = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) )
          & ( member8757157785044589968at_nat @ X2 @ A2 )
          & ( member8757157785044589968at_nat @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_524_equiv__class__eq__iff,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
        = ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) )
            = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) )
          & ( member3157015407008874800od_o_o @ X2 @ A2 )
          & ( member3157015407008874800od_o_o @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_525_equiv__class__eq__iff,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
        = ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) )
            = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) )
          & ( member2643936169264416010at_nat @ X2 @ A2 )
          & ( member2643936169264416010at_nat @ Y @ A2 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_526_map__option__eq__Some,axiom,
    ! [F: a > a,Xo: option_a,Y: a] :
      ( ( ( map_option_a_a @ F @ Xo )
        = ( some_a @ Y ) )
      = ( ? [Z3: a] :
            ( ( Xo
              = ( some_a @ Z3 ) )
            & ( ( F @ Z3 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_527_option_Omap__disc__iff,axiom,
    ! [F: a > a,A: option_a] :
      ( ( ( map_option_a_a @ F @ A )
        = none_a )
      = ( A = none_a ) ) ).

% option.map_disc_iff
thf(fact_528_map__option__is__None,axiom,
    ! [F: a > a,Opt: option_a] :
      ( ( ( map_option_a_a @ F @ Opt )
        = none_a )
      = ( Opt = none_a ) ) ).

% map_option_is_None
thf(fact_529_None__eq__map__option__iff,axiom,
    ! [F: a > a,X2: option_a] :
      ( ( none_a
        = ( map_option_a_a @ F @ X2 ) )
      = ( X2 = none_a ) ) ).

% None_eq_map_option_iff
thf(fact_530_option_Osimps_I9_J,axiom,
    ! [F: a > a,X22: a] :
      ( ( map_option_a_a @ F @ ( some_a @ X22 ) )
      = ( some_a @ ( F @ X22 ) ) ) ).

% option.simps(9)
thf(fact_531_option_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_option_a_a @ F @ none_a )
      = none_a ) ).

% option.simps(8)
thf(fact_532_subrelI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ S ) )
     => ( ord_le7586318239878589543od_o_o @ R2 @ S ) ) ).

% subrelI
thf(fact_533_subrelI,axiom,
    ! [R2: set_Pr457366540195662369at_nat,S: set_Pr457366540195662369at_nat] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ Y4 ) @ R2 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ Y4 ) @ S ) )
     => ( ord_le4841768313324958145at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_534_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_535_map__option__idI,axiom,
    ! [X2: option8293164206536526765od_o_o,F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o] :
      ( ! [Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ Y4 @ ( set_op3337521775309186165od_o_o @ X2 ) )
         => ( ( F @ Y4 )
            = Y4 ) )
     => ( ( map_op2750206784353119307od_o_o @ F @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_536_map__option__idI,axiom,
    ! [X2: option7754883786268931591at_nat,F: produc732395585841259969at_nat > produc732395585841259969at_nat] :
      ( ! [Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ Y4 @ ( set_op8338180575893423695at_nat @ X2 ) )
         => ( ( F @ Y4 )
            = Y4 ) )
     => ( ( map_op8542782130795478475at_nat @ F @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_537_map__option__idI,axiom,
    ! [X2: option8699557107911914541at_nat,F: produc3843707927480180839at_nat > produc3843707927480180839at_nat] :
      ( ! [Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ Y4 @ ( set_op5866586666149159413at_nat @ X2 ) )
         => ( ( F @ Y4 )
            = Y4 ) )
     => ( ( map_op4646528093940530251at_nat @ F @ X2 )
        = X2 ) ) ).

% map_option_idI
thf(fact_538_option_Omap__ident__strong,axiom,
    ! [T: option8293164206536526765od_o_o,F: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o] :
      ( ! [Z2: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ Z2 @ ( set_op3337521775309186165od_o_o @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_op2750206784353119307od_o_o @ F @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_539_option_Omap__ident__strong,axiom,
    ! [T: option7754883786268931591at_nat,F: produc732395585841259969at_nat > produc732395585841259969at_nat] :
      ( ! [Z2: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ Z2 @ ( set_op8338180575893423695at_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_op8542782130795478475at_nat @ F @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_540_option_Omap__ident__strong,axiom,
    ! [T: option8699557107911914541at_nat,F: produc3843707927480180839at_nat > produc3843707927480180839at_nat] :
      ( ! [Z2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ Z2 @ ( set_op5866586666149159413at_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_op4646528093940530251at_nat @ F @ T )
        = T ) ) ).

% option.map_ident_strong
thf(fact_541_wf__eq__minimal2,axiom,
    ( wf_o
    = ( ^ [R3: set_Product_prod_o_o] :
        ! [A7: set_o] :
          ( ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R3 ) )
            & ( A7 != bot_bot_set_o ) )
         => ? [X3: $o] :
              ( ( member_o @ X3 @ A7 )
              & ! [Y3: $o] :
                  ( ( member_o @ Y3 @ A7 )
                 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_542_wf__eq__minimal2,axiom,
    ( wf_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
        ! [A7: set_nat] :
          ( ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R3 ) )
            & ( A7 != bot_bot_set_nat ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ A7 )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A7 )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_543_wf__eq__minimal2,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [A7: set_se491199571785101031od_o_o] :
          ( ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R3 ) )
            & ( A7 != bot_bo8006819982598807891od_o_o ) )
         => ? [X3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X3 @ A7 )
              & ! [Y3: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                 => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_544_wf__eq__minimal2,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [A7: set_se7855581050983116737at_nat] :
          ( ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R3 ) )
            & ( A7 != bot_bo3083307316010499117at_nat ) )
         => ? [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ A7 )
              & ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                 => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_545_equiv__class__subset,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_subset
thf(fact_546_equiv__class__subset,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_547_subset__equiv__class,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,B: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_548_subset__equiv__class,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,B: produc732395585841259969at_nat,A: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_549_subset__equiv__class,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_550_subset__equiv__class,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,B: set_Product_prod_o_o,A: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
       => ( ( member3157015407008874800od_o_o @ B @ A2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_551_subset__equiv__class,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
       => ( ( member2643936169264416010at_nat @ B @ A2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_552_subset__Image__Image__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( ord_less_eq_set_o @ A2 @ ( field_o @ R2 ) )
       => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ A2 ) @ ( image_o_o @ R2 @ B2 ) )
            = ( ! [X3: $o] :
                  ( ( member_o @ X3 @ A2 )
                 => ? [Y3: $o] :
                      ( ( member_o @ Y3 @ B2 )
                      & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X3 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_553_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( field_nat @ R2 ) )
       => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A2 ) @ ( image_nat_nat @ R2 @ B2 ) )
            = ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ A2 )
                 => ? [Y3: nat] :
                      ( ( member_nat @ Y3 @ B2 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_554_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o,B2: set_se491199571785101031od_o_o] :
      ( ( order_6208692358724846110od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ A2 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ A2 ) @ ( image_3256184292080368531od_o_o @ R2 @ B2 ) )
            = ( ! [X3: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X3 @ A2 )
                 => ? [Y3: set_Product_prod_o_o] :
                      ( ( member3157015407008874800od_o_o @ Y3 @ B2 )
                      & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_555_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ A2 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ A2 ) @ ( image_3892192845344445203at_nat @ R2 @ B2 ) )
            = ( ! [X3: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X3 @ A2 )
                 => ? [Y3: set_Pr1261947904930325089at_nat] :
                      ( ( member2643936169264416010at_nat @ Y3 @ B2 )
                      & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_556_eq__equiv__class,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,A2: set_Pr1932065953672099015od_o_o] :
      ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) )
        = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
     => ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_557_eq__equiv__class,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,A2: set_Pr457366540195662369at_nat] :
      ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) )
        = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
     => ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_558_eq__equiv__class,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
     => ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_559_eq__equiv__class,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,A2: set_se491199571785101031od_o_o] :
      ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
        = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
     => ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
       => ( ( member3157015407008874800od_o_o @ B @ A2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_560_eq__equiv__class,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
        = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
     => ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
       => ( ( member2643936169264416010at_nat @ B @ A2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_561_equiv__class__eq,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
          = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_eq
thf(fact_562_equiv__class__eq,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
          = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_563_eq__equiv__class__iff,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ X2 @ bot_bo498831808395348787od_o_o ) )
              = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_564_eq__equiv__class__iff,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ X2 @ bot_bo1676454896636820109at_nat ) )
              = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_565_eq__equiv__class__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
              = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_566_eq__equiv__class__iff,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) )
              = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_567_eq__equiv__class__iff,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) )
              = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_568_wfE__min_H,axiom,
    ! [R4: set_Pr1013884327308616903od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R4 )
     => ( ( Q3 != bot_bo498831808395348787od_o_o )
       => ~ ! [Z2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z2 @ Q3 )
             => ~ ! [Y5: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member444158400953824016od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_569_wfE__min_H,axiom,
    ! [R4: set_Pr4427867301352232263at_nat,Q3: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R4 )
     => ( ( Q3 != bot_bo1676454896636820109at_nat )
       => ~ ! [Z2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member4095101504841534314at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_570_wfE__min_H,axiom,
    ! [R4: set_Pr5564308138774400199at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R4 )
     => ( ( Q3 != bot_bo228742789529271731at_nat )
       => ~ ! [Z2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member8757157785044589968at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_571_wfE__min_H,axiom,
    ! [R4: set_Pr1932065953672099015od_o_o,Q3: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R4 )
     => ( ( Q3 != bot_bo8006819982598807891od_o_o )
       => ~ ! [Z2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z2 @ Q3 )
             => ~ ! [Y5: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member3157015407008874800od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_572_wfE__min_H,axiom,
    ! [R4: set_Pr4329608150637261639at_nat,Q3: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R4 )
     => ( ( Q3 != bot_bo3083307316010499117at_nat )
       => ~ ! [Z2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member2643936169264416010at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_573_proj__iff,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) @ A2 )
       => ( ( ( equiv_6719278489894859850od_o_o @ R2 @ X2 )
            = ( equiv_6719278489894859850od_o_o @ R2 @ Y ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_574_proj__iff,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) @ A2 )
       => ( ( ( equiv_4785085088318667978at_nat @ R2 @ X2 )
            = ( equiv_4785085088318667978at_nat @ R2 @ Y ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_575_wf__def,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [P5: set_Product_prod_o_o > $o] :
          ( ! [X3: set_Product_prod_o_o] :
              ( ! [Y3: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 )
                 => ( P5 @ Y3 ) )
             => ( P5 @ X3 ) )
         => ! [X6: set_Product_prod_o_o] : ( P5 @ X6 ) ) ) ) ).

% wf_def
thf(fact_576_wf__def,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [P5: set_Pr1261947904930325089at_nat > $o] :
          ( ! [X3: set_Pr1261947904930325089at_nat] :
              ( ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 )
                 => ( P5 @ Y3 ) )
             => ( P5 @ X3 ) )
         => ! [X6: set_Pr1261947904930325089at_nat] : ( P5 @ X6 ) ) ) ) ).

% wf_def
thf(fact_577_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R2: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_578_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R2: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_579_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_580_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_581_wf__induct__rule,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ! [X: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_582_wf__induct__rule,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_583_wf__eq__minimal,axiom,
    ( wf_Pro3213319136295691984od_o_o
    = ( ^ [R3: set_Pr1013884327308616903od_o_o] :
        ! [Q4: set_Pr1932065953672099015od_o_o] :
          ( ? [X3: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X3 @ Q4 )
         => ? [X3: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X3 @ Q4 )
              & ! [Y3: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ X3 ) @ R3 )
                 => ~ ( member444158400953824016od_o_o @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_584_wf__eq__minimal,axiom,
    ( wf_Pro661967360799104810at_nat
    = ( ^ [R3: set_Pr4427867301352232263at_nat] :
        ! [Q4: set_Pr457366540195662369at_nat] :
          ( ? [X3: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X3 @ Q4 )
         => ? [X3: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X3 @ Q4 )
              & ! [Y3: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ X3 ) @ R3 )
                 => ~ ( member4095101504841534314at_nat @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_585_wf__eq__minimal,axiom,
    ( wf_Pro2357547232388600144at_nat
    = ( ^ [R3: set_Pr5564308138774400199at_nat] :
        ! [Q4: set_Pr4329608150637261639at_nat] :
          ( ? [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ Q4 )
         => ? [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ Q4 )
              & ! [Y3: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X3 ) @ R3 )
                 => ~ ( member8757157785044589968at_nat @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_586_wf__eq__minimal,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [Q4: set_se491199571785101031od_o_o] :
          ( ? [X3: set_Product_prod_o_o] : ( member3157015407008874800od_o_o @ X3 @ Q4 )
         => ? [X3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X3 @ Q4 )
              & ! [Y3: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 )
                 => ~ ( member3157015407008874800od_o_o @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_587_wf__eq__minimal,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [Q4: set_se7855581050983116737at_nat] :
          ( ? [X3: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X3 @ Q4 )
         => ? [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ Q4 )
              & ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 )
                 => ~ ( member2643936169264416010at_nat @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_588_wf__not__refl,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ).

% wf_not_refl
thf(fact_589_wf__not__refl,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ).

% wf_not_refl
thf(fact_590_wf__not__sym,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X2: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X2 ) @ R2 )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ A ) @ R2 ) ) ) ).

% wf_not_sym
thf(fact_591_wf__not__sym,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X2 ) @ R2 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ A ) @ R2 ) ) ) ).

% wf_not_sym
thf(fact_592_wf__irrefl,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ).

% wf_irrefl
thf(fact_593_wf__irrefl,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ).

% wf_irrefl
thf(fact_594_wf__induct,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ! [X: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_595_wf__induct,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_596_wf__asym,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X2: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X2 ) @ R2 )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ A ) @ R2 ) ) ) ).

% wf_asym
thf(fact_597_wf__asym,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X2 ) @ R2 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ A ) @ R2 ) ) ) ).

% wf_asym
thf(fact_598_wfUNIVI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [P6: set_Product_prod_o_o > $o,X: 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 ) @ R2 )
                 => ( P6 @ Y4 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X ) )
     => ( wf_set2384217916178216176od_o_o @ R2 ) ) ).

% wfUNIVI
thf(fact_599_wfUNIVI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [P6: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat] :
          ( ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa ) @ R2 )
                 => ( P6 @ Y4 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X ) )
     => ( wf_set7675013139414177994at_nat @ R2 ) ) ).

% wfUNIVI
thf(fact_600_wfI__min,axiom,
    ! [R4: set_Pr1013884327308616903od_o_o] :
      ( ! [X: produc2934264451710624999od_o_o,Q5: set_Pr1932065953672099015od_o_o] :
          ( ( member444158400953824016od_o_o @ X @ Q5 )
         => ? [Xa: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Xa @ Q5 )
              & ! [Y4: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ Xa ) @ R4 )
                 => ~ ( member444158400953824016od_o_o @ Y4 @ Q5 ) ) ) )
     => ( wf_Pro3213319136295691984od_o_o @ R4 ) ) ).

