TPTP Problem File: SLH0962^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Rewrite_Properties_Reduction/0018_Rewriting_GTRS/prob_00059_002064__13955278_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1680 ( 357 unt; 401 typ;   0 def)
%            Number of atoms       : 4159 ( 792 equ;   0 cnn)
%            Maximal formula atoms :    9 (   3 avg)
%            Number of connectives : 18130 ( 496   ~;  39   |; 278   &;15037   @)
%                                         (   0 <=>;2280  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   9 avg)
%            Number of types       :   45 (  44 usr)
%            Number of type conns  :  852 ( 852   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  360 ( 357 usr;  28 con; 0-4 aty)
%            Number of variables   : 4632 ( 310   ^;4139   !; 183   ?;4632   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:18:50.672
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
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thf(sy_c_Wfrec_Osame__fst_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
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    member_b: b > set_b > $o ).

thf(sy_v__092_060F_062,type,
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thf(sy_v__092_060R_062,type,
    r: set_Pr4386577575007340137rm_a_b ).

thf(sy_v_s,type,
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thf(sy_v_t,type,
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% Relevant facts (1275)
thf(fact_0__092_060open_062_092_060And_062t_As_A_092_060F_062_O_A_Is_M_At_J_A_092_060in_062_Asrrstep_A_092_060F_062_A_092_060R_062_A_092_060Longrightarrow_062_Aground_As_092_060close_062,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ r ) ) )
     => ( term_ground_a_b @ S ) ) ).

% \<open>\<And>t s \<F>. (s, t) \<in> srrstep \<F> \<R> \<Longrightarrow> ground s\<close>
thf(fact_1__092_060open_062_092_060And_062t_As_A_092_060F_062_O_A_Is_M_At_J_A_092_060in_062_Asrrstep_A_092_060F_062_A_092_060R_062_A_092_060Longrightarrow_062_Aground_At_092_060close_062,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ r ) ) )
     => ( term_ground_a_b @ T ) ) ).

% \<open>\<And>t s \<F>. (s, t) \<in> srrstep \<F> \<R> \<Longrightarrow> ground t\<close>
thf(fact_2__092_060open_062_092_060And_062t_As_A_092_060F_062_O_A_092_060lbrakk_062ground_As_059_A_Is_M_At_J_A_092_060in_062_A_Isrstep_A_092_060F_062_A_092_060R_062_J_092_060_094sup_062_K_092_060rbrakk_062_A_092_060Longrightarrow_062_Aground_At_092_060close_062,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( term_ground_a_b @ S )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ r ) ) ) )
       => ( term_ground_a_b @ T ) ) ) ).

% \<open>\<And>t s \<F>. \<lbrakk>ground s; (s, t) \<in> (srstep \<F> \<R>)\<^sup>*\<rbrakk> \<Longrightarrow> ground t\<close>
thf(fact_3__092_060open_062_092_060And_062t_As_A_092_060F_062_O_A_092_060lbrakk_062ground_At_059_A_Is_M_At_J_A_092_060in_062_A_Isrstep_A_092_060F_062_A_092_060R_062_J_092_060_094sup_062_K_092_060rbrakk_062_A_092_060Longrightarrow_062_Aground_As_092_060close_062,axiom,
    ! [T: term_a_b,S: term_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ r ) ) ) )
       => ( term_ground_a_b @ S ) ) ) ).

% \<open>\<And>t s \<F>. \<lbrakk>ground t; (s, t) \<in> (srstep \<F> \<R>)\<^sup>*\<rbrakk> \<Longrightarrow> ground s\<close>
thf(fact_4_assms_I2_J,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ).

% assms(2)
thf(fact_5_ground__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ ( term_a2780309480206857766_a_b_b @ T ) )
      = ( term_ground_a_b @ T ) ) ).

