%------------------------------------------------------------------------------
% 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    : Safe_Range_RC/0021_Relational_Calculus/prob_00607_022095__16660850_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2009 ( 458 unt; 724 typ;   0 def)
%            Number of atoms       : 4921 (1561 equ;   0 cnn)
%            Maximal formula atoms :   37 (   3 avg)
%            Number of connectives : 18863 ( 809   ~; 113   |; 386   &;15108   @)
%                                         (   0 <=>;2447  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   8 avg)
%            Number of types       :  108 ( 107 usr)
%            Number of type conns  : 1822 (1822   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  620 ( 617 usr;  49 con; 0-4 aty)
%            Number of variables   : 4889 ( 407   ^;4286   !; 196   ?;4889   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:25:24.656
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member2845199937591886608at_nat: produc5052769453400272615at_nat > set_Pr4654484256145968327at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4300370950623927696at_nat: produc7299876168958306407at_nat > set_Pr1604112560894399303at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member6436561159828024080at_nat: produc1296870326321989863at_nat > set_Pr5540436994911067335at_nat > $o ).

thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).

thf(sy_c_member_001t__Relational____Calculus__Oterm_Itf__a_J,type,
    member5466445112152945266term_a: relational_term_a > set_Re5178783185447174953term_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member3250002818903848048at_nat: set_Pr8693737435421807431at_nat > set_se6425085446898223655at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member2711111827193551600at_nat: set_Pr5564308138774400199at_nat > set_se4230842059925695655at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $o ).

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

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_Q,type,
    q: relational_fmla_a_b ).

% Relevant facts (1271)
thf(fact_0_qp__imp__qp__impl,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ Q )
     => ( relati3725921752842749053pl_a_b @ Q ) ) ).

% qp_imp_qp_impl
thf(fact_1_qp__impl__imp__qp,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ Q )
     => ( relational_qp_a_b @ Q ) ) ).

% qp_impl_imp_qp
thf(fact_2_ap,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ Q )
     => ( relational_qp_a_b @ Q ) ) ).

% ap
thf(fact_3_exists,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( relational_qp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) ) ) ).

% exists
thf(fact_4_qp__impl_Osimps_I6_J,axiom,
    ! [V: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Neg_a_b @ V ) ) ).

% qp_impl.simps(6)
thf(fact_5_qp__impl_Osimps_I8_J,axiom,
    ! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V @ Va ) ) ).

% qp_impl.simps(8)
thf(fact_6_qp__impl_Osimps_I7_J,axiom,
    ! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).

% qp_impl.simps(7)
thf(fact_7_qp__impl_Osimps_I4_J,axiom,
    ! [V: $o] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).

% qp_impl.simps(4)
thf(fact_8_qp__impl_Osimps_I2_J,axiom,
    ! [X: b,Ts: list_R6823256787227418703term_a] : ( relati3725921752842749053pl_a_b @ ( relational_Pred_b_a @ X @ Ts ) ) ).

% qp_impl.simps(2)
thf(fact_9_qp__impl_Osimps_I3_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_qp_a_b @ Q ) ) ) ).

% qp_impl.simps(3)
thf(fact_10_qp_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ A )
     => ( ~ ( relational_ap_a_b @ A )
       => ~ ! [Q2: relational_fmla_a_b] :
              ( ? [X2: nat] :
                  ( A
                  = ( relati3989891337220013914ts_a_b @ X2 @ Q2 ) )
             => ~ ( relational_qp_a_b @ Q2 ) ) ) ) ).

% qp.cases
thf(fact_11_qp_Osimps,axiom,
    ( relational_qp_a_b
    = ( ^ [A2: relational_fmla_a_b] :
          ( ? [Q3: relational_fmla_a_b] :
              ( ( A2 = Q3 )
              & ( relational_ap_a_b @ Q3 ) )
          | ? [Q3: relational_fmla_a_b,X3: nat] :
              ( ( A2
                = ( relati3989891337220013914ts_a_b @ X3 @ Q3 ) )
              & ( relational_qp_a_b @ Q3 ) ) ) ) ) ).

% qp.simps
thf(fact_12_qp__impl_Osimps_I1_J,axiom,
    ! [X: nat,C: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).

% qp_impl.simps(1)
thf(fact_13_fmla_Oinject_I4_J,axiom,
    ! [X4: relational_fmla_a_b,Y4: relational_fmla_a_b] :
      ( ( ( relational_Neg_a_b @ X4 )
        = ( relational_Neg_a_b @ Y4 ) )
      = ( X4 = Y4 ) ) ).

% fmla.inject(4)
thf(fact_14_fmla_Oinject_I6_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
      ( ( ( relational_Disj_a_b @ X61 @ X62 )
        = ( relational_Disj_a_b @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% fmla.inject(6)
thf(fact_15_fmla_Oinject_I7_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
      ( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
        = ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% fmla.inject(7)
thf(fact_16_fmla_Oinject_I5_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
      ( ( ( relational_Conj_a_b @ X51 @ X52 )
        = ( relational_Conj_a_b @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% fmla.inject(5)
thf(fact_17_fmla_Oinject_I2_J,axiom,
    ! [X22: $o,Y2: $o] :
      ( ( ( relational_Bool_a_b @ X22 )
        = ( relational_Bool_a_b @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% fmla.inject(2)
thf(fact_18_fmla_Oinject_I1_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
      ( ( ( relational_Pred_b_a @ X11 @ X12 )
        = ( relational_Pred_b_a @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% fmla.inject(1)
thf(fact_19_fv_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
      = ( relational_fv_a_b @ Phi ) ) ).

% fv.simps(4)
thf(fact_20_fmla_Odistinct_I41_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Disj_a_b @ X61 @ X62 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(41)
thf(fact_21_fmla_Odistinct_I39_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(39)
thf(fact_22_fmla_Odistinct_I37_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(37)
thf(fact_23_fmla_Odistinct_I35_J,axiom,
    ! [X4: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(35)
thf(fact_24_fmla_Odistinct_I33_J,axiom,
    ! [X4: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(33)
thf(fact_25_fmla_Odistinct_I31_J,axiom,
    ! [X4: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(31)
thf(fact_26_fmla_Odistinct_I29_J,axiom,
    ! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(29)
thf(fact_27_fmla_Odistinct_I27_J,axiom,
    ! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(27)
thf(fact_28_fmla_Odistinct_I25_J,axiom,
    ! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(25)
thf(fact_29_fmla_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: relational_term_a,X4: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(23)
thf(fact_30_fmla_Odistinct_I21_J,axiom,
    ! [X22: $o,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(21)
thf(fact_31_fmla_Odistinct_I19_J,axiom,
    ! [X22: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(19)
thf(fact_32_fmla_Odistinct_I17_J,axiom,
    ! [X22: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(17)
thf(fact_33_fmla_Odistinct_I15_J,axiom,
    ! [X22: $o,X4: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(15)
thf(fact_34_fmla_Odistinct_I13_J,axiom,
    ! [X22: $o,X31: nat,X32: relational_term_a] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(13)
thf(fact_35_fmla_Odistinct_I11_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(11)
thf(fact_36_fmla_Odistinct_I9_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(9)
thf(fact_37_fmla_Odistinct_I7_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(7)
thf(fact_38_fmla_Odistinct_I5_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X4: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(5)
thf(fact_39_fmla_Odistinct_I3_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(3)
thf(fact_40_fmla_Odistinct_I1_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X22: $o] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Bool_a_b @ X22 ) ) ).

% fmla.distinct(1)
thf(fact_41_qp__impl_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X )
     => ( ! [X2: nat,C2: a] :
            ( X
           != ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) )
       => ( ! [X2: b,Ts2: list_R6823256787227418703term_a] :
              ( X
             != ( relational_Pred_b_a @ X2 @ Ts2 ) )
         => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                ( ( X
                  = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
               => ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                    & ( relational_qp_a_b @ Q2 ) ) ) ) ) ) ).

% qp_impl.elims(2)
thf(fact_42_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_51_Eqc,axiom,
    ! [X: nat,C: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).

% Eqc
thf(fact_52_Pred,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a] : ( relational_ap_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ).

% Pred
thf(fact_53_ap_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ A )
     => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ P3 @ Ts2 ) )
       => ~ ! [X2: nat,C2: a] :
              ( A
             != ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) ) ) ) ).

% ap.cases
thf(fact_54_ap_Osimps,axiom,
    ( relational_ap_a_b
    = ( ^ [A2: relational_fmla_a_b] :
          ( ? [P4: b,Ts3: list_R6823256787227418703term_a] :
              ( A2
              = ( relational_Pred_b_a @ P4 @ Ts3 ) )
          | ? [X3: nat,C3: a] :
              ( A2
              = ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) ) ) ) ) ).

% ap.simps
thf(fact_55_cp_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [X2: nat,T: relational_term_a] :
          ( X
         != ( relational_Eq_a_b @ X2 @ T ) )
     => ( ! [Q2: relational_fmla_a_b] :
            ( X
           != ( relational_Neg_a_b @ Q2 ) )
       => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
              ( X
             != ( relational_Conj_a_b @ Q1 @ Q22 ) )
         => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                ( X
               != ( relational_Disj_a_b @ Q1 @ Q22 ) )
           => ( ! [X2: nat,Q2: relational_fmla_a_b] :
                  ( X
                 != ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
             => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                    ( X
                   != ( relational_Pred_b_a @ V2 @ Va2 ) )
               => ~ ! [V2: $o] :
                      ( X
                     != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).

% cp.cases
thf(fact_56_fv_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
          ( X
         != ( relational_Pred_b_a @ Uu @ Ts2 ) )
     => ( ! [B: $o] :
            ( X
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X2: nat,T2: relational_term_a] :
              ( X
             != ( relational_Eq_a_b @ X2 @ T2 ) )
         => ( ! [Phi2: relational_fmla_a_b] :
                ( X
               != ( relational_Neg_a_b @ Phi2 ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                  ( X
                 != ( relational_Conj_a_b @ Phi2 @ Psi ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( X
                   != ( relational_Disj_a_b @ Phi2 @ Psi ) )
               => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                      ( X
                     != ( relati591517084277583526ts_a_b @ Z @ Phi2 ) ) ) ) ) ) ) ) ).

% fv.cases
thf(fact_57_nocp_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [B: $o] :
          ( X
         != ( relational_Bool_a_b @ B ) )
     => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ P3 @ Ts2 ) )
       => ( ! [X2: nat,T: relational_term_a] :
              ( X
             != ( relational_Eq_a_b @ X2 @ T ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( X
               != ( relational_Neg_a_b @ Q2 ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( X
                 != ( relational_Conj_a_b @ Q1 @ Q22 ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( X
                   != ( relational_Disj_a_b @ Q1 @ Q22 ) )
               => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                      ( X
                     != ( relati591517084277583526ts_a_b @ X2 @ Q2 ) ) ) ) ) ) ) ) ).

% nocp.cases
thf(fact_58_fmla_Oexhaust,axiom,
    ! [Y: relational_fmla_a_b] :
      ( ! [X112: b,X122: list_R6823256787227418703term_a] :
          ( Y
         != ( relational_Pred_b_a @ X112 @ X122 ) )
     => ( ! [X23: $o] :
            ( Y
           != ( relational_Bool_a_b @ X23 ) )
       => ( ! [X312: nat,X322: relational_term_a] :
              ( Y
             != ( relational_Eq_a_b @ X312 @ X322 ) )
         => ( ! [X42: relational_fmla_a_b] :
                ( Y
               != ( relational_Neg_a_b @ X42 ) )
           => ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
                  ( Y
                 != ( relational_Conj_a_b @ X512 @ X522 ) )
             => ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
                    ( Y
                   != ( relational_Disj_a_b @ X612 @ X622 ) )
               => ~ ! [X712: nat,X722: relational_fmla_a_b] :
                      ( Y
                     != ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).

% fmla.exhaust
thf(fact_59_exists__def,axiom,
    ( relati3989891337220013914ts_a_b
    = ( ^ [X3: nat,Q3: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) ) @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) @ Q3 ) ) ) ).

% exists_def
thf(fact_60_exists__Exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( ( relati3989891337220013914ts_a_b @ X @ Q )
        = ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ).

% exists_Exists
thf(fact_61_qp__ExistsE,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
     => ~ ( ( relational_qp_a_b @ Q )
         => ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).

% qp_ExistsE
thf(fact_62_qp__Exists,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ) ).

% qp_Exists
thf(fact_63_qp__impl_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X )
        = Y )
     => ( ( ? [X2: nat,C2: a] :
              ( X
              = ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) )
         => ~ Y )
       => ( ( ? [X2: b,Ts2: list_R6823256787227418703term_a] :
                ( X
                = ( relational_Pred_b_a @ X2 @ Ts2 ) )
           => ~ Y )
         => ( ! [X2: nat,Q2: relational_fmla_a_b] :
                ( ( X
                  = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
               => ( Y
                  = ( ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                        & ( relational_qp_a_b @ Q2 ) ) ) ) )
           => ( ( ? [V2: $o] :
                    ( X
                    = ( relational_Bool_a_b @ V2 ) )
               => Y )
             => ( ( ? [V2: nat,Vb: nat] :
                      ( X
                      = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                 => Y )
               => ( ( ? [V2: relational_fmla_a_b] :
                        ( X
                        = ( relational_Neg_a_b @ V2 ) )
                   => Y )
                 => ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( X
                          = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => Y )
                   => ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( X
                            = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => Y ) ) ) ) ) ) ) ) ) ).

% qp_impl.elims(1)
thf(fact_64_qp__impl_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X )
     => ( ! [X2: nat,Q2: relational_fmla_a_b] :
            ( ( X
              = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
           => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
              & ( relational_qp_a_b @ Q2 ) ) )
       => ( ! [V2: $o] :
              ( X
             != ( relational_Bool_a_b @ V2 ) )
         => ( ! [V2: nat,Vb: nat] :
                ( X
               != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
           => ( ! [V2: relational_fmla_a_b] :
                  ( X
                 != ( relational_Neg_a_b @ V2 ) )
             => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                    ( X
                   != ( relational_Conj_a_b @ V2 @ Va2 ) )
               => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X
                     != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).

% qp_impl.elims(3)
thf(fact_65_qp__impl_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [X2: nat,C2: a] :
          ( X
         != ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) )
     => ( ! [X2: b,Ts2: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ X2 @ Ts2 ) )
       => ( ! [X2: nat,Q2: relational_fmla_a_b] :
              ( X
             != ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
         => ( ! [V2: $o] :
                ( X
               != ( relational_Bool_a_b @ V2 ) )
           => ( ! [V2: nat,Vb: nat] :
                  ( X
                 != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( X
                   != ( relational_Neg_a_b @ V2 ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X
                     != ( relational_Conj_a_b @ V2 @ Va2 ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( X
                       != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).

% qp_impl.cases
thf(fact_66_fmla_Oset__cases_I2_J,axiom,
    ! [E: b,A: relational_fmla_a_b] :
      ( ( member_b @ E @ ( relati8924981150291758614la_a_b @ A ) )
     => ( ! [Z2: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ E @ Z2 ) )
       => ( ! [Z: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ Z ) )
             => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z ) ) )
         => ( ! [Z1: relational_fmla_a_b] :
                ( ? [Z2: relational_fmla_a_b] :
                    ( A
                    = ( relational_Conj_a_b @ Z1 @ Z2 ) )
               => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
           => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                  ( ( A
                    = ( relational_Conj_a_b @ Z1 @ Z2 ) )
                 => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
             => ( ! [Z1: relational_fmla_a_b] :
                    ( ? [Z2: relational_fmla_a_b] :
                        ( A
                        = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                   => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
               => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                      ( ( A
                        = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                     => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
                 => ~ ! [Z1: nat,Z2: relational_fmla_a_b] :
                        ( ( A
                          = ( relati591517084277583526ts_a_b @ Z1 @ Z2 ) )
                       => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) ) ) ) ) ) ) ) ) ).

% fmla.set_cases(2)
thf(fact_67_qp__impl_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X2: nat,C2: a] :
              ( ( X
                = ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) ) )
         => ( ! [X2: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ X2 @ Ts2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X2 @ Ts2 ) ) )
           => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                   => ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                        & ( relational_qp_a_b @ Q2 ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(2)
thf(fact_68_fv_Osimps_I1_J,axiom,
    ! [Uu2: b,Ts: list_R6823256787227418703term_a] :
      ( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts ) )
      = ( relati4569515538964159125_set_a @ Ts ) ) ).

% fv.simps(1)
thf(fact_69_qp__impl_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X2: nat,C2: a] :
              ( ( X
                = ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) )
             => ( Y
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C2 ) ) ) ) )
         => ( ! [X2: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ X2 @ Ts2 ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X2 @ Ts2 ) ) ) )
           => ( ! [X2: nat,Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                 => ( ( Y
                      = ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                        & ( relational_qp_a_b @ Q2 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) ) ) )
             => ( ! [V2: $o] :
                    ( ( X
                      = ( relational_Bool_a_b @ V2 ) )
                   => ( ~ Y
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
               => ( ! [V2: nat,Vb: nat] :
                      ( ( X
                        = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                     => ( ~ Y
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) ) )
                 => ( ! [V2: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Neg_a_b @ V2 ) )
                       => ( ~ Y
                         => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
                   => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( ( X
                            = ( relational_Conj_a_b @ V2 @ Va2 ) )
                         => ( ~ Y
                           => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
                     => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( ( X
                              = ( relational_Disj_a_b @ V2 @ Va2 ) )
                           => ( ~ Y
                             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(1)
thf(fact_70_nocp_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X )
        = Y )
     => ( ( ? [B: $o] :
              ( X
              = ( relational_Bool_a_b @ B ) )
         => Y )
       => ( ( ? [P3: b,Ts2: list_R6823256787227418703term_a] :
                ( X
                = ( relational_Pred_b_a @ P3 @ Ts2 ) )
           => ~ Y )
         => ( ! [X2: nat,T: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T ) )
               => ( Y
                  = ( T
                    = ( relational_Var_a @ X2 ) ) ) )
           => ( ! [Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q2 ) )
                 => ( Y
                    = ( ~ ( relational_nocp_a_b @ Q2 ) ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                   => ( Y
                      = ( ~ ( ( relational_nocp_a_b @ Q1 )
                            & ( relational_nocp_a_b @ Q22 ) ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ( Y
                        = ( ~ ( ( relational_nocp_a_b @ Q1 )
                              & ( relational_nocp_a_b @ Q22 ) ) ) ) )
                 => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                       => ( Y
                          = ( ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                                & ( relational_nocp_a_b @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.elims(1)
thf(fact_71_cp__exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
      = ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).

% cp_exists
thf(fact_72_nocp__exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% nocp_exists
thf(fact_73_exists__cp__subst,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b] :
      ( ( X != Y )
     => ( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) )
        = ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).

% exists_cp_subst
thf(fact_74_subst_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X @ Y )
      = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ).

% subst.simps(4)
thf(fact_75_subst_Osimps_I6_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ X @ Y )
      = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q12 @ X @ Y ) @ ( relational_subst_a_b @ Q23 @ X @ Y ) ) ) ).

% subst.simps(6)
thf(fact_76_subst_Osimps_I5_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ X @ Y )
      = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q12 @ X @ Y ) @ ( relational_subst_a_b @ Q23 @ X @ Y ) ) ) ).

% subst.simps(5)
thf(fact_77_subst_Osimps_I1_J,axiom,
    ! [T3: $o,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Bool_a_b @ T3 ) @ X @ Y )
      = ( relational_Bool_a_b @ T3 ) ) ).

% subst.simps(1)
thf(fact_78_nocp_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% nocp.simps(4)
thf(fact_79_nocp_Osimps_I6_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
      = ( ( relational_nocp_a_b @ Q12 )
        & ( relational_nocp_a_b @ Q23 ) ) ) ).

% nocp.simps(6)
thf(fact_80_fmla_Oset__intros_I10_J,axiom,
    ! [Ys: b,X4: relational_fmla_a_b] :
      ( ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ X4 ) )
     => ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) ) ) ) ).

% fmla.set_intros(10)
thf(fact_81_fmla_Osimps_I130_J,axiom,
    ! [X4: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) )
      = ( relati8924981150291758614la_a_b @ X4 ) ) ).

% fmla.simps(130)
thf(fact_82_fmla_Oset__intros_I13_J,axiom,
    ! [Yy: b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ X61 ) )
     => ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(13)
thf(fact_83_fmla_Oset__intros_I14_J,axiom,
    ! [Za: b,X62: relational_fmla_a_b,X61: relational_fmla_a_b] :
      ( ( member_b @ Za @ ( relati8924981150291758614la_a_b @ X62 ) )
     => ( member_b @ Za @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(14)
thf(fact_84_nocp_Osimps_I5_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
      = ( ( relational_nocp_a_b @ Q12 )
        & ( relational_nocp_a_b @ Q23 ) ) ) ).

% nocp.simps(5)
thf(fact_85_cp_Osimps_I7_J,axiom,
    ! [V: $o] :
      ( ( relational_cp_a_b @ ( relational_Bool_a_b @ V ) )
      = ( relational_Bool_a_b @ V ) ) ).

% cp.simps(7)
thf(fact_86_fmla_Oset__intros_I15_J,axiom,
    ! [Zc: b,X72: relational_fmla_a_b,X71: nat] :
      ( ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ X72 ) )
     => ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ) ) ).

% fmla.set_intros(15)
thf(fact_87_fmla_Osimps_I133_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) )
      = ( relati8924981150291758614la_a_b @ X72 ) ) ).

% fmla.simps(133)
thf(fact_88_fmla_Oset__intros_I11_J,axiom,
    ! [Yu: b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ X51 ) )
     => ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(11)
thf(fact_89_fmla_Oset__intros_I12_J,axiom,
    ! [Yw: b,X52: relational_fmla_a_b,X51: relational_fmla_a_b] :
      ( ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ X52 ) )
     => ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(12)
thf(fact_90_nocp_Osimps_I1_J,axiom,
    ! [B2: $o] :
      ~ ( relational_nocp_a_b @ ( relational_Bool_a_b @ B2 ) ) ).

% nocp.simps(1)
thf(fact_91_cp_Osimps_I6_J,axiom,
    ! [V: b,Va: list_R6823256787227418703term_a] :
      ( ( relational_cp_a_b @ ( relational_Pred_b_a @ V @ Va ) )
      = ( relational_Pred_b_a @ V @ Va ) ) ).

% cp.simps(6)
thf(fact_92_nocp_Osimps_I2_J,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a] : ( relational_nocp_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ).

% nocp.simps(2)
thf(fact_93_fmla_Oset__intros_I9_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a] : ( member_b @ X11 @ ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) ) ) ).

% fmla.set_intros(9)
thf(fact_94_fresh2_I3_J,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b] :
      ~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_fv_a_b @ Q ) ) ).

% fresh2(3)
thf(fact_95_nocp_Osimps_I7_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% nocp.simps(7)
thf(fact_96_qp__impl_Osimps_I5_J,axiom,
    ! [V: nat,Vb2: nat] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ V @ ( relational_Var_a @ Vb2 ) ) ) ).

% qp_impl.simps(5)
thf(fact_97_cp_Osimps_I5_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).

% cp.simps(5)
thf(fact_98_qp__impl_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X2: nat,Q2: relational_fmla_a_b] :
              ( ( X
                = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
             => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
               => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                  & ( relational_qp_a_b @ Q2 ) ) ) )
         => ( ! [V2: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ V2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
           => ( ! [V2: nat,Vb: nat] :
                  ( ( X
                    = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ V2 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(3)
thf(fact_99_nocp_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X )
     => ( ! [B: $o] :
            ( X
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X2: nat,T: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X2 @ T ) )
             => ( T
               != ( relational_Var_a @ X2 ) ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q2 ) )
               => ( relational_nocp_a_b @ Q2 ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                 => ( ( relational_nocp_a_b @ Q1 )
                    & ( relational_nocp_a_b @ Q22 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                   => ( ( relational_nocp_a_b @ Q1 )
                      & ( relational_nocp_a_b @ Q22 ) ) )
               => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                        & ( relational_nocp_a_b @ Q2 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(3)
thf(fact_100_nocp_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X )
     => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ P3 @ Ts2 ) )
       => ( ! [X2: nat,T: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X2 @ T ) )
             => ( T
                = ( relational_Var_a @ X2 ) ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q2 ) )
               => ~ ( relational_nocp_a_b @ Q2 ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                 => ~ ( ( relational_nocp_a_b @ Q1 )
                      & ( relational_nocp_a_b @ Q22 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                   => ~ ( ( relational_nocp_a_b @ Q1 )
                        & ( relational_nocp_a_b @ Q22 ) ) )
               => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                     => ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                          & ( relational_nocp_a_b @ Q2 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(2)
thf(fact_101_nocp_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ( ~ Y
               => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
         => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P3 @ Ts2 ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
           => ( ! [X2: nat,T: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T ) )
                 => ( ( Y
                      = ( T
                       != ( relational_Var_a @ X2 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T ) ) ) )
             => ( ! [Q2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q2 ) )
                   => ( ( Y
                        = ( relational_nocp_a_b @ Q2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q2 ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                     => ( ( Y
                          = ( ( relational_nocp_a_b @ Q1 )
                            & ( relational_nocp_a_b @ Q22 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) )
                 => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                       => ( ( Y
                            = ( ( relational_nocp_a_b @ Q1 )
                              & ( relational_nocp_a_b @ Q22 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) )
                   => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                         => ( ( Y
                              = ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                                & ( relational_nocp_a_b @ Q2 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(1)
thf(fact_102_nocp_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P3 @ Ts2 ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) )
         => ( ! [X2: nat,T: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T ) )
                 => ( T
                    = ( relational_Var_a @ X2 ) ) ) )
           => ( ! [Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q2 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q2 ) )
                   => ~ ( relational_nocp_a_b @ Q2 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
                     => ~ ( ( relational_nocp_a_b @ Q1 )
                          & ( relational_nocp_a_b @ Q22 ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
                       => ~ ( ( relational_nocp_a_b @ Q1 )
                            & ( relational_nocp_a_b @ Q22 ) ) ) )
                 => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                         => ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                              & ( relational_nocp_a_b @ Q2 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(2)
thf(fact_103_nocp_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) )
         => ( ! [X2: nat,T: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T ) )
                 => ( T
                   != ( relational_Var_a @ X2 ) ) ) )
           => ( ! [Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q2 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q2 ) )
                   => ( relational_nocp_a_b @ Q2 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
                     => ( ( relational_nocp_a_b @ Q1 )
                        & ( relational_nocp_a_b @ Q22 ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
                       => ( ( relational_nocp_a_b @ Q1 )
                          & ( relational_nocp_a_b @ Q22 ) ) ) )
                 => ~ ! [X2: nat,Q2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                         => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q2 ) )
                            & ( relational_nocp_a_b @ Q2 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(3)
thf(fact_104_cpropagated__simps_I7_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_simps(7)
thf(fact_105_cpropagated__simps_I5_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
      = ( ( relational_nocp_a_b @ Q12 )
        & ( relational_nocp_a_b @ Q23 ) ) ) ).

% cpropagated_simps(5)
thf(fact_106_cpropagated__simps_I6_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
      = ( ( relational_nocp_a_b @ Q12 )
        & ( relational_nocp_a_b @ Q23 ) ) ) ).

% cpropagated_simps(6)
thf(fact_107_cpropagated__simps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% cpropagated_simps(4)
thf(fact_108_cpropagated__simps_I1_J,axiom,
    ! [B2: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B2 ) ) ).

% cpropagated_simps(1)
thf(fact_109_cpropagated__simps_I2_J,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ).

% cpropagated_simps(2)
thf(fact_110_cpropagated__nocp,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_nocp
thf(fact_111_subst_Ocases,axiom,
    ! [X: produc8867654947514737559at_nat] :
      ( ! [T: $o,X2: nat,Y3: nat] :
          ( X
         != ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
     => ( ! [P3: b,Ts2: list_R6823256787227418703term_a,X2: nat,Y3: nat] :
            ( X
           != ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
       => ( ! [Z: nat,T: relational_term_a,X2: nat,Y3: nat] :
              ( X
             != ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
         => ( ! [Q2: relational_fmla_a_b,X2: nat,Y3: nat] :
                ( X
               != ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y3: nat] :
                  ( X
                 != ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y3: nat] :
                    ( X
                   != ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
               => ~ ! [Z: nat,Q2: relational_fmla_a_b,X2: nat,Y3: nat] :
                      ( X
                     != ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ) ).

% subst.cases
thf(fact_112_sat_Ocases,axiom,
    ! [X: produc1132964494702330949_nat_a] :
      ( ! [R: b,Ts2: list_R6823256787227418703term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
          ( X
         != ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
     => ( ! [B: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
            ( X
           != ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
       => ( ! [X2: nat,T2: relational_term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
              ( X
             != ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X2 @ T2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
         => ( ! [Phi2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                ( X
               != ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                  ( X
                 != ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                    ( X
                   != ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
               => ~ ! [Z: nat,Phi2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                      ( X
                     != ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) ) ) ) ) ) ) ) ).

% sat.cases
thf(fact_113_subst_Opinduct,axiom,
    ! [A0: relational_fmla_a_b,A1: nat,A22: nat,P: relational_fmla_a_b > nat > nat > $o] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ A0 @ ( product_Pair_nat_nat @ A1 @ A22 ) ) )
     => ( ! [T: $o,X2: nat,Y3: nat] :
            ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
           => ( P @ ( relational_Bool_a_b @ T ) @ X2 @ Y3 ) )
       => ( ! [P3: b,Ts2: list_R6823256787227418703term_a,X2: nat,Y3: nat] :
              ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
             => ( P @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ X2 @ Y3 ) )
         => ( ! [Z: nat,T: relational_term_a,X2: nat,Y3: nat] :
                ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
               => ( P @ ( relational_Eq_a_b @ Z @ T ) @ X2 @ Y3 ) )
           => ( ! [Q2: relational_fmla_a_b,X2: nat,Y3: nat] :
                  ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
                 => ( ( P @ Q2 @ X2 @ Y3 )
                   => ( P @ ( relational_Neg_a_b @ Q2 ) @ X2 @ Y3 ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y3: nat] :
                    ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
                   => ( ( P @ Q1 @ X2 @ Y3 )
                     => ( ( P @ Q22 @ X2 @ Y3 )
                       => ( P @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X2 @ Y3 ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y3: nat] :
                      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
                     => ( ( P @ Q1 @ X2 @ Y3 )
                       => ( ( P @ Q22 @ X2 @ Y3 )
                         => ( P @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X2 @ Y3 ) ) ) )
                 => ( ! [Z: nat,Q2: relational_fmla_a_b,X2: nat,Y3: nat] :
                        ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
                       => ( ! [Xa: nat] :
                              ( ( X2 != Z )
                             => ( ( Z = Y3 )
                               => ( ( Xa
                                    = ( relati2677767559083392098h2_a_b @ X2 @ Y3 @ Q2 ) )
                                 => ( P @ Q2 @ Z @ Xa ) ) ) )
                         => ( ! [Xa: nat] :
                                ( ( X2 != Z )
                               => ( ( Z = Y3 )
                                 => ( ( Xa
                                      = ( relati2677767559083392098h2_a_b @ X2 @ Y3 @ Q2 ) )
                                   => ( P @ ( relational_subst_a_b @ Q2 @ Z @ Xa ) @ X2 @ Y3 ) ) ) )
                           => ( ( ( X2 != Z )
                               => ( ( Z != Y3 )
                                 => ( P @ Q2 @ X2 @ Y3 ) ) )
                             => ( P @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ X2 @ Y3 ) ) ) ) )
                   => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).

% subst.pinduct
thf(fact_114_old_Oprod_Oinject,axiom,
    ! [A: nat,B2: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B2 )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_115_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B2 )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_116_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B2 )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_117_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B2 )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_118_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_119_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_120_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_121_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X22 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_122_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: nat,C2: nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_123_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_124_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_125_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
          ( Y
         != ( produc9060074326276436823at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_127_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X2: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_128_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_129_surj__pair,axiom,
    ! [P2: produc3843707927480180839at_nat] :
    ? [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P2
      = ( produc2922128104949294807at_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_130_surj__pair,axiom,
    ! [P2: produc1319942482725812455at_nat] :
    ? [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P2
      = ( produc9060074326276436823at_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_131_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_132_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_133_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A5 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_134_prod__cases,axiom,
    ! [P: produc1319942482725812455at_nat > $o,P2: produc1319942482725812455at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( P @ ( produc9060074326276436823at_nat @ A5 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_135_Pair__inject,axiom,
    ! [A: nat,B2: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B2 )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_136_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B2 )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_137_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B2 )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_138_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B2 )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_139_subst_Opsimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X @ Y )
        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).

% subst.psimps(4)
thf(fact_140_subst_Opsimps_I6_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ X @ Y )
        = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q12 @ X @ Y ) @ ( relational_subst_a_b @ Q23 @ X @ Y ) ) ) ) ).

% subst.psimps(6)
thf(fact_141_subst_Opsimps_I5_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ X @ Y )
        = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q12 @ X @ Y ) @ ( relational_subst_a_b @ Q23 @ X @ Y ) ) ) ) ).

% subst.psimps(5)
thf(fact_142_subst_Opsimps_I1_J,axiom,
    ! [T3: $o,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T3 ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Bool_a_b @ T3 ) @ X @ Y )
        = ( relational_Bool_a_b @ T3 ) ) ) ).

% subst.psimps(1)
thf(fact_143_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B: nat,C2: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B @ C2 ) ) ) ).

% prod_cases3
thf(fact_144_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_145_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F: nat > nat > nat,A5: nat,B: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_146_subst_Opsimps_I7_J,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( ( X = Z3 )
         => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
            = ( relati591517084277583526ts_a_b @ X @ Q ) ) )
        & ( ( X != Z3 )
         => ( ( ( Z3 = Y )
             => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
                = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
            & ( ( Z3 != Y )
             => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
                = ( relati591517084277583526ts_a_b @ Z3 @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) ) ).

% subst.psimps(7)
thf(fact_147_gen_Ointros_I5_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G )
     => ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) @ G ) ) ).

% gen.intros(5)
thf(fact_148_gen_Ointros_I3_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).