% wfI_min
thf(fact_601_wfI__min,axiom,
    ! [R4: set_Pr4427867301352232263at_nat] :
      ( ! [X: produc732395585841259969at_nat,Q5: set_Pr457366540195662369at_nat] :
          ( ( member4095101504841534314at_nat @ X @ Q5 )
         => ? [Xa: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Xa @ Q5 )
              & ! [Y4: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ Xa ) @ R4 )
                 => ~ ( member4095101504841534314at_nat @ Y4 @ Q5 ) ) ) )
     => ( wf_Pro661967360799104810at_nat @ R4 ) ) ).

% wfI_min
thf(fact_602_wfI__min,axiom,
    ! [R4: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Q5: set_Pr4329608150637261639at_nat] :
          ( ( member8757157785044589968at_nat @ X @ Q5 )
         => ? [Xa: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Xa @ Q5 )
              & ! [Y4: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ Xa ) @ R4 )
                 => ~ ( member8757157785044589968at_nat @ Y4 @ Q5 ) ) ) )
     => ( wf_Pro2357547232388600144at_nat @ R4 ) ) ).

% wfI_min
thf(fact_603_wfI__min,axiom,
    ! [R4: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Q5: set_se491199571785101031od_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ Q5 )
         => ? [Xa: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa @ Q5 )
              & ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Xa ) @ R4 )
                 => ~ ( member3157015407008874800od_o_o @ Y4 @ Q5 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R4 ) ) ).

% wfI_min
thf(fact_604_wfI__min,axiom,
    ! [R4: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Q5: set_se7855581050983116737at_nat] :
          ( ( member2643936169264416010at_nat @ X @ Q5 )
         => ? [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ Q5 )
              & ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa ) @ R4 )
                 => ~ ( member2643936169264416010at_nat @ Y4 @ Q5 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R4 ) ) ).

% wfI_min
thf(fact_605_wfE__min,axiom,
    ! [R4: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R4 )
     => ( ( member444158400953824016od_o_o @ X2 @ Q3 )
       => ~ ! [Z2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z2 @ Q3 )
             => ~ ! [Y5: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member444158400953824016od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_606_wfE__min,axiom,
    ! [R4: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Q3: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R4 )
     => ( ( member4095101504841534314at_nat @ X2 @ Q3 )
       => ~ ! [Z2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member4095101504841534314at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_607_wfE__min,axiom,
    ! [R4: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R4 )
     => ( ( member8757157785044589968at_nat @ X2 @ Q3 )
       => ~ ! [Z2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member8757157785044589968at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_608_wfE__min,axiom,
    ! [R4: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Q3: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R4 )
     => ( ( member3157015407008874800od_o_o @ X2 @ Q3 )
       => ~ ! [Z2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z2 @ Q3 )
             => ~ ! [Y5: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member3157015407008874800od_o_o @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_609_wfE__min,axiom,
    ! [R4: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Q3: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R4 )
     => ( ( member2643936169264416010at_nat @ X2 @ Q3 )
       => ~ ! [Z2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z2 @ Q3 )
             => ~ ! [Y5: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z2 ) @ R4 )
                   => ~ ( member2643936169264416010at_nat @ Y5 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_610_Total__subset__Id,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( total_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( ord_le192195778054999463od_o_o @ R2 @ id_o )
       => ( ( R2 = bot_bo7073875226086086771od_o_o )
          | ? [A4: $o] :
              ( R2
              = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A4 @ A4 ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_611_Total__subset__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ R2 @ id_nat )
       => ( ( R2 = bot_bo2099793752762293965at_nat )
          | ? [A4: nat] :
              ( R2
              = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_612_Total__subset__Id,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( total_5784220159268842891od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ R2 @ id_set2647307015711338339od_o_o )
       => ( ( R2 = bot_bo498831808395348787od_o_o )
          | ? [A4: set_Product_prod_o_o] :
              ( R2
              = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A4 @ A4 ) @ bot_bo498831808395348787od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_613_Total__subset__Id,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( total_7212417256332265445at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ R2 @ id_set1796276052702428605at_nat )
       => ( ( R2 = bot_bo228742789529271731at_nat )
          | ? [A4: set_Pr1261947904930325089at_nat] :
              ( R2
              = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A4 @ A4 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_614_disjnt__equiv__class,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( disjnt556417628418099406od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
        = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_615_disjnt__equiv__class,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( disjnt2544082741087298984at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
        = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_616_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o,Phi: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
       => ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o ) )
              | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_617_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
       => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
              | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_618_wo__rel_Ocases__Total3,axiom,
    ! [R2: 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 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( minus_5704166846132356910od_o_o @ R2 @ id_set2647307015711338339od_o_o ) )
              | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R2 @ id_set2647307015711338339od_o_o ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_619_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ R2 @ id_set1796276052702428605at_nat ) )
              | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R2 @ id_set1796276052702428605at_nat ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_620_total__on__def,axiom,
    ( total_5784220159268842891od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ( ( X3 != Y3 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 )
                  | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_621_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( X3 != Y3 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_622_total__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( X != Y4 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y4 ) @ R2 )
                | ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X ) @ R2 ) ) ) ) )
     => ( total_3055748755304958059od_o_o @ A2 @ R2 ) ) ).

% total_onI
thf(fact_623_total__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( X != Y4 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y4 ) @ R2 )
                | ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X ) @ R2 ) ) ) ) )
     => ( total_8996415722781341509at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_624_total__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( X != Y4 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y4 ) @ R2 )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X ) @ R2 ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_625_total__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( X != Y4 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 ) ) ) ) )
     => ( total_5784220159268842891od_o_o @ A2 @ R2 ) ) ).

% total_onI
thf(fact_626_total__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( X != Y4 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_627_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o,I: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R2 ) )
           => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_628_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat,I: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R2 ) )
           => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_629_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat,I: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R2 ) )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_630_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Product_prod_o_o,J: $o,I: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member_o @ I @ ( field_o @ R2 ) )
         => ( ( member_o @ J @ ( field_o @ R2 ) )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_631_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,J: nat,I: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member_nat @ I @ ( field_nat @ R2 ) )
         => ( ( member_nat @ J @ ( field_nat @ R2 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_632_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o,I: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R2 ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_633_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R2 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_634_wo__rel_OTOTALS,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ! [X7: $o] :
          ( ( member_o @ X7 @ ( field_o @ R2 ) )
         => ! [Xa: $o] :
              ( ( member_o @ Xa @ ( field_o @ R2 ) )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X7 @ Xa ) @ R2 )
                | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_635_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ ( field_nat @ R2 ) )
         => ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( field_nat @ R2 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Xa ) @ R2 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_636_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ! [X7: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X7 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ! [Xa: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X7 @ Xa ) @ R2 )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_637_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ! [X7: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X7 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Xa ) @ R2 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_638_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ! [X: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( ( Y5 != X )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X ) @ R2 ) )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_639_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( ( Y5 != X )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X ) @ R2 ) )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_640_well__order__induct__imp,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,P2: produc2934264451710624999od_o_o > $o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ! [X: produc2934264451710624999od_o_o] :
            ( ! [Y5: produc2934264451710624999od_o_o] :
                ( ( ( Y5 != X )
                  & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ X ) @ R2 ) )
               => ( ( member444158400953824016od_o_o @ Y5 @ ( field_5121862139100741428od_o_o @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member444158400953824016od_o_o @ X @ ( field_5121862139100741428od_o_o @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_641_well__order__induct__imp,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,P2: produc732395585841259969at_nat > $o,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ! [X: produc732395585841259969at_nat] :
            ( ! [Y5: produc732395585841259969at_nat] :
                ( ( ( Y5 != X )
                  & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ X ) @ R2 ) )
               => ( ( member4095101504841534314at_nat @ Y5 @ ( field_7361900449976055182at_nat @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member4095101504841534314at_nat @ X @ ( field_7361900449976055182at_nat @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_642_well__order__induct__imp,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ! [X: produc3843707927480180839at_nat] :
            ( ! [Y5: produc3843707927480180839at_nat] :
                ( ( ( Y5 != X )
                  & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X ) @ R2 ) )
               => ( ( member8757157785044589968at_nat @ Y5 @ ( field_2862844103579723188at_nat @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member8757157785044589968at_nat @ X @ ( field_2862844103579723188at_nat @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_643_well__order__induct__imp,axiom,
    ! [R2: set_Product_prod_o_o,P2: $o > $o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ! [X: $o] :
            ( ! [Y5: $o] :
                ( ( ( Y5 != X )
                  & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X ) @ R2 ) )
               => ( ( member_o @ Y5 @ ( field_o @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_o @ X @ ( field_o @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member_o @ A @ ( field_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_644_well__order__induct__imp,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ! [X: nat] :
            ( ! [Y5: nat] :
                ( ( ( Y5 != X )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X ) @ R2 ) )
               => ( ( member_nat @ Y5 @ ( field_nat @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_nat @ X @ ( field_nat @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_645_well__order__induct__imp,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ! [X: set_Product_prod_o_o] :
            ( ! [Y5: set_Product_prod_o_o] :
                ( ( ( Y5 != X )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X ) @ R2 ) )
               => ( ( member3157015407008874800od_o_o @ Y5 @ ( field_6101175518749917524od_o_o @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_646_well__order__induct__imp,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( ( Y5 != X )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X ) @ R2 ) )
               => ( ( member2643936169264416010at_nat @ Y5 @ ( field_3145993188331922222at_nat @ R2 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( P2 @ X ) ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_647_totalI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( X != Y4 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
            | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 ) ) )
     => ( total_5784220159268842891od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% totalI
thf(fact_648_totalI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( X != Y4 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
            | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 ) ) )
     => ( total_7212417256332265445at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% totalI
thf(fact_649_totalI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y4: nat] :
          ( ( X != Y4 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
            | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R2 ) ) )
     => ( total_on_nat @ top_top_set_nat @ R2 ) ) ).