% ground_adapt_vars
thf(fact_6_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( term_a2780309480206857766_a_b_b @ ( term_a2780309480206857766_a_b_b @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_7_adapt__vars__inj,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( ( term_a2780309480206857766_a_b_b @ X )
        = ( term_a2780309480206857766_a_b_b @ Y ) )
     => ( ( term_ground_a_b @ X )
       => ( ( term_ground_a_b @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_8_adapt__vars__reverse,axiom,
    ! [T: term_a_b,T2: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( ( term_a2780309480206857766_a_b_b @ T2 )
          = T )
       => ( ( term_a2780309480206857766_a_b_b @ T )
          = T2 ) ) ) ).

% adapt_vars_reverse
thf(fact_9_ground__map__term,axiom,
    ! [F2: a > a,H: b > b,T: term_a_b] :
      ( ( term_ground_a_b @ ( map_term_a_a_b_b @ F2 @ H @ T ) )
      = ( term_ground_a_b @ T ) ) ).

% ground_map_term
thf(fact_10_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_a] : ( term_ground_a_b @ ( ground5786051259813020157rm_a_b @ S ) ) ).

% ground_term_of_gterm
thf(fact_11_ground__term__to__gtermD,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ? [T3: ground_gterm_a] :
          ( T
          = ( ground5786051259813020157rm_a_b @ T3 ) ) ) ).

% ground_term_to_gtermD
thf(fact_12_gound__linear,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( terms_5523818207328828897rm_a_b @ T ) ) ).

% gound_linear
thf(fact_13_ground_Osimps_I1_J,axiom,
    ! [X: b] :
      ~ ( term_ground_a_b @ ( var_b_a @ X ) ) ).

% ground.simps(1)
thf(fact_14_ground__subst__apply,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = T ) ) ).

% ground_subst_apply
thf(fact_15_subterm__eq__pres__ground,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( term_ground_a_b @ S )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( term_ground_a_b @ T ) ) ) ).

% subterm_eq_pres_ground
thf(fact_16_linear__term_Osimps_I1_J,axiom,
    ! [Uu: b] : ( terms_5523818207328828897rm_a_b @ ( var_b_a @ Uu ) ) ).

% linear_term.simps(1)
thf(fact_17_adapt__vars2,axiom,
    ! [T: term_a_b] :
      ( ( term_a2780309480206857766_a_b_b @ ( term_a2780309480206857766_a_b_b @ T ) )
      = ( term_a2780309480206857766_a_b_b @ T ) ) ).

% adapt_vars2
thf(fact_18_rstep__substI,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R ) ) ) ).

% rstep_substI
thf(fact_19_supteq__subst,axiom,
    ! [S: term_a7872195449889686814rm_a_b,T: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
      ( ( member3951536427926967622rm_a_b @ ( produc1288243494581591765rm_a_b @ S @ T ) @ subter4320051346911683477rm_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_4153759000330037794_a_b_b @ S @ Sigma ) @ ( subst_4153759000330037794_a_b_b @ T @ Sigma ) ) @ subter523971068842742411eq_a_b ) ) ).

% supteq_subst
thf(fact_20_supteq__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ subter523971068842742411eq_a_b ) ) ).

% supteq_subst
thf(fact_21_rrstep__basicI,axiom,
    ! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rrstep_a_b @ R ) ) ) ).

% rrstep_basicI
thf(fact_22_rtrancl__supteq,axiom,
    ( ( transi7742714808557438673rm_a_b @ subter523971068842742411eq_a_b )
    = subter523971068842742411eq_a_b ) ).

% rtrancl_supteq
thf(fact_23_subst__apply__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ T @ var_b_a )
      = T ) ).

% subst_apply_term_empty
thf(fact_24_rstep__ruleI,axiom,
    ! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rstep_a_b @ R ) ) ) ).

% rstep_ruleI
thf(fact_25_rrstep_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R ) )
     => ~ ! [L2: term_a_b,R3: term_a_b,Sigma2: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) )
           => ( ( A2
                = ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R ) ) ) ) ).