% gen.intros(3)
thf(fact_149_gen_Ointros_I7_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b @ X @ Q12 @ G )
        | ( relational_gen_a_b @ X @ Q23 @ G ) )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ G ) ) ).

% gen.intros(7)
thf(fact_150_subst_Osimps_I7_J,axiom,
    ! [X: nat,Z3: nat,Q: relational_fmla_a_b,Y: nat] :
      ( ( ( X = Z3 )
       => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
          = ( relati591517084277583526ts_a_b @ X @ Q ) ) )
      & ( ( X != Z3 )
       => ( ( ( Z3 = Y )
           => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
              = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
          & ( ( Z3 != Y )
           => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ X @ Y )
              = ( relati591517084277583526ts_a_b @ Z3 @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) ).

% subst.simps(7)
thf(fact_151_subst__exists,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
       => ( ( ( X = Z3 )
           => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
              = ( relati3989891337220013914ts_a_b @ X @ Q ) ) )
          & ( ( X != Z3 )
           => ( ( ( Z3 = Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
                  = ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
              & ( ( Z3 != Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
                  = ( relati3989891337220013914ts_a_b @ Z3 @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) )
      & ( ~ ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
          = ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).

% subst_exists
thf(fact_152_gen_Ointros_I4_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G )
     => ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) @ G ) ) ).

% gen.intros(4)
thf(fact_153_subst_Opelims,axiom,
    ! [X: relational_fmla_a_b,Xa2: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ X @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) )
       => ( ! [T: $o] :
              ( ( X
                = ( relational_Bool_a_b @ T ) )
             => ( ( Y
                  = ( relational_Bool_a_b @ T ) )
               => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
         => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P3 @ Ts2 ) )
               => ( ( Y
                    = ( relational_Pred_b_a @ P3
                      @ ( map_Re5736185711816362116term_a
                        @ ^ [T4: relational_term_a] : ( relati7175845559408349773term_a @ T4 @ Xa2 @ Xb )
                        @ Ts2 ) ) )
                 => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
           => ( ! [Z: nat,T: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ Z @ T ) )
                 => ( ( Y
                      = ( relational_Eq_a_b @ ( if_nat @ ( Z = Xa2 ) @ Xb @ Z ) @ ( relati7175845559408349773term_a @ T @ Xa2 @ Xb ) ) )
                   => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
             => ( ! [Q2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q2 ) )
                   => ( ( Y
                        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q2 @ Xa2 @ Xb ) ) )
                     => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q2 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                     => ( ( Y
                          = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa2 @ Xb ) ) )
                       => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
                 => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                       => ( ( Y
                            = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa2 @ Xb ) ) )
                         => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
                   => ~ ! [Z: nat,Q2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Q2 ) )
                         => ( ( ( ( Xa2 = Z )
                               => ( Y
                                  = ( relati591517084277583526ts_a_b @ Xa2 @ Q2 ) ) )
                              & ( ( Xa2 != Z )
                               => ( ( ( Z = Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q2 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q2 @ Z @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q2 ) ) @ Xa2 @ Xb ) ) ) )
                                  & ( ( Z != Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q2 @ Xa2 @ Xb ) ) ) ) ) ) )
                           => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.pelims
thf(fact_154_subst_Oelims,axiom,
    ! [X: relational_fmla_a_b,Xa2: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [T: $o] :
            ( ( X
              = ( relational_Bool_a_b @ T ) )
           => ( Y
             != ( relational_Bool_a_b @ T ) ) )
       => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P3 @ Ts2 ) )
             => ( Y
               != ( relational_Pred_b_a @ P3
                  @ ( map_Re5736185711816362116term_a
                    @ ^ [T4: relational_term_a] : ( relati7175845559408349773term_a @ T4 @ Xa2 @ Xb )
                    @ Ts2 ) ) ) )
         => ( ! [Z: nat,T: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ Z @ T ) )
               => ( Y
                 != ( relational_Eq_a_b @ ( if_nat @ ( Z = Xa2 ) @ Xb @ Z ) @ ( relati7175845559408349773term_a @ T @ Xa2 @ Xb ) ) ) )
           => ( ! [Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q2 ) )
                 => ( Y
                   != ( relational_Neg_a_b @ ( relational_subst_a_b @ Q2 @ Xa2 @ Xb ) ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                   => ( Y
                     != ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa2 @ Xb ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ( Y
                       != ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa2 @ Xb ) ) ) )
                 => ~ ! [Z: nat,Q2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Q2 ) )
                       => ~ ( ( ( Xa2 = Z )
                             => ( Y
                                = ( relati591517084277583526ts_a_b @ Xa2 @ Q2 ) ) )
                            & ( ( Xa2 != Z )
                             => ( ( ( Z = Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q2 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q2 @ Z @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q2 ) ) @ Xa2 @ Xb ) ) ) )
                                & ( ( Z != Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q2 @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.elims
thf(fact_155_cp_Osimps_I1_J,axiom,
    ! [X: nat,T3: relational_term_a] :
      ( ( relational_cp_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
      = ( relati582353067970734056la_a_b
        @ ^ [A2: a] : ( relational_Eq_a_b @ X @ T3 )
        @ ^ [Y5: nat] : ( if_Rel1279876242545935705la_a_b @ ( X = Y5 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y5 ) ) )
        @ T3 ) ) ).

% cp.simps(1)
thf(fact_156_subst_Opsimps_I2_J,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) @ X @ Y )
        = ( relational_Pred_b_a @ P2
          @ ( map_Re5736185711816362116term_a
            @ ^ [T4: relational_term_a] : ( relati7175845559408349773term_a @ T4 @ X @ Y )
            @ Ts ) ) ) ) ).

% subst.psimps(2)
thf(fact_157_gen_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen.intros(9)
thf(fact_158_gen_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen.intros(8)
thf(fact_159_gen_Ointros_I10_J,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_gen_a_b @ X @ Q @ G )
       => ( relational_gen_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).

% gen.intros(10)
thf(fact_160_subst_Osimps_I2_J,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) @ X @ Y )
      = ( relational_Pred_b_a @ P2
        @ ( map_Re5736185711816362116term_a
          @ ^ [T4: relational_term_a] : ( relati7175845559408349773term_a @ T4 @ X @ Y )
          @ Ts ) ) ) ).

% subst.simps(2)
thf(fact_161_cp_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X )
        = Y )
     => ( ! [X2: nat,T: relational_term_a] :
            ( ( X
              = ( relational_Eq_a_b @ X2 @ T ) )
           => ( Y
             != ( relati582353067970734056la_a_b
                @ ^ [A2: a] : ( relational_Eq_a_b @ X2 @ T )
                @ ^ [Y5: nat] : ( if_Rel1279876242545935705la_a_b @ ( X2 = Y5 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y5 ) ) )
                @ T ) ) )
       => ( ! [Q2: relational_fmla_a_b] :
              ( ( X
                = ( relational_Neg_a_b @ Q2 ) )
             => ( Y
               != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q2 ) )
                  @ ( relational_Bool_a_b
                    @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q2 ) ) )
                  @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q2 ) ) ) ) )
         => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Conj_a_b @ Q1 @ Q22 ) )
               => ( Y
                 != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_cp_a_b @ Q22 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_cp_a_b @ Q1 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                 => ( Y
                   != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) )
             => ( ! [X2: nat,Q2: relational_fmla_a_b] :
                    ( ( X
                      = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                   => ( Y
                     != ( relati3989891337220013914ts_a_b @ X2 @ ( relational_cp_a_b @ Q2 ) ) ) )
               => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                      ( ( X
                        = ( relational_Pred_b_a @ V2 @ Va2 ) )
                     => ( Y
                       != ( relational_Pred_b_a @ V2 @ Va2 ) ) )
                 => ~ ! [V2: $o] :
                        ( ( X
                          = ( relational_Bool_a_b @ V2 ) )
                       => ( Y
                         != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ).

% cp.elims
thf(fact_162_gen_H_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen'.intros(8)
thf(fact_163_gen_H_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen'.intros(9)
thf(fact_164_cp_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ X )
       => ( ! [X2: nat,T: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X2 @ T ) )
             => ( ( Y
                  = ( relati582353067970734056la_a_b
                    @ ^ [A2: a] : ( relational_Eq_a_b @ X2 @ T )
                    @ ^ [Y5: nat] : ( if_Rel1279876242545935705la_a_b @ ( X2 = Y5 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y5 ) ) )
                    @ T ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Eq_a_b @ X2 @ T ) ) ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q2 ) )
               => ( ( Y
                    = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q2 ) )
                      @ ( relational_Bool_a_b
                        @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q2 ) ) )
                      @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q2 ) ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Neg_a_b @ Q2 ) ) ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                 => ( ( Y
                      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_cp_a_b @ Q22 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_cp_a_b @ Q1 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                   => ( ( Y
                        = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) )
               => ( ! [X2: nat,Q2: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X2 @ Q2 ) )
                     => ( ( Y
                          = ( relati3989891337220013914ts_a_b @ X2 @ ( relational_cp_a_b @ Q2 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q2 ) ) ) )
                 => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                        ( ( X
                          = ( relational_Pred_b_a @ V2 @ Va2 ) )
                       => ( ( Y
                            = ( relational_Pred_b_a @ V2 @ Va2 ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
                   => ~ ! [V2: $o] :
                          ( ( X
                            = ( relational_Bool_a_b @ V2 ) )
                         => ( ( Y
                              = ( relational_Bool_a_b @ V2 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% cp.pelims
thf(fact_165_gen_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ A1 @ A22 @ A32 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A32 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A32
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
               => ~ ( relational_gen_a_b @ A1 @ Q2 @ A32 ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) )
                 => ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) )
                   => ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
               => ( ! [Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ! [G2: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) )
                         => ( ( relational_gen_a_b @ A1 @ Q1 @ G1 )
                           => ~ ( relational_gen_a_b @ A1 @ Q22 @ G2 ) ) ) )
                 => ( ! [Q1: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                       => ( ( A32 = G3 )
                         => ~ ( ( relational_gen_a_b @ A1 @ Q1 @ G3 )
                              | ( relational_gen_a_b @ A1 @ Q22 @ G3 ) ) ) )
                   => ( ! [Y3: nat,Q2: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G3: set_Re381260168593705685la_a_b] :
                              ( ( A32
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                  @ G3 ) )
                             => ~ ( relational_gen_a_b @ Y3 @ Q2 @ G3 ) ) )
                     => ( ! [Y3: nat,Q2: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G3: set_Re381260168593705685la_a_b] :
                                ( ( A32
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                    @ G3 ) )
                               => ~ ( relational_gen_a_b @ Y3 @ Q2 @ G3 ) ) )
                       => ~ ! [Y3: nat,Q2: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q2 ) )
                             => ! [G3: set_Re381260168593705685la_a_b] :
                                  ( ( A32
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G3 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b @ A1 @ Q2 @ G3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen.cases
thf(fact_166_gen_Osimps,axiom,
    ( relational_gen_a_b
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A33 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A33
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
              & ( relational_gen_a_b @ A12 @ Q3 @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q13 @ Q24 ) )
              & ? [G22: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G12 @ G22 ) )
                  & ( relational_gen_a_b @ A12 @ Q13 @ G12 )
                  & ( relational_gen_a_b @ A12 @ Q24 @ G22 ) ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q13 @ Q24 ) )
              & ( ( relational_gen_a_b @ A12 @ Q13 @ A33 )
                | ( relational_gen_a_b @ A12 @ Q24 @ A33 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y5 ) ) ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A12 ) )
                      @ G4 ) )
                  & ( relational_gen_a_b @ Y5 @ Q3 @ G4 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A12 ) )
                      @ G4 ) )
                  & ( relational_gen_a_b @ Y5 @ Q3 @ G4 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G4 ) )
                  & ( A12 != Y5 )
                  & ( relational_gen_a_b @ A12 @ Q3 @ G4 ) ) ) ) ) ) ).

% gen.simps
thf(fact_167_is__Bool__exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relati6551038146797045342ol_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
      = ( relati6551038146797045342ol_a_b @ Q ) ) ).

% is_Bool_exists
thf(fact_168_fmla_Ocollapse_I2_J,axiom,
    ! [Fmla: relational_fmla_a_b] :
      ( ( relati6551038146797045342ol_a_b @ Fmla )
     => ( ( relational_Bool_a_b @ ( relati2638701775882563405ol_a_b @ Fmla ) )
        = Fmla ) ) ).

% fmla.collapse(2)
thf(fact_169_gen_H_Ointros_I6_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ Q12 @ G13 )
     => ( ( relational_gen_a_b2 @ X @ Q23 @ G23 )
       => ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G23 ) ) ) ) ).

% gen'.intros(6)
thf(fact_170_gen_H_Ointros_I1_J,axiom,
    ! [X: nat] : ( relational_gen_a_b2 @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).

% gen'.intros(1)
thf(fact_171_gen_H_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen'.intros(2)
thf(fact_172_fmla_Osimps_I127_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) )
      = ( insert_b @ X11 @ bot_bot_set_b ) ) ).

% fmla.simps(127)
thf(fact_173_fmla_Odisc_I11_J,axiom,
    ! [X4: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Neg_a_b @ X4 ) ) ).

% fmla.disc(11)
thf(fact_174_fmla_Odisc_I13_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.disc(13)
thf(fact_175_fmla_Odisc_I14_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.disc(14)
thf(fact_176_fmla_Odisc_I12_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.disc(12)
thf(fact_177_is__Bool__def,axiom,
    ( relati6551038146797045342ol_a_b
    = ( ^ [Fmla2: relational_fmla_a_b] :
        ? [X24: $o] :
          ( Fmla2
          = ( relational_Bool_a_b @ X24 ) ) ) ) ).

% is_Bool_def
thf(fact_178_fmla_OdiscI_I2_J,axiom,
    ! [Fmla: relational_fmla_a_b,X22: $o] :
      ( ( Fmla
        = ( relational_Bool_a_b @ X22 ) )
     => ( relati6551038146797045342ol_a_b @ Fmla ) ) ).

% fmla.discI(2)
thf(fact_179_fmla_Odisc_I9_J,axiom,
    ! [X22: $o] : ( relati6551038146797045342ol_a_b @ ( relational_Bool_a_b @ X22 ) ) ).

% fmla.disc(9)
thf(fact_180_fmla_Odisc_I8_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) ) ).

% fmla.disc(8)
thf(fact_181_gen_H_Ointros_I3_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).

% gen'.intros(3)
thf(fact_182_gen_H_Ointros_I7_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b2 @ X @ Q12 @ G )
        | ( relational_gen_a_b2 @ X @ Q23 @ G ) )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ G ) ) ).

% gen'.intros(7)
thf(fact_183_fmla_Osel_I3_J,axiom,
    ! [X22: $o] :
      ( ( relati2638701775882563405ol_a_b @ ( relational_Bool_a_b @ X22 ) )
      = X22 ) ).

% fmla.sel(3)
thf(fact_184_gen_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen.intros(2)
thf(fact_185_cp_Osimps_I2_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q ) )
        @ ( relational_Bool_a_b
          @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q ) ) )
        @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q ) ) ) ) ).

% cp.simps(2)
thf(fact_186_cp_Osimps_I4_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q23 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q23 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q23 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q12 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q23 ) ) ) ) ) ).

% cp.simps(4)
thf(fact_187_cp_Osimps_I3_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( relational_cp_a_b @ Q23 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q23 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q23 ) ) @ ( relational_cp_a_b @ Q12 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q23 ) ) ) ) ) ).

% cp.simps(3)
thf(fact_188_fmla_Osimps_I132_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
      = ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X61 ) @ ( relati8924981150291758614la_a_b @ X62 ) ) ) ).

% fmla.simps(132)
thf(fact_189_fmla_Osimps_I131_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) )
      = ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X51 ) @ ( relati8924981150291758614la_a_b @ X52 ) ) ) ).

% fmla.simps(131)
thf(fact_190_gen_Ointros_I6_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ Q12 @ G13 )
     => ( ( relational_gen_a_b @ X @ Q23 @ G23 )
       => ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G23 ) ) ) ) ).

% gen.intros(6)
thf(fact_191_fmla_Osimps_I128_J,axiom,
    ! [X22: $o] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Bool_a_b @ X22 ) )
      = bot_bot_set_b ) ).

% fmla.simps(128)
thf(fact_192_gen_Ointros_I1_J,axiom,
    ! [X: nat] : ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).

% gen.intros(1)
thf(fact_193_gen_H_Osimps,axiom,
    ( relational_gen_a_b2
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A33 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A33
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
              & ( relational_gen_a_b2 @ A12 @ Q3 @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q13 @ Q24 ) )
              & ? [G22: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G12 @ G22 ) )
                  & ( relational_gen_a_b2 @ A12 @ Q13 @ G12 )
                  & ( relational_gen_a_b2 @ A12 @ Q24 @ G22 ) ) )
          | ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q13 @ Q24 ) )
              & ( ( relational_gen_a_b2 @ A12 @ Q13 @ A33 )
                | ( relational_gen_a_b2 @ A12 @ Q24 @ A33 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y5 ) ) ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ A12 )
                      @ G4 ) )
                  & ( relational_gen_a_b2 @ Y5 @ Q3 @ G4 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ A12 )
                      @ G4 ) )
                  & ( relational_gen_a_b2 @ Y5 @ Q3 @ G4 ) ) )
          | ? [Y5: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) )
              & ? [G4: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G4 ) )
                  & ( A12 != Y5 )
                  & ( relational_gen_a_b2 @ A12 @ Q3 @ G4 ) ) ) ) ) ) ).

% gen'.simps
thf(fact_194_gen_H_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ A1 @ A22 @ A32 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A32 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A32
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q2: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
               => ~ ( relational_gen_a_b2 @ A1 @ Q2 @ A32 ) )
           => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) )
                 => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) )
                   => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
               => ( ! [Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ! [G2: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) )
                         => ( ( relational_gen_a_b2 @ A1 @ Q1 @ G1 )
                           => ~ ( relational_gen_a_b2 @ A1 @ Q22 @ G2 ) ) ) )
                 => ( ! [Q1: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                       => ( ( A32 = G3 )
                         => ~ ( ( relational_gen_a_b2 @ A1 @ Q1 @ G3 )
                              | ( relational_gen_a_b2 @ A1 @ Q22 @ G3 ) ) ) )
                   => ( ! [Y3: nat,Q2: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G3: set_Re381260168593705685la_a_b] :
                              ( ( A32
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                  @ G3 ) )
                             => ~ ( relational_gen_a_b2 @ Y3 @ Q2 @ G3 ) ) )
                     => ( ! [Y3: nat,Q2: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G3: set_Re381260168593705685la_a_b] :
                                ( ( A32
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                    @ G3 ) )
                               => ~ ( relational_gen_a_b2 @ Y3 @ Q2 @ G3 ) ) )
                       => ~ ! [Y3: nat,Q2: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q2 ) )
                             => ! [G3: set_Re381260168593705685la_a_b] :
                                  ( ( A32
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G3 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b2 @ A1 @ Q2 @ G3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen'.cases
thf(fact_195_gen_H_Ointros_I4_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) @ G ) ) ).

% gen'.intros(4)
thf(fact_196_gen_H_Ointros_I5_J,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) @ G ) ) ).

% gen'.intros(5)
thf(fact_197_gen_H_Ointros_I10_J,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_gen_a_b2 @ X @ Q @ G )
       => ( relational_gen_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).

% gen'.intros(10)
thf(fact_198_sub_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X )
        = Y )
     => ( ! [T: $o] :
            ( ( X
              = ( relational_Bool_a_b @ T ) )
           => ( Y
             != ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T ) @ bot_bo4495933725496725865la_a_b ) ) )
       => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P3 @ Ts2 ) )
             => ( Y
               != ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ! [X2: nat,T: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T ) )
               => ( Y
                 != ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X2 @ T ) @ bot_bo4495933725496725865la_a_b ) ) )
           => ( ! [Q2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q2 ) )
                 => ( Y
                   != ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q2 ) @ ( relational_sub_a_b @ Q2 ) ) ) )
             => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                   => ( Y
                     != ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                     => ( Y
                       != ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
                 => ~ ! [Z: nat,Q2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Q2 ) )
                       => ( Y
                         != ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ) ) ) ) ) ) ) ).

% sub.elims
thf(fact_199_sub_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X )
       => ( ! [T: $o] :
              ( ( X
                = ( relational_Bool_a_b @ T ) )
             => ( ( Y
                  = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T ) @ bot_bo4495933725496725865la_a_b ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T ) ) ) )
         => ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P3 @ Ts2 ) )
               => ( ( Y
                    = ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
           => ( ! [X2: nat,T: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T ) )
                 => ( ( Y
                      = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X2 @ T ) @ bot_bo4495933725496725865la_a_b ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X2 @ T ) ) ) )
             => ( ! [Q2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q2 ) )
                   => ( ( Y
                        = ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q2 ) @ ( relational_sub_a_b @ Q2 ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q2 ) ) ) )
               => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q1 @ Q22 ) )
                     => ( ( Y
                          = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) )
                 => ( ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q1 @ Q22 ) )
                       => ( ( Y
                            = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) )
                   => ~ ! [Z: nat,Q2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Q2 ) )
                         => ( ( Y
                              = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) @ ( relational_sub_a_b @ Q2 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sub.pelims
thf(fact_200_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R2 ) )
        = ( ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_201_pred__equals__eq2,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R2 ) )
        = ( ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_202_pred__equals__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R2 ) )
        = ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_203_pred__equals__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R2 ) )
        = ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_204_fv__cp__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,A3: set_nat] :
      ( ! [X2: relational_fmla_a_b] :
          ( ( member4680049679412964150la_a_b @ X2 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ( relati1591879772219623554ed_a_b @ X2 )
            & ( ( relational_fv_a_b @ X2 )
              = A3 ) ) )
     => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
        = A3 ) ) ).

% fv_cp_foldr_Disj
thf(fact_205_Exists__in__sub__foldr__Disj,axiom,
    ! [X: nat,Q4: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
     => ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ Q ) )
        | ? [X2: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ X2 ) ) ) ) ) ).

% Exists_in_sub_foldr_Disj
thf(fact_206_Exists__in__sub__cp__foldr__Disj,axiom,
    ! [X: nat,Q4: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
     => ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) )
        | ? [X2: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X2 ) ) ) ) ) ) ).

% Exists_in_sub_cp_foldr_Disj
thf(fact_207_fv_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(6)
thf(fact_208_fv_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(5)
thf(fact_209_fv_Osimps_I2_J,axiom,
    ! [B2: $o] :
      ( ( relational_fv_a_b @ ( relational_Bool_a_b @ B2 ) )
      = bot_bot_set_nat ) ).

% fv.simps(2)
thf(fact_210_sub_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).

% sub.simps(4)
thf(fact_211_sub_Osimps_I7_J,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) )
      = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).

% sub.simps(7)
thf(fact_212_sub_Osimps_I1_J,axiom,
    ! [T3: $o] :
      ( ( relational_sub_a_b @ ( relational_Bool_a_b @ T3 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ).

% sub.simps(1)
thf(fact_213_sub_Osimps_I2_J,axiom,
    ! [P2: b,Ts: list_R6823256787227418703term_a] :
      ( ( relational_sub_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) ).

% sub.simps(2)
thf(fact_214_sub_Osimps_I6_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) ) ).

% sub.simps(6)
thf(fact_215_sub_Osimps_I5_J,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) ) ).

% sub.simps(5)
thf(fact_216_fv__cp__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,A3: set_nat] :
      ( ( ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) )
       != ( relational_Bool_a_b @ $false ) )
     => ( ! [X2: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
           => ( ( relati1591879772219623554ed_a_b @ X2 )
              & ( ( relational_fv_a_b @ X2 )
                = A3 ) ) )
       => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) ) )
          = A3 ) ) ) ).

% fv_cp_foldr1_Disj
thf(fact_217_sat__foldr__Disj,axiom,
    ! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Xs @ Q ) @ I2 @ Sigma2 )
      = ( ? [X3: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
            & ( relational_sat_a_b @ X3 @ I2 @ Sigma2 ) ) ) ) ).

% sat_foldr_Disj
thf(fact_218_sat_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X @ Xa2 @ Xb )
     => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts2 ) )
           => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => B )
         => ( ! [X2: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T2 ) )
               => ( ( Xb @ X2 )
                  = ( relati1177013128715261720term_a @ Xb @ T2 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                      & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                        | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X5 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(3)
thf(fact_219_sat_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X @ Xa2 @ Xb )
     => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts2 ) )
           => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ~ B )
         => ( ! [X2: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T2 ) )
               => ( ( Xb @ X2 )
                 != ( relati1177013128715261720term_a @ Xb @ T2 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                        & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                          | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ~ ? [X2: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X2 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(2)
thf(fact_220_sat_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts2 ) )
           => ( Y
              = ( ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ( Y = ~ B ) )
         => ( ! [X2: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T2 ) )
               => ( Y
                  = ( ( Xb @ X2 )
                   != ( relati1177013128715261720term_a @ Xb @ T2 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                    = ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                      = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                        = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ( Y
                          = ( ~ ? [X3: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(1)
thf(fact_221_fv_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X )
        = Y )
     => ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ Uu @ Ts2 ) )
           => ( Y
             != ( relati4569515538964159125_set_a @ Ts2 ) ) )
       => ( ( ? [B: $o] :
                ( X
                = ( relational_Bool_a_b @ B ) )
           => ( Y != bot_bot_set_nat ) )
         => ( ! [X2: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X2 @ T2 ) )
               => ( Y
                 != ( sup_sup_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T2 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                   != ( relational_fv_a_b @ Phi2 ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                     != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                       != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ( Y
                         != ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).

% fv.elims
thf(fact_222_sat__exists,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relati3989891337220013914ts_a_b @ N @ Q ) @ I2 @ Sigma2 )
      = ( ? [X3: a] : ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma2 @ N @ X3 ) ) ) ) ).

% sat_exists
thf(fact_223_fv__exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% fv_exists
thf(fact_224_sat_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi ) @ I2 @ Sigma2 )
      = ( ~ ( relational_sat_a_b @ Phi @ I2 @ Sigma2 ) ) ) ).

% sat.simps(4)
thf(fact_225_sat_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi2 ) @ I2 @ Sigma2 )
      = ( ( relational_sat_a_b @ Phi @ I2 @ Sigma2 )
        | ( relational_sat_a_b @ Psi2 @ I2 @ Sigma2 ) ) ) ).

% sat.simps(6)
thf(fact_226_sat_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) @ I2 @ Sigma2 )
      = ( ( relational_sat_a_b @ Phi @ I2 @ Sigma2 )
        & ( relational_sat_a_b @ Psi2 @ I2 @ Sigma2 ) ) ) ).

% sat.simps(5)
thf(fact_227_sat_Osimps_I2_J,axiom,
    ! [B2: $o,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Bool_a_b @ B2 ) @ I2 @ Sigma2 )
      = B2 ) ).

% sat.simps(2)
thf(fact_228_sat__fv__cong,axiom,
    ! [Phi: relational_fmla_a_b,Sigma2: nat > a,Sigma3: nat > a,I2: product_prod_b_nat > set_list_a] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ ( relational_fv_a_b @ Phi ) )
         => ( ( Sigma2 @ N2 )
            = ( Sigma3 @ N2 ) ) )
     => ( ( relational_sat_a_b @ Phi @ I2 @ Sigma2 )
        = ( relational_sat_a_b @ Phi @ I2 @ Sigma3 ) ) ) ).

% sat_fv_cong
thf(fact_229_sat_Osimps_I1_J,axiom,
    ! [R3: b,Ts: list_R6823256787227418703term_a,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Pred_b_a @ R3 @ Ts ) @ I2 @ Sigma2 )
      = ( member_list_a @ ( relati4772805863405912879erms_a @ Sigma2 @ Ts ) @ ( I2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) ).

% sat.simps(1)
thf(fact_230_fv_Osimps_I7_J,axiom,
    ! [Z3: nat,Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Phi ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) ).

% fv.simps(7)
thf(fact_231_sat_Osimps_I7_J,axiom,
    ! [Z3: nat,Phi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Phi ) @ I2 @ Sigma2 )
      = ( ? [X3: a] : ( relational_sat_a_b @ Phi @ I2 @ ( fun_upd_nat_a @ Sigma2 @ Z3 @ X3 ) ) ) ) ).

% sat.simps(7)
thf(fact_232_fv__subst,axiom,
    ! [X: nat,Q: relational_fmla_a_b,Y: nat] :
      ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
          = ( insert_nat @ Y @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
          = ( relational_fv_a_b @ Q ) ) ) ) ).

% fv_subst
thf(fact_233_sat__fun__upd,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a,Z3: a] :
      ( ~ ( member_nat @ N @ ( relational_fv_a_b @ Q ) )
     => ( ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma2 @ N @ Z3 ) )
        = ( relational_sat_a_b @ Q @ I2 @ Sigma2 ) ) ) ).

% sat_fun_upd
thf(fact_234_fv_Osimps_I3_J,axiom,
    ! [X: nat,T5: relational_term_a] :
      ( ( relational_fv_a_b @ ( relational_Eq_a_b @ X @ T5 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) ) ).

% fv.simps(3)
thf(fact_235_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_6706873702294942047at_nat @ F2 @ ( minus_8321449233255521966at_nat @ A3 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_6706873702294942047at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_236_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_7637461350386011077at_nat @ F2 @ ( minus_8321449233255521966at_nat @ A3 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_7637461350386011077at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_237_fun__upd__image,axiom,
    ! [X: nat,A3: set_nat,F2: nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( image_5846123807819985514at_nat @ ( fun_up4795876855168711666at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_5846123807819985514at_nat @ F2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( image_5846123807819985514at_nat @ ( fun_up4795876855168711666at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_5846123807819985514at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_238_fun__upd__image,axiom,
    ! [X: nat,A3: set_nat,F2: nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( image_2499013659892586576at_nat @ ( fun_up8749971096457065944at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_2499013659892586576at_nat @ F2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( image_2499013659892586576at_nat @ ( fun_up8749971096457065944at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_2499013659892586576at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_239_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_5168914502847457605at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_5168914502847457605at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_240_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_1672857615394513579at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_1672857615394513579at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_241_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_7325330684212377951at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_7325330684212377951at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_242_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F2 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_960689587819713477at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F2 @ X @ Y ) @ A3 )
          = ( image_960689587819713477at_nat @ F2 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_243_fv_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X )
       => ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ Uu @ Ts2 ) )
             => ( ( Y
                  = ( relati4569515538964159125_set_a @ Ts2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu @ Ts2 ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( Y = bot_bot_set_nat )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
           => ( ! [X2: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T2 ) )
                 => ( ( Y
                      = ( sup_sup_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T2 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X2 @ T2 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( relational_fv_a_b @ Phi2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( Y
                              = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% fv.pelims
thf(fact_244_sat_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts2 ) )
             => ( ( Y
                  = ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
               => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( Y = B )
                 => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
           => ( ! [X2: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T2 ) )
                 => ( ( Y
                      = ( ( Xb @ X2 )
                        = ( relati1177013128715261720term_a @ Xb @ T2 ) ) )
                   => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X2 @ T2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
                     => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
                       => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) )
                         => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( Y
                              = ( ? [X3: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X3 ) ) ) )
                           => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(1)
thf(fact_245_sat_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X @ Xa2 @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts2 ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
               => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                 => ~ B ) )
           => ( ! [X2: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T2 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X2 @ T2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                   => ( ( Xb @ X2 )
                     != ( relati1177013128715261720term_a @ Xb @ T2 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                     => ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                       => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                         => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                           => ~ ? [X2: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(2)
thf(fact_246_sat_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X @ Xa2 @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts2 ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
               => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa2 @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                 => B ) )
           => ( ! [X2: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X2 @ T2 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X2 @ T2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                   => ( ( Xb @ X2 )
                      = ( relati1177013128715261720term_a @ Xb @ T2 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                     => ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                       => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                          & ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                         => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            | ( relational_sat_a_b @ Psi @ Xa2 @ Xb ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                           => ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z @ X5 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(3)
thf(fact_247_Exists__in__sub__cp__foldr1__Disj,axiom,
    ! [X: nat,Q4: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
     => ( ( ( Qs = nil_Re6358386334527539737la_a_b )
          & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) ) )
        | ? [X2: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X2 ) ) ) ) ) ) ).

% Exists_in_sub_cp_foldr1_Disj
thf(fact_248_sat__foldr1__Disj,axiom,
    ! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Xs @ Q ) @ I2 @ Sigma2 )
      = ( ( ( Xs = nil_Re6358386334527539737la_a_b )
         => ( relational_sat_a_b @ Q @ I2 @ Sigma2 ) )
        & ( ( Xs != nil_Re6358386334527539737la_a_b )
         => ? [X3: relational_fmla_a_b] :
              ( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
              & ( relational_sat_a_b @ X3 @ I2 @ Sigma2 ) ) ) ) ) ).

% sat_foldr1_Disj
thf(fact_249_Exists__in__sub__foldr1__Disj,axiom,
    ! [X: nat,Q4: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
     => ( ( ( Qs = nil_Re6358386334527539737la_a_b )
          & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ Q ) ) )
        | ? [X2: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q4 ) @ ( relational_sub_a_b @ X2 ) ) ) ) ) ).

% Exists_in_sub_foldr1_Disj
thf(fact_250_fv__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( ( Qs = nil_Re6358386334527539737la_a_b )
       => ( ( relational_fv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
          = ( relational_fv_a_b @ Q ) ) )
      & ( ( Qs != nil_Re6358386334527539737la_a_b )
       => ( ( relational_fv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
          = ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ relational_fv_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) ) ) ) ) ) ).

% fv_foldr1_Disj
thf(fact_251_fv__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Q ) @ ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ relational_fv_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) ) ) ) ) ).

% fv_foldr_Disj
thf(fact_252_fv__EXISTS,axiom,
    ! [Xs: list_nat,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati7004044288120072026TS_a_b @ Xs @ Q ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( set_nat2 @ Xs ) ) ) ).