% totalI
thf(fact_650_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o,Phi: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
       => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_651_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_652_wo__rel_Ocases__Total,axiom,
    ! [R2: 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 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_653_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_654_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ A @ B2 )
         => ( ! [B3: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ B3 @ B2 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We4719373430050929331od_o_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_655_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ A @ B2 )
         => ( ! [B3: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ B3 @ B2 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We6249471513304245645at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_656_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ A @ B2 )
         => ( ! [B3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ B3 @ B2 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We7473068394052389427at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_657_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Product_prod_o_o,B2: set_o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R2 ) )
       => ( ( member_o @ A @ B2 )
         => ( ! [B3: $o] :
                ( ( member_o @ B3 @ B2 )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We537455116209922916inim_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_658_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B2: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R2 ) )
       => ( ( member_nat @ A @ B2 )
         => ( ! [B3: nat] :
                ( ( member_nat @ B3 @ B2 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_659_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ A @ B2 )
         => ( ! [B3: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ B3 @ B2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We6151898823597705171od_o_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_660_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ A @ B2 )
         => ( ! [B3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ B3 @ B2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ R2 ) )
           => ( A
              = ( bNF_We6832956061294468653at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_661_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ B2 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( bNF_We4719373430050929331od_o_o @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_662_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ B2 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( bNF_We6249471513304245645at_nat @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_663_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ B2 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We7473068394052389427at_nat @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_664_wo__rel_Ominim__least,axiom,
    ! [R2: set_Product_prod_o_o,B2: set_o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ B2 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_665_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B2: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ B2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_666_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ B2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We6151898823597705171od_o_o @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_667_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ B2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6832956061294468653at_nat @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_668_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member444158400953824016od_o_o @ ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B ) @ ( insert3855699742481506999od_o_o @ A @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_669_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member4095101504841534314at_nat @ ( bNF_We7253295970193759759at_nat @ R2 @ A @ B ) @ ( insert8109617542356073361at_nat @ A @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_670_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member8757157785044589968at_nat @ ( bNF_We1807050579049552309at_nat @ R2 @ A @ B ) @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_671_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R2 @ A @ B ) ) @ R2 )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A @ B ) ) @ R2 )
            & ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A @ B ) @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_672_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_673_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member3157015407008874800od_o_o @ ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B ) @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_674_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member2643936169264416010at_nat @ ( bNF_We6094461251688579247at_nat @ R2 @ A @ B ) @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_675_in__quotient__imp__in__rel,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X8: set_se491199571785101031od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X8 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
       => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) @ X8 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_676_in__quotient__imp__in__rel,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X8: set_se7855581050983116737at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8313122109188517482at_nat @ X8 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
       => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) @ X8 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_677_in__quotient__imp__closed,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X8: set_Pr1932065953672099015od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X8 @ ( equiv_1831625524721748142od_o_o @ A2 @ R2 ) )
       => ( ( member444158400953824016od_o_o @ X2 @ X8 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
           => ( member444158400953824016od_o_o @ Y @ X8 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_678_in__quotient__imp__closed,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X8: set_Pr457366540195662369at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member3243375828287704010at_nat @ X8 @ ( equiv_2323626691971332616at_nat @ A2 @ R2 ) )
       => ( ( member4095101504841534314at_nat @ X2 @ X8 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
           => ( member4095101504841534314at_nat @ Y @ X8 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_679_in__quotient__imp__closed,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X8: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member6273874620674182256at_nat @ X8 @ ( equiv_8408515162552411438at_nat @ A2 @ R2 ) )
       => ( ( member8757157785044589968at_nat @ X2 @ X8 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
           => ( member8757157785044589968at_nat @ Y @ X8 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_680_in__quotient__imp__closed,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X8: set_se491199571785101031od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X8 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ X2 @ X8 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
           => ( member3157015407008874800od_o_o @ Y @ X8 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_681_in__quotient__imp__closed,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X8: set_se7855581050983116737at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8313122109188517482at_nat @ X8 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
       => ( ( member2643936169264416010at_nat @ X2 @ X8 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
           => ( member2643936169264416010at_nat @ Y @ X8 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_682_quotient__eq__iff,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X8: set_Pr1932065953672099015od_o_o,Y6: set_Pr1932065953672099015od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X8 @ ( equiv_1831625524721748142od_o_o @ A2 @ R2 ) )
       => ( ( member4387393612980313840od_o_o @ Y6 @ ( equiv_1831625524721748142od_o_o @ A2 @ R2 ) )
         => ( ( member444158400953824016od_o_o @ X2 @ X8 )
           => ( ( member444158400953824016od_o_o @ Y @ Y6 )
             => ( ( X8 = Y6 )
                = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_683_quotient__eq__iff,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X8: set_Pr457366540195662369at_nat,Y6: set_Pr457366540195662369at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member3243375828287704010at_nat @ X8 @ ( equiv_2323626691971332616at_nat @ A2 @ R2 ) )
       => ( ( member3243375828287704010at_nat @ Y6 @ ( equiv_2323626691971332616at_nat @ A2 @ R2 ) )
         => ( ( member4095101504841534314at_nat @ X2 @ X8 )
           => ( ( member4095101504841534314at_nat @ Y @ Y6 )
             => ( ( X8 = Y6 )
                = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_684_quotient__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X8: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member6273874620674182256at_nat @ X8 @ ( equiv_8408515162552411438at_nat @ A2 @ R2 ) )
       => ( ( member6273874620674182256at_nat @ Y6 @ ( equiv_8408515162552411438at_nat @ A2 @ R2 ) )
         => ( ( member8757157785044589968at_nat @ X2 @ X8 )
           => ( ( member8757157785044589968at_nat @ Y @ Y6 )
             => ( ( X8 = Y6 )
                = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_685_quotient__eq__iff,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X8: set_se491199571785101031od_o_o,Y6: set_se491199571785101031od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X8 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
       => ( ( member2239101972932892176od_o_o @ Y6 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ X2 @ X8 )
           => ( ( member3157015407008874800od_o_o @ Y @ Y6 )
             => ( ( X8 = Y6 )
                = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_686_quotient__eq__iff,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X8: set_se7855581050983116737at_nat,Y6: set_se7855581050983116737at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8313122109188517482at_nat @ X8 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
       => ( ( member8313122109188517482at_nat @ Y6 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
         => ( ( member2643936169264416010at_nat @ X2 @ X8 )
           => ( ( member2643936169264416010at_nat @ Y @ Y6 )
             => ( ( X8 = Y6 )
                = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_687_quotient__eqI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X8: set_Pr1932065953672099015od_o_o,Y6: set_Pr1932065953672099015od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X8 @ ( equiv_1831625524721748142od_o_o @ A2 @ R2 ) )
       => ( ( member4387393612980313840od_o_o @ Y6 @ ( equiv_1831625524721748142od_o_o @ A2 @ R2 ) )
         => ( ( member444158400953824016od_o_o @ X2 @ X8 )
           => ( ( member444158400953824016od_o_o @ Y @ Y6 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
               => ( X8 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_688_quotient__eqI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X8: set_Pr457366540195662369at_nat,Y6: set_Pr457366540195662369at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member3243375828287704010at_nat @ X8 @ ( equiv_2323626691971332616at_nat @ A2 @ R2 ) )
       => ( ( member3243375828287704010at_nat @ Y6 @ ( equiv_2323626691971332616at_nat @ A2 @ R2 ) )
         => ( ( member4095101504841534314at_nat @ X2 @ X8 )
           => ( ( member4095101504841534314at_nat @ Y @ Y6 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
               => ( X8 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_689_quotient__eqI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X8: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member6273874620674182256at_nat @ X8 @ ( equiv_8408515162552411438at_nat @ A2 @ R2 ) )
       => ( ( member6273874620674182256at_nat @ Y6 @ ( equiv_8408515162552411438at_nat @ A2 @ R2 ) )
         => ( ( member8757157785044589968at_nat @ X2 @ X8 )
           => ( ( member8757157785044589968at_nat @ Y @ Y6 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
               => ( X8 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_690_quotient__eqI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X8: set_se491199571785101031od_o_o,Y6: set_se491199571785101031od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X8 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
       => ( ( member2239101972932892176od_o_o @ Y6 @ ( equiv_3134926782319234254od_o_o @ A2 @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ X2 @ X8 )
           => ( ( member3157015407008874800od_o_o @ Y @ Y6 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
               => ( X8 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_691_quotient__eqI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X8: set_se7855581050983116737at_nat,Y6: set_se7855581050983116737at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member8313122109188517482at_nat @ X8 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
       => ( ( member8313122109188517482at_nat @ Y6 @ ( equiv_7923377459618294184at_nat @ A2 @ R2 ) )
         => ( ( member2643936169264416010at_nat @ X2 @ X8 )
           => ( ( member2643936169264416010at_nat @ Y @ Y6 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
               => ( X8 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_692_wo__rel_Omax2__def,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
         => ( ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B )
            = B ) )
        & ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
         => ( ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_693_wo__rel_Omax2__def,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
         => ( ( bNF_We6094461251688579247at_nat @ R2 @ A @ B )
            = B ) )
        & ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
         => ( ( bNF_We6094461251688579247at_nat @ R2 @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_694_eq__equiv__class__iff2,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ X2 @ bot_bo498831808395348787od_o_o ) @ R2 )
              = ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) @ R2 ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_695_eq__equiv__class__iff2,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ X2 @ bot_bo1676454896636820109at_nat ) @ R2 )
              = ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) @ R2 ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_696_eq__equiv__class__iff2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) @ R2 )
              = ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) @ R2 ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_697_eq__equiv__class__iff2,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ X2 @ bot_bo8006819982598807891od_o_o ) @ R2 )
              = ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) @ R2 ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_698_eq__equiv__class__iff2,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ R2 )
              = ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) @ R2 ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_699_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B )
              = A )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_700_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R2 @ A @ B )
              = A )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_701_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R2 @ A @ B )
              = A )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_702_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A @ B )
              = A )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_703_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B )
              = A )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_704_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B )
              = A )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_705_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R2 @ A @ B )
              = A )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_706_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B )
              = B )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_707_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R2 @ A @ B )
              = B )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_708_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R2 @ A @ B )
              = B )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_709_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A @ B )
              = B )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_710_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B )
              = B )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_711_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B )
              = B )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_712_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R2 @ A @ B )
              = B )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_713_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_714_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_715_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_716_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R2 @ A @ B ) ) @ R2 )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_717_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_718_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B ) ) @ R2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_719_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R2 @ A @ B ) ) @ R2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_720_equiv__class__nondisjoint,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ ( inf_in6630186493572433077od_o_o @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_721_equiv__class__nondisjoint,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ ( inf_in3075509010647812879at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_722_equiv__class__nondisjoint,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_723_equiv__class__nondisjoint,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ ( inf_in8747346010413679829od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_724_equiv__class__nondisjoint,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ ( inf_in767997467307849391at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_725_under__incr,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ord_le2654036014095304327od_o_o @ ( order_8825441150112336748od_o_o @ R2 @ A ) @ ( order_8825441150112336748od_o_o @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_726_under__incr,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ord_le2077887516847798113at_nat @ ( order_8557040927991152966at_nat @ R2 @ A ) @ ( order_8557040927991152966at_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_727_under__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ord_less_eq_set_nat @ ( order_under_nat @ R2 @ A ) @ ( order_under_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_728_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        = ( ! [A7: set_o] :
              ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R2 ) )
             => ( ( A7 != bot_bot_set_o )
               => ? [X3: $o] :
                    ( ( member_o @ X3 @ A7 )
                    & ! [Y3: $o] :
                        ( ( member_o @ Y3 @ A7 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_729_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R2 ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                    & ! [Y3: nat] :
                        ( ( member_nat @ Y3 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_730_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
        = ( ! [A7: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( A7 != bot_bo8006819982598807891od_o_o )
               => ? [X3: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X3 @ A7 )
                    & ! [Y3: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_731_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X3: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X3 @ A7 )
                    & ! [Y3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y3 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_732_well__order__on__domain,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_2783320356396525957od_o_o @ A2 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( ( member444158400953824016od_o_o @ A @ A2 )
          & ( member444158400953824016od_o_o @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_733_well__order__on__domain,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_1951249980326167391at_nat @ A2 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( ( member4095101504841534314at_nat @ A @ A2 )
          & ( member4095101504841534314at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_734_well__order__on__domain,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ A2 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ( member8757157785044589968at_nat @ A @ A2 )
          & ( member8757157785044589968at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_735_well__order__on__domain,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( member3157015407008874800od_o_o @ A @ A2 )
          & ( member3157015407008874800od_o_o @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_736_well__order__on__domain,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( member2643936169264416010at_nat @ A @ A2 )
          & ( member2643936169264416010at_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_737_underS__incl__iff,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( order_1929855754104155309od_o_o @ R2 @ A ) @ ( order_1929855754104155309od_o_o @ R2 @ B ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_738_underS__incl__iff,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ord_le4841768313324958145at_nat @ ( order_6745690092771398279at_nat @ R2 @ A ) @ ( order_6745690092771398279at_nat @ R2 @ B ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_739_underS__incl__iff,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ord_le1268244103169919719at_nat @ ( order_3736018788886245933at_nat @ R2 @ A ) @ ( order_3736018788886245933at_nat @ R2 @ B ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_740_underS__incl__iff,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( order_underS_o @ R2 @ A ) @ ( order_underS_o @ R2 @ B ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_741_underS__incl__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A ) @ ( order_underS_nat @ R2 @ B ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_742_underS__incl__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R2 @ A ) @ ( order_3915675016455977933od_o_o @ R2 @ B ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_743_underS__incl__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R2 @ A ) @ ( order_2644262826035875111at_nat @ R2 @ B ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_744_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr1932065953672099015od_o_o,Y6: set_Pr1932065953672099015od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( finite820925532130942480od_o_o @ X8 )
     => ( ( finite820925532130942480od_o_o @ Y6 )
       => ( ( Y6 != bot_bo498831808395348787od_o_o )
         => ( ! [X: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ X @ X8 )
               => ? [Xa: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ Xa @ Y6 )
                    & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Xa ) @ R4 ) ) )
           => ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ X8 @ Y6 ) @ ( max_ex9141845325704764834od_o_o @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_745_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr457366540195662369at_nat,Y6: set_Pr457366540195662369at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( finite8072506237346428522at_nat @ X8 )
     => ( ( finite8072506237346428522at_nat @ Y6 )
       => ( ( Y6 != bot_bo1676454896636820109at_nat )
         => ( ! [X: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ X @ X8 )
               => ? [Xa: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ Xa @ Y6 )
                    & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Xa ) @ R4 ) ) )
           => ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ X8 @ Y6 ) @ ( max_ex1111939896555521276at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_746_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X8 )
     => ( ( finite4343798906461161616at_nat @ Y6 )
       => ( ( Y6 != bot_bo228742789529271731at_nat )
         => ( ! [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ X8 )
               => ? [Xa: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa @ Y6 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Xa ) @ R4 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X8 @ Y6 ) @ ( max_ex8833738776070432290at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_747_max__ext_Omax__extI,axiom,
    ! [X8: set_Product_prod_o_o,Y6: set_Product_prod_o_o,R4: set_Pr7011910401827753159od_o_o] :
      ( ( finite6120865539452801872od_o_o @ X8 )
     => ( ( finite6120865539452801872od_o_o @ Y6 )
       => ( ( Y6 != bot_bo7073875226086086771od_o_o )
         => ( ! [X: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ X @ X8 )
               => ? [Xa: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ Xa @ Y6 )
                    & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X @ Xa ) @ R4 ) ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X8 @ Y6 ) @ ( max_ex7572203241849361634od_o_o @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_748_max__ext_Omax__extI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X8 )
     => ( ( finite6177210948735845034at_nat @ Y6 )
       => ( ( Y6 != bot_bo2099793752762293965at_nat )
         => ( ! [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ X8 )
               => ? [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ Y6 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Xa ) @ R4 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y6 ) @ ( max_ex8135407076693332796at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_749_max__ext_Omax__extI,axiom,
    ! [X8: set_se491199571785101031od_o_o,Y6: set_se491199571785101031od_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( finite9011636195296603184od_o_o @ X8 )
     => ( ( finite9011636195296603184od_o_o @ Y6 )
       => ( ( Y6 != bot_bo8006819982598807891od_o_o )
         => ( ! [X: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ X @ X8 )
               => ? [Xa: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ Xa @ Y6 )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Xa ) @ R4 ) ) )
           => ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ X8 @ Y6 ) @ ( max_ex5401630698724402626od_o_o @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_750_max__ext_Omax__extI,axiom,
    ! [X8: set_se7855581050983116737at_nat,Y6: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X8 )
     => ( ( finite9047747110432174090at_nat @ Y6 )
       => ( ( Y6 != bot_bo3083307316010499117at_nat )
         => ( ! [X: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X @ X8 )
               => ? [Xa: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa @ Y6 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Xa ) @ R4 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X8 @ Y6 ) @ ( max_ex4535816423174879388at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_751_max__ext_Osimps,axiom,
    ! [A1: set_Pr1932065953672099015od_o_o,A22: set_Pr1932065953672099015od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A1 @ A22 ) @ ( max_ex9141845325704764834od_o_o @ R4 ) )
      = ( ( finite820925532130942480od_o_o @ A1 )
        & ( finite820925532130942480od_o_o @ A22 )
        & ( A22 != bot_bo498831808395348787od_o_o )
        & ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ A1 )
           => ? [Y3: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ Y3 @ A22 )
                & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_752_max__ext_Osimps,axiom,
    ! [A1: set_Pr457366540195662369at_nat,A22: set_Pr457366540195662369at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A1 @ A22 ) @ ( max_ex1111939896555521276at_nat @ R4 ) )
      = ( ( finite8072506237346428522at_nat @ A1 )
        & ( finite8072506237346428522at_nat @ A22 )
        & ( A22 != bot_bo1676454896636820109at_nat )
        & ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ A1 )
           => ? [Y3: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ Y3 @ A22 )
                & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_753_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A22 )
        & ( A22 != bot_bo228742789529271731at_nat )
        & ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A1 )
           => ? [Y3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y3 @ A22 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_754_max__ext_Osimps,axiom,
    ! [A1: set_se491199571785101031od_o_o,A22: set_se491199571785101031od_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A1 @ A22 ) @ ( max_ex5401630698724402626od_o_o @ R4 ) )
      = ( ( finite9011636195296603184od_o_o @ A1 )
        & ( finite9011636195296603184od_o_o @ A22 )
        & ( A22 != bot_bo8006819982598807891od_o_o )
        & ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A1 )
           => ? [Y3: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Y3 @ A22 )
                & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_755_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R4 ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A22 )
        & ( A22 != bot_bo3083307316010499117at_nat )
        & ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A1 )
           => ? [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ A22 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_756_max__ext_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R4: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R4 ) )
      = ( ( finite6120865539452801872od_o_o @ A1 )
        & ( finite6120865539452801872od_o_o @ A22 )
        & ( A22 != bot_bo7073875226086086771od_o_o )
        & ! [X3: product_prod_o_o] :
            ( ( member7466972457876170832od_o_o @ X3 @ A1 )
           => ? [Y3: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ Y3 @ A22 )
                & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_757_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A1 )
           => ? [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_758_underS__I,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( I != J )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R4 )
       => ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_759_underS__I,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( I != J )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R4 )
       => ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_760_underS__I,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( I != J )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 )
       => ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_761_underS__I,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( I != J )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R4 )
       => ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_762_underS__I,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( I != J )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 )
       => ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R4 @ J ) ) ) ) ).

% underS_I
thf(fact_763_underS__E,axiom,
    ! [I: produc2934264451710624999od_o_o,R4: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R4 @ J ) )
     => ( ( I != J )
        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_764_underS__E,axiom,
    ! [I: produc732395585841259969at_nat,R4: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R4 @ J ) )
     => ( ( I != J )
        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_765_underS__E,axiom,
    ! [I: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R4 @ J ) )
     => ( ( I != J )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_766_underS__E,axiom,
    ! [I: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R4 @ J ) )
     => ( ( I != J )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_767_underS__E,axiom,
    ! [I: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R4 @ J ) )
     => ( ( I != J )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 ) ) ) ).