% rrstep.cases
thf(fact_26_rrstep_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R ) )
      = ( ? [L3: term_a_b,R4: term_a_b,Sigma3: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L3 @ Sigma3 ) )
            & ( A2
              = ( subst_7999470309526761004_a_b_b @ R4 @ Sigma3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R4 ) @ R ) ) ) ) ).

% rrstep.simps
thf(fact_27_rrstep_Ointros,axiom,
    ! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma ) ) @ ( rrstep_a_b @ R ) ) ) ).

% rrstep.intros
thf(fact_28_rrstep__subst,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
     => ~ ! [L2: term_a_b,R3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) )
               => ( T
                 != ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ).

% rrstep_subst
thf(fact_29_supteq__var__imp__eq,axiom,
    ! [X: b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X ) @ T ) @ subter523971068842742411eq_a_b )
      = ( T
        = ( var_b_a @ X ) ) ) ).

% supteq_var_imp_eq
thf(fact_30_term_Oinject_I1_J,axiom,
    ! [X1: b,Y1: b] :
      ( ( ( var_b_a @ X1 )
        = ( var_b_a @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% term.inject(1)
thf(fact_31_subst__term__eqI,axiom,
    ! [Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ! [T4: term_a_b] :
          ( ( subst_7999470309526761004_a_b_b @ T4 @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ T4 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_32_subst__term__eqI,axiom,
    ! [Sigma: produc357393685978478089rm_a_b > term_a_b,Tau: produc357393685978478089rm_a_b > term_a_b] :
      ( ! [T4: term_a7872195449889686814rm_a_b] :
          ( ( subst_4153759000330037794_a_b_b @ T4 @ Sigma )
          = ( subst_4153759000330037794_a_b_b @ T4 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_33_supteq__antisym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
       => ( S = T ) ) ) ).

% supteq_antisym
thf(fact_34_supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subter523971068842742411eq_a_b ) ) ) ).

% supteq_trans
thf(fact_35_eq__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( S = T )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% eq_supteq
thf(fact_36_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_37_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: term_a_b,Z: term_a_b] : ( Y2 = Z ) )
    = ( ^ [A3: term_a_b,B2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_38_subterm_Odual__order_Otrans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.trans
thf(fact_39_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_40_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_41_subterm_Odual__order_Orefl,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subter523971068842742411eq_a_b ) ).

% subterm.dual_order.refl
thf(fact_42_subterm_Oorder__antisym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
       => ( X = Y ) ) ) ).

% subterm.order_antisym
thf(fact_43_subterm_Oorder_Oantisym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_44_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: term_a_b,Z: term_a_b] : ( Y2 = Z ) )
    = ( ^ [X2: term_a_b,Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_45_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: term_a_b,Z: term_a_b] : ( Y2 = Z ) )
    = ( ^ [A3: term_a_b,B2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: produc4523971788519308903rm_a_b,P: produc4523971788519308903rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ A @ ( collec1344989105561874898rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: produc9001763001631046773rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ A @ ( collec4780413679787665248rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: produc3421996462152255079rm_a_b,P: produc3421996462152255079rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ A @ ( collec1520397187239403986rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
    ! [A: term_a_b,P: term_a_b > $o] :
      ( ( member_term_a_b @ A @ ( collect_term_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( collec1344989105561874898rm_a_b
        @ ^ [X2: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A4: set_b] :
      ( ( collect_b
        @ ^ [X2: b] : ( member_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b] :
      ( ( collec4780413679787665248rm_a_b
        @ ^ [X2: produc9001763001631046773rm_a_b] : ( member8888773126115863070rm_a_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b] :
      ( ( collec1520397187239403986rm_a_b
        @ ^ [X2: produc3421996462152255079rm_a_b] : ( member1375104729371992464rm_a_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A4: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_58_Collect__cong,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec99177395925924084rm_a_b @ P )
        = ( collec99177395925924084rm_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_59_Collect__cong,axiom,
    ! [P: term_a_b > $o,Q: term_a_b > $o] :
      ( ! [X3: term_a_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_term_a_b @ P )
        = ( collect_term_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_60_subterm_Oantisym__conv,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv
thf(fact_61_subterm_Oorder__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order_trans
thf(fact_62_subterm_Oorder_Otrans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.trans
thf(fact_63_subterm_Oorder__refl,axiom,
    ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subter523971068842742411eq_a_b ) ).