% fv_EXISTS
thf(fact_253_foldr__Disj__equiv,axiom,
    ! [Q: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q4: relational_fmla_a_b,Qs2: list_R8263082107343818799la_a_b] :
      ( ( ( insert7010464514620295119la_a_b @ Q @ ( set_Re9104216502384355786la_a_b @ Qs ) )
        = ( insert7010464514620295119la_a_b @ Q4 @ ( set_Re9104216502384355786la_a_b @ Qs2 ) ) )
     => ( relational_equiv_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs2 @ Q4 ) ) ) ).

% foldr_Disj_equiv
thf(fact_254_foldr1__Disj__equiv,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Qs2: list_R8263082107343818799la_a_b] :
      ( ( ( set_Re9104216502384355786la_a_b @ Qs )
        = ( set_Re9104216502384355786la_a_b @ Qs2 ) )
     => ( relational_equiv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs2 @ ( relational_Bool_a_b @ $false ) ) ) ) ).

% foldr1_Disj_equiv
thf(fact_255_Exists__foldr__Disj,axiom,
    ! [X: nat,Xs: list_R8263082107343818799la_a_b,B2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Xs @ B2 ) ) @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ ( map_Re5057026939262599940la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ Xs ) @ ( relati3989891337220013914ts_a_b @ X @ B2 ) ) ) ).

% Exists_foldr_Disj
thf(fact_256_equiv__Neg__cong,axiom,
    ! [Q: relational_fmla_a_b,Q4: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q @ Q4 )
     => ( relational_equiv_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_Neg_a_b @ Q4 ) ) ) ).

% equiv_Neg_cong
thf(fact_257_equiv__Disj__Assoc,axiom,
    ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,Q32: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ Q32 ) @ ( relational_Disj_a_b @ Q12 @ ( relational_Disj_a_b @ Q23 @ Q32 ) ) ) ).

% equiv_Disj_Assoc
thf(fact_258_equiv__Disj__cong,axiom,
    ! [Q12: relational_fmla_a_b,Q14: relational_fmla_a_b,Q23: relational_fmla_a_b,Q25: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q12 @ Q14 )
     => ( ( relational_equiv_a_b @ Q23 @ Q25 )
       => ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( relational_Disj_a_b @ Q14 @ Q25 ) ) ) ) ).

% equiv_Disj_cong
thf(fact_259_equiv__Exists__cong,axiom,
    ! [Q: relational_fmla_a_b,Q4: relational_fmla_a_b,X: nat] :
      ( ( relational_equiv_a_b @ Q @ Q4 )
     => ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ ( relati591517084277583526ts_a_b @ X @ Q4 ) ) ) ).

% equiv_Exists_cong
thf(fact_260_equiv__Conj__cong,axiom,
    ! [Q12: relational_fmla_a_b,Q14: relational_fmla_a_b,Q23: relational_fmla_a_b,Q25: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q12 @ Q14 )
     => ( ( relational_equiv_a_b @ Q23 @ Q25 )
       => ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ ( relational_Conj_a_b @ Q14 @ Q25 ) ) ) ) ).

% equiv_Conj_cong
thf(fact_261_equiv__Exists__Disj,axiom,
    ! [X: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ Q12 ) @ ( relati591517084277583526ts_a_b @ X @ Q23 ) ) ) ).

% equiv_Exists_Disj
thf(fact_262_Exists__nonfree__equiv,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ).

% Exists_nonfree_equiv
thf(fact_263_equiv__Exists__exists__cong,axiom,
    ! [Q: relational_fmla_a_b,Q4: relational_fmla_a_b,X: nat] :
      ( ( relational_equiv_a_b @ Q @ Q4 )
     => ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ ( relati3989891337220013914ts_a_b @ X @ Q4 ) ) ) ).

% equiv_Exists_exists_cong
thf(fact_264_Exists__foldr1__Disj,axiom,
    ! [X: nat,Xs: list_R8263082107343818799la_a_b,B2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Xs @ B2 ) ) @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ ( map_Re5057026939262599940la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ Xs ) @ ( relati3989891337220013914ts_a_b @ X @ B2 ) ) ) ).

% Exists_foldr1_Disj
thf(fact_265_fv__close,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati7004044288120072026TS_a_b @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q ) ) @ Q ) )
      = bot_bot_set_nat ) ).

% fv_close
thf(fact_266_Field__insert,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R3 ) ) ) ).

% Field_insert
thf(fact_267_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% Field_insert
thf(fact_268_Field__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( field_1428494348145443988at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 ) )
      = ( sup_su474284620922681339at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B2 @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% Field_insert
thf(fact_269_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% Field_insert
thf(fact_270_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% Field_insert
thf(fact_271_inj__on__insert,axiom,
    ! [F2: product_prod_nat_nat > produc859450856879609959at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on3865918002419084311at_nat @ F2 @ A3 )
        & ~ ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_1119892238107900587at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_272_inj__on__insert,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on8969904277767023793at_nat @ F2 @ A3 )
        & ~ ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5168914502847457605at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_273_inj__on__insert,axiom,
    ! [F2: product_prod_nat_nat > produc3843707927480180839at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on1146879744823063575at_nat @ F2 @ A3 )
        & ~ ( member8757157785044589968at_nat @ ( F2 @ A ) @ ( image_1672857615394513579at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_274_inj__on__insert,axiom,
    ! [F2: product_prod_nat_nat > nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on2178005380612969504at_nat @ F2 @ A3 )
        & ~ ( member_nat @ ( F2 @ A ) @ ( image_2486076414777270412at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_275_inj__on__insert,axiom,
    ! [F2: produc3843707927480180839at_nat > produc859450856879609959at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7438016324537269041at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on7438016324537269041at_nat @ F2 @ A3 )
        & ~ ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_5653576389946358725at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_276_inj__on__insert,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on6799352813640927947at_nat @ F2 @ A3 )
        & ~ ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_7325330684212377951at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_277_inj__on__insert,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on8430303779476321073at_nat @ F2 @ A3 )
        & ~ ( member8757157785044589968at_nat @ ( F2 @ A ) @ ( image_960689587819713477at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_278_inj__on__insert,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7963584518437281414at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on7963584518437281414at_nat @ F2 @ A3 )
        & ~ ( member_nat @ ( F2 @ A ) @ ( image_3198525501578754290at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_279_fmla_Oset__cases_I1_J,axiom,
    ! [E: a,A: relational_fmla_a_b] :
      ( ( member_a @ E @ ( relati3071123380395136021la_a_b @ A ) )
     => ( ! [Z1: b,Z2: list_R6823256787227418703term_a] :
            ( ( A
              = ( relational_Pred_b_a @ Z1 @ Z2 ) )
           => ! [X2: relational_term_a] :
                ( ( member5466445112152945266term_a @ X2 @ ( set_Re3569617851344498910term_a @ Z2 ) )
               => ~ ( member_a @ E @ ( relati2096729434547099208term_a @ X2 ) ) ) )
       => ( ! [Z1: nat,Z2: relational_term_a] :
              ( ( A
                = ( relational_Eq_a_b @ Z1 @ Z2 ) )
             => ~ ( member_a @ E @ ( relati2096729434547099208term_a @ Z2 ) ) )
         => ( ! [Z: relational_fmla_a_b] :
                ( ( A
                  = ( relational_Neg_a_b @ Z ) )
               => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z ) ) )
           => ( ! [Z1: relational_fmla_a_b] :
                  ( ? [Z2: relational_fmla_a_b] :
                      ( A
                      = ( relational_Conj_a_b @ Z1 @ Z2 ) )
                 => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z1 ) ) )
             => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                    ( ( A
                      = ( relational_Conj_a_b @ Z1 @ Z2 ) )
                   => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z2 ) ) )
               => ( ! [Z1: relational_fmla_a_b] :
                      ( ? [Z2: relational_fmla_a_b] :
                          ( A
                          = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                     => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z1 ) ) )
                 => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                        ( ( A
                          = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                       => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z2 ) ) )
                   => ~ ! [Z1: nat,Z2: relational_fmla_a_b] :
                          ( ( A
                            = ( relati591517084277583526ts_a_b @ Z1 @ Z2 ) )
                         => ~ ( member_a @ E @ ( relati3071123380395136021la_a_b @ Z2 ) ) ) ) ) ) ) ) ) ) ) ).

% fmla.set_cases(1)
thf(fact_280_finite__set__term,axiom,
    ! [T3: relati8250976376238479564at_nat] : ( finite6177210948735845034at_nat @ ( relati3132094035298600779at_nat @ T3 ) ) ).

% finite_set_term
thf(fact_281_FieldI2,axiom,
    ! [I3: produc859450856879609959at_nat,J: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J ) @ R2 )
     => ( member8206827879206165904at_nat @ J @ ( field_3044959742847878580at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_282_FieldI2,axiom,
    ! [I3: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J ) @ R2 )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_283_FieldI2,axiom,
    ! [I3: nat,J: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R2 )
     => ( member_nat @ J @ ( field_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_284_FieldI2,axiom,
    ! [I3: product_prod_nat_nat,J: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R2 )
     => ( member8440522571783428010at_nat @ J @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_285_FieldI2,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J ) @ R2 )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_286_FieldI2,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J ) @ R2 )
     => ( member6273874620674182256at_nat @ J @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_287_FieldI1,axiom,
    ! [I3: produc859450856879609959at_nat,J: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J ) @ R2 )
     => ( member8206827879206165904at_nat @ I3 @ ( field_3044959742847878580at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_288_FieldI1,axiom,
    ! [I3: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J ) @ R2 )
     => ( member8757157785044589968at_nat @ I3 @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_289_FieldI1,axiom,
    ! [I3: nat,J: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R2 )
     => ( member_nat @ I3 @ ( field_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_290_FieldI1,axiom,
    ! [I3: product_prod_nat_nat,J: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R2 )
     => ( member8440522571783428010at_nat @ I3 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_291_FieldI1,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J ) @ R2 )
     => ( member2643936169264416010at_nat @ I3 @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_292_FieldI1,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J ) @ R2 )
     => ( member6273874620674182256at_nat @ I3 @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_293_fmla_Osimps_I123_J,axiom,
    ! [X4: relational_fmla_a_b] :
      ( ( relati3071123380395136021la_a_b @ ( relational_Neg_a_b @ X4 ) )
      = ( relati3071123380395136021la_a_b @ X4 ) ) ).

% fmla.simps(123)
thf(fact_294_fmla_Oset__intros_I3_J,axiom,
    ! [Yc: a,X4: relational_fmla_a_b] :
      ( ( member_a @ Yc @ ( relati3071123380395136021la_a_b @ X4 ) )
     => ( member_a @ Yc @ ( relati3071123380395136021la_a_b @ ( relational_Neg_a_b @ X4 ) ) ) ) ).

% fmla.set_intros(3)
thf(fact_295_fmla_Oset__intros_I7_J,axiom,
    ! [Yk: a,X62: relational_fmla_a_b,X61: relational_fmla_a_b] :
      ( ( member_a @ Yk @ ( relati3071123380395136021la_a_b @ X62 ) )
     => ( member_a @ Yk @ ( relati3071123380395136021la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(7)
thf(fact_296_fmla_Oset__intros_I6_J,axiom,
    ! [Yi: a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( member_a @ Yi @ ( relati3071123380395136021la_a_b @ X61 ) )
     => ( member_a @ Yi @ ( relati3071123380395136021la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(6)
thf(fact_297_fmla_Osimps_I126_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b] :
      ( ( relati3071123380395136021la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) )
      = ( relati3071123380395136021la_a_b @ X72 ) ) ).

% fmla.simps(126)
thf(fact_298_fmla_Oset__intros_I8_J,axiom,
    ! [Ym: a,X72: relational_fmla_a_b,X71: nat] :
      ( ( member_a @ Ym @ ( relati3071123380395136021la_a_b @ X72 ) )
     => ( member_a @ Ym @ ( relati3071123380395136021la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ) ) ).

% fmla.set_intros(8)
thf(fact_299_fmla_Oset__intros_I5_J,axiom,
    ! [Yg: a,X52: relational_fmla_a_b,X51: relational_fmla_a_b] :
      ( ( member_a @ Yg @ ( relati3071123380395136021la_a_b @ X52 ) )
     => ( member_a @ Yg @ ( relati3071123380395136021la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(5)
thf(fact_300_fmla_Oset__intros_I4_J,axiom,
    ! [Ye: a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( member_a @ Ye @ ( relati3071123380395136021la_a_b @ X51 ) )
     => ( member_a @ Ye @ ( relati3071123380395136021la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(4)
thf(fact_301_fmla_Oset__intros_I1_J,axiom,
    ! [Y: relational_term_a,X12: list_R6823256787227418703term_a,Ya: a,X11: b] :
      ( ( member5466445112152945266term_a @ Y @ ( set_Re3569617851344498910term_a @ X12 ) )
     => ( ( member_a @ Ya @ ( relati2096729434547099208term_a @ Y ) )
       => ( member_a @ Ya @ ( relati3071123380395136021la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) ) ) ) ) ).

% fmla.set_intros(1)
thf(fact_302_term_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A: relati1609568010824441138at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( relati8359993527032477617at_nat @ A ) )
     => ( A
        = ( relati1001352036157671836at_nat @ E ) ) ) ).

% term.set_cases
thf(fact_303_term_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A: relati8250976376238479564at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( relati3132094035298600779at_nat @ A ) )
     => ( A
        = ( relati6667065761635354038at_nat @ E ) ) ) ).

% term.set_cases
thf(fact_304_term_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A: relati2597207553810441522at_nat] :
      ( ( member8757157785044589968at_nat @ E @ ( relati360459484535582641at_nat @ A ) )
     => ( A
        = ( relati3582544344963639708at_nat @ E ) ) ) ).

% term.set_cases
thf(fact_305_term_Oset__cases,axiom,
    ! [E: nat,A: relational_term_nat] :
      ( ( member_nat @ E @ ( relati7416457470477713926rm_nat @ A ) )
     => ( A
        = ( relational_Const_nat @ E ) ) ) ).

% term.set_cases
thf(fact_306_term_Oset__intros,axiom,
    ! [X1: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X1 @ ( relati8359993527032477617at_nat @ ( relati1001352036157671836at_nat @ X1 ) ) ) ).

% term.set_intros
thf(fact_307_term_Oset__intros,axiom,
    ! [X1: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X1 @ ( relati3132094035298600779at_nat @ ( relati6667065761635354038at_nat @ X1 ) ) ) ).

% term.set_intros
thf(fact_308_term_Oset__intros,axiom,
    ! [X1: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X1 @ ( relati360459484535582641at_nat @ ( relati3582544344963639708at_nat @ X1 ) ) ) ).

% term.set_intros
thf(fact_309_term_Oset__intros,axiom,
    ! [X1: nat] : ( member_nat @ X1 @ ( relati7416457470477713926rm_nat @ ( relational_Const_nat @ X1 ) ) ) ).

% term.set_intros
thf(fact_310_inj__img__insertE,axiom,
    ! [F2: nat > nat,A3: set_nat,X: nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A3 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ B4 )
            = ( image_nat_nat @ F2 @ A3 ) )
         => ~ ! [X6: nat,A6: set_nat] :
                ( ~ ( member_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_nat_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_311_inj__img__insertE,axiom,
    ! [F2: nat > product_prod_nat_nat,A3: set_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B4 )
            = ( image_5846123807819985514at_nat @ F2 @ A3 ) )
         => ~ ! [X6: nat,A6: set_nat] :
                ( ~ ( member_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_5846123807819985514at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_312_inj__img__insertE,axiom,
    ! [F2: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,X: nat,B4: set_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ A3 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ B4 )
            = ( image_2486076414777270412at_nat @ F2 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_2486076414777270412at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_313_inj__img__insertE,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B4 )
            = ( image_5168914502847457605at_nat @ F2 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_5168914502847457605at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_314_inj__img__insertE,axiom,
    ! [F2: nat > produc859450856879609959at_nat,A3: set_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on6829720656803765732at_nat @ F2 @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ X @ B4 )
       => ( ( ( insert5050368324300391991at_nat @ X @ B4 )
            = ( image_3276603626957510736at_nat @ F2 @ A3 ) )
         => ~ ! [X6: nat,A6: set_nat] :
                ( ~ ( member_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_3276603626957510736at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_315_inj__img__insertE,axiom,
    ! [F2: produc859450856879609959at_nat > nat,A3: set_Pr8693737435421807431at_nat,X: nat,B4: set_nat] :
      ( ( inj_on490725545064285830at_nat @ F2 @ A3 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ B4 )
            = ( image_6160980552072806642at_nat @ F2 @ A3 ) )
         => ~ ! [X6: produc859450856879609959at_nat,A6: set_Pr8693737435421807431at_nat] :
                ( ~ ( member8206827879206165904at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert5050368324300391991at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_6160980552072806642at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_316_inj__img__insertE,axiom,
    ! [F2: product_prod_nat_nat > produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F2 @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ X @ B4 )
       => ( ( ( insert5050368324300391991at_nat @ X @ B4 )
            = ( image_1119892238107900587at_nat @ F2 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_1119892238107900587at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_317_inj__img__insertE,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B4 )
            = ( image_6706873702294942047at_nat @ F2 @ A3 ) )
         => ~ ! [X6: produc859450856879609959at_nat,A6: set_Pr8693737435421807431at_nat] :
                ( ~ ( member8206827879206165904at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert5050368324300391991at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_6706873702294942047at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_318_inj__img__insertE,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A3: set_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ A3 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B4 )
       => ( ( ( insert9069300056098147895at_nat @ X @ B4 )
            = ( image_2499013659892586576at_nat @ F2 @ A3 ) )
         => ~ ! [X6: nat,A6: set_nat] :
                ( ~ ( member_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_2499013659892586576at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_319_inj__img__insertE,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,A3: set_Pr4329608150637261639at_nat,X: nat,B4: set_nat] :
      ( ( inj_on7963584518437281414at_nat @ F2 @ A3 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ B4 )
            = ( image_3198525501578754290at_nat @ F2 @ A3 ) )
         => ~ ! [X6: produc3843707927480180839at_nat,A6: set_Pr4329608150637261639at_nat] :
                ( ~ ( member8757157785044589968at_nat @ X6 @ A6 )
               => ( ( A3
                    = ( insert9069300056098147895at_nat @ X6 @ A6 ) )
                 => ( ( X
                      = ( F2 @ X6 ) )
                   => ( B4
                     != ( image_3198525501578754290at_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_320_fmla_Osimps_I120_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a] :
      ( ( relati3071123380395136021la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) )
      = ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati2096729434547099208term_a @ ( set_Re3569617851344498910term_a @ X12 ) ) ) ) ).

% fmla.simps(120)
thf(fact_321_fmla_Osimps_I121_J,axiom,
    ! [X22: $o] :
      ( ( relati3071123380395136021la_a_b @ ( relational_Bool_a_b @ X22 ) )
      = bot_bot_set_a ) ).

% fmla.simps(121)
thf(fact_322_fmla_Osimps_I125_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relati3071123380395136021la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
      = ( sup_sup_set_a @ ( relati3071123380395136021la_a_b @ X61 ) @ ( relati3071123380395136021la_a_b @ X62 ) ) ) ).

% fmla.simps(125)
thf(fact_323_fmla_Osimps_I124_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relati3071123380395136021la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) )
      = ( sup_sup_set_a @ ( relati3071123380395136021la_a_b @ X51 ) @ ( relati3071123380395136021la_a_b @ X52 ) ) ) ).

% fmla.simps(124)
thf(fact_324_inj__singleton,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( inj_on8528517097230073361at_nat
      @ ^ [X3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      @ A3 ) ).

% inj_singleton
thf(fact_325_inj__singleton,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( inj_on104525631246638609at_nat
      @ ^ [X3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat )
      @ A3 ) ).

% inj_singleton
thf(fact_326_Relational__Calculus_Oterm_Osimps_I16_J,axiom,
    ! [X22: nat] :
      ( ( relati3132094035298600779at_nat @ ( relati1023757457726692818at_nat @ X22 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Relational_Calculus.term.simps(16)
thf(fact_327_Relational__Calculus_Oterm_Osimps_I16_J,axiom,
    ! [X22: nat] :
      ( ( relati360459484535582641at_nat @ ( relati2721891637006993336at_nat @ X22 ) )
      = bot_bo228742789529271731at_nat ) ).

% Relational_Calculus.term.simps(16)
thf(fact_328_Relational__Calculus_Oterm_Osimps_I15_J,axiom,
    ! [X1: product_prod_nat_nat] :
      ( ( relati3132094035298600779at_nat @ ( relati6667065761635354038at_nat @ X1 ) )
      = ( insert8211810215607154385at_nat @ X1 @ bot_bo2099793752762293965at_nat ) ) ).

% Relational_Calculus.term.simps(15)
thf(fact_329_Relational__Calculus_Oterm_Osimps_I15_J,axiom,
    ! [X1: produc3843707927480180839at_nat] :
      ( ( relati360459484535582641at_nat @ ( relati3582544344963639708at_nat @ X1 ) )
      = ( insert9069300056098147895at_nat @ X1 @ bot_bo228742789529271731at_nat ) ) ).

% Relational_Calculus.term.simps(15)
thf(fact_330_inj__on__Un,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A3 @ B4 ) )
      = ( ( inj_on6799352813640927947at_nat @ F2 @ A3 )
        & ( inj_on6799352813640927947at_nat @ F2 @ B4 )
        & ( ( inf_in2572325071724192079at_nat @ ( image_7325330684212377951at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ B4 ) ) @ ( image_7325330684212377951at_nat @ F2 @ ( minus_3314409938677909166at_nat @ B4 @ A3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_Un
thf(fact_331_inj__on__Un,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A3 @ B4 ) )
      = ( ( inj_on8430303779476321073at_nat @ F2 @ A3 )
        & ( inj_on8430303779476321073at_nat @ F2 @ B4 )
        & ( ( inf_in7913087082777306421at_nat @ ( image_960689587819713477at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A3 @ B4 ) ) @ ( image_960689587819713477at_nat @ F2 @ ( minus_3314409938677909166at_nat @ B4 @ A3 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% inj_on_Un
thf(fact_332_max__ext_Omax__extI,axiom,
    ! [X7: set_Pr8693737435421807431at_nat,Y6: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( finite4392333629123659920at_nat @ X7 )
     => ( ( finite4392333629123659920at_nat @ Y6 )
       => ( ( Y6 != bot_bo5327735625951526323at_nat )
         => ( ! [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ X7 )
               => ? [Xa: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Xa @ Y6 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ X7 @ Y6 ) @ ( max_ex4511810952740877858at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_333_max__ext_Omax__extI,axiom,
    ! [X7: set_nat,Y6: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( finite_finite_nat @ Y6 )
       => ( ( Y6 != bot_bot_set_nat )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ X7 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ Y6 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X7 @ Y6 ) @ ( max_ext_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_334_max__ext_Omax__extI,axiom,
    ! [X7: set_se7855581050983116737at_nat,Y6: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X7 )
     => ( ( finite9047747110432174090at_nat @ Y6 )
       => ( ( Y6 != bot_bo3083307316010499117at_nat )
         => ( ! [X2: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X2 @ X7 )
               => ? [Xa: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa @ Y6 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X7 @ Y6 ) @ ( max_ex4535816423174879388at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_335_max__ext_Omax__extI,axiom,
    ! [X7: set_se5479201105039950375at_nat,Y6: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( finite4768004253697144688at_nat @ X7 )
     => ( ( finite4768004253697144688at_nat @ Y6 )
       => ( ( Y6 != bot_bo1534825418459847827at_nat )
         => ( ! [X2: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ X2 @ X7 )
               => ? [Xa: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ Xa @ Y6 )
                    & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X7 @ Y6 ) @ ( max_ex4000160833828605186at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_336_max__ext_Omax__extI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X7 )
     => ( ( finite6177210948735845034at_nat @ Y6 )
       => ( ( Y6 != bot_bo2099793752762293965at_nat )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ X7 )
               => ? [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ Y6 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y6 ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_337_max__ext_Omax__extI,axiom,
    ! [X7: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X7 )
     => ( ( finite4343798906461161616at_nat @ Y6 )
       => ( ( Y6 != bot_bo228742789529271731at_nat )
         => ( ! [X2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X2 @ X7 )
               => ? [Xa: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa @ Y6 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Xa ) @ R2 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X7 @ Y6 ) @ ( max_ex8833738776070432290at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_338_finite__fv,axiom,
    ! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).

% finite_fv
thf(fact_339_max__ext__additive,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,C4: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B4 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C4 @ D ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ C4 ) @ ( sup_su6327502436637775413at_nat @ B4 @ D ) ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ).

% max_ext_additive
thf(fact_340_max__ext__additive,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,C4: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A3 @ B4 ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ C4 @ D ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ C4 ) @ ( sup_su5525570899277871387at_nat @ B4 @ D ) ) @ ( max_ex8833738776070432290at_nat @ R2 ) ) ) ) ).

% max_ext_additive
thf(fact_341_inj__on__image__mem__iff,axiom,
    ! [F2: nat > nat,B4: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ B4 )
     => ( ( member_nat @ A @ B4 )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_342_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > nat,B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ B4 )
     => ( ( member8440522571783428010at_nat @ A @ B4 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( ( member_nat @ ( F2 @ A ) @ ( image_2486076414777270412at_nat @ F2 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_343_inj__on__image__mem__iff,axiom,
    ! [F2: nat > product_prod_nat_nat,B4: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ B4 )
     => ( ( member_nat @ A @ B4 )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5846123807819985514at_nat @ F2 @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_344_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ B4 )
     => ( ( member8440522571783428010at_nat @ A @ B4 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5168914502847457605at_nat @ F2 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_345_inj__on__image__mem__iff,axiom,
    ! [F2: produc859450856879609959at_nat > nat,B4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on490725545064285830at_nat @ F2 @ B4 )
     => ( ( member8206827879206165904at_nat @ A @ B4 )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B4 )
         => ( ( member_nat @ ( F2 @ A ) @ ( image_6160980552072806642at_nat @ F2 @ A3 ) )
            = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_346_inj__on__image__mem__iff,axiom,
    ! [F2: nat > produc859450856879609959at_nat,B4: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on6829720656803765732at_nat @ F2 @ B4 )
     => ( ( member_nat @ A @ B4 )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_3276603626957510736at_nat @ F2 @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_347_inj__on__image__mem__iff,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,B4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ B4 )
     => ( ( member8206827879206165904at_nat @ A @ B4 )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B4 )
         => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_6706873702294942047at_nat @ F2 @ A3 ) )
            = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_348_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > produc859450856879609959at_nat,B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F2 @ B4 )
     => ( ( member8440522571783428010at_nat @ A @ B4 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_1119892238107900587at_nat @ F2 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_349_inj__on__image__mem__iff,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7963584518437281414at_nat @ F2 @ B4 )
     => ( ( member8757157785044589968at_nat @ A @ B4 )
       => ( ( ord_le1268244103169919719at_nat @ A3 @ B4 )
         => ( ( member_nat @ ( F2 @ A ) @ ( image_3198525501578754290at_nat @ F2 @ A3 ) )
            = ( member8757157785044589968at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_350_inj__on__image__mem__iff,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,B4: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ B4 )
     => ( ( member_nat @ A @ B4 )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( ( member8757157785044589968at_nat @ ( F2 @ A ) @ ( image_2499013659892586576at_nat @ F2 @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_351_fv__cp,axiom,
    ! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ).

% fv_cp
thf(fact_352_inj__on__disjoint__Un,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,G5: produc859450856879609959at_nat > product_prod_nat_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ A3 )
     => ( ( inj_on229527429751349963at_nat @ G5 @ B4 )
       => ( ( ( inf_in2572325071724192079at_nat @ ( image_6706873702294942047at_nat @ F2 @ A3 ) @ ( image_6706873702294942047at_nat @ G5 @ B4 ) )
            = bot_bo2099793752762293965at_nat )
         => ( inj_on229527429751349963at_nat
            @ ^ [X3: produc859450856879609959at_nat] : ( if_Pro6206227464963214023at_nat @ ( member8206827879206165904at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su718114333110466843at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_353_inj__on__disjoint__Un,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,G5: product_prod_nat_nat > product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ A3 )
     => ( ( inj_on8969904277767023793at_nat @ G5 @ B4 )
       => ( ( ( inf_in2572325071724192079at_nat @ ( image_5168914502847457605at_nat @ F2 @ A3 ) @ ( image_5168914502847457605at_nat @ G5 @ B4 ) )
            = bot_bo2099793752762293965at_nat )
         => ( inj_on8969904277767023793at_nat
            @ ^ [X3: product_prod_nat_nat] : ( if_Pro6206227464963214023at_nat @ ( member8440522571783428010at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su6327502436637775413at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_354_inj__on__disjoint__Un,axiom,
    ! [F2: nat > product_prod_nat_nat,A3: set_nat,G5: nat > product_prod_nat_nat,B4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A3 )
     => ( ( inj_on5538052773655684606at_nat @ G5 @ B4 )
       => ( ( ( inf_in2572325071724192079at_nat @ ( image_5846123807819985514at_nat @ F2 @ A3 ) @ ( image_5846123807819985514at_nat @ G5 @ B4 ) )
            = bot_bo2099793752762293965at_nat )
         => ( inj_on5538052773655684606at_nat
            @ ^ [X3: nat] : ( if_Pro6206227464963214023at_nat @ ( member_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_355_inj__on__disjoint__Un,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,G5: produc3843707927480180839at_nat > product_prod_nat_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F2 @ A3 )
     => ( ( inj_on6799352813640927947at_nat @ G5 @ B4 )
       => ( ( ( inf_in2572325071724192079at_nat @ ( image_7325330684212377951at_nat @ F2 @ A3 ) @ ( image_7325330684212377951at_nat @ G5 @ B4 ) )
            = bot_bo2099793752762293965at_nat )
         => ( inj_on6799352813640927947at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : ( if_Pro6206227464963214023at_nat @ ( member8757157785044589968at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su5525570899277871387at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_356_inj__on__disjoint__Un,axiom,
    ! [F2: produc859450856879609959at_nat > produc3843707927480180839at_nat,A3: set_Pr8693737435421807431at_nat,G5: produc859450856879609959at_nat > produc3843707927480180839at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on198529248122145585at_nat @ F2 @ A3 )
     => ( ( inj_on198529248122145585at_nat @ G5 @ B4 )
       => ( ( ( inf_in7913087082777306421at_nat @ ( image_7637461350386011077at_nat @ F2 @ A3 ) @ ( image_7637461350386011077at_nat @ G5 @ B4 ) )
            = bot_bo228742789529271731at_nat )
         => ( inj_on198529248122145585at_nat
            @ ^ [X3: produc859450856879609959at_nat] : ( if_Pro3594990065127090477at_nat @ ( member8206827879206165904at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su718114333110466843at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_357_inj__on__disjoint__Un,axiom,
    ! [F2: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,G5: product_prod_nat_nat > produc3843707927480180839at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F2 @ A3 )
     => ( ( inj_on1146879744823063575at_nat @ G5 @ B4 )
       => ( ( ( inf_in7913087082777306421at_nat @ ( image_1672857615394513579at_nat @ F2 @ A3 ) @ ( image_1672857615394513579at_nat @ G5 @ B4 ) )
            = bot_bo228742789529271731at_nat )
         => ( inj_on1146879744823063575at_nat
            @ ^ [X3: product_prod_nat_nat] : ( if_Pro3594990065127090477at_nat @ ( member8440522571783428010at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su6327502436637775413at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_358_inj__on__disjoint__Un,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A3: set_nat,G5: nat > produc3843707927480180839at_nat,B4: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ A3 )
     => ( ( inj_on7264072676751113700at_nat @ G5 @ B4 )
       => ( ( ( inf_in7913087082777306421at_nat @ ( image_2499013659892586576at_nat @ F2 @ A3 ) @ ( image_2499013659892586576at_nat @ G5 @ B4 ) )
            = bot_bo228742789529271731at_nat )
         => ( inj_on7264072676751113700at_nat
            @ ^ [X3: nat] : ( if_Pro3594990065127090477at_nat @ ( member_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_sup_set_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_359_inj__on__disjoint__Un,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,G5: produc3843707927480180839at_nat > produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F2 @ A3 )
     => ( ( inj_on8430303779476321073at_nat @ G5 @ B4 )
       => ( ( ( inf_in7913087082777306421at_nat @ ( image_960689587819713477at_nat @ F2 @ A3 ) @ ( image_960689587819713477at_nat @ G5 @ B4 ) )
            = bot_bo228742789529271731at_nat )
         => ( inj_on8430303779476321073at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : ( if_Pro3594990065127090477at_nat @ ( member8757157785044589968at_nat @ X3 @ A3 ) @ ( F2 @ X3 ) @ ( G5 @ X3 ) )
            @ ( sup_su5525570899277871387at_nat @ A3 @ B4 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_360_max__ext_Ocases,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R2 ) )
     => ~ ( ( finite4392333629123659920at_nat @ A1 )
         => ( ( finite4392333629123659920at_nat @ A22 )
           => ( ( A22 != bot_bo5327735625951526323at_nat )
             => ~ ! [X5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ X5 @ A1 )
                   => ? [Xa3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ Xa3 @ A22 )
                        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_361_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X5: nat] :
                    ( ( member_nat @ X5 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_362_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R2 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A22 )
           => ( ( A22 != bot_bo3083307316010499117at_nat )
             => ~ ! [X5: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X5 @ A1 )
                   => ? [Xa3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa3 @ A22 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_363_max__ext_Ocases,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R2 ) )
     => ~ ( ( finite4768004253697144688at_nat @ A1 )
         => ( ( finite4768004253697144688at_nat @ A22 )
           => ( ( A22 != bot_bo1534825418459847827at_nat )
             => ~ ! [X5: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X5 @ A1 )
                   => ? [Xa3: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Xa3 @ A22 )
                        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_364_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X5 @ A1 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_365_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A22 )
           => ( ( A22 != bot_bo228742789529271731at_nat )
             => ~ ! [X5: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X5 @ A1 )
                   => ? [Xa3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa3 @ A22 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_366_max__ext_Osimps,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R2 ) )
      = ( ( finite4392333629123659920at_nat @ A1 )
        & ( finite4392333629123659920at_nat @ A22 )
        & ( A22 != bot_bo5327735625951526323at_nat )
        & ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A1 )
           => ? [Y5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y5 @ A22 )
                & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_367_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ A1 )
           => ? [Y5: nat] :
                ( ( member_nat @ Y5 @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_368_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R2 ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A22 )
        & ( A22 != bot_bo3083307316010499117at_nat )
        & ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A1 )
           => ? [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y5 @ A22 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_369_max__ext_Osimps,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R2 ) )
      = ( ( finite4768004253697144688at_nat @ A1 )
        & ( finite4768004253697144688at_nat @ A22 )
        & ( A22 != bot_bo1534825418459847827at_nat )
        & ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A1 )
           => ? [Y5: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Y5 @ A22 )
                & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_370_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A1 )
           => ? [Y5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_371_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A22 )
        & ( A22 != bot_bo228742789529271731at_nat )
        & ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A1 )
           => ? [Y5: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y5 @ A22 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_372_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] :
      ( ( max_ex6719288698447201402at_nat
        @ ^ [X3: produc3843707927480180839at_nat,Y5: produc3843707927480180839at_nat] : ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y5 ) @ R2 ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ ( max_ex8833738776070432290at_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_373_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R2 ) )
      = ( ^ [X3: set_nat,Y5: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y5 ) @ ( max_ext_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_374_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( max_ex4845989503969680116at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R2 ) )
      = ( ^ [X3: set_se7855581050983116737at_nat,Y5: set_se7855581050983116737at_nat] : ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X3 @ Y5 ) @ ( max_ex4535816423174879388at_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_375_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( max_ex8849487260381072218at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R2 ) )
      = ( ^ [X3: set_se5479201105039950375at_nat,Y5: set_se5479201105039950375at_nat] : ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X3 @ Y5 ) @ ( max_ex4000160833828605186at_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_376_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R2 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_377_inj__on__convol__ident,axiom,
    ! [F2: nat > nat,X7: set_nat] :
      ( inj_on5538052773655684606at_nat
      @ ^ [X3: nat] : ( product_Pair_nat_nat @ X3 @ ( F2 @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_378_inj__on__convol__ident,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( inj_on3865918002419084311at_nat
      @ ^ [X3: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ X3 @ ( F2 @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_379_inj__on__convol__ident,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X7: set_se7855581050983116737at_nat] :
      ( inj_on699426629415580343at_nat
      @ ^ [X3: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X3 @ ( F2 @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_380_inj__on__convol__ident,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X7: set_se5479201105039950375at_nat] :
      ( inj_on1468925692805905105at_nat
      @ ^ [X3: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X3 @ ( F2 @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_381_Range__insert,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 ) )
      = ( insert_nat @ B2 @ ( range_nat_nat @ R3 ) ) ) ).