% underS_E
thf(fact_768_max__ext__additive,axiom,
    ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o,R4: 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 @ R4 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ C3 @ D ) @ ( max_ex7572203241849361634od_o_o @ R4 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( sup_su5769328420594410459od_o_o @ A2 @ C3 ) @ ( sup_su5769328420594410459od_o_o @ B2 @ D ) ) @ ( max_ex7572203241849361634od_o_o @ R4 ) ) ) ) ).

% max_ext_additive
thf(fact_769_max__ext__additive,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat,C3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C3 @ D ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ C3 ) @ ( sup_su6327502436637775413at_nat @ B2 @ D ) ) @ ( max_ex8135407076693332796at_nat @ R4 ) ) ) ) ).

% max_ext_additive
thf(fact_770_max__ext_Ocases,axiom,
    ! [A1: set_Pr1932065953672099015od_o_o,A22: set_Pr1932065953672099015od_o_o,R4: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A1 @ A22 ) @ ( max_ex9141845325704764834od_o_o @ R4 ) )
     => ~ ( ( finite820925532130942480od_o_o @ A1 )
         => ( ( finite820925532130942480od_o_o @ A22 )
           => ( ( A22 != bot_bo498831808395348787od_o_o )
             => ~ ! [X7: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ X7 @ A1 )
                   => ? [Xa2: produc2934264451710624999od_o_o] :
                        ( ( member444158400953824016od_o_o @ Xa2 @ A22 )
                        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_771_max__ext_Ocases,axiom,
    ! [A1: set_Pr457366540195662369at_nat,A22: set_Pr457366540195662369at_nat,R4: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A1 @ A22 ) @ ( max_ex1111939896555521276at_nat @ R4 ) )
     => ~ ( ( finite8072506237346428522at_nat @ A1 )
         => ( ( finite8072506237346428522at_nat @ A22 )
           => ( ( A22 != bot_bo1676454896636820109at_nat )
             => ~ ! [X7: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ X7 @ A1 )
                   => ? [Xa2: produc732395585841259969at_nat] :
                        ( ( member4095101504841534314at_nat @ Xa2 @ A22 )
                        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_772_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A22 )
           => ( ( A22 != bot_bo228742789529271731at_nat )
             => ~ ! [X7: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X7 @ A1 )
                   => ? [Xa2: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa2 @ A22 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_773_max__ext_Ocases,axiom,
    ! [A1: set_se491199571785101031od_o_o,A22: set_se491199571785101031od_o_o,R4: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A1 @ A22 ) @ ( max_ex5401630698724402626od_o_o @ R4 ) )
     => ~ ( ( finite9011636195296603184od_o_o @ A1 )
         => ( ( finite9011636195296603184od_o_o @ A22 )
           => ( ( A22 != bot_bo8006819982598807891od_o_o )
             => ~ ! [X7: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X7 @ A1 )
                   => ? [Xa2: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Xa2 @ A22 )
                        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_774_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R4 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A22 )
           => ( ( A22 != bot_bo3083307316010499117at_nat )
             => ~ ! [X7: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X7 @ A1 )
                   => ? [Xa2: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa2 @ A22 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_775_max__ext_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R4: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R4 ) )
     => ~ ( ( finite6120865539452801872od_o_o @ A1 )
         => ( ( finite6120865539452801872od_o_o @ A22 )
           => ( ( A22 != bot_bo7073875226086086771od_o_o )
             => ~ ! [X7: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X7 @ A1 )
                   => ? [Xa2: product_prod_o_o] :
                        ( ( member7466972457876170832od_o_o @ Xa2 @ A22 )
                        & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_776_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X7: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X7 @ A1 )
                   => ? [Xa2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa2 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X7 @ Xa2 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_777_underS__incr,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
         => ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R2 @ A ) @ ( order_3915675016455977933od_o_o @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_778_underS__incr,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
         => ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R2 @ A ) @ ( order_2644262826035875111at_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_779_underS__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( antisym_on_nat @ top_top_set_nat @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
         => ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A ) @ ( order_underS_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_780_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( bNF_We6886001019485585053od_o_o @ R2 @ A2 @ B )
        = ( ( member444158400953824016od_o_o @ B @ A2 )
          & ! [X3: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X3 @ A2 )
             => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ X3 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_781_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( bNF_We6170040095471856759at_nat @ R2 @ A2 @ B )
        = ( ( member4095101504841534314at_nat @ B @ A2 )
          & ! [X3: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X3 @ A2 )
             => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ X3 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_782_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( bNF_We2939539520719410205at_nat @ R2 @ A2 @ B )
        = ( ( member8757157785044589968at_nat @ B @ A2 )
          & ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ A2 )
             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ X3 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_783_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( bNF_We794377122088811965od_o_o @ R2 @ A2 @ B )
        = ( ( member3157015407008874800od_o_o @ B @ A2 )
          & ! [X3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X3 @ A2 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ X3 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_784_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( bNF_We943883530777384727at_nat @ R2 @ A2 @ B )
        = ( ( member2643936169264416010at_nat @ B @ A2 )
          & ! [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ A2 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ X3 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_785_bsqr__max2,axiom,
    ! [R2: set_Product_prod_o_o,A1: $o,A22: $o,B1: $o,B22: $o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( product_Pair_o_o @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr_o @ R2 ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A1 @ A22 ) @ ( bNF_We3104086342435940326max2_o @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).

% bsqr_max2
thf(fact_786_bsqr__max2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).

% bsqr_max2
thf(fact_787_bsqr__max2,axiom,
    ! [R2: 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 @ R2 ) @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( produc763777882069021527od_o_o @ B1 @ B22 ) ) @ ( bNF_We3935152771354962724od_o_o @ R2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We9141258785624724309od_o_o @ R2 @ A1 @ A22 ) @ ( bNF_We9141258785624724309od_o_o @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).

% bsqr_max2
thf(fact_788_bsqr__max2,axiom,
    ! [R2: 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 @ R2 ) @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( produc2922128104949294807at_nat @ B1 @ B22 ) ) @ ( bNF_We6127972852007642366at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6094461251688579247at_nat @ R2 @ A1 @ A22 ) @ ( bNF_We6094461251688579247at_nat @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).

% bsqr_max2
thf(fact_789_antisym__onD,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( antisy2259881126919591626od_o_o @ A2 @ R2 )
     => ( ( member444158400953824016od_o_o @ X2 @ A2 )
       => ( ( member444158400953824016od_o_o @ Y @ A2 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ R2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X2 ) @ R2 )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_790_antisym__onD,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( antisy1857962754316978724at_nat @ A2 @ R2 )
     => ( ( member4095101504841534314at_nat @ X2 @ A2 )
       => ( ( member4095101504841534314at_nat @ Y @ A2 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ R2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X2 ) @ R2 )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_791_antisym__onD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( antisy6856624793204979530at_nat @ A2 @ R2 )
     => ( ( member8757157785044589968at_nat @ X2 @ A2 )
       => ( ( member8757157785044589968at_nat @ Y @ A2 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ R2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X2 ) @ R2 )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_792_antisym__onD,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_o_o @ A2 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X2 @ A2 )
       => ( ( member3157015407008874800od_o_o @ Y @ A2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_793_antisym__onD,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ A2 @ R2 )
     => ( ( member2643936169264416010at_nat @ X2 @ A2 )
       => ( ( member2643936169264416010at_nat @ Y @ A2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 )
             => ( X2 = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_794_antisym__onI,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X @ A2 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y4 ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X ) @ R2 )
               => ( X = Y4 ) ) ) ) )
     => ( antisy2259881126919591626od_o_o @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_795_antisym__onI,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X @ A2 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y4 ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X ) @ R2 )
               => ( X = Y4 ) ) ) ) )
     => ( antisy1857962754316978724at_nat @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_796_antisym__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y4 ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X ) @ R2 )
               => ( X = Y4 ) ) ) ) )
     => ( antisy6856624793204979530at_nat @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_797_antisym__onI,axiom,
    ! [A2: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ A2 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 )
               => ( X = Y4 ) ) ) ) )
     => ( antisy777991898044409578od_o_o @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_798_antisym__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ A2 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 )
               => ( X = Y4 ) ) ) ) )
     => ( antisy1130263102092445124at_nat @ A2 @ R2 ) ) ).

% antisym_onI
thf(fact_799_antisym__on__def,axiom,
    ( antisy777991898044409578od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A7 )
         => ! [Y3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y3 @ A7 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R3 )
                 => ( X3 = Y3 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_800_antisym__on__def,axiom,
    ( antisy1130263102092445124at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 )
                 => ( X3 = Y3 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_801_antisymI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ R2 )
           => ( X = Y4 ) ) )
     => ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% antisymI
thf(fact_802_antisymI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ R2 )
           => ( X = Y4 ) ) )
     => ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% antisymI
thf(fact_803_antisymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R2 )
           => ( X = Y4 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R2 ) ) ).

% antisymI
thf(fact_804_antisymD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 )
         => ( X2 = Y ) ) ) ) ).

% antisymD
thf(fact_805_antisymD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 )
         => ( X2 = Y ) ) ) ) ).

% antisymD
thf(fact_806_antisymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat] :
      ( ( antisym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X2 ) @ R2 )
         => ( X2 = Y ) ) ) ) ).

% antisymD
thf(fact_807_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_808_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_809_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Product_prod_o_o,A5: set_Pr1261947904930325089at_nat,B5: set_Product_prod_o_o,R2: set_Pr4329608150637261639at_nat,S: set_Pr1932065953672099015od_o_o] :
      ( ( member625886800346054288od_o_o @ ( produc4770972565181134295od_o_o @ ( produc2070086378307464573od_o_o @ A @ B ) @ ( produc2070086378307464573od_o_o @ A5 @ B5 ) ) @ ( lex_pr3343786553352869269od_o_o @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_810_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A5: set_Product_prod_o_o,B5: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc763777882069021527od_o_o @ A5 @ B5 ) ) @ ( lex_pr2209867684007004015od_o_o @ R2 @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_811_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A5: set_Product_prod_o_o,B5: set_Pr1261947904930325089at_nat,R2: set_Pr1932065953672099015od_o_o,S: set_Pr4329608150637261639at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc8517790099723286449at_nat @ A5 @ B5 ) ) @ ( lex_pr568118237913915337at_nat @ R2 @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_812_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A5 @ B5 ) ) @ ( lex_pr8000699672184251119at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_813_same__fstI,axiom,
    ! [P2: set_Product_prod_o_o > $o,X2: set_Product_prod_o_o,Y7: set_Product_prod_o_o,Y: set_Product_prod_o_o,R4: set_Product_prod_o_o > set_Pr1932065953672099015od_o_o] :
      ( ( P2 @ X2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y7 @ Y ) @ ( R4 @ X2 ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y7 ) @ ( produc763777882069021527od_o_o @ X2 @ Y ) ) @ ( same_f884114232143252012od_o_o @ P2 @ R4 ) ) ) ) ).

% same_fstI
thf(fact_814_same__fstI,axiom,
    ! [P2: set_Product_prod_o_o > $o,X2: set_Product_prod_o_o,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R4: set_Product_prod_o_o > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R4 @ X2 ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y7 ) @ ( produc8517790099723286449at_nat @ X2 @ Y ) ) @ ( same_f6937299433074352134at_nat @ P2 @ R4 ) ) ) ) ).

% same_fstI
thf(fact_815_same__fstI,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,X2: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R4 @ X2 ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y7 ) @ ( produc2922128104949294807at_nat @ X2 @ Y ) ) @ ( same_f1460074399991292076at_nat @ P2 @ R4 ) ) ) ) ).

% same_fstI
thf(fact_816_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( ( order_909087248044410034od_o_o @ R2 @ B2 )
           != bot_bo498831808395348787od_o_o )
         => ( ( member444158400953824016od_o_o @ B @ B2 )
           => ( ( ( bNF_We7386194254115604808od_o_o @ R2 @ B2 )
               != B )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We7386194254115604808od_o_o @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_817_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( ( order_3539579856470909964at_nat @ R2 @ B2 )
           != bot_bo1676454896636820109at_nat )
         => ( ( member4095101504841534314at_nat @ B @ B2 )
           => ( ( ( bNF_We5419384366696408482at_nat @ R2 @ B2 )
               != B )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We5419384366696408482at_nat @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_818_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( ( order_5334716200017444146at_nat @ R2 @ B2 )
           != bot_bo228742789529271731at_nat )
         => ( ( member8757157785044589968at_nat @ B @ B2 )
           => ( ( ( bNF_We2397293392365222856at_nat @ R2 @ B2 )
               != B )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We2397293392365222856at_nat @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_819_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Product_prod_o_o,B2: set_o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R2 ) )
       => ( ( ( order_AboveS_o @ R2 @ B2 )
           != bot_bot_set_o )
         => ( ( member_o @ B @ B2 )
           => ( ( ( bNF_We8594000642646999417_suc_o @ R2 @ B2 )
               != B )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We8594000642646999417_suc_o @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_820_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B2: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R2 ) )
       => ( ( ( order_AboveS_nat @ R2 @ B2 )
           != bot_bot_set_nat )
         => ( ( member_nat @ B @ B2 )
           => ( ( ( bNF_We6243639482952820783uc_nat @ R2 @ B2 )
               != B )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We6243639482952820783uc_nat @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_821_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( order_5655281162628242642od_o_o @ R2 @ B2 )
           != bot_bo8006819982598807891od_o_o )
         => ( ( member3157015407008874800od_o_o @ B @ B2 )
           => ( ( ( bNF_We1257987224342916968od_o_o @ R2 @ B2 )
               != B )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We1257987224342916968od_o_o @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_822_wo__rel_Osuc__greater,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( order_9095727932502173612at_nat @ R2 @ B2 )
           != bot_bo3083307316010499117at_nat )
         => ( ( member2643936169264416010at_nat @ B @ B2 )
           => ( ( ( bNF_We3445218718054152002at_nat @ R2 @ B2 )
               != B )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We3445218718054152002at_nat @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_823_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B2: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( order_909087248044410034od_o_o @ R2 @ B2 ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( bNF_We7386194254115604808od_o_o @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_824_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( order_3539579856470909964at_nat @ R2 @ B2 ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( bNF_We5419384366696408482at_nat @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_825_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( order_5334716200017444146at_nat @ R2 @ B2 ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We2397293392365222856at_nat @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_826_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B2: set_se491199571785101031od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( order_5655281162628242642od_o_o @ R2 @ B2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We1257987224342916968od_o_o @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_827_wo__rel_Osuc__least__AboveS,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( order_9095727932502173612at_nat @ R2 @ B2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We3445218718054152002at_nat @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_828_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,B2: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ A @ ( order_909087248044410034od_o_o @ R2 @ B2 ) )
         => ( ! [A6: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ A6 @ ( order_909087248044410034od_o_o @ R2 @ B2 ) )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We7386194254115604808od_o_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_829_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,B2: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ B2 @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ A @ ( order_3539579856470909964at_nat @ R2 @ B2 ) )
         => ( ! [A6: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ A6 @ ( order_3539579856470909964at_nat @ R2 @ B2 ) )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We5419384366696408482at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_830_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ A @ ( order_5334716200017444146at_nat @ R2 @ B2 ) )
         => ( ! [A6: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ A6 @ ( order_5334716200017444146at_nat @ R2 @ B2 ) )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We2397293392365222856at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_831_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Product_prod_o_o,B2: set_o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ B2 @ ( field_o @ R2 ) )
       => ( ( member_o @ A @ ( order_AboveS_o @ R2 @ B2 ) )
         => ( ! [A6: $o] :
                ( ( member_o @ A6 @ ( order_AboveS_o @ R2 @ B2 ) )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We8594000642646999417_suc_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_832_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B2: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( field_nat @ R2 ) )
       => ( ( member_nat @ A @ ( order_AboveS_nat @ R2 @ B2 ) )
         => ( ! [A6: nat] :
                ( ( member_nat @ A6 @ ( order_AboveS_nat @ R2 @ B2 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We6243639482952820783uc_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_833_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,B2: set_se491199571785101031od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( order_5655281162628242642od_o_o @ R2 @ B2 ) )
         => ( ! [A6: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ A6 @ ( order_5655281162628242642od_o_o @ R2 @ B2 ) )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We1257987224342916968od_o_o @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_834_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,B2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ A @ ( order_9095727932502173612at_nat @ R2 @ B2 ) )
         => ( ! [A6: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ A6 @ ( order_9095727932502173612at_nat @ R2 @ B2 ) )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A6 ) @ R2 ) )
           => ( A
              = ( bNF_We3445218718054152002at_nat @ R2 @ B2 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_835_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A2: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( order_543634532542674813od_o_o @ R2 @ A2 )
       => ( ( ( order_909087248044410034od_o_o @ R2 @ A2 )
           != bot_bo498831808395348787od_o_o )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We7386194254115604808od_o_o @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We7386194254115604808od_o_o @ R2 @ A2 ) )
             => ( member444158400953824016od_o_o @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_836_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A2: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( order_3205064739578148183at_nat @ R2 @ A2 )
       => ( ( ( order_3539579856470909964at_nat @ R2 @ A2 )
           != bot_bo1676454896636820109at_nat )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We5419384366696408482at_nat @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We5419384366696408482at_nat @ R2 @ A2 ) )
             => ( member4095101504841534314at_nat @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_837_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( order_3142731893281451261at_nat @ R2 @ A2 )
       => ( ( ( order_5334716200017444146at_nat @ R2 @ A2 )
           != bot_bo228742789529271731at_nat )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We2397293392365222856at_nat @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We2397293392365222856at_nat @ R2 @ A2 ) )
             => ( member8757157785044589968at_nat @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_838_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A2: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( order_136921456694023837od_o_o @ R2 @ A2 )
       => ( ( ( order_5655281162628242642od_o_o @ R2 @ A2 )
           != bot_bo8006819982598807891od_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We1257987224342916968od_o_o @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We1257987224342916968od_o_o @ R2 @ A2 ) )
             => ( member3157015407008874800od_o_o @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_839_wo__rel_Osuc__ofilter__in,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( order_6860593127430111223at_nat @ R2 @ A2 )
       => ( ( ( order_9095727932502173612at_nat @ R2 @ A2 )
           != bot_bo3083307316010499117at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We3445218718054152002at_nat @ R2 @ A2 ) ) @ R2 )
           => ( ( B
               != ( bNF_We3445218718054152002at_nat @ R2 @ A2 ) )
             => ( member2643936169264416010at_nat @ B @ A2 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_840_wf__insert,axiom,
    ! [Y: set_Product_prod_o_o,X2: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 ) )
      = ( ( wf_set2384217916178216176od_o_o @ R2 )
        & ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ).