% subterm.order_refl
thf(fact_64_subterm_Oeq__refl,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( X = Y )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.eq_refl
thf(fact_65_supteq_Orefl,axiom,
    ! [T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subter523971068842742411eq_a_b ) ).

% supteq.refl
thf(fact_66_subst__apply__eq__Var,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b,X: b] :
      ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
        = ( var_b_a @ X ) )
     => ~ ! [Y4: b] :
            ( ( S
              = ( var_b_a @ Y4 ) )
           => ( ( Sigma @ Y4 )
             != ( var_b_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_67_subst__apply__eq__Var,axiom,
    ! [S: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b,X: b] :
      ( ( ( subst_4153759000330037794_a_b_b @ S @ Sigma )
        = ( var_b_a @ X ) )
     => ~ ! [Y4: produc357393685978478089rm_a_b] :
            ( ( S
              = ( var_Pr9194790515074578440_a_b_a @ Y4 ) )
           => ( ( Sigma @ Y4 )
             != ( var_b_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_68_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: b,Sigma: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ ( var_b_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_69_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: produc357393685978478089rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
      ( ( subst_4153759000330037794_a_b_b @ ( var_Pr9194790515074578440_a_b_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_70_Term_Oterm_Osimps_I9_J,axiom,
    ! [F1: a > a,F22: b > b,X1: b] :
      ( ( map_term_a_a_b_b @ F1 @ F22 @ ( var_b_a @ X1 ) )
      = ( var_b_a @ ( F22 @ X1 ) ) ) ).

% Term.term.simps(9)
thf(fact_71_r__into__rtrancl,axiom,
    ! [P2: produc9001763001631046773rm_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ P2 @ R2 )
     => ( member8888773126115863070rm_a_b @ P2 @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ).

% r_into_rtrancl
thf(fact_72_r__into__rtrancl,axiom,
    ! [P2: produc3421996462152255079rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ P2 @ R2 )
     => ( member1375104729371992464rm_a_b @ P2 @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ).

% r_into_rtrancl
thf(fact_73_r__into__rtrancl,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ R2 )
     => ( member8417600551952982416rm_a_b @ P2 @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).

% r_into_rtrancl
thf(fact_74_r__into__rtrancl,axiom,
    ! [P2: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ R2 )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% r_into_rtrancl
thf(fact_75_rtrancl__idemp,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) )
      = ( transi2615809358984392588rm_a_b @ R2 ) ) ).