% Range_insert
thf(fact_382_Range__insert,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( range_4455291928337903541at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 ) )
      = ( insert8211810215607154385at_nat @ B2 @ ( range_4455291928337903541at_nat @ R3 ) ) ) ).

% Range_insert
thf(fact_383_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 ) )
      = ( insert9200635055090092081at_nat @ B2 @ ( range_880292732526890933at_nat @ R3 ) ) ) ).

% Range_insert
thf(fact_384_Range__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( range_7218127889571902005at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 ) )
      = ( insert2290795007606087959at_nat @ B2 @ ( range_7218127889571902005at_nat @ R3 ) ) ) ).

% Range_insert
thf(fact_385_the__inv__into__into,axiom,
    ! [F2: nat > nat,A3: set_nat,X: nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A3 )
     => ( ( member_nat @ X @ ( image_nat_nat @ F2 @ A3 ) )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( member_nat @ ( the_inv_into_nat_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_386_the__inv__into__into,axiom,
    ! [F2: nat > product_prod_nat_nat,A3: set_nat,X: product_prod_nat_nat,B4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_5846123807819985514at_nat @ F2 @ A3 ) )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( member_nat @ ( the_in1052143192868928060at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_387_the__inv__into__into,axiom,
    ! [F2: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,X: nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ A3 )
     => ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ F2 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( member8440522571783428010at_nat @ ( the_in6915467836680988766at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_388_the__inv__into__into,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_5168914502847457605at_nat @ F2 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( member8440522571783428010at_nat @ ( the_in1699675864273310451at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_389_the__inv__into__into,axiom,
    ! [F2: nat > produc859450856879609959at_nat,A3: set_nat,X: produc859450856879609959at_nat,B4: set_nat] :
      ( ( inj_on6829720656803765732at_nat @ F2 @ A3 )
     => ( ( member8206827879206165904at_nat @ X @ ( image_3276603626957510736at_nat @ F2 @ A3 ) )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( member_nat @ ( the_in5925935293361986850at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_390_the__inv__into__into,axiom,
    ! [F2: produc859450856879609959at_nat > nat,A3: set_Pr8693737435421807431at_nat,X: nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on490725545064285830at_nat @ F2 @ A3 )
     => ( ( member_nat @ X @ ( image_6160980552072806642at_nat @ F2 @ A3 ) )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B4 )
         => ( member8206827879206165904at_nat @ ( the_in8810312218477282756at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_391_the__inv__into__into,axiom,
    ! [F2: product_prod_nat_nat > produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,X: produc859450856879609959at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F2 @ A3 )
     => ( ( member8206827879206165904at_nat @ X @ ( image_1119892238107900587at_nat @ F2 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
         => ( member8440522571783428010at_nat @ ( the_in4366950361296922457at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_392_the__inv__into__into,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_6706873702294942047at_nat @ F2 @ A3 ) )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B4 )
         => ( member8206827879206165904at_nat @ ( the_in730559788629188109at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_393_the__inv__into__into,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A3: set_nat,X: produc3843707927480180839at_nat,B4: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ A3 )
     => ( ( member8757157785044589968at_nat @ X @ ( image_2499013659892586576at_nat @ F2 @ A3 ) )
       => ( ( ord_less_eq_set_nat @ A3 @ B4 )
         => ( member_nat @ ( the_in107096465503030562at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_394_the__inv__into__into,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,A3: set_Pr4329608150637261639at_nat,X: nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7963584518437281414at_nat @ F2 @ A3 )
     => ( ( member_nat @ X @ ( image_3198525501578754290at_nat @ F2 @ A3 ) )
       => ( ( ord_le1268244103169919719at_nat @ A3 @ B4 )
         => ( member8757157785044589968at_nat @ ( the_in806608307189198276at_nat @ A3 @ F2 @ X ) @ B4 ) ) ) ) ).

% the_inv_into_into
thf(fact_395_subrelI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_396_subrelI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ S2 ) )
     => ( ord_le3000389064537975527at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_397_subrelI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ S2 ) )
     => ( ord_le1268244103169919719at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_398_subrelI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R3 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ S2 ) )
     => ( ord_le5997549366648089703at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_399_surj__def,axiom,
    ! [F2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y5: nat] :
          ? [X3: nat] :
            ( Y5
            = ( F2 @ X3 ) ) ) ) ).

% surj_def
thf(fact_400_surjI,axiom,
    ! [G5: nat > nat,F2: nat > nat] :
      ( ! [X2: nat] :
          ( ( G5 @ ( F2 @ X2 ) )
          = X2 )
     => ( ( image_nat_nat @ G5 @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_401_surjE,axiom,
    ! [F2: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X2: nat] :
            ( Y
           != ( F2 @ X2 ) ) ) ).

% surjE
thf(fact_402_surjD,axiom,
    ! [F2: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X2: nat] :
          ( Y
          = ( F2 @ X2 ) ) ) ).

% surjD
thf(fact_403_Range__iff,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R3 ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_404_Range__iff,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R3 ) )
      = ( ? [Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_405_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R3 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_406_Range__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R3 ) )
      = ( ? [Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_407_RangeE,axiom,
    ! [B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B2 @ ( range_nat_nat @ R3 ) )
     => ~ ! [A5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B2 ) @ R3 ) ) ).

% RangeE
thf(fact_408_RangeE,axiom,
    ! [B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( range_4455291928337903541at_nat @ R3 ) )
     => ~ ! [A5: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B2 ) @ R3 ) ) ).

% RangeE
thf(fact_409_RangeE,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ B2 @ ( range_880292732526890933at_nat @ R3 ) )
     => ~ ! [A5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B2 ) @ R3 ) ) ).

% RangeE
thf(fact_410_RangeE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ B2 @ ( range_7218127889571902005at_nat @ R3 ) )
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ B2 ) @ R3 ) ) ).

% RangeE
thf(fact_411_Range_Ointros,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
     => ( member_nat @ B2 @ ( range_nat_nat @ R3 ) ) ) ).

% Range.intros
thf(fact_412_Range_Ointros,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
     => ( member8440522571783428010at_nat @ B2 @ ( range_4455291928337903541at_nat @ R3 ) ) ) ).

% Range.intros
thf(fact_413_Range_Ointros,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
     => ( member2643936169264416010at_nat @ B2 @ ( range_880292732526890933at_nat @ R3 ) ) ) ).

% Range.intros
thf(fact_414_Range_Ointros,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
     => ( member6273874620674182256at_nat @ B2 @ ( range_7218127889571902005at_nat @ R3 ) ) ) ).

% Range.intros
thf(fact_415_Range_Osimps,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R3 ) )
      = ( ? [A2: nat,B5: nat] :
            ( ( A = B5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_416_Range_Osimps,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R3 ) )
      = ( ? [A2: product_prod_nat_nat,B5: product_prod_nat_nat] :
            ( ( A = B5 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_417_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R3 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
            ( ( A = B5 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_418_Range_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R3 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
            ( ( A = B5 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_419_Range_Ocases,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R3 ) )
     => ~ ! [A5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_420_Range_Ocases,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R3 ) )
     => ~ ! [A5: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_421_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R3 ) )
     => ~ ! [A5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_422_Range_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R3 ) )
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_423_range__ex1__eq,axiom,
    ! [F2: nat > produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( inj_on6829720656803765732at_nat @ F2 @ top_top_set_nat )
     => ( ( member8206827879206165904at_nat @ B2 @ ( image_3276603626957510736at_nat @ F2 @ top_top_set_nat ) )
        = ( ? [X3: nat] :
              ( ( B2
                = ( F2 @ X3 ) )
              & ! [Y5: nat] :
                  ( ( B2
                    = ( F2 @ Y5 ) )
                 => ( Y5 = X3 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_424_range__ex1__eq,axiom,
    ! [F2: nat > product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
     => ( ( member8440522571783428010at_nat @ B2 @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) )
        = ( ? [X3: nat] :
              ( ( B2
                = ( F2 @ X3 ) )
              & ! [Y5: nat] :
                  ( ( B2
                    = ( F2 @ Y5 ) )
                 => ( Y5 = X3 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_425_range__ex1__eq,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ B2 @ ( image_2499013659892586576at_nat @ F2 @ top_top_set_nat ) )
        = ( ? [X3: nat] :
              ( ( B2
                = ( F2 @ X3 ) )
              & ! [Y5: nat] :
                  ( ( B2
                    = ( F2 @ Y5 ) )
                 => ( Y5 = X3 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_426_range__ex1__eq,axiom,
    ! [F2: nat > nat,B2: nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( member_nat @ B2 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
        = ( ? [X3: nat] :
              ( ( B2
                = ( F2 @ X3 ) )
              & ! [Y5: nat] :
                  ( ( B2
                    = ( F2 @ Y5 ) )
                 => ( Y5 = X3 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_427_inj__image__mem__iff,axiom,
    ! [F2: nat > nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_428_inj__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ top_to4669805908274784177at_nat )
     => ( ( member_nat @ ( F2 @ A ) @ ( image_2486076414777270412at_nat @ F2 @ A3 ) )
        = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_429_inj__image__mem__iff,axiom,
    ! [F2: nat > product_prod_nat_nat,A: nat,A3: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
     => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5846123807819985514at_nat @ F2 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_430_inj__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ top_to4669805908274784177at_nat )
     => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5168914502847457605at_nat @ F2 @ A3 ) )
        = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_431_inj__image__mem__iff,axiom,
    ! [F2: produc859450856879609959at_nat > nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on490725545064285830at_nat @ F2 @ top_to8454300755403694231at_nat )
     => ( ( member_nat @ ( F2 @ A ) @ ( image_6160980552072806642at_nat @ F2 @ A3 ) )
        = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_432_inj__image__mem__iff,axiom,
    ! [F2: nat > produc859450856879609959at_nat,A: nat,A3: set_nat] :
      ( ( inj_on6829720656803765732at_nat @ F2 @ top_top_set_nat )
     => ( ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_3276603626957510736at_nat @ F2 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_433_inj__image__mem__iff,axiom,
    ! [F2: product_prod_nat_nat > produc859450856879609959at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F2 @ top_to4669805908274784177at_nat )
     => ( ( member8206827879206165904at_nat @ ( F2 @ A ) @ ( image_1119892238107900587at_nat @ F2 @ A3 ) )
        = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_434_inj__image__mem__iff,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ top_to8454300755403694231at_nat )
     => ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_6706873702294942047at_nat @ F2 @ A3 ) )
        = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_435_inj__image__mem__iff,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7963584518437281414at_nat @ F2 @ top_to6833984726390702231at_nat )
     => ( ( member_nat @ ( F2 @ A ) @ ( image_3198525501578754290at_nat @ F2 @ A3 ) )
        = ( member8757157785044589968at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_436_inj__image__mem__iff,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A: nat,A3: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ ( F2 @ A ) @ ( image_2499013659892586576at_nat @ F2 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_437_Rangep__Range__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( rangep_nat_nat
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R3 ) )
      = ( ^ [X3: nat] : ( member_nat @ X3 @ ( range_nat_nat @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_438_Rangep__Range__eq,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( rangep8386759446810740219at_nat
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R3 ) )
      = ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ ( range_4455291928337903541at_nat @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_439_Rangep__Range__eq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( rangep5542131597093287419at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R3 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X3 @ ( range_880292732526890933at_nat @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_440_Rangep__Range__eq,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( rangep5814831589058262651at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R3 ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ X3 @ ( range_7218127889571902005at_nat @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_441_Range__def,axiom,
    ( range_nat_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( rangep_nat_nat
            @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R4 ) ) ) ) ) ).

% Range_def
thf(fact_442_Range__def,axiom,
    ( range_4455291928337903541at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat] :
          ( collec3392354462482085612at_nat
          @ ( rangep8386759446810740219at_nat
            @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R4 ) ) ) ) ) ).

% Range_def
thf(fact_443_Range__def,axiom,
    ( range_880292732526890933at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat] :
          ( collec5514110066124741708at_nat
          @ ( rangep5542131597093287419at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R4 ) ) ) ) ) ).

% Range_def
thf(fact_444_Range__def,axiom,
    ( range_7218127889571902005at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat] :
          ( collec4573900263915890866at_nat
          @ ( rangep5814831589058262651at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R4 ) ) ) ) ) ).

% Range_def
thf(fact_445_prod__set__simps_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_446_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(1)
thf(fact_447_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_1274512288150691911at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(1)
thf(fact_448_prod__set__simps_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( basic_1775256158546629575at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) )
      = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% prod_set_simps(1)
thf(fact_449_apfst__conv,axiom,
    ! [F2: nat > nat,X: nat,Y: nat] :
      ( ( produc986720760941809198at_nat @ F2 @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ ( F2 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_450_apfst__conv,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( produc3057951030792887557at_nat @ F2 @ ( produc6161850002892822231at_nat @ X @ Y ) )
      = ( produc6161850002892822231at_nat @ ( F2 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_451_apfst__conv,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc5380969780118646373at_nat @ F2 @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F2 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_452_apfst__conv,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc6529295065807390923at_nat @ F2 @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ ( F2 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_453_apsnd__conv,axiom,
    ! [F2: nat > nat,X: nat,Y: nat] :
      ( ( produc3094765310956027504at_nat @ F2 @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ X @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_454_apsnd__conv,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( produc8987972918798826819at_nat @ F2 @ ( produc6161850002892822231at_nat @ X @ Y ) )
      = ( produc6161850002892822231at_nat @ X @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_455_apsnd__conv,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6468969725218530979at_nat @ F2 @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ X @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_456_apsnd__conv,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc8057050257526108681at_nat @ F2 @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ X @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_457_apfst__convE,axiom,
    ! [Q5: product_prod_nat_nat,F2: nat > nat,P2: product_prod_nat_nat] :
      ( ( Q5
        = ( produc986720760941809198at_nat @ F2 @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ( Q5
             != ( product_Pair_nat_nat @ ( F2 @ X2 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_458_apfst__convE,axiom,
    ! [Q5: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( Q5
        = ( produc3057951030792887557at_nat @ F2 @ P2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X2 @ Y3 ) )
           => ( Q5
             != ( produc6161850002892822231at_nat @ ( F2 @ X2 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_459_apfst__convE,axiom,
    ! [Q5: produc3843707927480180839at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( Q5
        = ( produc5380969780118646373at_nat @ F2 @ P2 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
           => ( Q5
             != ( produc2922128104949294807at_nat @ ( F2 @ X2 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_460_apfst__convE,axiom,
    ! [Q5: produc1319942482725812455at_nat,F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,P2: produc1319942482725812455at_nat] :
      ( ( Q5
        = ( produc6529295065807390923at_nat @ F2 @ P2 ) )
     => ~ ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P2
              = ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
           => ( Q5
             != ( produc9060074326276436823at_nat @ ( F2 @ X2 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_461_prod__set__simps_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( insert_nat @ Y @ bot_bot_set_nat ) ) ).

% prod_set_simps(2)
thf(fact_462_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_463_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_5713354656523905413at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(2)
thf(fact_464_prod__set__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( basic_2061583120998029061at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) )
      = ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ).

% prod_set_simps(2)
thf(fact_465_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B2: nat,B4: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_nat @ B2 @ ( B4 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_466_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member_nat @ B2 @ ( B4 @ A ) )
       => ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B2 ) @ ( produc7672662199629908489at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_467_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B2: product_prod_nat_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) )
       => ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B2 ) @ ( produc1809337555817847783at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_468_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc2761391749766926216at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_469_SigmaI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: nat,B4: produc859450856879609959at_nat > set_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( ( member_nat @ B2 @ ( B4 @ A ) )
       => ( member5422927253076579709at_nat @ ( produc1143935811122074848at_nat @ A @ B2 ) @ ( produc8706620304156252911at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_470_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B2: produc859450856879609959at_nat,B4: nat > set_Pr8693737435421807431at_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) )
       => ( member5754182594613576739at_nat @ ( produc7482930922861554750at_nat @ A @ B2 ) @ ( produc5822243379040957005at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_471_SigmaI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( ( member2643936169264416010at_nat @ B2 @ ( B4 @ A ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc7726050920695093128at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_472_SigmaI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: product_prod_nat_nat,B4: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) )
       => ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B2 ) @ ( produc3829605335166519074at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_473_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: produc859450856879609959at_nat,B4: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) )
       => ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B2 ) @ ( produc7465995907834253422at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_474_SigmaI,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: nat,B4: produc3843707927480180839at_nat > set_nat] :
      ( ( member8757157785044589968at_nat @ A @ A3 )
     => ( ( member_nat @ B2 @ ( B4 @ A ) )
       => ( member1490318826389990781at_nat @ ( produc564659340212411616at_nat @ A @ B2 ) @ ( produc1095920968030776047at_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_475_mem__Sigma__iff,axiom,
    ! [A: nat,B2: nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B4 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_476_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B2: nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B2 ) @ ( produc7672662199629908489at_nat @ A3 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ A @ A3 )
        & ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_477_mem__Sigma__iff,axiom,
    ! [A: nat,B2: product_prod_nat_nat,A3: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B2 ) @ ( produc1809337555817847783at_nat @ A3 @ B4 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_478_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc2761391749766926216at_nat @ A3 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_479_mem__Sigma__iff,axiom,
    ! [A: produc859450856879609959at_nat,B2: nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_nat] :
      ( ( member5422927253076579709at_nat @ ( produc1143935811122074848at_nat @ A @ B2 ) @ ( produc8706620304156252911at_nat @ A3 @ B4 ) )
      = ( ( member8206827879206165904at_nat @ A @ A3 )
        & ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_480_mem__Sigma__iff,axiom,
    ! [A: nat,B2: produc859450856879609959at_nat,A3: set_nat,B4: nat > set_Pr8693737435421807431at_nat] :
      ( ( member5754182594613576739at_nat @ ( produc7482930922861554750at_nat @ A @ B2 ) @ ( produc5822243379040957005at_nat @ A3 @ B4 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_481_mem__Sigma__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc7726050920695093128at_nat @ A3 @ B4 ) )
      = ( ( member2643936169264416010at_nat @ A @ A3 )
        & ( member2643936169264416010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_482_mem__Sigma__iff,axiom,
    ! [A: produc859450856879609959at_nat,B2: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B2 ) @ ( produc3829605335166519074at_nat @ A3 @ B4 ) )
      = ( ( member8206827879206165904at_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_483_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B2: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B2 ) @ ( produc7465995907834253422at_nat @ A3 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ A @ A3 )
        & ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_484_mem__Sigma__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B2: nat,A3: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat > set_nat] :
      ( ( member1490318826389990781at_nat @ ( produc564659340212411616at_nat @ A @ B2 ) @ ( produc1095920968030776047at_nat @ A3 @ B4 ) )
      = ( ( member8757157785044589968at_nat @ A @ A3 )
        & ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_485_Sigma__empty1,axiom,
    ! [B4: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B4 )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty1
thf(fact_486_Sigma__empty1,axiom,
    ! [B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ bot_bo3083307316010499117at_nat @ B4 )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty1
thf(fact_487_Times__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_488_Times__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 )
        = bot_bo3433114129128967449at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_489_Times__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : B4 )
        = bot_bo7908653319349560013at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_490_Times__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : B4 )
        = bot_bo7193250159860175667at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_491_Times__empty,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A3
          @ ^ [Uu3: nat] : B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ( B4 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_492_Times__empty,axiom,
    ! [A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 )
        = bot_bo228742789529271731at_nat )
      = ( ( A3 = bot_bo3083307316010499117at_nat )
        | ( B4 = bot_bo3083307316010499117at_nat ) ) ) ).

% Times_empty
thf(fact_493_Sigma__empty2,axiom,
    ! [A3: set_nat] :
      ( ( produc457027306803732586at_nat @ A3
        @ ^ [Uu3: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_494_Sigma__empty2,axiom,
    ! [A3: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ A3
        @ ^ [Uu3: set_Pr1261947904930325089at_nat] : bot_bo3083307316010499117at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty2
thf(fact_495_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu3: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_496_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
        @ ^ [Uu3: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B2 @ B4 ) )
      = ( insert8243672688981495325at_nat @ ( produc8594291983450332477at_nat @ A @ B2 )
        @ ( sup_su1595509577888697473at_nat
          @ ( produc3090783324798197870at_nat @ A3
            @ ^ [Uu3: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B2 @ B4 ) )
          @ ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
            @ ^ [Uu3: product_prod_nat_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_497_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
        @ ^ [Uu3: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B2 @ B4 ) )
      = ( insert373714669040926161at_nat @ ( produc5023393015413421041at_nat @ A @ B2 )
        @ ( sup_su6071048768109290037at_nat
          @ ( produc8743256393616062242at_nat @ A3
            @ ^ [Uu3: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B2 @ B4 ) )
          @ ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
            @ ^ [Uu3: produc3843707927480180839at_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_498_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
        @ ^ [Uu3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B2 @ B4 ) )
      = ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 )
        @ ( sup_su6664651903789602459at_nat
          @ ( produc4441300235185154312at_nat @ A3
            @ ^ [Uu3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B2 @ B4 ) )
          @ ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
            @ ^ [Uu3: produc3843707927480180839at_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_499_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B2: nat,B4: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
        @ ^ [Uu3: nat] : ( insert_nat @ B2 @ B4 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu3: nat] : ( insert_nat @ B2 @ B4 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
            @ ^ [Uu3: nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_500_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
        @ ^ [Uu3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B2 @ B4 ) )
      = ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 )
        @ ( sup_su718114333110466843at_nat
          @ ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B2 @ B4 ) )
          @ ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
            @ ^ [Uu3: product_prod_nat_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_501_insert__Times__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B2: set_Pr4329608150637261639at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A3 )
        @ ^ [Uu3: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B2 @ B4 ) )
      = ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 )
        @ ( sup_su5395584912770621083at_nat
          @ ( produc6258361011560675592at_nat @ A3
            @ ^ [Uu3: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B2 @ B4 ) )
          @ ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A3 )
            @ ^ [Uu3: set_Pr4329608150637261639at_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_502_insert__Times__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B2: set_Pr1261947904930325089at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A3 )
        @ ^ [Uu3: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B2 @ B4 ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 )
        @ ( sup_su5525570899277871387at_nat
          @ ( produc7726050920695093128at_nat @ A3
            @ ^ [Uu3: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B2 @ B4 ) )
          @ ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A3 )
            @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_503_SigmaE2,axiom,
    ! [A: nat,B2: nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B4 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_504_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B2: nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B2 ) @ ( produc7672662199629908489at_nat @ A3 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A3 )
         => ~ ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_505_SigmaE2,axiom,
    ! [A: nat,B2: product_prod_nat_nat,A3: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B2 ) @ ( produc1809337555817847783at_nat @ A3 @ B4 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_506_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc2761391749766926216at_nat @ A3 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_507_SigmaE2,axiom,
    ! [A: produc859450856879609959at_nat,B2: nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_nat] :
      ( ( member5422927253076579709at_nat @ ( produc1143935811122074848at_nat @ A @ B2 ) @ ( produc8706620304156252911at_nat @ A3 @ B4 ) )
     => ~ ( ( member8206827879206165904at_nat @ A @ A3 )
         => ~ ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_508_SigmaE2,axiom,
    ! [A: nat,B2: produc859450856879609959at_nat,A3: set_nat,B4: nat > set_Pr8693737435421807431at_nat] :
      ( ( member5754182594613576739at_nat @ ( produc7482930922861554750at_nat @ A @ B2 ) @ ( produc5822243379040957005at_nat @ A3 @ B4 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_509_SigmaE2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc7726050920695093128at_nat @ A3 @ B4 ) )
     => ~ ( ( member2643936169264416010at_nat @ A @ A3 )
         => ~ ( member2643936169264416010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_510_SigmaE2,axiom,
    ! [A: produc859450856879609959at_nat,B2: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B2 ) @ ( produc3829605335166519074at_nat @ A3 @ B4 ) )
     => ~ ( ( member8206827879206165904at_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_511_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B2: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B2 ) @ ( produc7465995907834253422at_nat @ A3 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A3 )
         => ~ ( member8206827879206165904at_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_512_SigmaE2,axiom,
    ! [A: produc3843707927480180839at_nat,B2: nat,A3: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat > set_nat] :
      ( ( member1490318826389990781at_nat @ ( produc564659340212411616at_nat @ A @ B2 ) @ ( produc1095920968030776047at_nat @ A3 @ B4 ) )
     => ~ ( ( member8757157785044589968at_nat @ A @ A3 )
         => ~ ( member_nat @ B2 @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_513_SigmaD2,axiom,
    ! [A: nat,B2: nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B4 ) )
     => ( member_nat @ B2 @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_514_SigmaD2,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc2761391749766926216at_nat @ A3 @ B4 ) )
     => ( member8440522571783428010at_nat @ B2 @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_515_SigmaD2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc7726050920695093128at_nat @ A3 @ B4 ) )
     => ( member2643936169264416010at_nat @ B2 @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_516_SigmaD2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B4: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( produc6258361011560675592at_nat @ A3 @ B4 ) )
     => ( member6273874620674182256at_nat @ B2 @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_517_SigmaD1,axiom,
    ! [A: nat,B2: nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B4 ) )
     => ( member_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_518_SigmaD1,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc2761391749766926216at_nat @ A3 @ B4 ) )
     => ( member8440522571783428010at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_519_SigmaD1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc7726050920695093128at_nat @ A3 @ B4 ) )
     => ( member2643936169264416010at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_520_SigmaD1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B4: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( produc6258361011560675592at_nat @ A3 @ B4 ) )
     => ( member6273874620674182256at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_521_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A3 @ B4 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_522_SigmaE,axiom,
    ! [C: produc8373899037510109440at_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ C @ ( produc7672662199629908489at_nat @ A3 @ B4 ) )
     => ~ ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc6350711070570205562at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_523_SigmaE,axiom,
    ! [C: produc7248412053542808358at_nat,A3: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ C @ ( produc1809337555817847783at_nat @ A3 @ B4 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc487386426758144856at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_524_SigmaE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( produc2761391749766926216at_nat @ A3 @ B4 ) )
     => ~ ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc6161850002892822231at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_525_SigmaE,axiom,
    ! [C: produc2156263037089731046at_nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_nat] :
      ( ( member5422927253076579709at_nat @ C @ ( produc8706620304156252911at_nat @ A3 @ B4 ) )
     => ~ ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc1143935811122074848at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_526_SigmaE,axiom,
    ! [C: produc2487518378626728076at_nat,A3: set_nat,B4: nat > set_Pr8693737435421807431at_nat] :
      ( ( member5754182594613576739at_nat @ C @ ( produc5822243379040957005at_nat @ A3 @ B4 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc7482930922861554750at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_527_SigmaE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( produc7726050920695093128at_nat @ A3 @ B4 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ A3 )
           => ! [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc2922128104949294807at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_528_SigmaE,axiom,
    ! [C: produc7148259590854449153at_nat,A3: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ C @ ( produc3829605335166519074at_nat @ A3 @ B4 ) )
     => ~ ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc1860397064299428849at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_529_SigmaE,axiom,
    ! [C: produc39072733474941005at_nat,A3: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ C @ ( produc7465995907834253422at_nat @ A3 @ B4 ) )
     => ~ ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc5496787636967163197at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_530_SigmaE,axiom,
    ! [C: produc5802417571888161254at_nat,A3: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat > set_nat] :
      ( ( member1490318826389990781at_nat @ C @ ( produc1095920968030776047at_nat @ A3 @ B4 ) )
     => ~ ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X2 ) )
               => ( C
                 != ( produc564659340212411616at_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_531_Sigma__Un__distrib1,axiom,
    ! [I2: set_se7855581050983116737at_nat,J2: set_se7855581050983116737at_nat,C4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ I2 @ J2 ) @ C4 )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ C4 ) @ ( produc7726050920695093128at_nat @ J2 @ C4 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_532_times__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 )
        = ( produc2761391749766926216at_nat @ C4
          @ ^ [Uu3: product_prod_nat_nat] : D ) )
      = ( ( ( A3 = C4 )
          & ( B4 = D ) )
        | ( ( ( A3 = bot_bo2099793752762293965at_nat )
            | ( B4 = bot_bo2099793752762293965at_nat ) )
          & ( ( C4 = bot_bo2099793752762293965at_nat )
            | ( D = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_533_times__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr4329608150637261639at_nat,C4: set_Pr1261947904930325089at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 )
        = ( produc3090783324798197870at_nat @ C4
          @ ^ [Uu3: product_prod_nat_nat] : D ) )
      = ( ( ( A3 = C4 )
          & ( B4 = D ) )
        | ( ( ( A3 = bot_bo2099793752762293965at_nat )
            | ( B4 = bot_bo228742789529271731at_nat ) )
          & ( ( C4 = bot_bo2099793752762293965at_nat )
            | ( D = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_534_times__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr1261947904930325089at_nat,C4: set_Pr4329608150637261639at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : B4 )
        = ( produc8743256393616062242at_nat @ C4
          @ ^ [Uu3: produc3843707927480180839at_nat] : D ) )
      = ( ( ( A3 = C4 )
          & ( B4 = D ) )
        | ( ( ( A3 = bot_bo228742789529271731at_nat )
            | ( B4 = bot_bo2099793752762293965at_nat ) )
          & ( ( C4 = bot_bo228742789529271731at_nat )
            | ( D = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_535_times__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : B4 )
        = ( produc4441300235185154312at_nat @ C4
          @ ^ [Uu3: produc3843707927480180839at_nat] : D ) )
      = ( ( ( A3 = C4 )
          & ( B4 = D ) )
        | ( ( ( A3 = bot_bo228742789529271731at_nat )
            | ( B4 = bot_bo228742789529271731at_nat ) )
          & ( ( C4 = bot_bo228742789529271731at_nat )
            | ( D = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_536_Sigma__empty__iff,axiom,
    ! [I2: set_nat,X7: nat > set_nat] :
      ( ( ( produc457027306803732586at_nat @ I2 @ X7 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ I2 )
           => ( ( X7 @ X3 )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_537_Sigma__empty__iff,axiom,
    ! [I2: set_se7855581050983116737at_nat,X7: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ I2 @ X7 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ I2 )
           => ( ( X7 @ X3 )
              = bot_bo3083307316010499117at_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_538_Sigma__Un__distrib2,axiom,
    ! [I2: set_se7855581050983116737at_nat,A3: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat,B4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ I2
        @ ^ [I4: set_Pr1261947904930325089at_nat] : ( sup_su3642409539654194069at_nat @ ( A3 @ I4 ) @ ( B4 @ I4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ A3 ) @ ( produc7726050920695093128at_nat @ I2 @ B4 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_539_Times__Un__distrib1,axiom,
    ! [A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat,C4: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ A3 @ B4 )
        @ ^ [Uu3: set_Pr1261947904930325089at_nat] : C4 )
      = ( sup_su5525570899277871387at_nat
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : C4 )
        @ ( produc7726050920695093128at_nat @ B4
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : C4 ) ) ) ).

% Times_Un_distrib1
thf(fact_540_Sigma__def,axiom,
    ( produc2761391749766926216at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
          ( comple2748534634209756852at_nat
          @ ( image_5803077947706746251at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( comple2748534634209756852at_nat
                @ ( image_5803077947706746251at_nat
                  @ ^ [Y5: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ bot_bo5327735625951526323at_nat )
                  @ ( B6 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_541_Sigma__def,axiom,
    ( produc6258361011560675592at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,B6: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat] :
                ( comple7342073961973170484at_nat
                @ ( image_1124969593608450885at_nat
                  @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ bot_bo4948859079157340979at_nat )
                  @ ( B6 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_542_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A7: set_nat,B6: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X3: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ bot_bo2099793752762293965at_nat )
                  @ ( B6 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_543_Sigma__def,axiom,
    ( produc7726050920695093128at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,B6: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] :
                ( comple4364388969068068532at_nat
                @ ( image_6898982319499645995at_nat
                  @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ bot_bo228742789529271731at_nat )
                  @ ( B6 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_544_times__subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : C4 )
        @ ( produc2761391749766926216at_nat @ B4
          @ ^ [Uu3: product_prod_nat_nat] : D ) )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( C4 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
          & ( ord_le3146513528884898305at_nat @ C4 @ D ) ) ) ) ).