% wf_insert
thf(fact_841_wf__insert,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 ) )
      = ( ( wf_set7675013139414177994at_nat @ R2 )
        & ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ).

% wf_insert
thf(fact_842_Not__Domain__rtrancl,axiom,
    ! [X2: produc2934264451710624999od_o_o,R4: set_Pr1013884327308616903od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ~ ( member444158400953824016od_o_o @ X2 @ ( domain2187190646820045772od_o_o @ R4 ) )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y ) @ ( transi3610599525415160810od_o_o @ R4 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_843_Not__Domain__rtrancl,axiom,
    ! [X2: produc732395585841259969at_nat,R4: set_Pr4427867301352232263at_nat,Y: produc732395585841259969at_nat] :
      ( ~ ( member4095101504841534314at_nat @ X2 @ ( domain8659765193365367884at_nat @ R4 ) )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y ) @ ( transi4536970581772660548at_nat @ R4 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_844_Not__Domain__rtrancl,axiom,
    ! [X2: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X2 @ ( domain1102310642530566604at_nat @ R4 ) )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y ) @ ( transi5049001397978532970at_nat @ R4 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_845_Not__Domain__rtrancl,axiom,
    ! [X2: set_Product_prod_o_o,R4: set_Pr1932065953672099015od_o_o,Y: set_Product_prod_o_o] :
      ( ~ ( member3157015407008874800od_o_o @ X2 @ ( domain8908270702770587084od_o_o @ R4 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( transi8975321201238539274od_o_o @ R4 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_846_Not__Domain__rtrancl,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ~ ( member2643936169264416010at_nat @ X2 @ ( domain8802589944165416012at_nat @ R4 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( transi6170703685534862052at_nat @ R4 ) )
        = ( X2 = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_847_rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Product_prod_o_o,Bx: set_Product_prod_o_o,By: set_Product_prod_o_o,R2: set_Pr1013884327308616903od_o_o,P2: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o,Aa: set_Product_prod_o_o,Ba: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ A4 @ B3 ) ) @ ( transi3610599525415160810od_o_o @ R2 ) )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A4 @ B3 ) @ ( produc763777882069021527od_o_o @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_848_rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Pr1261947904930325089at_nat,Bx: set_Product_prod_o_o,By: set_Pr1261947904930325089at_nat,R2: set_Pr4427867301352232263at_nat,P2: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat,Aa: set_Product_prod_o_o,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ A4 @ B3 ) ) @ ( transi4536970581772660548at_nat @ R2 ) )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A4 @ B3 ) @ ( produc8517790099723286449at_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_849_rtrancl__induct2,axiom,
    ! [Ax: set_Pr1261947904930325089at_nat,Ay: set_Pr1261947904930325089at_nat,Bx: set_Pr1261947904930325089at_nat,By: set_Pr1261947904930325089at_nat,R2: set_Pr5564308138774400199at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ A4 @ B3 ) ) @ ( transi5049001397978532970at_nat @ R2 ) )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A4 @ B3 ) @ ( produc2922128104949294807at_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_850_converse__rtranclE2,axiom,
    ! [Xa3: set_Product_prod_o_o,Xb: set_Product_prod_o_o,Za: set_Product_prod_o_o,Zb: set_Product_prod_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Xa3 @ Xb ) @ ( produc763777882069021527od_o_o @ Za @ Zb ) ) @ ( transi3610599525415160810od_o_o @ R2 ) )
     => ( ( ( produc763777882069021527od_o_o @ Xa3 @ Xb )
         != ( produc763777882069021527od_o_o @ Za @ Zb ) )
       => ~ ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Xa3 @ Xb ) @ ( produc763777882069021527od_o_o @ A4 @ B3 ) ) @ R2 )
             => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A4 @ B3 ) @ ( produc763777882069021527od_o_o @ Za @ Zb ) ) @ ( transi3610599525415160810od_o_o @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_851_converse__rtranclE2,axiom,
    ! [Xa3: set_Product_prod_o_o,Xb: set_Pr1261947904930325089at_nat,Za: set_Product_prod_o_o,Zb: set_Pr1261947904930325089at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Xa3 @ Xb ) @ ( produc8517790099723286449at_nat @ Za @ Zb ) ) @ ( transi4536970581772660548at_nat @ R2 ) )
     => ( ( ( produc8517790099723286449at_nat @ Xa3 @ Xb )
         != ( produc8517790099723286449at_nat @ Za @ Zb ) )
       => ~ ! [A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Xa3 @ Xb ) @ ( produc8517790099723286449at_nat @ A4 @ B3 ) ) @ R2 )
             => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A4 @ B3 ) @ ( produc8517790099723286449at_nat @ Za @ Zb ) ) @ ( transi4536970581772660548at_nat @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_852_converse__rtranclE2,axiom,
    ! [Xa3: set_Pr1261947904930325089at_nat,Xb: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat,Zb: set_Pr1261947904930325089at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Xa3 @ Xb ) @ ( produc2922128104949294807at_nat @ Za @ Zb ) ) @ ( transi5049001397978532970at_nat @ R2 ) )
     => ( ( ( produc2922128104949294807at_nat @ Xa3 @ Xb )
         != ( produc2922128104949294807at_nat @ Za @ Zb ) )
       => ~ ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Xa3 @ Xb ) @ ( produc2922128104949294807at_nat @ A4 @ B3 ) ) @ R2 )
             => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A4 @ B3 ) @ ( produc2922128104949294807at_nat @ Za @ Zb ) ) @ ( transi5049001397978532970at_nat @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_853_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Product_prod_o_o,Bx: set_Product_prod_o_o,By: set_Product_prod_o_o,R2: set_Pr1013884327308616903od_o_o,P2: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o,Aa: set_Product_prod_o_o,Ba: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A4 @ B3 ) @ ( produc763777882069021527od_o_o @ Aa @ Ba ) ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Aa @ Ba ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_854_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Pr1261947904930325089at_nat,Bx: set_Product_prod_o_o,By: set_Pr1261947904930325089at_nat,R2: set_Pr4427867301352232263at_nat,P2: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat,Aa: set_Product_prod_o_o,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A4 @ B3 ) @ ( produc8517790099723286449at_nat @ Aa @ Ba ) ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Aa @ Ba ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_855_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Pr1261947904930325089at_nat,Ay: set_Pr1261947904930325089at_nat,Bx: set_Pr1261947904930325089at_nat,By: set_Pr1261947904930325089at_nat,R2: set_Pr5564308138774400199at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A4 @ B3 ) @ ( produc2922128104949294807at_nat @ Aa @ Ba ) ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Aa @ Ba ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_856_rtrancl__Un__separatorE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,P2: set_Pr1932065953672099015od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ ( sup_su4078869789063261851od_o_o @ P2 @ Q3 ) ) )
     => ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X ) @ ( transi8975321201238539274od_o_o @ P2 ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ Q3 )
             => ( X = Y4 ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_857_rtrancl__Un__separatorE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P2: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ ( sup_su5525570899277871387at_nat @ P2 @ Q3 ) ) )
     => ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ ( transi6170703685534862052at_nat @ P2 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ Q3 )
             => ( X = Y4 ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_858_rtrancl__Un__separator__converseE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,P2: set_Pr1932065953672099015od_o_o,Q3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ ( sup_su4078869789063261851od_o_o @ P2 @ Q3 ) ) )
     => ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ B ) @ ( transi8975321201238539274od_o_o @ P2 ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X ) @ Q3 )
             => ( Y4 = X ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_859_rtrancl__Un__separator__converseE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P2: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ ( sup_su5525570899277871387at_nat @ P2 @ Q3 ) ) )
     => ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ B ) @ ( transi6170703685534862052at_nat @ P2 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X ) @ Q3 )
             => ( Y4 = X ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_860_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( transi8975321201238539274od_o_o @ R2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_861_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( transi6170703685534862052at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_862_converse__rtrancl__induct,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( P2 @ B )
       => ( ! [Y4: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z2 ) @ R2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z2 @ B ) @ ( transi8975321201238539274od_o_o @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y4 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_863_converse__rtrancl__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( P2 @ B )
       => ( ! [Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z2 ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z2 @ B ) @ ( transi6170703685534862052at_nat @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y4 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_864_converse__rtranclE,axiom,
    ! [X2: set_Product_prod_o_o,Z: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( X2 != Z )
       => ~ ! [Y4: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R2 )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_865_converse__rtranclE,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( X2 != Z )
       => ~ ! [Y4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R2 )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_866_rtrancl__induct,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( P2 @ A )
       => ( ! [Y4: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y4 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z2 ) @ R2 )
               => ( ( P2 @ Y4 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_867_rtrancl__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( P2 @ A )
       => ( ! [Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y4 ) @ ( transi6170703685534862052at_nat @ R2 ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z2 ) @ R2 )
               => ( ( P2 @ Y4 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_868_rtrancl__trans,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,Z: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_869_rtrancl__trans,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_870_rtranclE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y4 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_871_rtranclE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y4 ) @ ( transi6170703685534862052at_nat @ R2 ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_872_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ R2 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_873_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ R2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_874_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_875_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( transi6170703685534862052at_nat @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_876_rtrancl_Osimps,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
      = ( ? [A3: set_Product_prod_o_o] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: set_Product_prod_o_o,B4: set_Product_prod_o_o,C2: set_Product_prod_o_o] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B4 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B4 @ C2 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_877_rtrancl_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( transi6170703685534862052at_nat @ R2 ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B4 ) @ ( transi6170703685534862052at_nat @ R2 ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ C2 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_878_rtrancl_Ocases,axiom,
    ! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ B3 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B3 @ A22 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_879_rtrancl_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( transi6170703685534862052at_nat @ R2 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ B3 ) @ ( transi6170703685534862052at_nat @ R2 ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_880_acyclic__insert,axiom,
    ! [Y: set_Product_prod_o_o,X2: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( transi6732886088652640214od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ Y @ X2 ) @ R2 ) )
      = ( ( transi6732886088652640214od_o_o @ R2 )
        & ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ).

% acyclic_insert
thf(fact_881_acyclic__insert,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( transi5940216143853766832at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ Y @ X2 ) @ R2 ) )
      = ( ( transi5940216143853766832at_nat @ R2 )
        & ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ).

% acyclic_insert
thf(fact_882_comp__the__Some,axiom,
    ( ( comp_option_a_a_a @ the_a @ some_a )
    = id_a ) ).

% comp_the_Some
thf(fact_883_single__valued__confluent,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( single7370670047157432131od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( transi8975321201238539274od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ ( transi8975321201238539274od_o_o @ R2 ) )
            | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z @ Y ) @ ( transi8975321201238539274od_o_o @ R2 ) ) ) ) ) ) ).

% single_valued_confluent
thf(fact_884_single__valued__confluent,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( single6503460853754975939at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( transi6170703685534862052at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ ( transi6170703685534862052at_nat @ R2 ) )
            | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z @ Y ) @ ( transi6170703685534862052at_nat @ R2 ) ) ) ) ) ) ).

% single_valued_confluent
thf(fact_885_finite__ordLess__infinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( finite_finite_nat @ ( field_nat @ R2 ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R5 ) )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_886_finite__ordLess__infinite,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( finite_finite_o @ ( field_o @ R2 ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R5 ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_887_finite__ordLess__infinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( finite_finite_nat @ ( field_nat @ R2 ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R5 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_888_finite__ordLess__infinite,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( finite_finite_o @ ( field_o @ R2 ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R5 ) )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_889_ordLess__irreflexive,axiom,
    ! [R2: set_Product_prod_o_o] :
      ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ).

% ordLess_irreflexive
thf(fact_890_ordLess__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R2 ) @ bNF_We3448473429330749420at_nat ) ).