% rtrancl_idemp
thf(fact_76_rtrancl__idemp,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) )
      = ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% rtrancl_idemp
thf(fact_77_old_Oprod_Oinject,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ A @ B )
        = ( produc1763473618796451543rm_a_b @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_78_old_Oprod_Oinject,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A5: set_term_a_b,B3: set_term_a_b] :
      ( ( ( produc8517751122659631533rm_a_b @ A @ B )
        = ( produc8517751122659631533rm_a_b @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_79_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
      ( ( ( produc3977138704197412567rm_a_b @ A @ B )
        = ( produc3977138704197412567rm_a_b @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_80_old_Oprod_Oinject,axiom,
    ! [A: term_a_b,B: term_a_b,A5: term_a_b,B3: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A5 @ B3 ) )
      = ( ( A = A5 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_81_prod_Oinject,axiom,
    ! [X1: produc357393685978478089rm_a_b,X22: produc357393685978478089rm_a_b,Y1: produc357393685978478089rm_a_b,Y22: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ X1 @ X22 )
        = ( produc1763473618796451543rm_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_82_prod_Oinject,axiom,
    ! [X1: set_term_a_b,X22: set_term_a_b,Y1: set_term_a_b,Y22: set_term_a_b] :
      ( ( ( produc8517751122659631533rm_a_b @ X1 @ X22 )
        = ( produc8517751122659631533rm_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_83_prod_Oinject,axiom,
    ! [X1: set_Pr4386577575007340137rm_a_b,X22: set_Pr4386577575007340137rm_a_b,Y1: set_Pr4386577575007340137rm_a_b,Y22: set_Pr4386577575007340137rm_a_b] :
      ( ( ( produc3977138704197412567rm_a_b @ X1 @ X22 )
        = ( produc3977138704197412567rm_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_84_prod_Oinject,axiom,
    ! [X1: term_a_b,X22: term_a_b,Y1: term_a_b,Y22: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X1 @ X22 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_85_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_86_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_87_converse__rtrancl__into__rtrancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_88_converse__rtrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_89_converse__rtrancl__induct,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,P: set_term_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( P @ B )
       => ( ! [Y4: set_term_a_b,Z3: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z3 @ B ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_90_converse__rtrancl__induct,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,P: set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( P @ B )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z3 @ B ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_91_converse__rtrancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( P @ B )
       => ( ! [Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z3 @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_92_converse__rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( P @ B )
       => ( ! [Y4: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_93_converse__rtranclE,axiom,
    ! [X: set_term_a_b,Z2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( X != Z2 )
       => ~ ! [Y4: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_94_converse__rtranclE,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( X != Z2 )
       => ~ ! [Y4: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_95_converse__rtranclE,axiom,
    ! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( X != Z2 )
       => ~ ! [Y4: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_96_converse__rtranclE,axiom,
    ! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( X != Z2 )
       => ~ ! [Y4: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_97_rtrancl__induct,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,P: set_term_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( P @ A )
       => ( ! [Y4: set_term_a_b,Z3: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y4 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
             => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_98_rtrancl__induct,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,P: set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( P @ A )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y4 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
             => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_99_rtrancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( P @ A )
       => ( ! [Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_100_rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( P @ A )
       => ( ! [Y4: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_101_rtrancl__trans,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,Z2: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_102_rtrancl__trans,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_103_rtrancl__trans,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_104_rtrancl__trans,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_105_rtranclE,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y4 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_106_rtranclE,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y4 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_107_rtranclE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_108_rtranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y4: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_109_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ R2 )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_110_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ R2 )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_111_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ R2 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_112_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_113_rtrancl__induct2,axiom,
    ! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b,P: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
              ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
             => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A6 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_114_rtrancl__induct2,axiom,
    ! [Ax: set_term_a_b,Ay: set_term_a_b,Bx: set_term_a_b,By: set_term_a_b,R2: set_Pr7061646551094443591rm_a_b,P: set_term_a_b > set_term_a_b > $o] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A6: set_term_a_b,B4: set_term_a_b,Aa: set_term_a_b,Ba: set_term_a_b] :
              ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
             => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A6 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_115_rtrancl__induct2,axiom,
    ! [Ax: set_Pr4386577575007340137rm_a_b,Ay: set_Pr4386577575007340137rm_a_b,Bx: set_Pr4386577575007340137rm_a_b,By: set_Pr4386577575007340137rm_a_b,R2: set_Pr4284312964791606471rm_a_b,P: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,Aa: set_Pr4386577575007340137rm_a_b,Ba: set_Pr4386577575007340137rm_a_b] :
              ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
             => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A6 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_116_rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A6: term_a_b,B4: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A6 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_117_converse__rtranclE2,axiom,
    ! [Xa: produc357393685978478089rm_a_b,Xb: produc357393685978478089rm_a_b,Za: produc357393685978478089rm_a_b,Zb: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
     => ( ( ( produc1763473618796451543rm_a_b @ Xa @ Xb )
         != ( produc1763473618796451543rm_a_b @ Za @ Zb ) )
       => ~ ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
              ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) ) @ R2 )
             => ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_118_converse__rtranclE2,axiom,
    ! [Xa: set_term_a_b,Xb: set_term_a_b,Za: set_term_a_b,Zb: set_term_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Xa @ Xb ) @ ( produc8517751122659631533rm_a_b @ Za @ Zb ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
     => ( ( ( produc8517751122659631533rm_a_b @ Xa @ Xb )
         != ( produc8517751122659631533rm_a_b @ Za @ Zb ) )
       => ~ ! [A6: set_term_a_b,B4: set_term_a_b] :
              ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Xa @ Xb ) @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) ) @ R2 )
             => ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( produc8517751122659631533rm_a_b @ Za @ Zb ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_119_converse__rtranclE2,axiom,
    ! [Xa: set_Pr4386577575007340137rm_a_b,Xb: set_Pr4386577575007340137rm_a_b,Za: set_Pr4386577575007340137rm_a_b,Zb: set_Pr4386577575007340137rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Xa @ Xb ) @ ( produc3977138704197412567rm_a_b @ Za @ Zb ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
     => ( ( ( produc3977138704197412567rm_a_b @ Xa @ Xb )
         != ( produc3977138704197412567rm_a_b @ Za @ Zb ) )
       => ~ ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
              ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Xa @ Xb ) @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) ) @ R2 )
             => ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( produc3977138704197412567rm_a_b @ Za @ Zb ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_120_converse__rtranclE2,axiom,
    ! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
         != ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
       => ~ ! [A6: term_a_b,B4: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) @ R2 )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_121_converse__rtrancl__induct2,axiom,
    ! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b,P: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
     => ( ( P @ Bx @ By )
       => ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
              ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
             => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A6 @ B4 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_122_converse__rtrancl__induct2,axiom,
    ! [Ax: set_term_a_b,Ay: set_term_a_b,Bx: set_term_a_b,By: set_term_a_b,R2: set_Pr7061646551094443591rm_a_b,P: set_term_a_b > set_term_a_b > $o] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
     => ( ( P @ Bx @ By )
       => ( ! [A6: set_term_a_b,B4: set_term_a_b,Aa: set_term_a_b,Ba: set_term_a_b] :
              ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) ) @ R2 )
             => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A6 @ B4 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_123_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Pr4386577575007340137rm_a_b,Ay: set_Pr4386577575007340137rm_a_b,Bx: set_Pr4386577575007340137rm_a_b,By: set_Pr4386577575007340137rm_a_b,R2: set_Pr4284312964791606471rm_a_b,P: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
     => ( ( P @ Bx @ By )
       => ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,Aa: set_Pr4386577575007340137rm_a_b,Ba: set_Pr4386577575007340137rm_a_b] :
              ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) ) @ R2 )
             => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A6 @ B4 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_124_converse__rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( P @ Bx @ By )
       => ( ! [A6: term_a_b,B4: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A6 @ B4 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_125_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4523971788519308903rm_a_b] :