% times_subset_iff
thf(fact_545_times__subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C4: set_Pr4329608150637261639at_nat,B4: set_Pr1261947904930325089at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le7005211468645301325at_nat
        @ ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : C4 )
        @ ( produc3090783324798197870at_nat @ B4
          @ ^ [Uu3: product_prod_nat_nat] : D ) )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( C4 = bot_bo228742789529271731at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A3 @ B4 )
          & ( ord_le1268244103169919719at_nat @ C4 @ D ) ) ) ) ).

% times_subset_iff
thf(fact_546_times__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C4: set_Pr1261947904930325089at_nat,B4: set_Pr4329608150637261639at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2257378622011118081at_nat
        @ ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : C4 )
        @ ( produc8743256393616062242at_nat @ B4
          @ ^ [Uu3: produc3843707927480180839at_nat] : D ) )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( C4 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A3 @ B4 )
          & ( ord_le3146513528884898305at_nat @ C4 @ D ) ) ) ) ).

% times_subset_iff
thf(fact_547_times__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat
        @ ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : C4 )
        @ ( produc4441300235185154312at_nat @ B4
          @ ^ [Uu3: produc3843707927480180839at_nat] : D ) )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( C4 = bot_bo228742789529271731at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A3 @ B4 )
          & ( ord_le1268244103169919719at_nat @ C4 @ D ) ) ) ) ).

% times_subset_iff
thf(fact_548_member__product,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( produc6773009516019950738at_nat @ A3 @ B4 ) )
      = ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 ) ) ) ).

% member_product
thf(fact_549_member__product,axiom,
    ! [X: product_prod_nat_nat,A3: set_nat,B4: set_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( produc929334515565554804at_nat @ A3 @ B4 ) )
      = ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu3: nat] : B4 ) ) ) ).

% member_product
thf(fact_550_member__product,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( produc3241716407206042770at_nat @ A3 @ B4 ) )
      = ( member8757157785044589968at_nat @ X
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 ) ) ) ).

% member_product
thf(fact_551_refl__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( refl_o6272981002401706822at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% refl_on_singleton
thf(fact_552_refl__on__singleton,axiom,
    ! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% refl_on_singleton
thf(fact_553_refl__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_554_refl__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( refl_o7122697186039111552at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% refl_on_singleton
thf(fact_555_refl__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( refl_o7446464703481863270at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% refl_on_singleton
thf(fact_556_refl__onD2,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R3 )
       => ( member8206827879206165904at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_557_refl__onD2,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R3 )
       => ( member8757157785044589968at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_558_refl__onD2,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
       => ( member_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_559_refl__onD2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
       => ( member8440522571783428010at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_560_refl__onD2,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
       => ( member2643936169264416010at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_561_refl__onD2,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
       => ( member6273874620674182256at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_562_refl__onD1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R3 )
       => ( member8206827879206165904at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_563_refl__onD1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R3 )
       => ( member8757157785044589968at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_564_refl__onD1,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
       => ( member_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_565_refl__onD1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
       => ( member8440522571783428010at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_566_refl__onD1,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
       => ( member2643936169264416010at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_567_refl__onD1,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
       => ( member6273874620674182256at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_568_refl__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_569_refl__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_570_refl__onD,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A3 @ R3 )
     => ( ( member_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_571_refl__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_572_refl__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_573_refl__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ A3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_574_surj__image__vimage__eq,axiom,
    ! [F2: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( image_nat_nat @ F2 @ ( vimage_nat_nat @ F2 @ A3 ) )
        = A3 ) ) ).

% surj_image_vimage_eq
thf(fact_575_reflI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R3 )
     => ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R3 ) ) ).

% reflI
thf(fact_576_reflI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ R3 )
     => ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R3 ) ) ).

% reflI
thf(fact_577_reflI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ R3 )
     => ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R3 ) ) ).

% reflI
thf(fact_578_reflI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R3 )
     => ( refl_on_nat @ top_top_set_nat @ R3 ) ) ).

% reflI
thf(fact_579_reflD,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R3 ) ) ).

% reflD
thf(fact_580_reflD,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R3 ) ) ).

% reflD
thf(fact_581_reflD,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R3 ) ) ).

% reflD
thf(fact_582_reflD,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R3 ) ) ).

% reflD
thf(fact_583_surj__vimage__empty,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_5168914502847457605at_nat @ F2 @ top_to4669805908274784177at_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage2449269961533847803at_nat @ F2 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_584_surj__vimage__empty,axiom,
    ! [F2: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_1672857615394513579at_nat @ F2 @ top_to4669805908274784177at_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage1489446588345997665at_nat @ F2 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_585_surj__vimage__empty,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_7325330684212377951at_nat @ F2 @ top_to6833984726390702231at_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage7141919657163862037at_nat @ F2 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_586_surj__vimage__empty,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_960689587819713477at_nat @ F2 @ top_to6833984726390702231at_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage6896080417876799867at_nat @ F2 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_587_surj__vimage__empty,axiom,
    ! [F2: product_prod_nat_nat > nat,A3: set_nat] :
      ( ( ( image_2486076414777270412at_nat @ F2 @ top_to4669805908274784177at_nat )
        = top_top_set_nat )
     => ( ( ( vimage4653281326611754070at_nat @ F2 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_588_surj__vimage__empty,axiom,
    ! [F2: produc3843707927480180839at_nat > nat,A3: set_nat] :
      ( ( ( image_3198525501578754290at_nat @ F2 @ top_to6833984726390702231at_nat )
        = top_top_set_nat )
     => ( ( ( vimage7134676753939176892at_nat @ F2 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_589_surj__vimage__empty,axiom,
    ! [F2: nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage8013328719654469172at_nat @ F2 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_590_surj__vimage__empty,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_2499013659892586576at_nat @ F2 @ top_top_set_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage6435164912253009178at_nat @ F2 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_591_surj__vimage__empty,axiom,
    ! [F2: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( vimage_nat_nat @ F2 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_592_vimage__subsetD,axiom,
    ! [F2: nat > nat,B4: set_nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ B4 ) @ A3 )
       => ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A3 ) ) ) ) ).

% vimage_subsetD
thf(fact_593_Pair__vimage__Sigma,axiom,
    ! [X: nat,A3: set_nat,F2: nat > set_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A3 @ F2 ) )
          = bot_bot_set_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_594_Pair__vimage__Sigma,axiom,
    ! [X: nat,A3: set_nat,F2: nat > set_Pr1261947904930325089at_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( vimage5148388808775819948at_nat @ ( produc487386426758144856at_nat @ X ) @ ( produc1809337555817847783at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( vimage5148388808775819948at_nat @ ( produc487386426758144856at_nat @ X ) @ ( produc1809337555817847783at_nat @ A3 @ F2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_595_Pair__vimage__Sigma,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( vimage3674627534083940705at_nat @ ( produc6161850002892822231at_nat @ X ) @ ( produc2761391749766926216at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( vimage3674627534083940705at_nat @ ( produc6161850002892822231at_nat @ X ) @ ( produc2761391749766926216at_nat @ A3 @ F2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_596_Pair__vimage__Sigma,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,F2: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( vimage2590017774394568705at_nat @ ( produc2922128104949294807at_nat @ X ) @ ( produc7726050920695093128at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( vimage2590017774394568705at_nat @ ( produc2922128104949294807at_nat @ X ) @ ( produc7726050920695093128at_nat @ A3 @ F2 ) )
          = bot_bo3083307316010499117at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_597_Pair__vimage__Sigma,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( vimage34882777259934203at_nat @ ( produc1860397064299428849at_nat @ X ) @ ( produc3829605335166519074at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( vimage34882777259934203at_nat @ ( produc1860397064299428849at_nat @ X ) @ ( produc3829605335166519074at_nat @ A3 @ F2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_598_Pair__vimage__Sigma,axiom,
    ! [X: nat,A3: set_nat,F2: nat > set_Pr4329608150637261639at_nat] :
      ( ( ( member_nat @ X @ A3 )
       => ( ( vimage5635820159961579128at_nat @ ( produc9088519535381019710at_nat @ X ) @ ( produc396409126344608333at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member_nat @ X @ A3 )
       => ( ( vimage5635820159961579128at_nat @ ( produc9088519535381019710at_nat @ X ) @ ( produc396409126344608333at_nat @ A3 @ F2 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_599_Pair__vimage__Sigma,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( vimage6015846564984431099at_nat @ ( produc5023393015413421041at_nat @ X ) @ ( produc8743256393616062242at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( vimage6015846564984431099at_nat @ ( produc5023393015413421041at_nat @ X ) @ ( produc8743256393616062242at_nat @ A3 @ F2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_600_Pair__vimage__Sigma,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( vimage7889098223007972193at_nat @ ( produc8594291983450332477at_nat @ X ) @ ( produc3090783324798197870at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( vimage7889098223007972193at_nat @ ( produc8594291983450332477at_nat @ X ) @ ( produc3090783324798197870at_nat @ A3 @ F2 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_601_Pair__vimage__Sigma,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( vimage3196295220526820859at_nat @ ( produc636707851812660055at_nat @ X ) @ ( produc8155347593902061832at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( vimage3196295220526820859at_nat @ ( produc636707851812660055at_nat @ X ) @ ( produc8155347593902061832at_nat @ A3 @ F2 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_602_Pair__vimage__Sigma,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( vimage4336369872271834619at_nat @ ( produc2335810615602371415at_nat @ X ) @ ( produc4441300235185154312at_nat @ A3 @ F2 ) )
          = ( F2 @ X ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( vimage4336369872271834619at_nat @ ( produc2335810615602371415at_nat @ X ) @ ( produc4441300235185154312at_nat @ A3 @ F2 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_603_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A7: set_nat,R4: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R4
            @ ( produc457027306803732586at_nat @ A7
              @ ^ [Uu3: nat] : A7 ) )
          & ! [X3: nat] :
              ( ( member_nat @ X3 @ A7 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R4 ) ) ) ) ) ).

% refl_on_def
thf(fact_604_refl__on__def,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
          ( ( ord_le3000389064537975527at_nat @ R4
            @ ( produc2761391749766926216at_nat @ A7
              @ ^ [Uu3: product_prod_nat_nat] : A7 ) )
          & ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A7 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R4 ) ) ) ) ) ).

% refl_on_def
thf(fact_605_refl__on__def,axiom,
    ( refl_o6955724340487615712at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
          ( ( ord_le1268244103169919719at_nat @ R4
            @ ( produc7726050920695093128at_nat @ A7
              @ ^ [Uu3: set_Pr1261947904930325089at_nat] : A7 ) )
          & ! [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ A7 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R4 ) ) ) ) ) ).

% refl_on_def
thf(fact_606_refl__on__def,axiom,
    ( refl_o6272981002401706822at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R4: set_Pr7459493094073627847at_nat] :
          ( ( ord_le5997549366648089703at_nat @ R4
            @ ( produc6258361011560675592at_nat @ A7
              @ ^ [Uu3: set_Pr4329608150637261639at_nat] : A7 ) )
          & ! [X3: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X3 @ A7 )
             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ R4 ) ) ) ) ) ).

% refl_on_def
thf(fact_607_refl__onI,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le2286497330737841255at_nat @ R3
        @ ( produc18557253517588744at_nat @ A3
          @ ^ [Uu3: produc859450856879609959at_nat] : A3 ) )
     => ( ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_o6679620997798956134at_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_608_refl__onI,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat @ R3
        @ ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : A3 ) )
     => ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A3 )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_o7446464703481863270at_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_609_refl__onI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R3
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu3: nat] : A3 ) )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_on_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_610_refl__onI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R3
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : A3 ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_o7122697186039111552at_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_611_refl__onI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( ord_le1268244103169919719at_nat @ R3
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : A3 ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ A3 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_o6955724340487615712at_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_612_refl__onI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( ord_le5997549366648089703at_nat @ R3
        @ ( produc6258361011560675592at_nat @ A3
          @ ^ [Uu3: set_Pr4329608150637261639at_nat] : A3 ) )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X2 @ A3 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ R3 ) )
       => ( refl_o6272981002401706822at_nat @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_613_Id__on__def,axiom,
    ( id_on_2554058798563519774at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat] :
          ( comple2748534634209756852at_nat
          @ ( image_5803077947706746251at_nat
            @ ^ [X3: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ bot_bo5327735625951526323at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_614_Id__on__def,axiom,
    ( id_on_3275838214495744996at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ bot_bo4948859079157340979at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_615_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A7: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_616_Id__on__def,axiom,
    ( id_on_6713504163309144702at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ bot_bo228742789529271731at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_617_fold__union__pair,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( sup_su718114333110466843at_nat
          @ ( comple2748534634209756852at_nat
            @ ( image_5803077947706746251at_nat
              @ ^ [Y5: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y5 ) @ bot_bo5327735625951526323at_nat )
              @ B4 ) )
          @ A3 )
        = ( finite8071398511639125710at_nat
          @ ^ [Y5: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y5 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_618_fold__union__pair,axiom,
    ! [B4: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,A3: set_Pr7459493094073627847at_nat] :
      ( ( finite4768004253697144688at_nat @ B4 )
     => ( ( sup_su5395584912770621083at_nat
          @ ( comple7342073961973170484at_nat
            @ ( image_1124969593608450885at_nat
              @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y5 ) @ bot_bo4948859079157340979at_nat )
              @ B4 ) )
          @ A3 )
        = ( finite3577408862935371528at_nat
          @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y5 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_619_fold__union__pair,axiom,
    ! [B4: set_nat,X: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ bot_bo2099793752762293965at_nat )
              @ B4 ) )
          @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_620_fold__union__pair,axiom,
    ! [B4: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ B4 )
     => ( ( sup_su5525570899277871387at_nat
          @ ( comple4364388969068068532at_nat
            @ ( image_6898982319499645995at_nat
              @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y5 ) @ bot_bo228742789529271731at_nat )
              @ B4 ) )
          @ A3 )
        = ( finite3179781904389220718at_nat
          @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y5 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_621_refl__on__domain,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ R3 )
       => ( ( member8206827879206165904at_nat @ A @ A3 )
          & ( member8206827879206165904at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_622_refl__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_623_refl__on__domain,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( refl_on_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
       => ( ( member_nat @ A @ A3 )
          & ( member_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_624_refl__on__domain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
          & ( member8440522571783428010at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_625_refl__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_626_refl__on__domain,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
       => ( ( member6273874620674182256at_nat @ A @ A3 )
          & ( member6273874620674182256at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_627_inj__on__vimage__singleton,axiom,
    ! [F2: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( inj_on229527429751349963at_nat @ F2 @ A3 )
     => ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ ( vimage38236961416206357at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ A3 )
        @ ( insert5050368324300391991at_nat
          @ ( the_Pr8653942702660118126at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo5327735625951526323at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_628_inj__on__vimage__singleton,axiom,
    ! [F2: nat > product_prod_nat_nat,A3: set_nat,A: product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ A3 )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_629_inj__on__vimage__singleton,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ A3 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage2449269961533847803at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ A3 )
        @ ( insert8211810215607154385at_nat
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_630_inj__on__vimage__singleton,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,A: product_prod_nat_nat] :
      ( ( inj_on6799352813640927947at_nat @ F2 @ A3 )
     => ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ ( vimage7141919657163862037at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ A3 )
        @ ( insert9069300056098147895at_nat
          @ ( the_Pr4604535484834969198at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo228742789529271731at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_631_inj__on__vimage__singleton,axiom,
    ! [F2: produc859450856879609959at_nat > produc3843707927480180839at_nat,A3: set_Pr8693737435421807431at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on198529248122145585at_nat @ F2 @ A3 )
     => ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ ( vimage1392009304509651323at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ A3 )
        @ ( insert5050368324300391991at_nat
          @ ( the_Pr8653942702660118126at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo5327735625951526323at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_632_inj__on__vimage__singleton,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A3: set_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ A3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( vimage6435164912253009178at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ A3 )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_633_inj__on__vimage__singleton,axiom,
    ! [F2: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F2 @ A3 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage1489446588345997665at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ A3 )
        @ ( insert8211810215607154385at_nat
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_634_inj__on__vimage__singleton,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F2 @ A3 )
     => ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ ( vimage6896080417876799867at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ A3 )
        @ ( insert9069300056098147895at_nat
          @ ( the_Pr4604535484834969198at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ A3 )
                & ( ( F2 @ X3 )
                  = A ) ) )
          @ bot_bo228742789529271731at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_635_Id__onI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( id_on_7917584626360399108at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_636_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A3 )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_637_Id__onI,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_638_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_639_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_640_Id__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ A @ A3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( id_on_3275838214495744996at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_641_Id__on__iff,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( id_on_7917584626360399108at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8206827879206165904at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_642_Id__on__iff,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ ( id_on_7869290023980668164at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8757157785044589968at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_643_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A3 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_644_Id__on__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_645_Id__on__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( id_on_6713504163309144702at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member2643936169264416010at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_646_Id__on__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( id_on_3275838214495744996at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member6273874620674182256at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_647_Id__on__eqI,axiom,
    ! [A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( A = B2 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ ( id_on_7917584626360399108at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_648_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( A = B2 )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ ( id_on_7869290023980668164at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_649_Id__on__eqI,axiom,
    ! [A: nat,B2: nat,A3: set_nat] :
      ( ( A = B2 )
     => ( ( member_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( id_on_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_650_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( A = B2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( id_on_2554058798563519774at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_651_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( A = B2 )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( id_on_6713504163309144702at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_652_Id__on__eqI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( A = B2 )
     => ( ( member6273874620674182256at_nat @ A @ A3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( id_on_3275838214495744996at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_653_Id__onE,axiom,
    ! [C: produc349518998152878311at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C @ ( id_on_7917584626360399108at_nat @ A3 ) )
     => ~ ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
           => ( C
             != ( produc4662710985925991255at_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_654_Id__onE,axiom,
    ! [C: produc1906917888121415911at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ C @ ( id_on_7869290023980668164at_nat @ A3 ) )
     => ~ ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A3 )
           => ( C
             != ( produc2335810615602371415at_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_655_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ( C
             != ( product_Pair_nat_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_656_Id__onE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A3 ) )
     => ~ ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ( C
             != ( produc6161850002892822231at_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_657_Id__onE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( id_on_6713504163309144702at_nat @ A3 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ A3 )
           => ( C
             != ( produc2922128104949294807at_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_658_Id__onE,axiom,
    ! [C: produc1319942482725812455at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ C @ ( id_on_3275838214495744996at_nat @ A3 ) )
     => ~ ! [X2: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X2 @ A3 )
           => ( C
             != ( produc9060074326276436823at_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_659_Id__on__fold,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( id_on_2554058798563519774at_nat @ A3 )
        = ( finite8071398511639125710at_nat
          @ ^ [X3: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) )
          @ bot_bo5327735625951526323at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_660_Id__on__fold,axiom,
    ! [A3: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A3 )
     => ( ( id_on_3275838214495744996at_nat @ A3 )
        = ( finite3577408862935371528at_nat
          @ ^ [X3: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) )
          @ bot_bo4948859079157340979at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_661_Id__on__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( id_on_nat @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) )
          @ bot_bo2099793752762293965at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_662_Id__on__fold,axiom,
    ! [A3: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( id_on_6713504163309144702at_nat @ A3 )
        = ( finite3179781904389220718at_nat
          @ ^ [X3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) )
          @ bot_bo228742789529271731at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_663_Id__on__set,axiom,
    ! [Xs: list_nat] :
      ( ( id_on_nat @ ( set_nat2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X3: nat] : ( product_Pair_nat_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_664_Id__on__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( id_on_2554058798563519774at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( set_Pr5518436109238095868at_nat
        @ ( map_Pr9105590726435236202at_nat
          @ ^ [X3: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_665_Id__on__set,axiom,
    ! [Xs: list_s1210847774152347623at_nat] :
      ( ( id_on_6713504163309144702at_nat @ ( set_se5049602875457034614at_nat @ Xs ) )
      = ( set_Pr3765526544606949372at_nat
        @ ( map_se8673859202641857546at_nat
          @ ^ [X3: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_666_Id__on__set,axiom,
    ! [Xs: list_s4242580248156217677at_nat] :
      ( ( id_on_3275838214495744996at_nat @ ( set_se3524105090824187612at_nat @ Xs ) )
      = ( set_Pr2074889473033032060at_nat
        @ ( map_se8572903940320514724at_nat
          @ ^ [X3: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_667_product__fold,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu3: product_prod_nat_nat] : B4 )
          = ( finite8071398511639125710at_nat
            @ ^ [X3: product_prod_nat_nat,Z4: set_Pr8693737435421807431at_nat] :
                ( finite8071398511639125710at_nat
                @ ^ [Y5: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) )
                @ Z4
                @ B4 )
            @ bot_bo5327735625951526323at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_668_product__fold,axiom,
    ! [A3: set_se5479201105039950375at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A3 )
     => ( ( finite4768004253697144688at_nat @ B4 )
       => ( ( produc6258361011560675592at_nat @ A3
            @ ^ [Uu3: set_Pr4329608150637261639at_nat] : B4 )
          = ( finite3577408862935371528at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat,Z4: set_Pr7459493094073627847at_nat] :
                ( finite3577408862935371528at_nat
                @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) )
                @ Z4
                @ B4 )
            @ bot_bo4948859079157340979at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_669_product__fold,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( produc457027306803732586at_nat @ A3
            @ ^ [Uu3: nat] : B4 )
          = ( finite3745491028973389255at_nat
            @ ^ [X3: nat,Z4: set_Pr1261947904930325089at_nat] :
                ( finite3745491028973389255at_nat
                @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) )
                @ Z4
                @ B4 )
            @ bot_bo2099793752762293965at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_670_product__fold,axiom,
    ! [A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( finite9047747110432174090at_nat @ B4 )
       => ( ( produc7726050920695093128at_nat @ A3
            @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 )
          = ( finite3179781904389220718at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat,Z4: set_Pr4329608150637261639at_nat] :
                ( finite3179781904389220718at_nat
                @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) )
                @ Z4
                @ B4 )
            @ bot_bo228742789529271731at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_671_inj__vimage__singleton,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( inj_on8969904277767023793at_nat @ F2 @ top_to4669805908274784177at_nat )
     => ( ord_le3146513528884898305at_nat @ ( vimage2449269961533847803at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
        @ ( insert8211810215607154385at_nat
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bo2099793752762293965at_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_672_inj__vimage__singleton,axiom,
    ! [F2: produc3843707927480180839at_nat > product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( inj_on6799352813640927947at_nat @ F2 @ top_to6833984726390702231at_nat )
     => ( ord_le1268244103169919719at_nat @ ( vimage7141919657163862037at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
        @ ( insert9069300056098147895at_nat
          @ ( the_Pr4604535484834969198at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bo228742789529271731at_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_673_inj__vimage__singleton,axiom,
    ! [F2: product_prod_nat_nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F2 @ top_to4669805908274784177at_nat )
     => ( ord_le3146513528884898305at_nat @ ( vimage1489446588345997665at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        @ ( insert8211810215607154385at_nat
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bo2099793752762293965at_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_674_inj__vimage__singleton,axiom,
    ! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F2 @ top_to6833984726390702231at_nat )
     => ( ord_le1268244103169919719at_nat @ ( vimage6896080417876799867at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        @ ( insert9069300056098147895at_nat
          @ ( the_Pr4604535484834969198at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bo228742789529271731at_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_675_inj__vimage__singleton,axiom,
    ! [F2: nat > product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X3: nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_676_inj__vimage__singleton,axiom,
    ! [F2: nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F2 @ top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( vimage6435164912253009178at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X3: nat] :
                ( ( F2 @ X3 )
                = A ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_677_comp__fun__commute__product__fold,axiom,
    ! [B4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( finite1174777150170982526at_nat
        @ ^ [X3: nat,Z4: set_Pr1261947904930325089at_nat] :
            ( finite3745491028973389255at_nat
            @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) )
            @ Z4
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_678_comp__fun__commute__product__fold,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( finite3773130646654257623at_nat
        @ ^ [X3: product_prod_nat_nat,Z4: set_Pr8693737435421807431at_nat] :
            ( finite8071398511639125710at_nat
            @ ^ [Y5: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) )
            @ Z4
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_679_comp__fun__commute__product__fold,axiom,
    ! [B4: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ B4 )
     => ( finite9202498192941607031at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Z4: set_Pr4329608150637261639at_nat] :
            ( finite3179781904389220718at_nat
            @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) )
            @ Z4
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_680_comp__fun__commute__product__fold,axiom,
    ! [B4: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ B4 )
     => ( finite4517476712773419409at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Z4: set_Pr7459493094073627847at_nat] :
            ( finite3577408862935371528at_nat
            @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) )
            @ Z4
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_681_linear__order__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( order_1878278401035043452at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% linear_order_on_singleton
thf(fact_682_linear__order__on__singleton,axiom,
    ! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% linear_order_on_singleton
thf(fact_683_linear__order__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_684_linear__order__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( order_7480980593162297270at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% linear_order_on_singleton
thf(fact_685_linear__order__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( order_2711500226769404828at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% linear_order_on_singleton
thf(fact_686_in__lex__prod,axiom,
    ! [A: nat,B2: nat,A4: nat,B3: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( lex_prod_nat_nat @ R3 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_687_in__lex__prod,axiom,
    ! [A: nat,B2: product_prod_nat_nat,A4: nat,B3: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B2 ) @ ( produc487386426758144856at_nat @ A4 @ B3 ) ) @ ( lex_pr8029265285556086080at_nat @ R3 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_688_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B2: nat,A4: product_prod_nat_nat,B3: nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B2 ) @ ( produc6350711070570205562at_nat @ A4 @ B3 ) ) @ ( lex_pr4669217892513370978at_nat @ R3 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_689_in__lex__prod,axiom,
    ! [A: nat,B2: set_Pr1261947904930325089at_nat,A4: nat,B3: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member4107301959146914710at_nat @ ( produc8824474337961625893at_nat @ ( produc5463602008962177208at_nat @ A @ B2 ) @ ( produc5463602008962177208at_nat @ A4 @ B3 ) ) @ ( lex_pr4265550801952275104at_nat @ R3 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_690_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: nat,A4: set_Pr1261947904930325089at_nat,B3: nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member3480088626692287306at_nat @ ( produc3585720979125124569at_nat @ ( produc3149970401713881818at_nat @ A @ B2 ) @ ( produc3149970401713881818at_nat @ A4 @ B3 ) ) @ ( lex_pr1951919194703979714at_nat @ R3 @ S2 ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_691_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( produc6161850002892822231at_nat @ A4 @ B3 ) ) @ ( lex_pr8801849515957261039at_nat @ R3 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_692_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member4971006070143632144at_nat @ ( produc3335200116822430551at_nat @ ( produc5966227314700062775at_nat @ A @ B2 ) @ ( produc5966227314700062775at_nat @ A4 @ B3 ) ) @ ( lex_pr864846108736269391at_nat @ R3 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_693_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member6233145392104960784at_nat @ ( produc9001802813318374231at_nat @ ( produc4320700058201173367at_nat @ A @ B2 ) @ ( produc4320700058201173367at_nat @ A4 @ B3 ) ) @ ( lex_pr8442690889092155791at_nat @ R3 @ S2 ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_694_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( produc2922128104949294807at_nat @ A4 @ B3 ) ) @ ( lex_pr8000699672184251119at_nat @ R3 @ S2 ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_695_in__lex__prod,axiom,
    ! [A: nat,B2: set_Pr4329608150637261639at_nat,A4: nat,B3: set_Pr4329608150637261639at_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member4210486617425595874at_nat @ ( produc908947232176754289at_nat @ ( produc6484155927498994462at_nat @ A @ B2 ) @ ( produc6484155927498994462at_nat @ A4 @ B3 ) ) @ ( lex_pr1368066972704168198at_nat @ R3 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ B3 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_696_product__code,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( produc929334515565554804at_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X3: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_697_product__code,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( produc6773009516019950738at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( set_Pr5648618587558075414at_nat @ Ys ) )
      = ( set_Pr5518436109238095868at_nat
        @ ( concat5616020275982107560at_nat
          @ ( map_Pr2325594728238001648at_nat
            @ ^ [X3: product_prod_nat_nat] : ( map_Pr9105590726435236202at_nat @ ( produc6161850002892822231at_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_698_product__code,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( produc3241716407206042770at_nat @ ( set_se5049602875457034614at_nat @ Xs ) @ ( set_se5049602875457034614at_nat @ Ys ) )
      = ( set_Pr3765526544606949372at_nat
        @ ( concat2387994006493283240at_nat
          @ ( map_se3732254603470645392at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] : ( map_se8673859202641857546at_nat @ ( produc2922128104949294807at_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_699_product__code,axiom,
    ! [Xs: list_s4242580248156217677at_nat,Ys: list_s4242580248156217677at_nat] :
      ( ( produc4669607784529162258at_nat @ ( set_se3524105090824187612at_nat @ Xs ) @ ( set_se3524105090824187612at_nat @ Ys ) )
      = ( set_Pr2074889473033032060at_nat
        @ ( concat6901911743312625960at_nat
          @ ( map_se721323744775520554at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat] : ( map_se8572903940320514724at_nat @ ( produc9060074326276436823at_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_700_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( order_7200634732473852828at_nat @ ( field_3044959742847878580at_nat @ R3 ) @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ A ) @ ( minus_350630037651270446at_nat @ R3 @ id_Pro618200070886599107at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_701_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ ( minus_8235382265540049710at_nat @ R3 @ id_Pro5428700290092274115at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_702_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_703_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ ( minus_8321449233255521966at_nat @ R3 @ id_Pro2258643101195443293at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_704_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ ( minus_3314409938677909166at_nat @ R3 @ id_set1796276052702428605at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_705_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
            = ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ ( minus_8674369445256047406at_nat @ R3 @ id_set4888139568991089827at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_706_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B5: nat] :
              ( ( B5 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_707_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B5: product_prod_nat_nat] :
              ( ( B5 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B5 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_708_aboveS__def,axiom,
    ( order_4440095836999870412at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B5: set_Pr1261947904930325089at_nat] :
              ( ( B5 != A2 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_709_aboveS__def,axiom,
    ( order_2442354982230848562at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B5: set_Pr4329608150637261639at_nat] :
              ( ( B5 != A2 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B5 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_710_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R3 ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_711_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R3 ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X3: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X3 @ A7 )
                    & ! [Y5: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y5 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_712_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( order_423636163458052581at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
        = ( ! [A7: set_se5479201105039950375at_nat] :
              ( ( ord_le7966958417815640519at_nat @ A7 @ ( field_1428494348145443988at_nat @ R3 ) )
             => ( ( A7 != bot_bo1534825418459847827at_nat )
               => ? [X3: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X3 @ A7 )
                    & ! [Y5: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Y5 @ A7 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_713_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
        = ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ A7 @ ( field_4706345496283612622at_nat @ R3 ) )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ? [X3: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X3 @ A7 )
                    & ! [Y5: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_714_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr5564308138774400199at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
     => ( ( order_2339240034117818117at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
        = ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ A7 @ ( field_2862844103579723188at_nat @ R3 ) )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ? [X3: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X3 @ A7 )
                    & ! [Y5: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Y5 @ A7 )
                       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_715_pair__in__Id__conv,axiom,
    ! [A: nat,B2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ id_nat )
      = ( A = B2 ) ) ).

% pair_in_Id_conv
thf(fact_716_pair__in__Id__conv,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ id_Pro2258643101195443293at_nat )
      = ( A = B2 ) ) ).

% pair_in_Id_conv
thf(fact_717_pair__in__Id__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ id_set1796276052702428605at_nat )
      = ( A = B2 ) ) ).

% pair_in_Id_conv
thf(fact_718_pair__in__Id__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ id_set4888139568991089827at_nat )
      = ( A = B2 ) ) ).

% pair_in_Id_conv
thf(fact_719_IdI,axiom,
    ! [A: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ id_nat ) ).

% IdI
thf(fact_720_IdI,axiom,
    ! [A: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ id_Pro2258643101195443293at_nat ) ).

% IdI
thf(fact_721_IdI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ id_set1796276052702428605at_nat ) ).

% IdI
thf(fact_722_IdI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ id_set4888139568991089827at_nat ) ).