% ordLess_irreflexive
thf(fact_891_ordLess__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_892_ordLess__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_893_ordLess__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_894_ordLess__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_895_ordLess__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_896_ordLess__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_897_single__valuedD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( single7370670047157432131od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ R2 )
         => ( Y = Z ) ) ) ) ).

% single_valuedD
thf(fact_898_single__valuedD,axiom,
    ! [R2: set_Pr457366540195662369at_nat,X2: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( single7212042597521132957at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y ) @ R2 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Z ) @ R2 )
         => ( Y = Z ) ) ) ) ).

% single_valuedD
thf(fact_899_single__valuedD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( single6503460853754975939at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ R2 )
         => ( Y = Z ) ) ) ) ).

% single_valuedD
thf(fact_900_single__valuedI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y4 ) @ R2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 )
           => ( Y4 = Z2 ) ) )
     => ( single7370670047157432131od_o_o @ R2 ) ) ).

% single_valuedI
thf(fact_901_single__valuedI,axiom,
    ! [R2: set_Pr457366540195662369at_nat] :
      ( ! [X: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ Y4 ) @ R2 )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ Z2 ) @ R2 )
           => ( Y4 = Z2 ) ) )
     => ( single7212042597521132957at_nat @ R2 ) ) ).

% single_valuedI
thf(fact_902_single__valuedI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ R2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 )
           => ( Y4 = Z2 ) ) )
     => ( single6503460853754975939at_nat @ R2 ) ) ).

% single_valuedI
thf(fact_903_single__valued__def,axiom,
    ( single7370670047157432131od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R3 )
         => ! [Z3: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z3 ) @ R3 )
             => ( Y3 = Z3 ) ) ) ) ) ).

% single_valued_def
thf(fact_904_single__valued__def,axiom,
    ( single7212042597521132957at_nat
    = ( ^ [R3: set_Pr457366540195662369at_nat] :
        ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) @ R3 )
         => ! [Z3: set_Pr1261947904930325089at_nat] :
              ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ Z3 ) @ R3 )
             => ( Y3 = Z3 ) ) ) ) ) ).

% single_valued_def
thf(fact_905_single__valued__def,axiom,
    ( single6503460853754975939at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
         => ! [Z3: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ R3 )
             => ( Y3 = Z3 ) ) ) ) ) ).

% single_valued_def
thf(fact_906_wf__bounded__measure,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,Ub: set_Product_prod_o_o > nat,F: set_Product_prod_o_o > nat] :
      ( ! [A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B3 @ A4 ) @ R2 )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R2 ) ) ).

% wf_bounded_measure
thf(fact_907_wf__bounded__measure,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,Ub: set_Pr1261947904930325089at_nat > nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ! [A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ A4 ) @ R2 )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R2 ) ) ).

% wf_bounded_measure
thf(fact_908_in__measure,axiom,
    ! [X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,F: set_Product_prod_o_o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ ( measur5705974739304014543od_o_o @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_909_in__measure,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ ( measur1827424007717751593at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_910_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_o2 @ 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_911_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_nat2 @ F2 @ A2 ) @ B2 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We3448473429330749420at_nat ) ) ).

% card_of_ordLess
thf(fact_912_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_913_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_914_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R2 ) ) @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_915_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_916_card__of__ordLess2,axiom,
    ! [B2: set_o,A2: set_o] :
      ( ( B2 != bot_bot_set_o )
     => ( ( ~ ? [F2: $o > $o] :
                ( ( image_o_o2 @ 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_917_card__of__ordLess2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( B2 != bot_bot_set_nat )
     => ( ( ~ ? [F2: nat > nat] :
                ( ( image_nat_nat2 @ F2 @ A2 )
                = B2 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% card_of_ordLess2
thf(fact_918_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_919_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_920_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_921_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_922_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_923_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_924_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_925_card__of__Un__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ) ).

% card_of_Un_ordLess_infinite_Field
thf(fact_926_card__of__underS,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( member2182030262393914128od_o_o @ ( produc23025780167396183od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ ( order_1929855754104155309od_o_o @ R2 @ A ) ) @ R2 ) @ bNF_We9142809693765176714od_o_o ) ) ) ).

% card_of_underS
thf(fact_927_card__of__underS,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( member4079835277323035408at_nat @ ( produc8612946581105484631at_nat @ ( bNF_Ca8664924986033767453at_nat @ ( order_6745690092771398279at_nat @ R2 @ A ) ) @ R2 ) @ bNF_We1318246335796365322at_nat ) ) ) ).

% card_of_underS
thf(fact_928_card__of__underS,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( member2845199937591886608at_nat @ ( produc7545183385222921047at_nat @ ( bNF_Ca9194600574093320387at_nat @ ( order_3736018788886245933at_nat @ R2 @ A ) ) @ R2 ) @ bNF_We2594139806602030986at_nat ) ) ) ).

% card_of_underS
thf(fact_929_card__of__underS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( order_underS_nat @ R2 @ A ) ) @ R2 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% card_of_underS
thf(fact_930_card__of__underS,axiom,
    ! [R2: set_Product_prod_o_o,A: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( order_underS_o @ R2 @ A ) ) @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% card_of_underS
thf(fact_931_Card__order__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat,Z: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( X2 != Y )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 )
         => ( ( Y != Z )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
             => ( ( X2 != Z )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_932_Card__order__trans,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X2: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( X2 != Y )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y ) @ R2 )
         => ( ( Y != Z )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
             => ( ( X2 != Z )
                & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_933_Card__order__trans,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( X2 != Y )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y ) @ R2 )
         => ( ( Y != Z )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
             => ( ( X2 != Z )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_934_Card__order__trans,axiom,
    ! [R2: set_Product_prod_o_o,X2: $o,Y: $o,Z: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( X2 != Y )
       => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y ) @ R2 )
         => ( ( Y != Z )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z ) @ R2 )
             => ( ( X2 != Z )
                & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z ) @ R2 ) ) ) ) ) ) ) ).

% Card_order_trans
thf(fact_935_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ~ ( finite820925532130942480od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ? [X: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X @ ( field_5121862139100741428od_o_o @ R2 ) )
              & ( A != X )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_936_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ~ ( finite8072506237346428522at_nat @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
         => ? [X: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X @ ( field_7361900449976055182at_nat @ R2 ) )
              & ( A != X )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_937_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ~ ( finite4343798906461161616at_nat @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
         => ? [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ ( field_2862844103579723188at_nat @ R2 ) )
              & ( A != X )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_938_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
       => ( ( member_nat @ A @ ( field_nat @ R2 ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ ( field_nat @ R2 ) )
              & ( A != X )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_939_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ~ ( finite9011636195296603184od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ? [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R2 ) )
              & ( A != X )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_940_infinite__Card__order__limit,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ~ ( finite9047747110432174090at_nat @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
         => ? [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R2 ) )
              & ( A != X )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_941_infinite__Card__order__limit,axiom,
    ! [R2: set_Product_prod_o_o,A: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
       => ( ( member_o @ A @ ( field_o @ R2 ) )
         => ? [X: $o] :
              ( ( member_o @ X @ ( field_o @ R2 ) )
              & ( A = (~ X) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ X ) @ R2 ) ) ) ) ) ).

% infinite_Card_order_limit
thf(fact_942_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
           => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We7041958966232365723at_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_943_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
           => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We4935586570105487173_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_944_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
           => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We1194315142955236767at_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_945_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
         => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
           => ( member989342599902505446at_nat @ ( produc4966347635449911085at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We7066948511106741057_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_946_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
           => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We7847786823715778023_o_o_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_947_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
           => ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We2130777192033087497_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_948_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
           => ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We1503459268740929763at_o_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_949_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
         => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
           => ( member3285728679383846360od_o_o @ ( produc9174727453167716639od_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We9222787287886609421_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
thf(fact_950_Cinfinite__gt__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We3448473429330749420at_nat ) ) ).

% Cinfinite_gt_empty
thf(fact_951_Cinfinite__gt__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We8182288985678559134_o_nat ) ) ).

% Cinfinite_gt_empty
thf(fact_952_Cinfinite__gt__empty,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% Cinfinite_gt_empty
thf(fact_953_Cinfinite__gt__empty,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We2061588309931968636_nat_o ) ) ).

% Cinfinite_gt_empty
thf(fact_954_card__of__empty1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        | ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_empty1
thf(fact_955_card__of__empty1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        | ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_empty1
thf(fact_956_card__of__empty1,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        | ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_empty1
thf(fact_957_card__of__empty1,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        | ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ).

% card_of_empty1
thf(fact_958_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member989342599902505446at_nat @ ( produc4966347635449911085at_nat @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We3900823710022635230_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_959_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We7395407223230661442at_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_960_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We1913306613526136040_o_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_961_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We5086521355153908664at_nat ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_962_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We3513324756044439818_o_o_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_963_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We8188024427803757478_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_964_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We7560706504511599744at_o_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_965_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member3285728679383846360od_o_o @ ( produc9174727453167716639od_o_o @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We6200507331307258288_nat_o ) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
thf(fact_966_ordLeq__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_967_ordLeq__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_968_ordLeq__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_969_ordLeq__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_970_ordLeq__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_971_ordLeq__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_ordLeq__Plus__mono,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We7217180388603359757um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_982_ordLeq__Plus__mono,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We1079009337437905725um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_983_ordLeq__Plus__mono,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We2094493454282013185_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_984_ordLeq__Plus__mono,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7239028013353664305_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_985_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8976728017661452183um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_986_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8987810345020088115um_o_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_987_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We3854041083340105611_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_988_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We8273322497166968871_o_nat ) ) ) ).

% ordLeq_Plus_mono
thf(fact_989_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We2558484984367686093_nat_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_990_ordLeq__Plus__mono,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We7362575607395598641_nat_o ) ) ) ).

% ordLeq_Plus_mono
thf(fact_991_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_992_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_993_not__ordLeq__ordLess,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% not_ordLeq_ordLess
thf(fact_994_not__ordLeq__ordLess,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We2061588309931968636_nat_o ) ) ).

% not_ordLeq_ordLess
thf(fact_995_not__ordLeq__ordLess,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We3448473429330749420at_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_996_not__ordLeq__ordLess,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We8182288985678559134_o_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_997_not__ordLess__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_998_not__ordLess__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% not_ordLess_ordLeq
thf(fact_999_not__ordLess__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_1000_not__ordLess__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ).

% not_ordLess_ordLeq
thf(fact_1001_ordLess__imp__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLess_imp_ordLeq
thf(fact_1002_ordLess__imp__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1003_ordLess__imp__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1004_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1005_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1006_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1007_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1008_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1009_ordLeq__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1010_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1011_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1012_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1013_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1014_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1015_ordLess__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1016_card__of__image,axiom,
    ! [F: $o > $o,A2: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( image_o_o2 @ F @ A2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) ) @ bNF_We8115293805659342695eq_o_o ) ).

% card_of_image
thf(fact_1017_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_1018_card__of__image,axiom,
    ! [F: nat > nat,A2: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( image_nat_nat2 @ F @ A2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ).

% card_of_image
thf(fact_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_card__of__mono2,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R5 ) ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_mono2
thf(fact_1034_card__of__mono2,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R5 ) ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_mono2
thf(fact_1035_card__of__mono2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R2 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R5 ) ) ) @ bNF_We6538615181293537951_nat_o ) ) ).

% card_of_mono2
thf(fact_1036_card__of__mono2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R2 ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R5 ) ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_mono2
thf(fact_1037_exists__minim__Card__order,axiom,
    ! [R4: set_se7855581050983116737at_nat] :
      ( ( R4 != bot_bo3083307316010499117at_nat )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ R4 )
           => ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ X ) @ X ) )
       => ? [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ R4 )
            & ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R4 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Xa ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Card_order
thf(fact_1038_exists__minim__Card__order,axiom,
    ! [R4: set_se491199571785101031od_o_o] :
      ( ( R4 != bot_bo8006819982598807891od_o_o )
     => ( ! [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ R4 )
           => ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ X ) @ X ) )
       => ? [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ R4 )
            & ! [Xa: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Xa @ R4 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Xa ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% exists_minim_Card_order
thf(fact_1039_Cinfinite__limit,axiom,
    ! [X2: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ X2 @ ( field_5121862139100741428od_o_o @ R2 ) )
     => ( ( ( bNF_Ca9205897942416659591od_o_o @ R2 )
          & ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 ) )
       => ? [X: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X @ ( field_5121862139100741428od_o_o @ R2 ) )
            & ( X2 != X )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1040_Cinfinite__limit,axiom,
    ! [X2: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ X2 @ ( field_7361900449976055182at_nat @ R2 ) )
     => ( ( ( bNF_Ca6670480423259178337at_nat @ R2 )
          & ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 ) )
       => ? [X: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X @ ( field_7361900449976055182at_nat @ R2 ) )
            & ( X2 != X )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1041_Cinfinite__limit,axiom,
    ! [X2: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ X2 @ ( field_2862844103579723188at_nat @ R2 ) )
     => ( ( ( bNF_Ca8759257893432750087at_nat @ R2 )
          & ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 ) )
       => ? [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ ( field_2862844103579723188at_nat @ R2 ) )
            & ( X2 != X )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1042_Cinfinite__limit,axiom,
    ! [X2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ X2 @ ( field_nat @ R2 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
       => ? [X: nat] :
            ( ( member_nat @ X @ ( field_nat @ R2 ) )
            & ( X2 != X )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1043_Cinfinite__limit,axiom,
    ! [X2: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ X2 @ ( field_6101175518749917524od_o_o @ R2 ) )
     => ( ( ( bNF_Ca140147938854208935od_o_o @ R2 )
          & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 ) )
       => ? [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R2 ) )
            & ( X2 != X )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1044_Cinfinite__limit,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ X2 @ ( field_3145993188331922222at_nat @ R2 ) )
     => ( ( ( bNF_Ca4089899758288986625at_nat @ R2 )
          & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 ) )
       => ? [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R2 ) )
            & ( X2 != X )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1045_Cinfinite__limit,axiom,
    ! [X2: $o,R2: set_Product_prod_o_o] :
      ( ( member_o @ X2 @ ( field_o @ R2 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
       => ? [X: $o] :
            ( ( member_o @ X @ ( field_o @ R2 ) )
            & ( X2 = (~ X) )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X ) @ R2 ) ) ) ) ).