      ~ ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
          ( Y
         != ( produc1763473618796451543rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9001763001631046773rm_a_b] :
      ~ ! [A6: set_term_a_b,B4: set_term_a_b] :
          ( Y
         != ( produc8517751122659631533rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_127_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3421996462152255079rm_a_b] :
      ~ ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
          ( Y
         != ( produc3977138704197412567rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_128_old_Oprod_Oexhaust,axiom,
    ! [Y: produc357393685978478089rm_a_b] :
      ~ ! [A6: term_a_b,B4: term_a_b] :
          ( Y
         != ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_129_surj__pair,axiom,
    ! [P2: produc4523971788519308903rm_a_b] :
    ? [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
      ( P2
      = ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_130_surj__pair,axiom,
    ! [P2: produc9001763001631046773rm_a_b] :
    ? [X3: set_term_a_b,Y4: set_term_a_b] :
      ( P2
      = ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_131_surj__pair,axiom,
    ! [P2: produc3421996462152255079rm_a_b] :
    ? [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
      ( P2
      = ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_132_surj__pair,axiom,
    ! [P2: produc357393685978478089rm_a_b] :
    ? [X3: term_a_b,Y4: term_a_b] :
      ( P2
      = ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_133_prod__cases,axiom,
    ! [P: produc4523971788519308903rm_a_b > $o,P2: produc4523971788519308903rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] : ( P @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_134_prod__cases,axiom,
    ! [P: produc9001763001631046773rm_a_b > $o,P2: produc9001763001631046773rm_a_b] :
      ( ! [A6: set_term_a_b,B4: set_term_a_b] : ( P @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_135_prod__cases,axiom,
    ! [P: produc3421996462152255079rm_a_b > $o,P2: produc3421996462152255079rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] : ( P @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_136_prod__cases,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_137_Pair__inject,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ A @ B )
        = ( produc1763473618796451543rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_138_Pair__inject,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A5: set_term_a_b,B3: set_term_a_b] :
      ( ( ( produc8517751122659631533rm_a_b @ A @ B )
        = ( produc8517751122659631533rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_139_Pair__inject,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
      ( ( ( produc3977138704197412567rm_a_b @ A @ B )
        = ( produc3977138704197412567rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_140_Pair__inject,axiom,
    ! [A: term_a_b,B: term_a_b,A5: term_a_b,B3: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A = A5 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_141_prod__cases3,axiom,
    ! [Y: produc4523971788519308903rm_a_b] :
      ~ ! [A6: produc357393685978478089rm_a_b,B4: term_a_b,C2: term_a_b] :
          ( Y
         != ( produc1763473618796451543rm_a_b @ A6 @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_142_prod__induct3,axiom,
    ! [P: produc4523971788519308903rm_a_b > $o,X: produc4523971788519308903rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: term_a_b,C2: term_a_b] : ( P @ ( produc1763473618796451543rm_a_b @ A6 @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_143_rtrancl_Ocases,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( A2 != A1 )
       => ~ ! [B4: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B4 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_144_rtrancl_Ocases,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( A2 != A1 )
       => ~ ! [B4: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B4 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_145_rtrancl_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( A2 != A1 )
       => ~ ! [B4: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_146_rtrancl_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( A2 != A1 )
       => ~ ! [B4: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_147_rtrancl_Osimps,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
      = ( ? [A3: set_term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = A3 ) )
        | ? [A3: set_term_a_b,B2: set_term_a_b,C3: set_term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_148_rtrancl_Osimps,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
      = ( ? [A3: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = A3 ) )
        | ? [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_149_rtrancl_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
      = ( ? [A3: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = A3 ) )
        | ? [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_150_rtrancl_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
      = ( ? [A3: term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = A3 ) )
        | ? [A3: term_a_b,B2: term_a_b,C3: term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_151_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ A ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_152_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ A ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_153_rtrancl_Ortrancl__refl,axiom,
    ! [A: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_154_rtrancl_Ortrancl__refl,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_155_srsteps__with__root__step__def,axiom,
    ( srstep7844470518422762656ep_a_b
    = ( ^ [F3: set_Pr4934435412358123699_a_nat,R5: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F3 @ ( rrstep_a_b @ R5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) ) ) ) ) ).