% IdI
thf(fact_723_IdE,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ id_nat )
     => ~ ! [X2: nat] :
            ( P2
           != ( product_Pair_nat_nat @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_724_IdE,axiom,
    ! [P2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ P2 @ id_Pro2258643101195443293at_nat )
     => ~ ! [X2: product_prod_nat_nat] :
            ( P2
           != ( produc6161850002892822231at_nat @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_725_IdE,axiom,
    ! [P2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ P2 @ id_set1796276052702428605at_nat )
     => ~ ! [X2: set_Pr1261947904930325089at_nat] :
            ( P2
           != ( produc2922128104949294807at_nat @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_726_IdE,axiom,
    ! [P2: produc1319942482725812455at_nat] :
      ( ( member1466754251312161552at_nat @ P2 @ id_set4888139568991089827at_nat )
     => ~ ! [X2: set_Pr4329608150637261639at_nat] :
            ( P2
           != ( produc9060074326276436823at_nat @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_727_well__order__on__domain,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( order_8549180852824817413at_nat @ A3 @ R3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ R3 )
       => ( ( member8206827879206165904at_nat @ A @ A3 )
          & ( member8206827879206165904at_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_728_well__order__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ A3 @ R3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_729_well__order__on__domain,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( order_2888998067076097458on_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
       => ( ( member_nat @ A @ A3 )
          & ( member_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_730_well__order__on__domain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ A3 @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
          & ( member8440522571783428010at_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_731_well__order__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ A3 @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_732_well__order__on__domain,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( order_423636163458052581at_nat @ A3 @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
       => ( ( member6273874620674182256at_nat @ A @ A3 )
          & ( member6273874620674182256at_nat @ B2 @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_733_product__concat__map,axiom,
    ( product_nat_nat
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X3: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_734_product__concat__map,axiom,
    ( produc3544356994491977349at_nat
    = ( ^ [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
          ( concat5616020275982107560at_nat
          @ ( map_Pr2325594728238001648at_nat
            @ ^ [X3: product_prod_nat_nat] : ( map_Pr9105590726435236202at_nat @ ( produc6161850002892822231at_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_735_product__concat__map,axiom,
    ( produc8019046267696426629at_nat
    = ( ^ [Xs2: list_s1210847774152347623at_nat,Ys2: list_s1210847774152347623at_nat] :
          ( concat2387994006493283240at_nat
          @ ( map_se3732254603470645392at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] : ( map_se8673859202641857546at_nat @ ( produc2922128104949294807at_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_736_product__concat__map,axiom,
    ( produc224253269364318469at_nat
    = ( ^ [Xs2: list_s4242580248156217677at_nat,Ys2: list_s4242580248156217677at_nat] :
          ( concat6901911743312625960at_nat
          @ ( map_se721323744775520554at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat] : ( map_se8572903940320514724at_nat @ ( produc9060074326276436823at_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_737_Total__subset__Id,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( total_3592101749530773125at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( ord_le3000389064537975527at_nat @ R3 @ id_Pro2258643101195443293at_nat )
       => ( ( R3 = bot_bo5327735625951526323at_nat )
          | ? [A5: product_prod_nat_nat] :
              ( R3
              = ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A5 @ A5 ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_738_Total__subset__Id,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( total_5117575981256876747at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( ord_le5997549366648089703at_nat @ R3 @ id_set4888139568991089827at_nat )
       => ( ( R3 = bot_bo4948859079157340979at_nat )
          | ? [A5: set_Pr4329608150637261639at_nat] :
              ( R3
              = ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A5 @ A5 ) @ bot_bo4948859079157340979at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_739_Total__subset__Id,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ R3 @ id_nat )
       => ( ( R3 = bot_bo2099793752762293965at_nat )
          | ? [A5: nat] :
              ( R3
              = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A5 @ A5 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_740_Total__subset__Id,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( total_7212417256332265445at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( ord_le1268244103169919719at_nat @ R3 @ id_set1796276052702428605at_nat )
       => ( ( R3 = bot_bo228742789529271731at_nat )
          | ? [A5: set_Pr1261947904930325089at_nat] :
              ( R3
              = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A5 @ A5 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_741_IdD,axiom,
    ! [A: nat,B2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ id_nat )
     => ( A = B2 ) ) ).

% IdD
thf(fact_742_IdD,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ id_Pro2258643101195443293at_nat )
     => ( A = B2 ) ) ).

% IdD
thf(fact_743_IdD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ id_set1796276052702428605at_nat )
     => ( A = B2 ) ) ).

% IdD
thf(fact_744_IdD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ id_set4888139568991089827at_nat )
     => ( A = B2 ) ) ).

% IdD
thf(fact_745_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R3 @ id_nat ) )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R3 ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_746_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( wf_set7675013139414177994at_nat @ ( minus_3314409938677909166at_nat @ R3 @ id_set1796276052702428605at_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 )
                    & ! [Y5: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y5 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_747_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( wf_set680811228771762224at_nat @ ( minus_8674369445256047406at_nat @ R3 @ id_set4888139568991089827at_nat ) )
        = ( ! [A7: set_se5479201105039950375at_nat] :
              ( ( ord_le7966958417815640519at_nat @ A7 @ ( field_1428494348145443988at_nat @ R3 ) )
             => ( ( A7 != bot_bo1534825418459847827at_nat )
               => ? [X3: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X3 @ A7 )
                    & ! [Y5: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Y5 @ A7 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_748_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( wf_Pro7803398752247294826at_nat @ ( minus_8321449233255521966at_nat @ R3 @ id_Pro2258643101195443293at_nat ) )
        = ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ A7 @ ( field_4706345496283612622at_nat @ R3 ) )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ? [X3: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X3 @ A7 )
                    & ! [Y5: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_749_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr5564308138774400199at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
     => ( ( wf_Pro2357547232388600144at_nat @ ( minus_8235382265540049710at_nat @ R3 @ id_Pro5428700290092274115at_nat ) )
        = ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ A7 @ ( field_2862844103579723188at_nat @ R3 ) )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ? [X3: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X3 @ A7 )
                    & ! [Y5: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Y5 @ A7 )
                       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y5 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_750_wf__empty,axiom,
    wf_nat @ bot_bo2099793752762293965at_nat ).

% wf_empty
thf(fact_751_wf__empty,axiom,
    wf_set7675013139414177994at_nat @ bot_bo228742789529271731at_nat ).

% wf_empty
thf(fact_752_wf__lex__prod,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ Ra )
     => ( ( wf_nat @ Rb )
       => ( wf_Pro7803398752247294826at_nat @ ( lex_prod_nat_nat @ Ra @ Rb ) ) ) ) ).

% wf_lex_prod
thf(fact_753_total__on__def,axiom,
    ( total_3592101749530773125at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
        ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A7 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A7 )
             => ( ( X3 != Y5 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R4 )
                  | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_754_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A7 )
             => ( ( X3 != Y5 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R4 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_755_total__on__def,axiom,
    ( total_5117575981256876747at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R4: set_Pr7459493094073627847at_nat] :
        ! [X3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A7 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A7 )
             => ( ( X3 != Y5 )
               => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R4 )
                  | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_756_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A7: set_nat,R4: set_Pr1261947904930325089at_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A7 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A7 )
             => ( ( X3 != Y5 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R4 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_757_total__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ! [X2: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X2 @ A3 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Y3 ) @ R3 )
                | ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_3247520894076811243at_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_758_total__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ! [X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y3 ) @ R3 )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_759_total__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_3592101749530773125at_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_760_total__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_761_total__onI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A3 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R3 )
                | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_5117575981256876747at_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_762_total__onI,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y3: nat] :
          ( ( member_nat @ X2 @ A3 )
         => ( ( member_nat @ Y3 @ A3 )
           => ( ( X2 != Y3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R3 ) ) ) ) )
     => ( total_on_nat @ A3 @ R3 ) ) ).

% total_onI
thf(fact_763_wf__def,axiom,
    ( wf_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
        ! [P5: nat > $o] :
          ( ! [X3: nat] :
              ( ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R4 )
                 => ( P5 @ Y5 ) )
             => ( P5 @ X3 ) )
         => ! [X8: nat] : ( P5 @ X8 ) ) ) ) ).

% wf_def
thf(fact_764_wf__def,axiom,
    ( wf_Pro7803398752247294826at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat] :
        ! [P5: product_prod_nat_nat > $o] :
          ( ! [X3: product_prod_nat_nat] :
              ( ! [Y5: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X3 ) @ R4 )
                 => ( P5 @ Y5 ) )
             => ( P5 @ X3 ) )
         => ! [X8: product_prod_nat_nat] : ( P5 @ X8 ) ) ) ) ).

% wf_def
thf(fact_765_wf__def,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat] :
        ! [P5: set_Pr1261947904930325089at_nat > $o] :
          ( ! [X3: set_Pr1261947904930325089at_nat] :
              ( ! [Y5: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X3 ) @ R4 )
                 => ( P5 @ Y5 ) )
             => ( P5 @ X3 ) )
         => ! [X8: set_Pr1261947904930325089at_nat] : ( P5 @ X8 ) ) ) ) ).

% wf_def
thf(fact_766_wf__def,axiom,
    ( wf_set680811228771762224at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat] :
        ! [P5: set_Pr4329608150637261639at_nat > $o] :
          ( ! [X3: set_Pr4329608150637261639at_nat] :
              ( ! [Y5: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X3 ) @ R4 )
                 => ( P5 @ Y5 ) )
             => ( P5 @ X3 ) )
         => ! [X8: set_Pr4329608150637261639at_nat] : ( P5 @ X8 ) ) ) ) ).

% wf_def
thf(fact_767_wfE__min,axiom,
    ! [R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Q: set_Pr8693737435421807431at_nat] :
      ( ( wf_Pro1994782231525216592at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ Q )
       => ~ ! [Z: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Z @ Q )
             => ~ ! [Y7: produc859450856879609959at_nat] :
                    ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8206827879206165904at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_768_wfE__min,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ Q )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q )
             => ~ ! [Y7: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8757157785044589968at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_769_wfE__min,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Q: set_nat] :
      ( ( wf_nat @ R2 )
     => ( ( member_nat @ X @ Q )
       => ~ ! [Z: nat] :
              ( ( member_nat @ Z @ Q )
             => ~ ! [Y7: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_770_wfE__min,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ Q )
       => ~ ! [Z: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Z @ Q )
             => ~ ! [Y7: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8440522571783428010at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_771_wfE__min,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Q: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ Q )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q )
             => ~ ! [Y7: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member2643936169264416010at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_772_wfE__min,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Q: set_se5479201105039950375at_nat] :
      ( ( wf_set680811228771762224at_nat @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ Q )
       => ~ ! [Z: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Z @ Q )
             => ~ ! [Y7: set_Pr4329608150637261639at_nat] :
                    ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member6273874620674182256at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_773_wfI__min,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ! [X2: produc859450856879609959at_nat,Q2: set_Pr8693737435421807431at_nat] :
          ( ( member8206827879206165904at_nat @ X2 @ Q2 )
         => ? [Xa: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Xa @ Q2 )
              & ! [Y3: produc859450856879609959at_nat] :
                  ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member8206827879206165904at_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_Pro1994782231525216592at_nat @ R2 ) ) ).

% wfI_min
thf(fact_774_wfI__min,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] :
      ( ! [X2: produc3843707927480180839at_nat,Q2: set_Pr4329608150637261639at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ Q2 )
         => ? [Xa: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Xa @ Q2 )
              & ! [Y3: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member8757157785044589968at_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_Pro2357547232388600144at_nat @ R2 ) ) ).

% wfI_min
thf(fact_775_wfI__min,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Q2: set_nat] :
          ( ( member_nat @ X2 @ Q2 )
         => ? [Xa: nat] :
              ( ( member_nat @ Xa @ Q2 )
              & ! [Y3: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_nat @ R2 ) ) ).

% wfI_min
thf(fact_776_wfI__min,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Q2: set_Pr1261947904930325089at_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ Q2 )
         => ? [Xa: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa @ Q2 )
              & ! [Y3: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member8440522571783428010at_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_Pro7803398752247294826at_nat @ R2 ) ) ).

% wfI_min
thf(fact_777_wfI__min,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Q2: set_se7855581050983116737at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ Q2 )
         => ? [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ Q2 )
              & ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member2643936169264416010at_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R2 ) ) ).

% wfI_min
thf(fact_778_wfI__min,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Q2: set_se5479201105039950375at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ Q2 )
         => ? [Xa: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Xa @ Q2 )
              & ! [Y3: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Xa ) @ R2 )
                 => ~ ( member6273874620674182256at_nat @ Y3 @ Q2 ) ) ) )
     => ( wf_set680811228771762224at_nat @ R2 ) ) ).

% wfI_min
thf(fact_779_wfUNIVI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ! [P6: nat > $o,X2: nat] :
          ( ! [Xa: nat] :
              ( ! [Y3: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa ) @ R3 )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X2 ) )
     => ( wf_nat @ R3 ) ) ).

% wfUNIVI
thf(fact_780_wfUNIVI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ! [P6: product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
          ( ! [Xa: product_prod_nat_nat] :
              ( ! [Y3: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Xa ) @ R3 )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X2 ) )
     => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ).

% wfUNIVI
thf(fact_781_wfUNIVI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ! [P6: set_Pr1261947904930325089at_nat > $o,X2: set_Pr1261947904930325089at_nat] :
          ( ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ! [Y3: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Xa ) @ R3 )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X2 ) )
     => ( wf_set7675013139414177994at_nat @ R3 ) ) ).

% wfUNIVI
thf(fact_782_wfUNIVI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ! [P6: set_Pr4329608150637261639at_nat > $o,X2: set_Pr4329608150637261639at_nat] :
          ( ! [Xa: set_Pr4329608150637261639at_nat] :
              ( ! [Y3: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Xa ) @ R3 )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa ) )
         => ( P6 @ X2 ) )
     => ( wf_set680811228771762224at_nat @ R3 ) ) ).

% wfUNIVI
thf(fact_783_wf__asym,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R3 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R3 ) ) ) ).

% wf_asym
thf(fact_784_wf__asym,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ X ) @ R3 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_asym
thf(fact_785_wf__asym,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ R3 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_asym
thf(fact_786_wf__asym,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ X ) @ R3 )
       => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_asym
thf(fact_787_wf__induct,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P: nat > $o,A: nat] :
      ( ( wf_nat @ R3 )
     => ( ! [X2: nat] :
            ( ! [Y7: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_788_wf__induct,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ! [Y7: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_789_wf__induct,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y7: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_790_wf__induct,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,P: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ! [Y7: set_Pr4329608150637261639at_nat] :
                ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_791_wf__irrefl,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R3 )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R3 ) ) ).

% wf_irrefl
thf(fact_792_wf__irrefl,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R3 ) ) ).

% wf_irrefl
thf(fact_793_wf__irrefl,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R3 ) ) ).

% wf_irrefl
thf(fact_794_wf__irrefl,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R3 ) ) ).

% wf_irrefl
thf(fact_795_wf__not__sym,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R3 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R3 ) ) ) ).

% wf_not_sym
thf(fact_796_wf__not__sym,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ X ) @ R3 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_not_sym
thf(fact_797_wf__not__sym,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X ) @ R3 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_not_sym
thf(fact_798_wf__not__sym,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ X ) @ R3 )
       => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ A ) @ R3 ) ) ) ).

% wf_not_sym
thf(fact_799_wf__not__refl,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R3 )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R3 ) ) ).

% wf_not_refl
thf(fact_800_wf__not__refl,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R3 ) ) ).

% wf_not_refl
thf(fact_801_wf__not__refl,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R3 ) ) ).

% wf_not_refl
thf(fact_802_wf__not__refl,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R3 ) ) ).

% wf_not_refl
thf(fact_803_wf__eq__minimal,axiom,
    ( wf_Pro1994782231525216592at_nat
    = ( ^ [R4: set_Pr553994874890374343at_nat] :
        ! [Q3: set_Pr8693737435421807431at_nat] :
          ( ? [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ Q3 )
         => ? [X3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X3 @ Q3 )
              & ! [Y5: produc859450856879609959at_nat] :
                  ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member8206827879206165904at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_804_wf__eq__minimal,axiom,
    ( wf_Pro2357547232388600144at_nat
    = ( ^ [R4: set_Pr5564308138774400199at_nat] :
        ! [Q3: set_Pr4329608150637261639at_nat] :
          ( ? [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ Q3 )
         => ? [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ Q3 )
              & ! [Y5: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member8757157785044589968at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_805_wf__eq__minimal,axiom,
    ( wf_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
        ! [Q3: set_nat] :
          ( ? [X3: nat] : ( member_nat @ X3 @ Q3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ Q3 )
              & ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_806_wf__eq__minimal,axiom,
    ( wf_Pro7803398752247294826at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat] :
        ! [Q3: set_Pr1261947904930325089at_nat] :
          ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ Q3 )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ Q3 )
              & ! [Y5: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member8440522571783428010at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_807_wf__eq__minimal,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat] :
        ! [Q3: set_se7855581050983116737at_nat] :
          ( ? [X3: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X3 @ Q3 )
         => ? [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ Q3 )
              & ! [Y5: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member2643936169264416010at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_808_wf__eq__minimal,axiom,
    ( wf_set680811228771762224at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat] :
        ! [Q3: set_se5479201105039950375at_nat] :
          ( ? [X3: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ X3 @ Q3 )
         => ? [X3: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X3 @ Q3 )
              & ! [Y5: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X3 ) @ R4 )
                 => ~ ( member6273874620674182256at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_809_wf__induct__rule,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P: nat > $o,A: nat] :
      ( ( wf_nat @ R3 )
     => ( ! [X2: nat] :
            ( ! [Y7: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_810_wf__induct__rule,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ! [Y7: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_811_wf__induct__rule,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y7: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_812_wf__induct__rule,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,P: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ R3 )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ! [Y7: set_Pr4329608150637261639at_nat] :
                ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X2 ) @ R3 )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_813_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( wf_nat @ S2 )
     => ( ! [X2: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_814_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,R3: set_Pr1261947904930325089at_nat,F2: nat > product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ S2 )
     => ( ! [X2: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_815_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > nat] :
      ( ( wf_nat @ S2 )
     => ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_816_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,R3: set_Pr1261947904930325089at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S2 )
     => ( ! [X2: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_817_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ( wf_nat @ S2 )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_set7675013139414177994at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_818_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,R3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ S2 )
     => ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_819_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,R3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S2 )
     => ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_820_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,R3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ S2 )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_set7675013139414177994at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_821_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S2 )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_set7675013139414177994at_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_822_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr7459493094073627847at_nat,R3: set_Pr1261947904930325089at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
      ( ( wf_set680811228771762224at_nat @ S2 )
     => ( ! [X2: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ S2 ) )
       => ( wf_nat @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_823_max__ext__wf,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( wf_set_nat @ ( max_ext_nat @ R3 ) ) ) ).

% max_ext_wf
thf(fact_824_max__ext__wf,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( wf_set7675013139414177994at_nat @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ).

% max_ext_wf
thf(fact_825_wf__subset,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ P2 @ R3 )
       => ( wf_nat @ P2 ) ) ) ).

% wf_subset
thf(fact_826_wf__Int1,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R3 @ R5 ) ) ) ).

% wf_Int1
thf(fact_827_wf__Int2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R5 @ R3 ) ) ) ).

% wf_Int2
thf(fact_828_total__on__lex__prod,axiom,
    ! [A3: set_nat,R_A: set_Pr1261947904930325089at_nat,B4: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ A3 @ R_A )
     => ( ( total_on_nat @ B4 @ R_B )
       => ( total_3592101749530773125at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu3: nat] : B4 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_829_wfE__min_H,axiom,
    ! [R2: set_Pr553994874890374343at_nat,Q: set_Pr8693737435421807431at_nat] :
      ( ( wf_Pro1994782231525216592at_nat @ R2 )
     => ( ( Q != bot_bo5327735625951526323at_nat )
       => ~ ! [Z: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Z @ Q )
             => ~ ! [Y7: produc859450856879609959at_nat] :
                    ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8206827879206165904at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_830_wfE__min_H,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Q: set_nat] :
      ( ( wf_nat @ R2 )
     => ( ( Q != bot_bot_set_nat )
       => ~ ! [Z: nat] :
              ( ( member_nat @ Z @ Q )
             => ~ ! [Y7: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_831_wfE__min_H,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,Q: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( Q != bot_bo3083307316010499117at_nat )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q )
             => ~ ! [Y7: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member2643936169264416010at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_832_wfE__min_H,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,Q: set_se5479201105039950375at_nat] :
      ( ( wf_set680811228771762224at_nat @ R2 )
     => ( ( Q != bot_bo1534825418459847827at_nat )
       => ~ ! [Z: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Z @ Q )
             => ~ ! [Y7: set_Pr4329608150637261639at_nat] :
                    ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member6273874620674182256at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_833_wfE__min_H,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R2 )
     => ( ( Q != bot_bo2099793752762293965at_nat )
       => ~ ! [Z: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Z @ Q )
             => ~ ! [Y7: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8440522571783428010at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_834_wfE__min_H,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R2 )
     => ( ( Q != bot_bo228742789529271731at_nat )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q )
             => ~ ! [Y7: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y7 @ Z ) @ R2 )
                   => ~ ( member8757157785044589968at_nat @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_835_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_on_nat @ top_top_set_nat @ R_B )
       => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_836_totalI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( X2 != Y3 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
            | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X2 ) @ R3 ) ) )
     => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R3 ) ) ).

% totalI
thf(fact_837_totalI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( X2 != Y3 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
            | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X2 ) @ R3 ) ) )
     => ( total_7212417256332265445at_nat @ top_to7629004291339433233at_nat @ R3 ) ) ).

% totalI
thf(fact_838_totalI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( X2 != Y3 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R3 )
            | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X2 ) @ R3 ) ) )
     => ( total_5117575981256876747at_nat @ top_to1754375979021635447at_nat @ R3 ) ) ).

% totalI
thf(fact_839_totalI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y3: nat] :
          ( ( X2 != Y3 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
            | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R3 ) ) )
     => ( total_on_nat @ top_top_set_nat @ R3 ) ) ).

% totalI
thf(fact_840_wfI,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A3: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le2286497330737841255at_nat @ R3
        @ ( produc18557253517588744at_nat @ A3
          @ ^ [Uu3: produc859450856879609959at_nat] : B4 ) )
     => ( ! [X2: produc859450856879609959at_nat,P6: produc859450856879609959at_nat > $o] :
            ( ! [Xa: produc859450856879609959at_nat] :
                ( ! [Y3: produc859450856879609959at_nat] :
                    ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member8206827879206165904at_nat @ X2 @ A3 )
             => ( ( member8206827879206165904at_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_Pro1994782231525216592at_nat @ R3 ) ) ) ).

% wfI
thf(fact_841_wfI,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat @ R3
        @ ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu3: produc3843707927480180839at_nat] : B4 ) )
     => ( ! [X2: produc3843707927480180839at_nat,P6: produc3843707927480180839at_nat > $o] :
            ( ! [Xa: produc3843707927480180839at_nat] :
                ( ! [Y3: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member8757157785044589968at_nat @ X2 @ A3 )
             => ( ( member8757157785044589968at_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_Pro2357547232388600144at_nat @ R3 ) ) ) ).

% wfI
thf(fact_842_wfI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat,B4: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R3
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu3: nat] : B4 ) )
     => ( ! [X2: nat,P6: nat > $o] :
            ( ! [Xa: nat] :
                ( ! [Y3: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member_nat @ X2 @ A3 )
             => ( ( member_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_nat @ R3 ) ) ) ).

% wfI
thf(fact_843_wfI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R3
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu3: product_prod_nat_nat] : B4 ) )
     => ( ! [X2: product_prod_nat_nat,P6: product_prod_nat_nat > $o] :
            ( ! [Xa: product_prod_nat_nat] :
                ( ! [Y3: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member8440522571783428010at_nat @ X2 @ A3 )
             => ( ( member8440522571783428010at_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ) ).

% wfI
thf(fact_844_wfI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( ord_le1268244103169919719at_nat @ R3
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat,P6: set_Pr1261947904930325089at_nat > $o] :
            ( ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ! [Y3: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member2643936169264416010at_nat @ X2 @ A3 )
             => ( ( member2643936169264416010at_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_set7675013139414177994at_nat @ R3 ) ) ) ).

% wfI
thf(fact_845_wfI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A3: set_se5479201105039950375at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( ord_le5997549366648089703at_nat @ R3
        @ ( produc6258361011560675592at_nat @ A3
          @ ^ [Uu3: set_Pr4329608150637261639at_nat] : B4 ) )
     => ( ! [X2: set_Pr4329608150637261639at_nat,P6: set_Pr4329608150637261639at_nat > $o] :
            ( ! [Xa: set_Pr4329608150637261639at_nat] :
                ( ! [Y3: set_Pr4329608150637261639at_nat] :
                    ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Xa ) @ R3 )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa ) )
           => ( ( member6273874620674182256at_nat @ X2 @ A3 )
             => ( ( member6273874620674182256at_nat @ X2 @ B4 )
               => ( P6 @ X2 ) ) ) )
       => ( wf_set680811228771762224at_nat @ R3 ) ) ) ).

% wfI
thf(fact_846_wf__eq__minimal2,axiom,
    ( wf_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
        ! [A7: set_nat] :
          ( ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R4 ) )
            & ( A7 != bot_bot_set_nat ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ A7 )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A7 )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_847_wf__eq__minimal2,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat] :
        ! [A7: set_se7855581050983116737at_nat] :
          ( ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R4 ) )
            & ( A7 != bot_bo3083307316010499117at_nat ) )
         => ? [X3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X3 @ A7 )
              & ! [Y5: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Y5 @ A7 )
                 => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_848_wf__eq__minimal2,axiom,
    ( wf_set680811228771762224at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat] :
        ! [A7: set_se5479201105039950375at_nat] :
          ( ( ( ord_le7966958417815640519at_nat @ A7 @ ( field_1428494348145443988at_nat @ R4 ) )
            & ( A7 != bot_bo1534825418459847827at_nat ) )
         => ? [X3: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X3 @ A7 )
              & ! [Y5: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ Y5 @ A7 )
                 => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_849_wf__eq__minimal2,axiom,
    ( wf_Pro7803398752247294826at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat] :
        ! [A7: set_Pr1261947904930325089at_nat] :
          ( ( ( ord_le3146513528884898305at_nat @ A7 @ ( field_4706345496283612622at_nat @ R4 ) )
            & ( A7 != bot_bo2099793752762293965at_nat ) )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A7 )
              & ! [Y5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                 => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_850_wf__eq__minimal2,axiom,
    ( wf_Pro2357547232388600144at_nat
    = ( ^ [R4: set_Pr5564308138774400199at_nat] :
        ! [A7: set_Pr4329608150637261639at_nat] :
          ( ( ( ord_le1268244103169919719at_nat @ A7 @ ( field_2862844103579723188at_nat @ R4 ) )
            & ( A7 != bot_bo228742789529271731at_nat ) )
         => ? [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ A7 )
              & ! [Y5: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ Y5 @ A7 )
                 => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_851_wf__measure,axiom,
    ! [F2: nat > nat] : ( wf_nat @ ( measure_nat @ F2 ) ) ).

% wf_measure
thf(fact_852_wf__UN,axiom,
    ! [I2: set_nat,R3: nat > set_Pr1261947904930325089at_nat] :
      ( ! [I5: nat] :
          ( ( member_nat @ I5 @ I2 )
         => ( wf_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: nat,J3: nat] :
            ( ( member_nat @ I5 @ I2 )
           => ( ( member_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R3 @ I5 ) ) @ ( range_nat_nat @ ( R3 @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_853_wf__UN,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [I5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I5 @ I2 )
         => ( wf_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: product_prod_nat_nat,J3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I5 @ I2 )
           => ( ( member8440522571783428010at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R3 @ I5 ) ) @ ( range_nat_nat @ ( R3 @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_854_wf__UN,axiom,
    ! [I2: set_nat,R3: nat > set_Pr8693737435421807431at_nat] :
      ( ! [I5: nat] :
          ( ( member_nat @ I5 @ I2 )
         => ( wf_Pro7803398752247294826at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: nat,J3: nat] :
            ( ( member_nat @ I5 @ I2 )
           => ( ( member_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ ( R3 @ I5 ) ) @ ( range_4455291928337903541at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo2099793752762293965at_nat ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_855_wf__UN,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ! [I5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I5 @ I2 )
         => ( wf_Pro7803398752247294826at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: product_prod_nat_nat,J3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I5 @ I2 )
           => ( ( member8440522571783428010at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ ( R3 @ I5 ) ) @ ( range_4455291928337903541at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo2099793752762293965at_nat ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_856_wf__UN,axiom,
    ! [I2: set_Pr8693737435421807431at_nat,R3: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [I5: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ I5 @ I2 )
         => ( wf_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: produc859450856879609959at_nat,J3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ I5 @ I2 )
           => ( ( member8206827879206165904at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R3 @ I5 ) ) @ ( range_nat_nat @ ( R3 @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_1384942819424619199at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_857_wf__UN,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,R3: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [I5: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ I5 @ I2 )
         => ( wf_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ I5 @ I2 )
           => ( ( member8757157785044589968at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R3 @ I5 ) ) @ ( range_nat_nat @ ( R3 @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_858_wf__UN,axiom,
    ! [I2: set_Pr8693737435421807431at_nat,R3: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ! [I5: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ I5 @ I2 )
         => ( wf_Pro7803398752247294826at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: produc859450856879609959at_nat,J3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ I5 @ I2 )
           => ( ( member8206827879206165904at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ ( R3 @ I5 ) ) @ ( range_4455291928337903541at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo2099793752762293965at_nat ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_859_wf__UN,axiom,
    ! [I2: set_nat,R3: nat > set_Pr5564308138774400199at_nat] :
      ( ! [I5: nat] :
          ( ( member_nat @ I5 @ I2 )
         => ( wf_Pro2357547232388600144at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: nat,J3: nat] :
            ( ( member_nat @ I5 @ I2 )
           => ( ( member_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in7913087082777306421at_nat @ ( domain1102310642530566604at_nat @ ( R3 @ I5 ) ) @ ( range_1175081151104272949at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo228742789529271731at_nat ) ) ) )
       => ( wf_Pro2357547232388600144at_nat @ ( comple5861762854689456436at_nat @ ( image_4210561975642625456at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_860_wf__UN,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,R3: produc3843707927480180839at_nat > set_Pr8693737435421807431at_nat] :
      ( ! [I5: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ I5 @ I2 )
         => ( wf_Pro7803398752247294826at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ I5 @ I2 )
           => ( ( member8757157785044589968at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ ( R3 @ I5 ) ) @ ( range_4455291928337903541at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo2099793752762293965at_nat ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_861_wf__UN,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > set_Pr5564308138774400199at_nat] :
      ( ! [I5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I5 @ I2 )
         => ( wf_Pro2357547232388600144at_nat @ ( R3 @ I5 ) ) )
     => ( ! [I5: product_prod_nat_nat,J3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I5 @ I2 )
           => ( ( member8440522571783428010at_nat @ J3 @ I2 )
             => ( ( ( R3 @ I5 )
                 != ( R3 @ J3 ) )
               => ( ( inf_in7913087082777306421at_nat @ ( domain1102310642530566604at_nat @ ( R3 @ I5 ) ) @ ( range_1175081151104272949at_nat @ ( R3 @ J3 ) ) )
                  = bot_bo228742789529271731at_nat ) ) ) )
       => ( wf_Pro2357547232388600144at_nat @ ( comple5861762854689456436at_nat @ ( image_3137182557746950923at_nat @ R3 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_862_wf__Union,axiom,
    ! [R2: set_se7855581050983116737at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ R2 )
         => ( wf_nat @ X2 ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R2 )
           => ! [Xa3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa3 @ R2 )
               => ( ( X2 != Xa3 )
                 => ( ( inf_inf_set_nat @ ( domain_nat_nat @ X2 ) @ ( range_nat_nat @ Xa3 ) )
                    = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ R2 ) ) ) ) ).

% wf_Union
thf(fact_863_wf__Union,axiom,
    ! [R2: set_se6425085446898223655at_nat] :
      ( ! [X2: set_Pr8693737435421807431at_nat] :
          ( ( member3250002818903848048at_nat @ X2 @ R2 )
         => ( wf_Pro7803398752247294826at_nat @ X2 ) )
     => ( ! [X2: set_Pr8693737435421807431at_nat] :
            ( ( member3250002818903848048at_nat @ X2 @ R2 )
           => ! [Xa3: set_Pr8693737435421807431at_nat] :
                ( ( member3250002818903848048at_nat @ Xa3 @ R2 )
               => ( ( X2 != Xa3 )
                 => ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ X2 ) @ ( range_4455291928337903541at_nat @ Xa3 ) )
                    = bot_bo2099793752762293965at_nat ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ ( comple2748534634209756852at_nat @ R2 ) ) ) ) ).

% wf_Union
thf(fact_864_wf__Union,axiom,
    ! [R2: set_se4230842059925695655at_nat] :
      ( ! [X2: set_Pr5564308138774400199at_nat] :
          ( ( member2711111827193551600at_nat @ X2 @ R2 )
         => ( wf_Pro2357547232388600144at_nat @ X2 ) )
     => ( ! [X2: set_Pr5564308138774400199at_nat] :
            ( ( member2711111827193551600at_nat @ X2 @ R2 )
           => ! [Xa3: set_Pr5564308138774400199at_nat] :
                ( ( member2711111827193551600at_nat @ Xa3 @ R2 )
               => ( ( X2 != Xa3 )
                 => ( ( inf_in7913087082777306421at_nat @ ( domain1102310642530566604at_nat @ X2 ) @ ( range_1175081151104272949at_nat @ Xa3 ) )
                    = bot_bo228742789529271731at_nat ) ) ) )
       => ( wf_Pro2357547232388600144at_nat @ ( comple5861762854689456436at_nat @ R2 ) ) ) ) ).

% wf_Union
thf(fact_865_Domain__insert,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 ) )
      = ( insert_nat @ A @ ( domain_nat_nat @ R3 ) ) ) ).

% Domain_insert
thf(fact_866_Domain__insert,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( domain2386162569883866188at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 ) )
      = ( insert8211810215607154385at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) ) ) ).

% Domain_insert
thf(fact_867_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) ) ) ).

% Domain_insert
thf(fact_868_Domain__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( domain6426273929576919500at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 ) )
      = ( insert2290795007606087959at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) ) ) ).

% Domain_insert
thf(fact_869_Domain__iff,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R3 ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y5 ) @ R3 ) ) ) ).

% Domain_iff
thf(fact_870_Domain__iff,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) )
      = ( ? [Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y5 ) @ R3 ) ) ) ).

% Domain_iff
thf(fact_871_Domain__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y5 ) @ R3 ) ) ) ).

% Domain_iff
thf(fact_872_Domain__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) )
      = ( ? [Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ Y5 ) @ R3 ) ) ) ).

% Domain_iff
thf(fact_873_DomainE,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R3 ) )
     => ~ ! [B: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 ) ) ).

% DomainE
thf(fact_874_DomainE,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) )
     => ~ ! [B: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 ) ) ).

% DomainE
thf(fact_875_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) )
     => ~ ! [B: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R3 ) ) ).

% DomainE
thf(fact_876_DomainE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) )
     => ~ ! [B: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R3 ) ) ).

% DomainE
thf(fact_877_Domain_ODomainI,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
     => ( member_nat @ A @ ( domain_nat_nat @ R3 ) ) ) ).

% Domain.DomainI
thf(fact_878_Domain_ODomainI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
     => ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) ) ) ).

% Domain.DomainI
thf(fact_879_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) ) ) ).

% Domain.DomainI
thf(fact_880_Domain_ODomainI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
     => ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) ) ) ).