% Cinfinite_limit
thf(fact_1046_Cinfinite__limit2,axiom,
    ! [X1: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,X22: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ X1 @ ( field_5121862139100741428od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ X22 @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( ( bNF_Ca9205897942416659591od_o_o @ R2 )
            & ( bNF_Ca4198200952365436517od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 ) )
         => ? [X: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X @ ( field_5121862139100741428od_o_o @ R2 ) )
              & ( X1 != X )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1047_Cinfinite__limit2,axiom,
    ! [X1: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,X22: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ X1 @ ( field_7361900449976055182at_nat @ R2 ) )
     => ( ( member4095101504841534314at_nat @ X22 @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( ( bNF_Ca6670480423259178337at_nat @ R2 )
            & ( bNF_Ca7438713789483989055at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 ) )
         => ? [X: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X @ ( field_7361900449976055182at_nat @ R2 ) )
              & ( X1 != X )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1048_Cinfinite__limit2,axiom,
    ! [X1: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,X22: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ X1 @ ( field_2862844103579723188at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ X22 @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( ( bNF_Ca8759257893432750087at_nat @ R2 )
            & ( bNF_Ca8230877790521596901at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 ) )
         => ? [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ ( field_2862844103579723188at_nat @ R2 ) )
              & ( X1 != X )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1049_Cinfinite__limit2,axiom,
    ! [X1: nat,R2: set_Pr1261947904930325089at_nat,X22: nat] :
      ( ( member_nat @ X1 @ ( field_nat @ R2 ) )
     => ( ( member_nat @ X22 @ ( field_nat @ R2 ) )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ ( field_nat @ R2 ) )
              & ( X1 != X )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1050_Cinfinite__limit2,axiom,
    ! [X1: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,X22: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ X1 @ ( field_6101175518749917524od_o_o @ R2 ) )
     => ( ( member3157015407008874800od_o_o @ X22 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( bNF_Ca140147938854208935od_o_o @ R2 )
            & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 ) )
         => ? [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R2 ) )
              & ( X1 != X )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1051_Cinfinite__limit2,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,X22: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ X1 @ ( field_3145993188331922222at_nat @ R2 ) )
     => ( ( member2643936169264416010at_nat @ X22 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( bNF_Ca4089899758288986625at_nat @ R2 )
            & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 ) )
         => ? [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R2 ) )
              & ( X1 != X )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X1 @ X ) @ R2 )
              & ( X22 != X )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1052_Cinfinite__limit2,axiom,
    ! [X1: $o,R2: set_Product_prod_o_o,X22: $o] :
      ( ( member_o @ X1 @ ( field_o @ R2 ) )
     => ( ( member_o @ X22 @ ( field_o @ R2 ) )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ? [X: $o] :
              ( ( member_o @ X @ ( field_o @ R2 ) )
              & ( X1 = (~ X) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X1 @ X ) @ R2 )
              & ( X22 = (~ X) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X22 @ X ) @ R2 ) ) ) ) ) ).

% Cinfinite_limit2
thf(fact_1053_card__of__ordLeqI,axiom,
    ! [F: $o > $o,A2: set_o,B2: set_o] :
      ( ( inj_on_o_o @ F @ A2 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A2 )
           => ( member_o @ ( F @ A4 ) @ 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_1054_card__of__ordLeqI,axiom,
    ! [F: $o > nat,A2: set_o,B2: set_nat] :
      ( ( inj_on_o_nat @ F @ A2 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A2 )
           => ( member_nat @ ( F @ A4 ) @ B2 ) )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1055_card__of__ordLeqI,axiom,
    ! [F: nat > nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( member_nat @ ( F @ A4 ) @ B2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1056_card__of__ordLeqI,axiom,
    ! [F: produc2934264451710624999od_o_o > nat,A2: set_Pr1932065953672099015od_o_o,B2: set_nat] :
      ( ( inj_on9163735697967710470_o_nat @ F @ A2 )
     => ( ! [A4: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ A4 @ A2 )
           => ( member_nat @ ( F @ A4 ) @ B2 ) )
       => ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We7173072801150825232_o_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1057_card__of__ordLeqI,axiom,
    ! [F: produc732395585841259969at_nat > nat,A2: set_Pr457366540195662369at_nat,B2: set_nat] :
      ( ( inj_on9206849020120836832at_nat @ F @ A2 )
     => ( ! [A4: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ A4 @ A2 )
           => ( member_nat @ ( F @ A4 ) @ B2 ) )
       => ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We8470213457150283754at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1058_card__of__ordLeqI,axiom,
    ! [F: produc3843707927480180839at_nat > nat,A2: set_Pr4329608150637261639at_nat,B2: set_nat] :
      ( ( inj_on7963584518437281414at_nat @ F @ A2 )
     => ( ! [A4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ A4 @ A2 )
           => ( member_nat @ ( F @ A4 ) @ B2 ) )
       => ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We8384597595354199696at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1059_card__of__ordLeqI,axiom,
    ! [F: nat > produc2934264451710624999od_o_o,A2: set_nat,B2: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on2663835283014092644od_o_o @ F @ A2 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( member444158400953824016od_o_o @ ( F @ A4 ) @ 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_1060_card__of__ordLeqI,axiom,
    ! [F: nat > produc732395585841259969at_nat,A2: set_nat,B2: set_Pr457366540195662369at_nat] :
      ( ( inj_on400939309715711422at_nat @ F @ A2 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( member4095101504841534314at_nat @ ( F @ A4 ) @ B2 ) )
       => ( member156447988302929782at_nat @ ( produc5313346920894465213at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca8664924986033767453at_nat @ B2 ) ) @ bNF_We8887675783599934152at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1061_card__of__ordLeqI,axiom,
    ! [F: nat > produc3843707927480180839at_nat,A2: set_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F @ A2 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( member8757157785044589968at_nat @ ( F @ A4 ) @ B2 ) )
       => ( member8293877710313108214at_nat @ ( produc5916770540428940861at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca9194600574093320387at_nat @ B2 ) ) @ bNF_We7685085753668031982at_nat ) ) ) ).

% card_of_ordLeqI
thf(fact_1062_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 )
     => ( ! [A4: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ A4 @ A2 )
           => ( member444158400953824016od_o_o @ ( F @ A4 ) @ 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_1063_exists__minim__Well__order,axiom,
    ! [R4: set_se491199571785101031od_o_o] :
      ( ( R4 != bot_bo8006819982598807891od_o_o )
     => ( ! [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ R4 )
           => ( order_2615775362358406454r_on_o @ ( field_o @ X ) @ X ) )
       => ? [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ R4 )
            & ! [Xa: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Xa @ R4 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Xa ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1064_exists__minim__Well__order,axiom,
    ! [R4: set_se7855581050983116737at_nat] :
      ( ( R4 != bot_bo3083307316010499117at_nat )
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ R4 )
           => ( order_2888998067076097458on_nat @ ( field_nat @ X ) @ X ) )
       => ? [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ R4 )
            & ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R4 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Xa ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1065_ordLeq__Well__order__simp,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1066_ordLeq__Well__order__simp,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1067_ordLeq__Well__order__simp,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1068_ordLeq__Well__order__simp,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1069_ordLeq__reflexive,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLeq_reflexive
thf(fact_1070_ordLeq__reflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_reflexive
thf(fact_1071_ordLeq__total,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1072_ordLeq__total,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLeq_total
thf(fact_1073_ordLeq__total,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLeq_total
thf(fact_1074_ordLeq__total,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1075_card__of__least,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ A2 @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_least
thf(fact_1076_card__of__least,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ A2 @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_least
thf(fact_1077_card__order__on__def,axiom,
    ( bNF_Ca1281551314933786834on_nat
    = ( ^ [A7: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ( order_2888998067076097458on_nat @ A7 @ R3 )
          & ! [R7: set_Pr1261947904930325089at_nat] :
              ( ( order_2888998067076097458on_nat @ A7 @ R7 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% card_order_on_def
thf(fact_1078_card__order__on__def,axiom,
    ( bNF_Ca8331644756375544342r_on_o
    = ( ^ [A7: set_o,R3: set_Product_prod_o_o] :
          ( ( order_2615775362358406454r_on_o @ A7 @ R3 )
          & ! [R7: set_Product_prod_o_o] :
              ( ( order_2615775362358406454r_on_o @ A7 @ R7 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R3 @ R7 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% card_order_on_def
thf(fact_1079_card__of__ordLeq2,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( ? [G2: $o > $o] :
              ( ( image_o_o2 @ 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_1080_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_1081_card__of__ordLeq2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ( ? [G2: nat > nat] :
              ( ( image_nat_nat2 @ G2 @ B2 )
              = A2 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% card_of_ordLeq2
thf(fact_1082_surj__imp__ordLeq,axiom,
    ! [B2: set_o,F: $o > $o,A2: set_o] :
      ( ( ord_less_eq_set_o @ B2 @ ( image_o_o2 @ 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_1083_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_1084_surj__imp__ordLeq,axiom,
    ! [B2: set_nat,F: nat > nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat2 @ F @ A2 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ ( bNF_Ca3793111618940312692of_nat @ A2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% surj_imp_ordLeq
thf(fact_1085_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_1086_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_1087_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_1088_Card__order__iff__ordLeq__card__of,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R2 ) ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% Card_order_iff_ordLeq_card_of
thf(fact_1089_Card__order__iff__ordLeq__card__of,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% Card_order_iff_ordLeq_card_of
thf(fact_1090_card__of__Field__ordLess,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R2 ) ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_Field_ordLess
thf(fact_1091_card__of__Field__ordLess,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R2 ) ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_Field_ordLess
thf(fact_1092_ordLess__or__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1093_ordLess__or__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1094_ordLess__or__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1095_ordLess__or__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1096_not__ordLeq__iff__ordLess,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1097_not__ordLeq__iff__ordLess,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We3435943820185352641_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1098_not__ordLeq__iff__ordLess,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We2031159114419172489at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1099_not__ordLeq__iff__ordLess,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We6538615181293537951_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1100_not__ordLess__iff__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We2061588309931968636_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1101_not__ordLess__iff__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We2568191115274586186ss_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1102_not__ordLess__iff__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We3448473429330749420at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1103_not__ordLess__iff__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We8182288985678559134_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_o2 @ 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_1112_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_1113_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_nat2 @ F2 @ A2 ) @ B2 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ ( bNF_Ca3793111618940312692of_nat @ B2 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_ordLeq
thf(fact_1114_Card__order__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% Card_order_empty
thf(fact_1115_Card__order__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% Card_order_empty
thf(fact_1116_Card__order__empty,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% Card_order_empty
thf(fact_1117_Card__order__empty,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ).

% Card_order_empty
thf(fact_1118_Cinfinite__limit__finite,axiom,
    ! [X8: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X8 )
     => ( ( ord_less_eq_set_nat @ X8 @ ( field_nat @ R2 ) )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ ( field_nat @ R2 ) )
              & ! [Xa: nat] :
                  ( ( member_nat @ Xa @ X8 )
                 => ( ( Xa != X )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X ) @ R2 ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_1119_Cinfinite__limit__finite,axiom,
    ! [X8: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( finite9011636195296603184od_o_o @ X8 )
     => ( ( ord_le2654036014095304327od_o_o @ X8 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( bNF_Ca140147938854208935od_o_o @ R2 )
            & ( bNF_Ca4526578624105657733od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 ) )
         => ? [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R2 ) )
              & ! [Xa: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Xa @ X8 )
                 => ( ( Xa != X )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa @ X ) @ R2 ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_1120_Cinfinite__limit__finite,axiom,
    ! [X8: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X8 )
     => ( ( ord_le2077887516847798113at_nat @ X8 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( bNF_Ca4089899758288986625at_nat @ R2 )
            & ( bNF_Ca4062989147270679263at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 ) )
         => ? [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R2 ) )
              & ! [Xa: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Xa @ X8 )
                 => ( ( Xa != X )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa @ X ) @ R2 ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_1121_Cinfinite__limit__finite,axiom,
    ! [X8: set_o,R2: set_Product_prod_o_o] :
      ( ( finite_finite_o @ X8 )
     => ( ( ord_less_eq_set_o @ X8 @ ( field_o @ R2 ) )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ? [X: $o] :
              ( ( member_o @ X @ ( field_o @ R2 ) )
              & ! [Xa: $o] :
                  ( ( member_o @ Xa @ X8 )
                 => ( ( Xa = (~ X) )
                    & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa @ X ) @ R2 ) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
thf(fact_1122_Card__order__singl__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B: $o] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( ( field_nat @ R2 )
         != bot_bot_set_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_1123_Card__order__singl__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( ( field_nat @ R2 )
         != bot_bot_set_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_1124_Card__order__singl__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,B: $o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( ( field_o @ R2 )
         != bot_bot_set_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( insert_o @ B @ bot_bot_set_o ) ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_1125_Card__order__singl__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,B: nat] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( ( field_o @ R2 )
         != bot_bot_set_o )
       => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ).

% Card_order_singl_ordLeq
thf(fact_1126_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_1127_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
         => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_1128_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_o,B2: set_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_1129_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [R2: set_Product_prod_o_o,A2: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
         => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
thf(fact_1130_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We3448473429330749420at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_1131_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat,B2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8182288985678559134_o_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_1132_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_1133_Un__Cinfinite__bound__strict,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o,B2: set_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2061588309931968636_nat_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ).

% Un_Cinfinite_bound_strict
thf(fact_1134_Un__Cinfinite__bound,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat,B2: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_1135_Un__Cinfinite__bound,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat,B2: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
       => ( ( ( bNF_Ca7751713578713111600te_nat @ R2 )
            & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_1136_Un__Cinfinite__bound,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o,B2: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A2 @ B2 ) ) @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_1137_Un__Cinfinite__bound,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o,B2: set_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ B2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
       => ( ( ( bNF_Ca7082918126634695480nite_o @ R2 )
            & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
         => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% Un_Cinfinite_bound
thf(fact_1138_ordLeq__refl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_refl
thf(fact_1139_ordLeq__refl,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLeq_refl
thf(fact_1140_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_1141_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_1142_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_1143_ex__toCard__pred,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ? [X_1: $o > nat] : ( bNF_Gr988296774780055034_o_nat @ A2 @ R2 @ X_1 ) ) ) ).

% ex_toCard_pred
thf(fact_1144_ex__toCard__pred,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ? [X_1: nat > nat] : ( bNF_Gr4367463400984379536at_nat @ A2 @ R2 @ X_1 ) ) ) ).

% ex_toCard_pred
thf(fact_1145_ex__toCard__pred,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ? [X_1: $o > $o] : ( bNF_Gr3194703283472089838ed_o_o @ A2 @ R2 @ X_1 ) ) ) ).

% ex_toCard_pred
thf(fact_1146_ex__toCard__pred,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ? [X_1: nat > $o] : ( bNF_Gr4090968135888240344_nat_o @ A2 @ R2 @ X_1 ) ) ) ).