% Domain.DomainI
thf(fact_881_Domain_Osimps,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R3 ) )
      = ( ? [A2: nat,B5: nat] :
            ( ( A = A2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Domain.simps
thf(fact_882_Domain_Osimps,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) )
      = ( ? [A2: product_prod_nat_nat,B5: product_prod_nat_nat] :
            ( ( A = A2 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Domain.simps
thf(fact_883_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
            ( ( A = A2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Domain.simps
thf(fact_884_Domain_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
            ( ( A = A2 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B5 ) @ R3 ) ) ) ) ).

% Domain.simps
thf(fact_885_Domain_Ocases,axiom,
    ! [A: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R3 ) )
     => ~ ! [B: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 ) ) ).

% Domain.cases
thf(fact_886_Domain_Ocases,axiom,
    ! [A: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R3 ) )
     => ~ ! [B: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 ) ) ).

% Domain.cases
thf(fact_887_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R3 ) )
     => ~ ! [B: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R3 ) ) ).

% Domain.cases
thf(fact_888_Domain_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R3 ) )
     => ~ ! [B: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R3 ) ) ).

% Domain.cases
thf(fact_889_wf__Un,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( ( wf_nat @ S2 )
       => ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ S2 ) )
            = bot_bot_set_nat )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S2 ) ) ) ) ) ).

% wf_Un
thf(fact_890_wf__Un,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R3 )
     => ( ( wf_set7675013139414177994at_nat @ S2 )
       => ( ( ( inf_in767997467307849391at_nat @ ( domain8802589944165416012at_nat @ R3 ) @ ( range_880292732526890933at_nat @ S2 ) )
            = bot_bo3083307316010499117at_nat )
         => ( wf_set7675013139414177994at_nat @ ( sup_su5525570899277871387at_nat @ R3 @ S2 ) ) ) ) ) ).

% wf_Un
thf(fact_891_wf__Un,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( ( wf_Pro7803398752247294826at_nat @ S2 )
       => ( ( ( inf_in2572325071724192079at_nat @ ( domain2386162569883866188at_nat @ R3 ) @ ( range_4455291928337903541at_nat @ S2 ) )
            = bot_bo2099793752762293965at_nat )
         => ( wf_Pro7803398752247294826at_nat @ ( sup_su718114333110466843at_nat @ R3 @ S2 ) ) ) ) ) ).

% wf_Un
thf(fact_892_wf__Un,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,S2: set_Pr5564308138774400199at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R3 )
     => ( ( wf_Pro2357547232388600144at_nat @ S2 )
       => ( ( ( inf_in7913087082777306421at_nat @ ( domain1102310642530566604at_nat @ R3 ) @ ( range_1175081151104272949at_nat @ S2 ) )
            = bot_bo228742789529271731at_nat )
         => ( wf_Pro2357547232388600144at_nat @ ( sup_su6664651903789602459at_nat @ R3 @ S2 ) ) ) ) ) ).

% wf_Un
thf(fact_893_min__ext__wf,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( wf_set_nat @ ( min_ext_nat @ R3 ) ) ) ).

% min_ext_wf
thf(fact_894_min__ext__wf,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R3 )
     => ( wf_set7675013139414177994at_nat @ ( min_ex6901939911449802026at_nat @ R3 ) ) ) ).

% min_ext_wf
thf(fact_895_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat,Phi3: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R3 ) )
       => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat ) )
              | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat ) ) )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( A = B2 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_896_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Phi3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( minus_3314409938677909166at_nat @ R3 @ id_set1796276052702428605at_nat ) )
              | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ ( minus_3314409938677909166at_nat @ R3 @ id_set1796276052702428605at_nat ) ) )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( A = B2 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_897_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,Phi3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ord_le7966958417815640519at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B2 @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( minus_8674369445256047406at_nat @ R3 @ id_set4888139568991089827at_nat ) )
              | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ ( minus_8674369445256047406at_nat @ R3 @ id_set4888139568991089827at_nat ) ) )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( A = B2 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_898_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat,Phi3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( minus_8321449233255521966at_nat @ R3 @ id_Pro2258643101195443293at_nat ) )
              | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ ( minus_8321449233255521966at_nat @ R3 @ id_Pro2258643101195443293at_nat ) ) )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( A = B2 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_899_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Phi3: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ ( minus_8235382265540049710at_nat @ R3 @ id_Pro5428700290092274115at_nat ) )
              | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ ( minus_8235382265540049710at_nat @ R3 @ id_Pro5428700290092274115at_nat ) ) )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( A = B2 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_900_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr553994874890374343at_nat,J: produc859450856879609959at_nat,I3: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member8206827879206165904at_nat @ I3 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( member8206827879206165904at_nat @ J @ ( field_3044959742847878580at_nat @ R3 ) )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_901_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat,I3: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member8757157785044589968at_nat @ I3 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R3 ) )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_902_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,J: nat,I3: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member_nat @ I3 @ ( field_nat @ R3 ) )
         => ( ( member_nat @ J @ ( field_nat @ R3 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_903_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,J: product_prod_nat_nat,I3: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member8440522571783428010at_nat @ I3 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( member8440522571783428010at_nat @ J @ ( field_4706345496283612622at_nat @ R3 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_904_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat,I3: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member2643936169264416010at_nat @ I3 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R3 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_905_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,J: set_Pr4329608150637261639at_nat,I3: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ J @ I3 ) @ R3 )
          | ( J = I3 ) )
       => ( ( member6273874620674182256at_nat @ I3 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( member6273874620674182256at_nat @ J @ ( field_1428494348145443988at_nat @ R3 ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_906_wo__rel_OTOTALS,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ ( field_nat @ R3 ) )
         => ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( field_nat @ R3 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Xa ) @ R3 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X5 ) @ R3 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_907_wo__rel_OTOTALS,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ! [Xa: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa @ ( field_4706345496283612622at_nat @ R3 ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Xa ) @ R3 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Xa @ X5 ) @ R3 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_908_wo__rel_OTOTALS,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ! [X5: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X5 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ! [Xa: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa @ ( field_3145993188331922222at_nat @ R3 ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Xa ) @ R3 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa @ X5 ) @ R3 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_909_wo__rel_OTOTALS,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ! [X5: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X5 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ! [Xa: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Xa @ ( field_1428494348145443988at_nat @ R3 ) )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Xa ) @ R3 )
                | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Xa @ X5 ) @ R3 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_910_wo__rel_Owell__order__induct,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ! [X2: nat] :
            ( ! [Y7: nat] :
                ( ( ( Y7 != X2 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X2 ) @ R3 ) )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_911_wo__rel_Owell__order__induct,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ! [Y7: product_prod_nat_nat] :
                ( ( ( Y7 != X2 )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_912_wo__rel_Owell__order__induct,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y7: set_Pr1261947904930325089at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_913_wo__rel_Owell__order__induct,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,P: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ! [Y7: set_Pr4329608150637261639at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( P @ Y7 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_914_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat,Phi3: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R3 ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ R3 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_915_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Phi3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ R3 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_916_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,Phi3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ord_le7966958417815640519at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B2 @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ R3 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_917_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat,Phi3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ R3 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_918_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Phi3: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 )
           => ( Phi3 @ A @ B2 ) )
         => ( ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ R3 )
             => ( Phi3 @ A @ B2 ) )
           => ( Phi3 @ A @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_919_well__order__induct__imp,axiom,
    ! [R3: set_Pr553994874890374343at_nat,P: produc859450856879609959at_nat > $o,A: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ! [X2: produc859450856879609959at_nat] :
            ( ! [Y7: produc859450856879609959at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member8206827879206165904at_nat @ Y7 @ ( field_3044959742847878580at_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member8206827879206165904at_nat @ X2 @ ( field_3044959742847878580at_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_920_well__order__induct__imp,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ! [X2: produc3843707927480180839at_nat] :
            ( ! [Y7: produc3843707927480180839at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member8757157785044589968at_nat @ Y7 @ ( field_2862844103579723188at_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member8757157785044589968at_nat @ X2 @ ( field_2862844103579723188at_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_921_well__order__induct__imp,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ! [X2: nat] :
            ( ! [Y7: nat] :
                ( ( ( Y7 != X2 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member_nat @ Y7 @ ( field_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member_nat @ X2 @ ( field_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_922_well__order__induct__imp,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ! [Y7: product_prod_nat_nat] :
                ( ( ( Y7 != X2 )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member8440522571783428010at_nat @ Y7 @ ( field_4706345496283612622at_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member8440522571783428010at_nat @ X2 @ ( field_4706345496283612622at_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_923_well__order__induct__imp,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y7: set_Pr1261947904930325089at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member2643936169264416010at_nat @ Y7 @ ( field_3145993188331922222at_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member2643936169264416010at_nat @ X2 @ ( field_3145993188331922222at_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_924_well__order__induct__imp,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,P: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ! [Y7: set_Pr4329608150637261639at_nat] :
                ( ( ( Y7 != X2 )
                  & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X2 ) @ R3 ) )
               => ( ( member6273874620674182256at_nat @ Y7 @ ( field_1428494348145443988at_nat @ R3 ) )
                 => ( P @ Y7 ) ) )
           => ( ( member6273874620674182256at_nat @ X2 @ ( field_1428494348145443988at_nat @ R3 ) )
             => ( P @ X2 ) ) )
       => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_925_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr553994874890374343at_nat,B4: set_Pr8693737435421807431at_nat,B2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( ord_le3000389064537975527at_nat @ B4 @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ B4 )
         => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( bNF_We1477774224477332019at_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_926_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,B4: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ B4 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We7473068394052389427at_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_927_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,B4: set_nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ B4 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_928_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ B4 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( bNF_We4841632414944885965at_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_929_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,B4: set_se7855581050983116737at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ord_le2077887516847798113at_nat @ B4 @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ B4 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6832956061294468653at_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_930_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,B4: set_se5479201105039950375at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ord_le7966958417815640519at_nat @ B4 @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ B4 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( bNF_We1162954353113525523at_nat @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_931_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr553994874890374343at_nat,B4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( ord_le3000389064537975527at_nat @ B4 @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ A @ B4 )
         => ( ! [B: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ B @ B4 )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We1477774224477332019at_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_932_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ A @ B4 )
         => ( ! [B: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ B @ B4 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We7473068394052389427at_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_933_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,B4: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R3 ) )
       => ( ( member_nat @ A @ B4 )
         => ( ! [B: nat] :
                ( ( member_nat @ B @ B4 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_934_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ A @ B4 )
         => ( ! [B: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B @ B4 )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We4841632414944885965at_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_935_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,B4: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ord_le2077887516847798113at_nat @ B4 @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ A @ B4 )
         => ( ! [B: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ B @ B4 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We6832956061294468653at_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_936_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,B4: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ord_le7966958417815640519at_nat @ B4 @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ A @ B4 )
         => ( ! [B: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ B @ B4 )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R3 ) )
           => ( A
              = ( bNF_We1162954353113525523at_nat @ R3 @ B4 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_937_ssubst__Pair__rhs,axiom,
    ! [R3: nat,S2: nat,R2: set_Pr1261947904930325089at_nat,S3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_938_ssubst__Pair__rhs,axiom,
    ! [R3: product_prod_nat_nat,S2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R3 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R3 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_939_ssubst__Pair__rhs,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_940_ssubst__Pair__rhs,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_941_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8206827879206165904at_nat @ ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 ) @ ( insert5050368324300391991at_nat @ A @ ( insert5050368324300391991at_nat @ B2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_942_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 ) @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_943_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member2643936169264416010at_nat @ ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 ) @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_944_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member6273874620674182256at_nat @ ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 ) @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B2 @ bot_bo1534825418459847827at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_945_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8440522571783428010at_nat @ ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 ) @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_946_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8757157785044589968at_nat @ ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 ) @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_947_transymcl__insert__both__new,axiom,
    ! [A: produc859450856879609959at_nat,Qeq: set_Pr553994874890374343at_nat,B2: produc859450856879609959at_nat] :
      ( ~ ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ Qeq ) )
     => ( ~ ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ Qeq ) )
       => ( ( transy7518170843122434953at_nat @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su593301525918095003at_nat @ ( transy7518170843122434953at_nat @ Qeq ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ B2 @ B2 ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ B2 @ A ) @ bot_bo8719193631845471027at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_948_transymcl__insert__both__new,axiom,
    ! [A: produc3843707927480180839at_nat,Qeq: set_Pr5564308138774400199at_nat,B2: produc3843707927480180839at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ Qeq ) )
     => ( ~ ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ Qeq ) )
       => ( ( transy4668831892376664969at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su6664651903789602459at_nat @ ( transy4668831892376664969at_nat @ Qeq ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ B2 @ B2 ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ bot_bo7193250159860175667at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_949_transymcl__insert__both__new,axiom,
    ! [A: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat,B2: product_prod_nat_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ Qeq ) )
     => ( ~ ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ Qeq ) )
       => ( ( transy7867444541631690531at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su718114333110466843at_nat @ ( transy7867444541631690531at_nat @ Qeq ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ B2 ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_950_transymcl__insert__both__new,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ Qeq ) )
     => ( ~ ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ Qeq ) )
       => ( ( transy4884439702966519401at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su5395584912770621083at_nat @ ( transy4884439702966519401at_nat @ Qeq ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ B2 ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ bot_bo4948859079157340979at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_951_transymcl__insert__both__new,axiom,
    ! [A: nat,Qeq: set_Pr1261947904930325089at_nat,B2: nat] :
      ( ~ ( member_nat @ A @ ( field_nat @ Qeq ) )
     => ( ~ ( member_nat @ B2 @ ( field_nat @ Qeq ) )
       => ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_952_transymcl__insert__both__new,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ Qeq ) )
     => ( ~ ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ Qeq ) )
       => ( ( transy8868430128242704515at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su5525570899277871387at_nat @ ( transy8868430128242704515at_nat @ Qeq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ B2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_953_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B4 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A3 @ B4 )
        & ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_954_in__finite__psubset,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A3 @ B4 ) @ finite4695646753290404266at_nat )
      = ( ( ord_le2604355607129572851at_nat @ A3 @ B4 )
        & ( finite4343798906461161616at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_955_Field__transymcl__self,axiom,
    ! [A: produc859450856879609959at_nat,Qeq: set_Pr553994874890374343at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ Qeq ) )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( transy7518170843122434953at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_956_Field__transymcl__self,axiom,
    ! [A: produc3843707927480180839at_nat,Qeq: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ Qeq ) )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( transy4668831892376664969at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_957_Field__transymcl__self,axiom,
    ! [A: nat,Qeq: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( field_nat @ Qeq ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transymcl_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_958_Field__transymcl__self,axiom,
    ! [A: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ Qeq ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( transy7867444541631690531at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_959_Field__transymcl__self,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ Qeq ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( transy8868430128242704515at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_960_Field__transymcl__self,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ Qeq ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( transy4884439702966519401at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_961_transymcl__trans,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Z3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ ( transymcl_nat @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transymcl_nat @ R3 ) ) ) ) ).

% transymcl_trans
thf(fact_962_transymcl__trans,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,Z3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R3 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ ( transy7867444541631690531at_nat @ R3 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z3 ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ) ).

% transymcl_trans
thf(fact_963_transymcl__trans,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R3 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ ( transy8868430128242704515at_nat @ R3 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ) ).

% transymcl_trans
thf(fact_964_transymcl__trans,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R3 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z3 ) @ ( transy4884439702966519401at_nat @ R3 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z3 ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ) ).

% transymcl_trans
thf(fact_965_transymcl__sym,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R3 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transymcl_nat @ R3 ) ) ) ).

% transymcl_sym
thf(fact_966_transymcl__sym,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R3 ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ).

% transymcl_sym
thf(fact_967_transymcl__sym,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R3 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ).

% transymcl_sym
thf(fact_968_transymcl__sym,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R3 ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ).

% transymcl_sym
thf(fact_969_transymcl__self_I1_J,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transymcl_nat @ R3 ) ) ) ).

% transymcl_self(1)
thf(fact_970_transymcl__self_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ).

% transymcl_self(1)
thf(fact_971_transymcl__self_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ).

% transymcl_self(1)
thf(fact_972_transymcl__self_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ).

% transymcl_self(1)
thf(fact_973_transymcl__self_I2_J,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y ) @ ( transymcl_nat @ R3 ) ) ) ).

% transymcl_self(2)
thf(fact_974_transymcl__self_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ).

% transymcl_self(2)
thf(fact_975_transymcl__self_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Y ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ).

% transymcl_self(2)
thf(fact_976_transymcl__self_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Y ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ).

% transymcl_self(2)
thf(fact_977_transymcl__into_I1_J,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R3 ) ) ) ).

% transymcl_into(1)
thf(fact_978_transymcl__into_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ).

% transymcl_into(1)
thf(fact_979_transymcl__into_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ).

% transymcl_into(1)
thf(fact_980_transymcl__into_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ).

% transymcl_into(1)
thf(fact_981_transymcl__into_I2_J,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transymcl_nat @ R3 ) ) ) ).

% transymcl_into(2)
thf(fact_982_transymcl__into_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( transy7867444541631690531at_nat @ R3 ) ) ) ).

% transymcl_into(2)
thf(fact_983_transymcl__into_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( transy8868430128242704515at_nat @ R3 ) ) ) ).

% transymcl_into(2)
thf(fact_984_transymcl__into_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( transy4884439702966519401at_nat @ R3 ) ) ) ).

% transymcl_into(2)
thf(fact_985_transymcl__insert__same__class,axiom,
    ! [X: nat,Y: nat,Qeq: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ Qeq ) )
     => ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Qeq ) )
        = ( transymcl_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_986_transymcl__insert__same__class,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ Qeq ) )
     => ( ( transy7867444541631690531at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ Qeq ) )
        = ( transy7867444541631690531at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_987_transymcl__insert__same__class,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ Qeq ) )
     => ( ( transy8868430128242704515at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ Qeq ) )
        = ( transy8868430128242704515at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_988_transymcl__insert__same__class,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ Qeq ) )
     => ( ( transy4884439702966519401at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ Qeq ) )
        = ( transy4884439702966519401at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_989_wo__rel_Omax2__def,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 )
            = B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_990_wo__rel_Omax2__def,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 )
            = B2 ) )
        & ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_991_wo__rel_Omax2__def,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 )
            = B2 ) )
        & ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_992_wo__rel_Omax2__def,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 )
            = B2 ) )
        & ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
         => ( ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_993_wf__bounded__set,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Ub: nat > set_Pr1261947904930325089at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A5 ) @ R3 )
         => ( ( finite6177210948735845034at_nat @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_le7866589430770878221at_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_nat @ R3 ) ) ).

% wf_bounded_set
thf(fact_994_wf__bounded__set,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,Ub: product_prod_nat_nat > set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A5 ) @ R3 )
         => ( ( finite6177210948735845034at_nat @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_le7866589430770878221at_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ).

% wf_bounded_set
thf(fact_995_wf__bounded__set,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,Ub: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A5 ) @ R3 )
         => ( ( finite6177210948735845034at_nat @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_le7866589430770878221at_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R3 ) ) ).

% wf_bounded_set
thf(fact_996_wf__bounded__set,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,Ub: set_Pr4329608150637261639at_nat > set_Pr1261947904930325089at_nat,F2: set_Pr4329608150637261639at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A5 ) @ R3 )
         => ( ( finite6177210948735845034at_nat @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_le3146513528884898305at_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_le7866589430770878221at_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_set680811228771762224at_nat @ R3 ) ) ).

% wf_bounded_set
thf(fact_997_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 )
              = A )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_998_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 )
              = A )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_999_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 )
              = A )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1000_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 )
              = A )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1001_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 )
              = A )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1002_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 )
              = A )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1003_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1004_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1005_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1006_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1007_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1008_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 )
              = B2 )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1009_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ ( bNF_We7452524311743152565at_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1010_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ ( bNF_We1807050579049552309at_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1011_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ ( bNF_We2380939065524664066x2_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1012_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ ( bNF_We6793920340776068943at_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1013_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ ( bNF_We6094461251688579247at_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1014_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 ) ) @ R3 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ ( bNF_We1331222558317278357at_nat @ R3 @ A @ B2 ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1015_transymcl__insert,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat] :
      ( ( transy7867444541631690531at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ Qeq ) )
      = ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ ( transy7867444541631690531at_nat @ Qeq ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ B2 ) @ bot_bo5327735625951526323at_nat ) ) ) @ ( minus_8321449233255521966at_nat @ ( relcom7295539661566034944at_nat @ ( sup_su718114333110466843at_nat @ ( transy7867444541631690531at_nat @ Qeq ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ B2 ) @ bot_bo5327735625951526323at_nat ) ) ) @ ( relcom7295539661566034944at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ bot_bo5327735625951526323at_nat ) ) @ ( sup_su718114333110466843at_nat @ ( transy7867444541631690531at_nat @ Qeq ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ B2 ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) @ ( transy7867444541631690531at_nat @ Qeq ) ) ) ) ).

% transymcl_insert
thf(fact_1016_transymcl__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat] :
      ( ( transy4884439702966519401at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ Qeq ) )
      = ( sup_su5395584912770621083at_nat @ ( sup_su5395584912770621083at_nat @ ( transy4884439702966519401at_nat @ Qeq ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ B2 ) @ bot_bo4948859079157340979at_nat ) ) ) @ ( minus_8674369445256047406at_nat @ ( relcom1576913146843059782at_nat @ ( sup_su5395584912770621083at_nat @ ( transy4884439702966519401at_nat @ Qeq ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ B2 ) @ bot_bo4948859079157340979at_nat ) ) ) @ ( relcom1576913146843059782at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ bot_bo4948859079157340979at_nat ) ) @ ( sup_su5395584912770621083at_nat @ ( transy4884439702966519401at_nat @ Qeq ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ B2 ) @ bot_bo4948859079157340979at_nat ) ) ) ) ) @ ( transy4884439702966519401at_nat @ Qeq ) ) ) ) ).

% transymcl_insert
thf(fact_1017_transymcl__insert,axiom,
    ! [A: nat,B2: nat,Qeq: set_Pr1261947904930325089at_nat] :
      ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ Qeq ) )
      = ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) @ bot_bo2099793752762293965at_nat ) ) ) @ ( minus_1356011639430497352at_nat @ ( relcomp_nat_nat_nat @ ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) @ bot_bo2099793752762293965at_nat ) ) ) @ ( relcomp_nat_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) @ ( transymcl_nat @ Qeq ) ) ) ) ).

% transymcl_insert
thf(fact_1018_transymcl__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat] :
      ( ( transy8868430128242704515at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ Qeq ) )
      = ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( transy8868430128242704515at_nat @ Qeq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ B2 ) @ bot_bo228742789529271731at_nat ) ) ) @ ( minus_3314409938677909166at_nat @ ( relcom6558574857482195296at_nat @ ( sup_su5525570899277871387at_nat @ ( transy8868430128242704515at_nat @ Qeq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ B2 ) @ bot_bo228742789529271731at_nat ) ) ) @ ( relcom6558574857482195296at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ bot_bo228742789529271731at_nat ) ) @ ( sup_su5525570899277871387at_nat @ ( transy8868430128242704515at_nat @ Qeq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ B2 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) @ ( transy8868430128242704515at_nat @ Qeq ) ) ) ) ).

% transymcl_insert
thf(fact_1019_bsqr__max2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A1: product_prod_nat_nat,A22: product_prod_nat_nat,B1: product_prod_nat_nat,B22: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( produc6161850002892822231at_nat @ B1 @ B22 ) ) @ ( bNF_We3318688506120914846at_nat @ R3 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( bNF_We6793920340776068943at_nat @ R3 @ A1 @ A22 ) @ ( bNF_We6793920340776068943at_nat @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).

% bsqr_max2
thf(fact_1020_bsqr__max2,axiom,
    ! [R3: 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 @ R3 ) @ R3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( produc2922128104949294807at_nat @ B1 @ B22 ) ) @ ( bNF_We6127972852007642366at_nat @ R3 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6094461251688579247at_nat @ R3 @ A1 @ A22 ) @ ( bNF_We6094461251688579247at_nat @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).

% bsqr_max2
thf(fact_1021_bsqr__max2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,B1: set_Pr4329608150637261639at_nat,B22: set_Pr4329608150637261639at_nat] :
      ( ( order_423636163458052581at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( member8215962448328497936at_nat @ ( produc2220582066590712663at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( produc9060074326276436823at_nat @ B1 @ B22 ) ) @ ( bNF_We4343417614439468644at_nat @ R3 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( bNF_We1331222558317278357at_nat @ R3 @ A1 @ A22 ) @ ( bNF_We1331222558317278357at_nat @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).

% bsqr_max2
thf(fact_1022_bsqr__max2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R3 @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).

% bsqr_max2
thf(fact_1023_underS__incl__iff,axiom,
    ! [R3: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
      ( ( order_7200634732473852828at_nat @ ( field_3044959742847878580at_nat @ R3 ) @ R3 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R3 ) )
       => ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R3 ) )
         => ( ( ord_le3000389064537975527at_nat @ ( order_4712251856133020205at_nat @ R3 @ A ) @ ( order_4712251856133020205at_nat @ R3 @ B2 ) )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1024_underS__incl__iff,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R3 ) )
       => ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R3 ) )
         => ( ( ord_le1268244103169919719at_nat @ ( order_3736018788886245933at_nat @ R3 @ A ) @ ( order_3736018788886245933at_nat @ R3 @ B2 ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1025_underS__incl__iff,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( member_nat @ A @ ( field_nat @ R3 ) )
       => ( ( member_nat @ B2 @ ( field_nat @ R3 ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R3 @ A ) @ ( order_underS_nat @ R3 @ B2 ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1026_underS__incl__iff,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R3 ) )
       => ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ( ord_le3146513528884898305at_nat @ ( order_4046151456452760519at_nat @ R3 @ A ) @ ( order_4046151456452760519at_nat @ R3 @ B2 ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1027_underS__incl__iff,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R3 @ A ) @ ( order_2644262826035875111at_nat @ R3 @ B2 ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1028_underS__incl__iff,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R3 ) @ R3 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R3 ) )
       => ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R3 ) )
         => ( ( ord_le7966958417815640519at_nat @ ( order_4357248199484075277at_nat @ R3 @ A ) @ ( order_4357248199484075277at_nat @ R3 @ B2 ) )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1029_in__measure,axiom,
    ! [X: nat,Y: nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1030_in__measure,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1031_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1032_in__measure,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F2: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( measur4922264856574889999at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1033_wf__bounded__measure,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Ub: nat > nat,F2: nat > nat] :
      ( ! [A5: nat,B: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A5 ) @ R3 )
         => ( ( ord_less_eq_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_eq_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_nat @ R3 ) ) ).

% wf_bounded_measure
thf(fact_1034_wf__bounded__measure,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,Ub: product_prod_nat_nat > nat,F2: product_prod_nat_nat > nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A5 ) @ R3 )
         => ( ( ord_less_eq_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_eq_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_Pro7803398752247294826at_nat @ R3 ) ) ).

% wf_bounded_measure
thf(fact_1035_wf__bounded__measure,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,Ub: set_Pr1261947904930325089at_nat > nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A5 ) @ R3 )
         => ( ( ord_less_eq_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_eq_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R3 ) ) ).

% wf_bounded_measure
thf(fact_1036_wf__bounded__measure,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,Ub: set_Pr4329608150637261639at_nat > nat,F2: set_Pr4329608150637261639at_nat > nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A5 ) @ R3 )
         => ( ( ord_less_eq_nat @ ( Ub @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_eq_nat @ ( F2 @ B ) @ ( Ub @ A5 ) )
            & ( ord_less_nat @ ( F2 @ A5 ) @ ( F2 @ B ) ) ) )
     => ( wf_set680811228771762224at_nat @ R3 ) ) ).

% wf_bounded_measure
thf(fact_1037_relcomp_Ocases,axiom,
    ! [A1: nat,A22: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R3 @ S2 ) )
     => ~ ! [B: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1038_relcomp_Ocases,axiom,
    ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) )
     => ~ ! [B: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ B ) @ R3 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1039_relcomp_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom6558574857482195296at_nat @ R3 @ S2 ) )
     => ~ ! [B: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ B ) @ R3 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1040_relcomp_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( relcom1576913146843059782at_nat @ R3 @ S2 ) )
     => ~ ! [B: set_Pr4329608150637261639at_nat] :
            ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ B ) @ R3 )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1041_relcomp_Osimps,axiom,
    ! [A1: nat,A22: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R3 @ S2 ) )
      = ( ? [A2: nat,B5: nat,C3: nat] :
            ( ( A1 = A2 )
            & ( A22 = C3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1042_relcomp_Osimps,axiom,
    ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) )
      = ( ? [A2: product_prod_nat_nat,B5: product_prod_nat_nat,C3: product_prod_nat_nat] :
            ( ( A1 = A2 )
            & ( A22 = C3 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B5 ) @ R3 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B5 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1043_relcomp_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( relcom6558574857482195296at_nat @ R3 @ S2 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A2 )
            & ( A22 = C3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ R3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1044_relcomp_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( relcom1576913146843059782at_nat @ R3 @ S2 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
            ( ( A1 = A2 )
            & ( A22 = C3 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B5 ) @ R3 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B5 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1045_relcomp_OrelcompI,axiom,
    ! [A: nat,B2: nat,R3: set_Pr1261947904930325089at_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R3 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1046_relcomp_OrelcompI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,C: product_prod_nat_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ C ) @ S2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1047_relcomp_OrelcompI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ S2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R3 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1048_relcomp_OrelcompI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,C: set_Pr4329608150637261639at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ C ) @ S2 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ C ) @ ( relcom1576913146843059782at_nat @ R3 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1049_relcompE,axiom,
    ! [Xz: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R3 @ S2 ) )
     => ~ ! [X2: nat,Y3: nat,Z: nat] :
            ( ( Xz
              = ( product_Pair_nat_nat @ X2 @ Z ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1050_relcompE,axiom,
    ! [Xz: produc859450856879609959at_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ Xz @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
            ( ( Xz
              = ( produc6161850002892822231at_nat @ X2 @ Z ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R3 )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1051_relcompE,axiom,
    ! [Xz: produc3843707927480180839at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ Xz @ ( relcom6558574857482195296at_nat @ R3 @ S2 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc2922128104949294807at_nat @ X2 @ Z ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R3 )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1052_relcompE,axiom,
    ! [Xz: produc1319942482725812455at_nat,R3: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ Xz @ ( relcom1576913146843059782at_nat @ R3 @ S2 ) )
     => ~ ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
            ( ( Xz
              = ( produc9060074326276436823at_nat @ X2 @ Z ) )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R3 )
             => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1053_relcompEpair,axiom,
    ! [A: nat,C: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R3 @ S2 ) )
     => ~ ! [B: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1054_relcompEpair,axiom,
    ! [A: product_prod_nat_nat,C: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( relcom7295539661566034944at_nat @ R3 @ S2 ) )
     => ~ ! [B: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1055_relcompEpair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R3 @ S2 ) )
     => ~ ! [B: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R3 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1056_relcompEpair,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ C ) @ ( relcom1576913146843059782at_nat @ R3 @ S2 ) )
     => ~ ! [B: set_Pr4329608150637261639at_nat] :
            ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R3 )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1057_underS__def,axiom,
    ( order_underS_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B5: nat] :
              ( ( B5 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A2 ) @ R4 ) ) ) ) ) ).

% underS_def
thf(fact_1058_underS__def,axiom,
    ( order_4046151456452760519at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B5: product_prod_nat_nat] :
              ( ( B5 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B5 @ A2 ) @ R4 ) ) ) ) ) ).

% underS_def
thf(fact_1059_underS__def,axiom,
    ( order_2644262826035875111at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B5: set_Pr1261947904930325089at_nat] :
              ( ( B5 != A2 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B5 @ A2 ) @ R4 ) ) ) ) ) ).

% underS_def
thf(fact_1060_underS__def,axiom,
    ( order_4357248199484075277at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B5: set_Pr4329608150637261639at_nat] :
              ( ( B5 != A2 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B5 @ A2 ) @ R4 ) ) ) ) ) ).

% underS_def
thf(fact_1061_wf__comp__self,axiom,
    ( wf_nat
    = ( ^ [R6: set_Pr1261947904930325089at_nat] : ( wf_nat @ ( relcomp_nat_nat_nat @ R6 @ R6 ) ) ) ) ).

% wf_comp_self
thf(fact_1062_underS__I,axiom,
    ! [I3: produc859450856879609959at_nat,J: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( I3 != J )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J ) @ R2 )
       => ( member8206827879206165904at_nat @ I3 @ ( order_4712251856133020205at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1063_underS__I,axiom,
    ! [I3: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( I3 != J )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J ) @ R2 )
       => ( member8757157785044589968at_nat @ I3 @ ( order_3736018788886245933at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1064_underS__I,axiom,
    ! [I3: nat,J: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( I3 != J )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R2 )
       => ( member_nat @ I3 @ ( order_underS_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1065_underS__I,axiom,
    ! [I3: product_prod_nat_nat,J: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( I3 != J )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R2 )
       => ( member8440522571783428010at_nat @ I3 @ ( order_4046151456452760519at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1066_underS__I,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( I3 != J )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J ) @ R2 )
       => ( member2643936169264416010at_nat @ I3 @ ( order_2644262826035875111at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1067_underS__I,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( I3 != J )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J ) @ R2 )
       => ( member6273874620674182256at_nat @ I3 @ ( order_4357248199484075277at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1068_underS__E,axiom,
    ! [I3: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat,J: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ I3 @ ( order_4712251856133020205at_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1069_underS__E,axiom,
    ! [I3: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I3 @ ( order_3736018788886245933at_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1070_underS__E,axiom,
    ! [I3: nat,R2: set_Pr1261947904930325089at_nat,J: nat] :
      ( ( member_nat @ I3 @ ( order_underS_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1071_underS__E,axiom,
    ! [I3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,J: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I3 @ ( order_4046151456452760519at_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1072_underS__E,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I3 @ ( order_2644262826035875111at_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1073_underS__E,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,J: set_Pr4329608150637261639at_nat] :
      ( ( member6273874620674182256at_nat @ I3 @ ( order_4357248199484075277at_nat @ R2 @ J ) )
     => ( ( I3 != J )
        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1074_wf__relcomp__compatible,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S ) @ ( relcomp_nat_nat_nat @ S @ R2 ) )
       => ( wf_nat @ ( relcomp_nat_nat_nat @ S @ R2 ) ) ) ) ).

% wf_relcomp_compatible
thf(fact_1075_wf__union__merge,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S ) )
      = ( wf_nat @ ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ R2 @ R2 ) @ ( relcomp_nat_nat_nat @ S @ R2 ) ) @ S ) ) ) ).

% wf_union_merge
thf(fact_1076_wf__union__merge,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ ( sup_su5525570899277871387at_nat @ R2 @ S ) )
      = ( wf_set7675013139414177994at_nat @ ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( relcom6558574857482195296at_nat @ R2 @ R2 ) @ ( relcom6558574857482195296at_nat @ S @ R2 ) ) @ S ) ) ) ).

% wf_union_merge
thf(fact_1077_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R2 )
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1078_pred__subset__eq2,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R2 )
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ S ) )
      = ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1079_pred__subset__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R2 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ S ) )
      = ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1080_pred__subset__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R2 )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ S ) )
      = ( ord_le5997549366648089703at_nat @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1081_fresh2__gt_I3_J,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
     => ( ord_less_nat @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) ) ).

% fresh2_gt(3)
thf(fact_1082_wf__union__compatible,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( wf_nat @ S )
       => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S ) @ R2 )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S ) ) ) ) ) ).

% wf_union_compatible
thf(fact_1083_wf__union__compatible,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( wf_set7675013139414177994at_nat @ S )
       => ( ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ R2 @ S ) @ R2 )
         => ( wf_set7675013139414177994at_nat @ ( sup_su5525570899277871387at_nat @ R2 @ S ) ) ) ) ) ).

% wf_union_compatible
thf(fact_1084_min__ext__compat,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R2 @ S ) @ R2 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( min_ex6901939911449802026at_nat @ R2 ) @ ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ S ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( min_ex6901939911449802026at_nat @ R2 ) ) ) ).

% min_ext_compat
thf(fact_1085_min__ext__compat,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,S: set_Pr5564308138774400199at_nat] :
      ( ( ord_le3091953645718357095at_nat @ ( relcom9041858334329474918at_nat @ R2 @ S ) @ R2 )
     => ( ord_le5997549366648089703at_nat @ ( relcom1576913146843059782at_nat @ ( min_ex2032397954771319568at_nat @ R2 ) @ ( sup_su5395584912770621083at_nat @ ( min_ex2032397954771319568at_nat @ S ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ bot_bo228742789529271731at_nat @ bot_bo228742789529271731at_nat ) @ bot_bo4948859079157340979at_nat ) ) ) @ ( min_ex2032397954771319568at_nat @ R2 ) ) ) ).

% min_ext_compat
thf(fact_1086_max__ext__compat,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R2 @ S ) @ R2 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( max_ex8135407076693332796at_nat @ R2 ) @ ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ S ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ).

% max_ext_compat
thf(fact_1087_max__ext__compat,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,S: set_Pr5564308138774400199at_nat] :
      ( ( ord_le3091953645718357095at_nat @ ( relcom9041858334329474918at_nat @ R2 @ S ) @ R2 )
     => ( ord_le5997549366648089703at_nat @ ( relcom1576913146843059782at_nat @ ( max_ex8833738776070432290at_nat @ R2 ) @ ( sup_su5395584912770621083at_nat @ ( max_ex8833738776070432290at_nat @ S ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ bot_bo228742789529271731at_nat @ bot_bo228742789529271731at_nat ) @ bot_bo4948859079157340979at_nat ) ) ) @ ( max_ex8833738776070432290at_nat @ R2 ) ) ) ).

% max_ext_compat
thf(fact_1088_underS__Restr__ordLess,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( ( field_nat @ R3 )
         != bot_bot_set_nat )
       => ( member8757157785044589968at_nat
          @ ( produc2922128104949294807at_nat
            @ ( inf_in2572325071724192079at_nat @ R3
              @ ( produc457027306803732586at_nat @ ( order_underS_nat @ R3 @ A )
                @ ^ [Uu3: nat] : ( order_underS_nat @ R3 @ A ) ) )
            @ R3 )
          @ bNF_We3448473429330749420at_nat ) ) ) ).

% underS_Restr_ordLess
thf(fact_1089_underS__Restr__ordLess,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( ( field_3145993188331922222at_nat @ R3 )
         != bot_bo3083307316010499117at_nat )
       => ( member1466754251312161552at_nat
          @ ( produc9060074326276436823at_nat
            @ ( inf_in7913087082777306421at_nat @ R3
              @ ( produc7726050920695093128at_nat @ ( order_2644262826035875111at_nat @ R3 @ A )
                @ ^ [Uu3: set_Pr1261947904930325089at_nat] : ( order_2644262826035875111at_nat @ R3 @ A ) ) )
            @ R3 )
          @ bNF_We4049965099390796810at_nat ) ) ) ).

% underS_Restr_ordLess
thf(fact_1090_underS__Restr__ordLess,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( ( field_4706345496283612622at_nat @ R3 )
         != bot_bo2099793752762293965at_nat )
       => ( member5855424355840516880at_nat
          @ ( produc3236233026405413719at_nat
            @ ( inf_in4302113700860409141at_nat @ R3
              @ ( produc2761391749766926216at_nat @ ( order_4046151456452760519at_nat @ R3 @ A )
                @ ^ [Uu3: product_prod_nat_nat] : ( order_4046151456452760519at_nat @ R3 @ A ) ) )
            @ R3 )
          @ bNF_We1714537421611338762at_nat ) ) ) ).

% underS_Restr_ordLess
thf(fact_1091_underS__Restr__ordLess,axiom,
    ! [R3: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ ( field_2862844103579723188at_nat @ R3 ) @ R3 )
     => ( ( ( field_2862844103579723188at_nat @ R3 )
         != bot_bo228742789529271731at_nat )
       => ( member2845199937591886608at_nat
          @ ( produc7545183385222921047at_nat
            @ ( inf_in3690739966772221109at_nat @ R3
              @ ( produc4441300235185154312at_nat @ ( order_3736018788886245933at_nat @ R3 @ A )
                @ ^ [Uu3: produc3843707927480180839at_nat] : ( order_3736018788886245933at_nat @ R3 @ A ) ) )
            @ R3 )
          @ bNF_We2594139806602030986at_nat ) ) ) ).

% underS_Restr_ordLess
thf(fact_1092_ordLess__transitive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_1093_ordLess__transitive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4049965099390796810at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4049965099390796810at_nat ) ) ) ).

% ordLess_transitive
thf(fact_1094_ordLess__irreflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R3 ) @ bNF_We3448473429330749420at_nat ) ).

% ordLess_irreflexive
thf(fact_1095_ordLess__irreflexive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R3 ) @ bNF_We4049965099390796810at_nat ) ).

% ordLess_irreflexive
thf(fact_1096_finite__ordLess__infinite,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,R5: set_Pr8693737435421807431at_nat] :
      ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R3 ) @ R3 )
     => ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R5 ) @ R5 )
       => ( ( finite6177210948735845034at_nat @ ( field_4706345496283612622at_nat @ R3 ) )
         => ( ~ ( finite6177210948735845034at_nat @ ( field_4706345496283612622at_nat @ R5 ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ R3 @ R5 ) @ bNF_We1714537421611338762at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_1097_finite__ordLess__infinite,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( finite_finite_nat @ ( field_nat @ R3 ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R5 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_1098_finite__ordLess__infinite,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( finite9047747110432174090at_nat @ ( field_3145993188331922222at_nat @ R3 ) )
         => ( ~ ( finite9047747110432174090at_nat @ ( field_3145993188331922222at_nat @ R5 ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_1099_reduction__pairI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S ) @ R2 )
       => ( fun_re1951171424197210636ir_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) ) ) ) ).

% reduction_pairI
thf(fact_1100_reduction__pairI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ R2 @ S ) @ R2 )
       => ( fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ R2 @ S ) ) ) ) ).

% reduction_pairI
thf(fact_1101_ofilter__ordLess,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_ofilter_nat @ R3 @ A3 )
       => ( ( ord_less_set_nat @ A3 @ ( field_nat @ R3 ) )
          = ( member8757157785044589968at_nat
            @ ( produc2922128104949294807at_nat
              @ ( inf_in2572325071724192079at_nat @ R3
                @ ( produc457027306803732586at_nat @ A3
                  @ ^ [Uu3: nat] : A3 ) )
              @ R3 )
            @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% ofilter_ordLess
thf(fact_1102_ofilter__ordLess,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_6860593127430111223at_nat @ R3 @ A3 )
       => ( ( ord_le3519150613208228973at_nat @ A3 @ ( field_3145993188331922222at_nat @ R3 ) )
          = ( member1466754251312161552at_nat
            @ ( produc9060074326276436823at_nat
              @ ( inf_in7913087082777306421at_nat @ R3
                @ ( produc7726050920695093128at_nat @ A3
                  @ ^ [Uu3: set_Pr1261947904930325089at_nat] : A3 ) )
              @ R3 )
            @ bNF_We4049965099390796810at_nat ) ) ) ) ).

% ofilter_ordLess
thf(fact_1103_ofilter__subset__ordLess,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat,B4: set_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_ofilter_nat @ R3 @ A3 )
       => ( ( order_ofilter_nat @ R3 @ B4 )
         => ( ( ord_less_set_nat @ A3 @ B4 )
            = ( member8757157785044589968at_nat
              @ ( produc2922128104949294807at_nat
                @ ( inf_in2572325071724192079at_nat @ R3
                  @ ( produc457027306803732586at_nat @ A3
                    @ ^ [Uu3: nat] : A3 ) )
                @ ( inf_in2572325071724192079at_nat @ R3
                  @ ( produc457027306803732586at_nat @ B4
                    @ ^ [Uu3: nat] : B4 ) ) )
              @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% ofilter_subset_ordLess
thf(fact_1104_ofilter__subset__ordLess,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_6860593127430111223at_nat @ R3 @ A3 )
       => ( ( order_6860593127430111223at_nat @ R3 @ B4 )
         => ( ( ord_le3519150613208228973at_nat @ A3 @ B4 )
            = ( member1466754251312161552at_nat
              @ ( produc9060074326276436823at_nat
                @ ( inf_in7913087082777306421at_nat @ R3
                  @ ( produc7726050920695093128at_nat @ A3
                    @ ^ [Uu3: set_Pr1261947904930325089at_nat] : A3 ) )
                @ ( inf_in7913087082777306421at_nat @ R3
                  @ ( produc7726050920695093128at_nat @ B4
                    @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 ) ) )
              @ bNF_We4049965099390796810at_nat ) ) ) ) ) ).

% ofilter_subset_ordLess
thf(fact_1105_ordLeq__iff__ordLess__Restr,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ ( field_nat @ R3 ) )
               => ( member8757157785044589968at_nat
                  @ ( produc2922128104949294807at_nat
                    @ ( inf_in2572325071724192079at_nat @ R3
                      @ ( produc457027306803732586at_nat @ ( order_underS_nat @ R3 @ X3 )
                        @ ^ [Uu3: nat] : ( order_underS_nat @ R3 @ X3 ) ) )
                    @ R5 )
                  @ bNF_We3448473429330749420at_nat ) ) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
thf(fact_1106_ordLeq__iff__ordLess__Restr,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
          = ( ! [X3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X3 @ ( field_3145993188331922222at_nat @ R3 ) )
               => ( member1466754251312161552at_nat
                  @ ( produc9060074326276436823at_nat
                    @ ( inf_in7913087082777306421at_nat @ R3
                      @ ( produc7726050920695093128at_nat @ ( order_2644262826035875111at_nat @ R3 @ X3 )
                        @ ^ [Uu3: set_Pr1261947904930325089at_nat] : ( order_2644262826035875111at_nat @ R3 @ X3 ) ) )
                    @ R5 )
                  @ bNF_We4049965099390796810at_nat ) ) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
thf(fact_1107_ofilter__subset__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat,B4: set_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_ofilter_nat @ R3 @ A3 )
       => ( ( order_ofilter_nat @ R3 @ B4 )
         => ( ( ord_less_eq_set_nat @ A3 @ B4 )
            = ( member8757157785044589968at_nat
              @ ( produc2922128104949294807at_nat
                @ ( inf_in2572325071724192079at_nat @ R3
                  @ ( produc457027306803732586at_nat @ A3
                    @ ^ [Uu3: nat] : A3 ) )
                @ ( inf_in2572325071724192079at_nat @ R3
                  @ ( produc457027306803732586at_nat @ B4
                    @ ^ [Uu3: nat] : B4 ) ) )
              @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% ofilter_subset_ordLeq
thf(fact_1108_ofilter__subset__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_6860593127430111223at_nat @ R3 @ A3 )
       => ( ( order_6860593127430111223at_nat @ R3 @ B4 )
         => ( ( ord_le2077887516847798113at_nat @ A3 @ B4 )
            = ( member1466754251312161552at_nat
              @ ( produc9060074326276436823at_nat
                @ ( inf_in7913087082777306421at_nat @ R3
                  @ ( produc7726050920695093128at_nat @ A3
                    @ ^ [Uu3: set_Pr1261947904930325089at_nat] : A3 ) )
                @ ( inf_in7913087082777306421at_nat @ R3
                  @ ( produc7726050920695093128at_nat @ B4
                    @ ^ [Uu3: set_Pr1261947904930325089at_nat] : B4 ) ) )
              @ bNF_We4758492634085796135at_nat ) ) ) ) ) ).

% ofilter_subset_ordLeq
thf(fact_1109_ordLess__iff__ordIso__Restr,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We3448473429330749420at_nat )
          = ( ? [X3: nat] :
                ( ( member_nat @ X3 @ ( field_nat @ R3 ) )
                & ( member8757157785044589968at_nat
                  @ ( produc2922128104949294807at_nat @ R5
                    @ ( inf_in2572325071724192079at_nat @ R3
                      @ ( produc457027306803732586at_nat @ ( order_underS_nat @ R3 @ X3 )
                        @ ^ [Uu3: nat] : ( order_underS_nat @ R3 @ X3 ) ) ) )
                  @ bNF_We5258908940166488438at_nat ) ) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
thf(fact_1110_ordLess__iff__ordIso__Restr,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4049965099390796810at_nat )
          = ( ? [X3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X3 @ ( field_3145993188331922222at_nat @ R3 ) )
                & ( member1466754251312161552at_nat
                  @ ( produc9060074326276436823at_nat @ R5
                    @ ( inf_in7913087082777306421at_nat @ R3
                      @ ( produc7726050920695093128at_nat @ ( order_2644262826035875111at_nat @ R3 @ X3 )
                        @ ^ [Uu3: set_Pr1261947904930325089at_nat] : ( order_2644262826035875111at_nat @ R3 @ X3 ) ) ) )
                  @ bNF_We1279280605989566100at_nat ) ) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
thf(fact_1111_internalize__ordLeq,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat )
      = ( ? [P4: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_eq_set_nat @ ( field_nat @ P4 ) @ ( field_nat @ R3 ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ P4 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P4 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1112_internalize__ordLeq,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat )
      = ( ? [P4: set_Pr4329608150637261639at_nat] :
            ( ( ord_le2077887516847798113at_nat @ ( field_3145993188331922222at_nat @ P4 ) @ ( field_3145993188331922222at_nat @ R3 ) )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ P4 ) @ bNF_We1279280605989566100at_nat )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ P4 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1113_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
        | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
thf(fact_1114_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
      = ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
        | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat ) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
thf(fact_1115_ordLeq__ordIso__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1116_ordLeq__ordIso__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We1279280605989566100at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4758492634085796135at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1117_ordIso__ordLeq__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1118_ordIso__ordLeq__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4758492634085796135at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4758492634085796135at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1119_ordLeq__transitive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1120_ordLeq__transitive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4758492634085796135at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4758492634085796135at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1121_ordIso__transitive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1122_ordIso__transitive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We1279280605989566100at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We1279280605989566100at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1123_ordIso__imp__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1124_ordIso__imp__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1125_ordIso__iff__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1126_ordIso__iff__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
      = ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1127_ordIso__symmetric,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_symmetric
thf(fact_1128_ordIso__symmetric,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We1279280605989566100at_nat ) ) ).

% ordIso_symmetric
thf(fact_1129_not__ordLess__ordIso,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1130_not__ordLess__ordIso,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1131_ordIso__ordLess__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1132_ordIso__ordLess__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4049965099390796810at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4049965099390796810at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1133_ordLess__ordIso__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1134_ordLess__ordIso__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We1279280605989566100at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4049965099390796810at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1135_not__ordLeq__ordLess,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We3448473429330749420at_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_1136_not__ordLeq__ordLess,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
     => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4049965099390796810at_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_1137_not__ordLess__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_1138_not__ordLess__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_1139_ordLess__imp__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1140_ordLess__imp__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1141_ordLeq__ordLess__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1142_ordLeq__ordLess__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4049965099390796810at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4049965099390796810at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1143_ordLess__ordLeq__trans,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R7 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R7 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1144_ordLess__ordLeq__trans,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat,R7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R7 ) @ bNF_We4758492634085796135at_nat )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R7 ) @ bNF_We4049965099390796810at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1145_finite__well__order__on__ordIso,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,R5: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( order_8331247224601271455at_nat @ A3 @ R3 )
       => ( ( order_8331247224601271455at_nat @ A3 @ R5 )
         => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ R3 @ R5 ) @ bNF_We4319090096570846868at_nat ) ) ) ) ).

% finite_well_order_on_ordIso
thf(fact_1146_finite__well__order__on__ordIso,axiom,
    ! [A3: set_nat,R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( order_2888998067076097458on_nat @ A3 @ R3 )
       => ( ( order_2888998067076097458on_nat @ A3 @ R5 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ) ) ).

% finite_well_order_on_ordIso
thf(fact_1147_finite__well__order__on__ordIso,axiom,
    ! [A3: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( order_1968183844437341695at_nat @ A3 @ R3 )
       => ( ( order_1968183844437341695at_nat @ A3 @ R5 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We1279280605989566100at_nat ) ) ) ) ).

% finite_well_order_on_ordIso
thf(fact_1148_ordIso__reflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R3 ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_reflexive
thf(fact_1149_ordIso__reflexive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R3 ) @ bNF_We1279280605989566100at_nat ) ) ).

% ordIso_reflexive
thf(fact_1150_exists__minim__Well__order,axiom,
    ! [R2: set_se7855581050983116737at_nat] :
      ( ( R2 != bot_bo3083307316010499117at_nat )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R2 )
           => ( order_2888998067076097458on_nat @ ( field_nat @ X2 ) @ X2 ) )
       => ? [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R2 )
            & ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Xa ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1151_exists__minim__Well__order,axiom,
    ! [R2: set_se5479201105039950375at_nat] :
      ( ( R2 != bot_bo1534825418459847827at_nat )
     => ( ! [X2: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X2 @ R2 )
           => ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ X2 ) @ X2 ) )
       => ? [X2: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X2 @ R2 )
            & ! [Xa: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Xa @ R2 )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Xa ) @ bNF_We4758492634085796135at_nat ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1152_ordLeq__total,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1153_ordLeq__total,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
          | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1154_ordLeq__reflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_reflexive
thf(fact_1155_ordLeq__reflexive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ).

% ordLeq_reflexive
thf(fact_1156_ordLeq__Well__order__simp,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1157_ordLeq__Well__order__simp,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
        & ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1158_internalize__ordLess,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We3448473429330749420at_nat )
      = ( ? [P4: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_set_nat @ ( field_nat @ P4 ) @ ( field_nat @ R3 ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ P4 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P4 @ R3 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1159_internalize__ordLess,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4049965099390796810at_nat )
      = ( ? [P4: set_Pr4329608150637261639at_nat] :
            ( ( ord_le3519150613208228973at_nat @ ( field_3145993188331922222at_nat @ P4 ) @ ( field_3145993188331922222at_nat @ R3 ) )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ P4 ) @ bNF_We1279280605989566100at_nat )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ P4 @ R3 ) @ bNF_We4049965099390796810at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1160_not__ordLess__iff__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We3448473429330749420at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1161_not__ordLess__iff__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4049965099390796810at_nat ) )
          = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4758492634085796135at_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1162_not__ordLeq__iff__ordLess,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1163_not__ordLeq__iff__ordLess,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat ) )
          = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1164_ordLess__or__ordLeq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ R5 ) @ bNF_We3448473429330749420at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R3 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1165_ordLess__or__ordLeq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,R5: set_Pr4329608150637261639at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R5 ) @ R5 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ R5 ) @ bNF_We4049965099390796810at_nat )
          | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R5 @ R3 ) @ bNF_We4758492634085796135at_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1166_dir__image__ordIso,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R3 ) @ R3 )
     => ( ( inj_on_nat_nat @ F2 @ ( field_nat @ R3 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R3 @ ( bNF_We1800842361606551104at_nat @ R3 @ F2 ) ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% dir_image_ordIso
thf(fact_1167_dir__image__ordIso,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R3 ) @ R3 )
     => ( ( inj_on4785126279584951473at_nat @ F2 @ ( field_3145993188331922222at_nat @ R3 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R3 @ ( bNF_We7693244424125670494at_nat @ R3 @ F2 ) ) @ bNF_We1279280605989566100at_nat ) ) ) ).

% dir_image_ordIso
thf(fact_1168_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S2: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ S2 @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T3 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1169_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_1170_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S2: nat,T3: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T3 ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1171_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S2: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ S2 @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T3 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1172_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S2: nat,T3: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T3 ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1173_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y6 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y6 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X7 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1174_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y6 )
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_1175_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1176_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1177_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1178_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1179_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1180_smin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( X7 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_1181_wmax__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X7 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_1182_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1183_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_1184_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_1185_relcomp__fold,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( ( finite6177210948735845034at_nat @ S )
       => ( ( relcomp_nat_nat_nat @ R2 @ S )
          = ( finite5618522546757629032at_nat
            @ ( produc7697293333247743475at_nat
              @ ^ [X3: nat,Y5: nat,A7: set_Pr1261947904930325089at_nat] :
                  ( finite5618522546757629032at_nat
                  @ ( produc7697293333247743475at_nat
                    @ ^ [W: nat,Z4: nat,A8: set_Pr1261947904930325089at_nat] : ( if_set4744415642439091239at_nat @ ( Y5 = W ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Z4 ) @ A8 ) @ A8 ) )
                  @ A7
                  @ S ) )
            @ bot_bo2099793752762293965at_nat
            @ R2 ) ) ) ) ).

% relcomp_fold
thf(fact_1186_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_1187_wf__less__than,axiom,
    wf_nat @ less_than ).

% wf_less_than
thf(fact_1188_case__prod__conv,axiom,
    ! [F2: nat > nat > $o,A: nat,B2: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B2 ) )
      = ( F2 @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_1189_mem__case__prodI,axiom,
    ! [Z3: nat,C: nat > nat > set_nat,A: nat,B2: nat] :
      ( ( member_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1190_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1191_mem__case__prodI,axiom,
    ! [Z3: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C @ ( produc6161850002892822231at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1192_mem__case__prodI,axiom,
    ! [Z3: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,A: nat,B2: nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1193_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1194_mem__case__prodI,axiom,
    ! [Z3: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member_nat @ Z3 @ ( produc5395831842199599730et_nat @ C @ ( produc2922128104949294807at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1195_mem__case__prodI,axiom,
    ! [Z3: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,A: nat,B2: nat] :
      ( ( member8757157785044589968at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8757157785044589968at_nat @ Z3 @ ( produc3705604114177432185at_nat @ C @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1196_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc1433696742876942453at_nat @ C @ ( produc2922128104949294807at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1197_mem__case__prodI,axiom,
    ! [Z3: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1198_mem__case__prodI,axiom,
    ! [Z3: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z3 @ ( C @ A @ B2 ) )
     => ( member8757157785044589968at_nat @ Z3 @ ( produc1422205692209202011at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1199_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: nat,C: nat > nat > set_nat] :
      ( ! [A5: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) )
         => ( member_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1200_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1201_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) )
         => ( member_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1202_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) )
         => ( member8206827879206165904at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1203_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1204_mem__case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,Z3: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A5 @ B ) )
         => ( member_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member_nat @ Z3 @ ( produc5395831842199599730et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1205_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat] :
      ( ! [A5: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) )
         => ( member8757157785044589968at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8757157785044589968at_nat @ Z3 @ ( produc3705604114177432185at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1206_mem__case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,Z3: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A5 @ B ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc1433696742876942453at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1207_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) )
         => ( member8206827879206165904at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1208_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) )
         => ( member8757157785044589968at_nat @ Z3 @ ( C @ A5 @ B ) ) )
     => ( member8757157785044589968at_nat @ Z3 @ ( produc1422205692209202011at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1209_pair__imageI,axiom,
    ! [A: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F2: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ A3 )
     => ( member_nat @ ( F2 @ A @ B2 ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1210_pair__imageI,axiom,
    ! [A: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ A3 )
     => ( member_o @ ( F2 @ A @ B2 ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1211_pair__imageI,axiom,
    ! [A: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F2: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F2 @ A @ B2 ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1212_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ A3 )
     => ( member_nat @ ( F2 @ A @ B2 ) @ ( image_6160980552072806642at_nat @ ( produc6237124055692578492at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1213_pair__imageI,axiom,
    ! [A: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F2: nat > nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ A3 )
     => ( member8206827879206165904at_nat @ ( F2 @ A @ B2 ) @ ( image_1119892238107900587at_nat @ ( produc2382474379627796377at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1214_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F2 @ A @ B2 ) @ ( image_6706873702294942047at_nat @ ( produc2982569246487150357at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1215_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ A3 )
     => ( member_nat @ ( F2 @ A @ B2 ) @ ( image_3198525501578754290at_nat @ ( produc4001655692221972156at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1216_pair__imageI,axiom,
    ! [A: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F2: nat > nat > produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ A3 )
     => ( member8757157785044589968at_nat @ ( F2 @ A @ B2 ) @ ( image_1672857615394513579at_nat @ ( produc153460504799203225at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1217_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F2 @ A @ B2 ) @ ( image_7325330684212377951at_nat @ ( produc2376066862417776405at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1218_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ A3 )
     => ( member8206827879206165904at_nat @ ( F2 @ A @ B2 ) @ ( image_6321802852832981957at_nat @ ( produc6744312701629110395at_nat @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1219_total__on__less__than,axiom,
    ! [A3: set_nat] : ( total_on_nat @ A3 @ less_than ) ).

% total_on_less_than
thf(fact_1220_total__less__than,axiom,
    total_on_nat @ top_top_set_nat @ less_than ).

% total_less_than
thf(fact_1221_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F2: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F2 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X13: nat,X24: nat] : ( H @ ( F2 @ X13 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_1222_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > $o,G5: product_prod_nat_nat > $o] :
      ( ! [X2: nat,Y3: nat] :
          ( ( F2 @ X2 @ Y3 )
          = ( G5 @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F2 )
        = G5 ) ) ).

% cond_case_prod_eta
thf(fact_1223_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X3: nat,Y5: nat] : ( F2 @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_1224_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z3: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z3 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( Z3
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_1225_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_1226_mem__case__prodE,axiom,
    ! [Z3: nat,C: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1227_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1228_mem__case__prodE,axiom,
    ! [Z3: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat,P2: produc859450856879609959at_nat] :
      ( ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C @ P2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X2 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1229_mem__case__prodE,axiom,
    ! [Z3: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1230_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C @ P2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X2 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1231_mem__case__prodE,axiom,
    ! [Z3: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat,P2: produc3843707927480180839at_nat] :
      ( ( member_nat @ Z3 @ ( produc5395831842199599730et_nat @ C @ P2 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1232_mem__case__prodE,axiom,
    ! [Z3: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,P2: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z3 @ ( produc3705604114177432185at_nat @ C @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X2 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1233_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc1433696742876942453at_nat @ C @ P2 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1234_mem__case__prodE,axiom,
    ! [Z3: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C @ P2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X2 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1235_mem__case__prodE,axiom,
    ! [Z3: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8757157785044589968at_nat @ Z3 @ ( produc1422205692209202011at_nat @ C @ P2 ) )
     => ~ ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X2 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z3 @ ( C @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1236_split__cong,axiom,
    ! [Q5: product_prod_nat_nat,F2: nat > nat > $o,G5: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ! [X2: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X2 @ Y3 )
            = Q5 )
         => ( ( F2 @ X2 @ Y3 )
            = ( G5 @ X2 @ Y3 ) ) )
     => ( ( P2 = Q5 )
       => ( ( produc6081775807080527818_nat_o @ F2 @ P2 )
          = ( produc6081775807080527818_nat_o @ G5 @ Q5 ) ) ) ) ).

% split_cong
thf(fact_1237_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_1238_comp__fun__commute__relcomp__fold,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( finite5656229966737060081at_nat
        @ ( produc7697293333247743475at_nat
          @ ^ [X3: nat,Y5: nat,A7: set_Pr1261947904930325089at_nat] :
              ( finite5618522546757629032at_nat
              @ ( produc7697293333247743475at_nat
                @ ^ [W: nat,Z4: nat,A8: set_Pr1261947904930325089at_nat] : ( if_set4744415642439091239at_nat @ ( Y5 = W ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Z4 ) @ A8 ) @ A8 ) )
              @ A7
              @ S ) ) ) ) ).

% comp_fun_commute_relcomp_fold
thf(fact_1239_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ S ) @ X7 )
        = ( finite5618522546757629032at_nat
          @ ( produc7697293333247743475at_nat
            @ ^ [W: nat,Z4: nat,A8: set_Pr1261947904930325089at_nat] :
                ( if_set4744415642439091239at_nat
                @ ( ( product_snd_nat_nat @ X )
                  = W )
                @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ ( product_fst_nat_nat @ X ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ X7
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1240_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,X7: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( sup_su5525570899277871387at_nat @ ( relcom6558574857482195296at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ S ) @ X7 )
        = ( finite2056329533557663592at_nat
          @ ( produc2761740369720999185at_nat
            @ ^ [W: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat,A8: set_Pr4329608150637261639at_nat] :
                ( if_set3208018901896979469at_nat
                @ ( ( produc2937545259423399429at_nat @ X )
                  = W )
                @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ X ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ X7
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1241_finite__subset__wf,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( wf_set7675013139414177994at_nat
        @ ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [X8: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat] :
                ( ( ord_le7866589430770878221at_nat @ X8 @ Y8 )
                & ( ord_le3146513528884898305at_nat @ Y8 @ A3 ) ) ) ) ) ) ).

% finite_subset_wf
thf(fact_1242_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,B5: nat] :
            ( P
            & ( Q @ A2 @ B5 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_1243_in__inv__image,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R3 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1244_in__inv__image,axiom,
    ! [X: nat,Y: nat,R3: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im8721129524585065343at_nat @ R3 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1245_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im2857804880773004637at_nat @ R3 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1246_in__inv__image,axiom,
    ! [X: nat,Y: nat,R3: set_Pr4329608150637261639at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im4437241714082187487at_nat @ R3 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1247_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im6750873321330482877at_nat @ R3 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1248_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im5896302226278515346at_nat @ R3 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1249_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr4329608150637261639at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im8562125120543730994at_nat @ R3 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1250_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,F2: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im984280340187844594at_nat @ R3 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1251_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im7023619800666061458at_nat @ R3 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1252_in__inv__image,axiom,
    ! [X: nat,Y: nat,R3: set_Pr7459493094073627847at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im2903009620893676101at_nat @ R3 @ F2 ) )
      = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_1253_wf__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( wf_nat @ R3 )
     => ( wf_nat @ ( inv_image_nat_nat @ R3 @ F2 ) ) ) ).

% wf_inv_image
thf(fact_1254_case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) )
         => ( C @ A5 @ B ) )
     => ( produc6590410687421337004_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1255_case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A5 @ B ) )
         => ( C @ A5 @ B ) )
     => ( produc410239310623530412_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1256_case__prodI2,axiom,
    ! [P2: produc1319942482725812455at_nat,C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
          ( ( P2
            = ( produc9060074326276436823at_nat @ A5 @ B ) )
         => ( C @ A5 @ B ) )
     => ( produc9020218426428501292_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1257_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) )
         => ( C @ A5 @ B ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1258_case__prodI,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( F2 @ A @ B2 )
     => ( produc6590410687421337004_nat_o @ F2 @ ( produc6161850002892822231at_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_1259_case__prodI,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( F2 @ A @ B2 )
     => ( produc410239310623530412_nat_o @ F2 @ ( produc2922128104949294807at_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_1260_case__prodI,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( F2 @ A @ B2 )
     => ( produc9020218426428501292_nat_o @ F2 @ ( produc9060074326276436823at_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_1261_case__prodI,axiom,
    ! [F2: nat > nat > $o,A: nat,B2: nat] :
      ( ( F2 @ A @ B2 )
     => ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_1262_The__split__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( the_Pr8653942702660118126at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X9: product_prod_nat_nat,Y9: product_prod_nat_nat] :
              ( ( X = X9 )
              & ( Y = Y9 ) ) ) )
      = ( produc6161850002892822231at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_1263_The__split__eq,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( the_Pr4604535484834969198at_nat
        @ ( produc410239310623530412_nat_o
          @ ^ [X9: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat] :
              ( ( X = X9 )
              & ( Y = Y9 ) ) ) )
      = ( produc2922128104949294807at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_1264_The__split__eq,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( the_Pr4877703158196064750at_nat
        @ ( produc9020218426428501292_nat_o
          @ ^ [X9: set_Pr4329608150637261639at_nat,Y9: set_Pr4329608150637261639at_nat] :
              ( ( X = X9 )
              & ( Y = Y9 ) ) ) )
      = ( produc9060074326276436823at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_1265_The__split__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X9: nat,Y9: nat] :
              ( ( X = X9 )
              & ( Y = Y9 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_1266_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A2: nat,B5: nat] :
              ( ( P @ A2 )
              & ( Q @ B5 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu3: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1267_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_1268_wf__pred__nat,axiom,
    wf_nat @ pred_nat ).

% wf_pred_nat
thf(fact_1269_irrefl__less__than,axiom,
    irrefl_on_nat @ top_top_set_nat @ less_than ).

% irrefl_less_than
thf(fact_1270_trans__less__than,axiom,
    trans_on_nat @ top_top_set_nat @ less_than ).

% trans_less_than

% Helper facts (13)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( if_Pro3594990065127090477at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( if_Pro3594990065127090477at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( if_set3208018901896979469at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( if_set3208018901896979469at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( relational_qp_a_b @ q )
    = ( relati3725921752842749053pl_a_b @ q ) ) ).

%------------------------------------------------------------------------------