% ex_toCard_pred
thf(fact_1147_toCard__inj,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1261947904930325089at_nat,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R2 ) @ bNF_We7173072801150825232_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ X2 @ A2 )
         => ( ( member444158400953824016od_o_o @ Y @ A2 )
           => ( ( ( bNF_Gr2687907518133588746_o_nat @ A2 @ R2 @ X2 )
                = ( bNF_Gr2687907518133588746_o_nat @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1148_toCard__inj,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr1261947904930325089at_nat,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R2 ) @ bNF_We8470213457150283754at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ X2 @ A2 )
         => ( ( member4095101504841534314at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr859799933468256996at_nat @ A2 @ R2 @ X2 )
                = ( bNF_Gr859799933468256996at_nat @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1149_toCard__inj,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr1261947904930325089at_nat,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R2 ) @ bNF_We8384597595354199696at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ X2 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr8319706499545975946at_nat @ A2 @ R2 @ X2 )
                = ( bNF_Gr8319706499545975946at_nat @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1150_toCard__inj,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat,X2: $o,Y: $o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member_o @ X2 @ A2 )
         => ( ( member_o @ Y @ A2 )
           => ( ( ( bNF_Gr1235702096034787515_o_nat @ A2 @ R2 @ X2 )
                = ( bNF_Gr1235702096034787515_o_nat @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1151_toCard__inj,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat,X2: nat,Y: nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member_nat @ X2 @ A2 )
         => ( ( member_nat @ Y @ A2 )
           => ( ( ( bNF_Gr5321390424033619983at_nat @ A2 @ R2 @ X2 )
                = ( bNF_Gr5321390424033619983at_nat @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1152_toCard__inj,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Product_prod_o_o,X2: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( member5998968164356017744od_o_o @ ( produc8369371535328480407od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R2 ) @ bNF_We5212295730361461848_o_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ X2 @ A2 )
         => ( ( member444158400953824016od_o_o @ Y @ A2 )
           => ( ( ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R2 @ X2 )
                = ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1153_toCard__inj,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Product_prod_o_o,X2: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( member1980351565637979088od_o_o @ ( produc8102740133251482647od_o_o @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R2 ) @ bNF_We6477715483879687422_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ X2 @ A2 )
         => ( ( member4095101504841534314at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr7020601193219610500_nat_o @ A2 @ R2 @ X2 )
                = ( bNF_Gr7020601193219610500_nat_o @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1154_toCard__inj,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Product_prod_o_o,X2: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( member8532720315100152400od_o_o @ ( produc3565432567573771415od_o_o @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R2 ) @ bNF_We3095294111201691864_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ X2 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y @ A2 )
           => ( ( ( bNF_Gr9013816254238203998_nat_o @ A2 @ R2 @ X2 )
                = ( bNF_Gr9013816254238203998_nat_o @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1155_toCard__inj,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o,X2: $o,Y: $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member_o @ X2 @ A2 )
         => ( ( member_o @ Y @ A2 )
           => ( ( ( bNF_Gr748490023993654765rd_o_o @ A2 @ R2 @ X2 )
                = ( bNF_Gr748490023993654765rd_o_o @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1156_toCard__inj,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o,X2: nat,Y: nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member_nat @ X2 @ A2 )
         => ( ( member_nat @ Y @ A2 )
           => ( ( ( bNF_Gr4338373457142972825_nat_o @ A2 @ R2 @ X2 )
                = ( bNF_Gr4338373457142972825_nat_o @ A2 @ R2 @ Y ) )
              = ( X2 = Y ) ) ) ) ) ) ).

% toCard_inj
thf(fact_1157_Cfinite__ordLess__Cinfinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R2 )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1158_Cfinite__ordLess__Cinfinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca6649290325350769387te_nat @ R2 )
        & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ S ) @ bNF_We2061588309931968636_nat_o ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1159_Cfinite__ordLess__Cinfinite,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R2 )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( ( ( bNF_Ca7751713578713111600te_nat @ S )
          & ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ S ) @ S ) )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ S ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1160_Cfinite__ordLess__Cinfinite,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( ( bNF_Ca4027112139517550525nite_o @ R2 )
        & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 ) )
     => ( ( ( bNF_Ca7082918126634695480nite_o @ S )
          & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ S ) @ S ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ S ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% Cfinite_ordLess_Cinfinite
thf(fact_1161_toCard__pred__toCard,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( bNF_Gr988296774780055034_o_nat @ A2 @ R2 @ ( bNF_Gr1235702096034787515_o_nat @ A2 @ R2 ) ) ) ) ).

% toCard_pred_toCard
thf(fact_1162_toCard__pred__toCard,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( bNF_Gr4367463400984379536at_nat @ A2 @ R2 @ ( bNF_Gr5321390424033619983at_nat @ A2 @ R2 ) ) ) ) ).

% toCard_pred_toCard
thf(fact_1163_toCard__pred__toCard,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( bNF_Gr3194703283472089838ed_o_o @ A2 @ R2 @ ( bNF_Gr748490023993654765rd_o_o @ A2 @ R2 ) ) ) ) ).

% toCard_pred_toCard
thf(fact_1164_toCard__pred__toCard,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( bNF_Gr4090968135888240344_nat_o @ A2 @ R2 @ ( bNF_Gr4338373457142972825_nat_o @ A2 @ R2 ) ) ) ) ).

% toCard_pred_toCard
thf(fact_1165_fromCard__toCard,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Pr1261947904930325089at_nat,B: produc2934264451710624999od_o_o] :
      ( ( member7927422385807836842at_nat @ ( produc1598809612926693617at_nat @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R2 ) @ bNF_We7173072801150825232_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( ( bNF_Gr6179234852794809723_o_nat @ A2 @ R2 @ ( bNF_Gr2687907518133588746_o_nat @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1166_fromCard__toCard,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Pr1261947904930325089at_nat,B: produc732395585841259969at_nat] :
      ( ( member8415039729195402282at_nat @ ( produc4982878492540823665at_nat @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R2 ) @ bNF_We8470213457150283754at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( ( bNF_Gr6550272361737169109at_nat @ A2 @ R2 @ ( bNF_Gr859799933468256996at_nat @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1167_fromCard__toCard,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat] :
      ( ( member7359442244455152298at_nat @ ( produc5407148829662812401at_nat @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R2 ) @ bNF_We8384597595354199696at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( ( bNF_Gr2810406598121987579at_nat @ A2 @ R2 @ ( bNF_Gr8319706499545975946at_nat @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1168_fromCard__toCard,axiom,
    ! [A2: set_o,R2: set_Pr1261947904930325089at_nat,B: $o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member_o @ B @ A2 )
         => ( ( bNF_Gr6736587135095141548_o_nat @ A2 @ R2 @ ( bNF_Gr1235702096034787515_o_nat @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1169_fromCard__toCard,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We2031159114419172489at_nat )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member_nat @ B @ A2 )
         => ( ( bNF_Gr6953169559552526942at_nat @ A2 @ R2 @ ( bNF_Gr5321390424033619983at_nat @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1170_fromCard__toCard,axiom,
    ! [A2: set_Pr1932065953672099015od_o_o,R2: set_Product_prod_o_o,B: produc2934264451710624999od_o_o] :
      ( ( member5998968164356017744od_o_o @ ( produc8369371535328480407od_o_o @ ( bNF_Ca6121500631330824515od_o_o @ A2 ) @ R2 ) @ bNF_We5212295730361461848_o_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ B @ A2 )
         => ( ( bNF_Gr3541370040355373357_o_o_o @ A2 @ R2 @ ( bNF_Gr1716226591470254558_o_o_o @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1171_fromCard__toCard,axiom,
    ! [A2: set_Pr457366540195662369at_nat,R2: set_Product_prod_o_o,B: produc732395585841259969at_nat] :
      ( ( member1980351565637979088od_o_o @ ( produc8102740133251482647od_o_o @ ( bNF_Ca8664924986033767453at_nat @ A2 ) @ R2 ) @ bNF_We6477715483879687422_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ B @ A2 )
         => ( ( bNF_Gr2632823273528503123_nat_o @ A2 @ R2 @ ( bNF_Gr7020601193219610500_nat_o @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1172_fromCard__toCard,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Product_prod_o_o,B: produc3843707927480180839at_nat] :
      ( ( member8532720315100152400od_o_o @ ( produc3565432567573771415od_o_o @ ( bNF_Ca9194600574093320387at_nat @ A2 ) @ R2 ) @ bNF_We3095294111201691864_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ B @ A2 )
         => ( ( bNF_Gr5397592485871042733_nat_o @ A2 @ R2 @ ( bNF_Gr9013816254238203998_nat_o @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1173_fromCard__toCard,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o,B: $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A2 ) @ R2 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member_o @ B @ A2 )
         => ( ( bNF_Gr5731841836078664636rd_o_o @ A2 @ R2 @ ( bNF_Gr748490023993654765rd_o_o @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1174_fromCard__toCard,axiom,
    ! [A2: set_nat,R2: set_Product_prod_o_o,B: nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A2 ) @ R2 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member_nat @ B @ A2 )
         => ( ( bNF_Gr615886459348551050_nat_o @ A2 @ R2 @ ( bNF_Gr4338373457142972825_nat_o @ A2 @ R2 @ B ) )
            = B ) ) ) ) ).

% fromCard_toCard
thf(fact_1175_Card__order__Plus__infinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ P @ R2 ) @ bNF_We3435943820185352641_o_nat )
         => ( ( member4812592190795650218at_nat @ ( produc5868257188881964785at_nat @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R2 ) @ ( field_o @ P ) ) ) @ R2 ) @ bNF_We5202279374879649339_o_nat )
            & ( member6178483945413987626at_nat @ ( produc7234148943500302193at_nat @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ P ) @ ( field_nat @ R2 ) ) ) @ R2 ) @ bNF_We1461007947729398933at_nat ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1176_Card__order__Plus__infinite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_nat @ ( field_nat @ R2 ) )
     => ( ( bNF_Ca1281551314933786834on_nat @ ( field_nat @ R2 ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P @ R2 ) @ bNF_We2031159114419172489at_nat )
         => ( ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R2 ) @ ( field_nat @ P ) ) ) @ R2 ) @ bNF_We4717344193235055909at_nat )
            & ( member3047918529990910514at_nat @ ( produc6460772959230282105at_nat @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ P ) @ ( field_nat @ R2 ) ) ) @ R2 ) @ bNF_We4717344193235055909at_nat ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1177_Card__order__Plus__infinite,axiom,
    ! [R2: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ P @ R2 ) @ bNF_We6538615181293537951_nat_o )
         => ( ( member121125266993124560od_o_o @ ( produc8931464078238085911od_o_o @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R2 ) @ ( field_nat @ P ) ) ) @ R2 ) @ bNF_We8455499808419870611_nat_o )
            & ( member4590549062032530512od_o_o @ ( produc4177515836422716055od_o_o @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ P ) @ ( field_o @ R2 ) ) ) @ R2 ) @ bNF_We7828181885127712877at_o_o ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1178_Card__order__Plus__infinite,axiom,
    ! [R2: set_Product_prod_o_o,P: set_Product_prod_o_o] :
      ( ~ ( finite_finite_o @ ( field_o @ R2 ) )
     => ( ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ R2 ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P @ R2 ) @ bNF_We8115293805659342695eq_o_o )
         => ( ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R2 ) @ ( field_o @ P ) ) ) @ R2 ) @ bNF_We5455623833347018717_o_o_o )
            & ( member8030178841557499788od_o_o @ ( produc5317466613051064531od_o_o @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ P ) @ ( field_o @ R2 ) ) ) @ R2 ) @ bNF_We5455623833347018717_o_o_o ) ) ) ) ) ).

% Card_order_Plus_infinite
thf(fact_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_ordIso__symmetric,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We4116350175183677332_o_nat ) ) ).

% ordIso_symmetric
thf(fact_1189_ordIso__symmetric,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We7219021536291862642_nat_o ) ) ).

% ordIso_symmetric
thf(fact_1190_ordIso__symmetric,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We2654380646378065620so_o_o ) ) ).

% ordIso_symmetric
thf(fact_1191_ordIso__symmetric,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_symmetric
thf(fact_1192_ordIso__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1193_ordIso__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1194_ordIso__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1195_ordIso__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1196_ordIso__transitive,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1197_ordIso__transitive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1198_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_1199_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_1200_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_1201_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_1202_card__order__on__ordIso,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_Ca1281551314933786834on_nat @ A2 @ R2 )
     => ( ( bNF_Ca1281551314933786834on_nat @ A2 @ R5 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% card_order_on_ordIso
thf(fact_1203_card__order__on__ordIso,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( bNF_Ca8331644756375544342r_on_o @ A2 @ R2 )
     => ( ( bNF_Ca8331644756375544342r_on_o @ A2 @ R5 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% card_order_on_ordIso
thf(fact_1204_ordIso__iff__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We7219021536291862642_nat_o )
      = ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
        & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R2 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1205_ordIso__iff__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
      = ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
        & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R2 ) @ bNF_We6538615181293537951_nat_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1206_ordIso__iff__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1207_ordIso__iff__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R2 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1208_ordIso__imp__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1209_ordIso__imp__ordLeq,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordIso_imp_ordLeq
thf(fact_1210_ordIso__imp__ordLeq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1211_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1212_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1213_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1214_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1215_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1216_ordIso__ordLeq__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1217_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1218_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1219_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1220_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1221_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1222_ordLeq__ordIso__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1223_ordLess__ordIso__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1224_ordLess__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1225_ordLess__ordIso__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1226_ordLess__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1227_ordLess__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1228_ordLess__ordIso__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1229_ordIso__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1230_ordIso__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1231_ordIso__ordLess__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1232_ordIso__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1233_ordIso__ordLess__trans,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1234_ordIso__ordLess__trans,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1235_not__ordLess__ordIso,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ bNF_We2654380646378065620so_o_o ) ) ).

% not_ordLess_ordIso
thf(fact_1236_not__ordLess__ordIso,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1237_not__ordLess__ordIso,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ bNF_We4116350175183677332_o_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_ordIso__Plus__cong,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We3692099834942423674um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1248_ordIso__Plus__cong,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7792784937475852910_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1249_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We8256776467561525818_nat_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1250_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We7027991437926027642at_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1251_ordIso__Plus__cong,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We5813203082145254800um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1252_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Product_prod_o_o,P7: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca4833655811768152859_nat_o @ ( sum_Plus_nat_o @ ( field_nat @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We2873397315248171908_nat_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1253_ordIso__Plus__cong,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We6180393435421844484_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1254_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca786952496063569315at_nat @ ( sum_Plus_nat_nat @ ( field_nat @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We6373243813402207300at_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1255_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Pr1261947904930325089at_nat,P7: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_nat @ P ) ) ) @ ( bNF_Ca5391278176305430217um_o_o @ ( sum_Plus_o_o @ ( field_o @ R5 ) @ ( field_o @ P7 ) ) ) ) @ bNF_We4487549725514025450um_o_o ) ) ) ).

% ordIso_Plus_cong
thf(fact_1256_ordIso__Plus__cong,axiom,
    ! [R2: set_Product_prod_o_o,R5: set_Product_prod_o_o,P: set_Product_prod_o_o,P7: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ R5 ) @ 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 @ R2 ) @ ( field_o @ P ) ) ) @ ( bNF_Ca1325121287712659829_o_nat @ ( sum_Plus_o_nat @ ( field_o @ R5 ) @ ( field_nat @ P7 ) ) ) ) @ bNF_We8588234828047454686_o_nat ) ) ) ).

% ordIso_Plus_cong
thf(fact_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_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,
    ! [X2: option_a,Y: option_a] :
      ( ( if_option_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X2: option_a,Y: option_a] :
      ( ( if_option_a @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    sat_a_b_d_c @ plus @ mult @ valid @ sigma @ s @ delta @ ( mult_b_a_c_d @ p @ b2 ) ).

%------------------------------------------------------------------------------