% srsteps_with_root_step_def
thf(fact_156_gterm__of__term__inv,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( ground5786051259813020157rm_a_b @ ( ground3594116404241818627rm_a_b @ T ) )
        = T ) ) ).

% gterm_of_term_inv
thf(fact_157_restrict__subst__domain__Var,axiom,
    ! [V: set_b] :
      ( ( restri22458263168500592in_b_a @ V @ var_b_a )
      = var_b_a ) ).

% restrict_subst_domain_Var
thf(fact_158_rename__subst__domain__range__Var__lhs,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( rename5388498879213883313_b_a_b @ var_b_a @ Sigma )
      = Sigma ) ).

% rename_subst_domain_range_Var_lhs
thf(fact_159_rename__subst__domain__range__Var__rhs,axiom,
    ! [Rho: b > term_a_b] :
      ( ( rename5388498879213883313_b_a_b @ Rho @ var_b_a )
      = var_b_a ) ).

% rename_subst_domain_range_Var_rhs
thf(fact_160_rstepI,axiom,
    ! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: term_a_b,C4: subterm_and_ctxt_a_b,Sigma: b > term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
     => ( ( S
          = ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
       => ( ( T
            = ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma ) ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) ) ) ) ) ).

% rstepI
thf(fact_161_ctxt__eq,axiom,
    ! [C4: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( ( subter2376574525758040790rm_a_b @ C4 @ S )
        = ( subter2376574525758040790rm_a_b @ C4 @ T ) )
      = ( S = T ) ) ).

% ctxt_eq
thf(fact_162_rtrancl__idemp__self__comp,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R6 ) @ ( transi2615809358984392588rm_a_b @ R6 ) )
      = ( transi2615809358984392588rm_a_b @ R6 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_163_rtrancl__idemp__self__comp,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = ( transi7742714808557438673rm_a_b @ R6 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_164_term__of__gterm__inv,axiom,
    ! [T: ground_gterm_a] :
      ( ( ground3594116404241818627rm_a_b @ ( ground5786051259813020157rm_a_b @ T ) )
      = T ) ).

% term_of_gterm_inv
thf(fact_165_ctxt__supteq,axiom,
    ! [S: term_a_b,C4: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( S
        = ( subter2376574525758040790rm_a_b @ C4 @ T ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% ctxt_supteq
thf(fact_166_rstep__ctxtI,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R ) ) ) ).

% rstep_ctxtI
thf(fact_167_r__comp__rtrancl__eq,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ R2 @ ( transi2615809358984392588rm_a_b @ R2 ) )
      = ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ R2 ) ) ).

% r_comp_rtrancl_eq
thf(fact_168_r__comp__rtrancl__eq,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ R2 ) )
      = ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 ) ) ).

% r_comp_rtrancl_eq
thf(fact_169_supteq__ctxt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
      = ( ? [C5: subterm_and_ctxt_a_b] :
            ( S
            = ( subter2376574525758040790rm_a_b @ C5 @ T ) ) ) ) ).

% supteq_ctxt_conv
thf(fact_170_ctxt__imp__supteq,axiom,
    ! [C4: subterm_and_ctxt_a_b,T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) @ T ) @ subter523971068842742411eq_a_b ) ).

% ctxt_imp_supteq
thf(fact_171_supteq__ctxtE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ~ ! [C6: subterm_and_ctxt_a_b] :
            ( S
           != ( subter2376574525758040790rm_a_b @ C6 @ T ) ) ) ).

% supteq_ctxtE
thf(fact_172_restrict__subst__domain__def,axiom,
    ( restri22458263168500592in_b_a
    = ( ^ [V2: set_b,Sigma3: b > term_a_b,X2: b] : ( if_term_a_b @ ( member_b @ X2 @ V2 ) @ ( Sigma3 @ X2 ) @ ( var_b_a @ X2 ) ) ) ) ).

% restrict_subst_domain_def
thf(fact_173_rstep__imp__C__s__r_H,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ~ ! [C6: subterm_and_ctxt_a_b,L2: term_a_b,R3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subter2376574525758040790rm_a_b @ C6 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) ) )
               => ( T
                 != ( subter2376574525758040790rm_a_b @ C6 @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ) ).

% rstep_imp_C_s_r'
thf(fact_174_rstep__imp__C__s__r,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ? [C6: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,L2: term_a_b,R3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
          & ( S
            = ( subter2376574525758040790rm_a_b @ C6 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) ) )
          & ( T
            = ( subter2376574525758040790rm_a_b @ C6 @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ).

% rstep_imp_C_s_r
thf(fact_175_rstep_Ointros,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R ) ) ) ).

% rstep.intros
thf(fact_176_rstep_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rstep_a_b @ R ) )
      = ( ? [S2: term_a_b,T5: term_a_b,C5: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C5 @ S2 ) )
            & ( A2
              = ( subter2376574525758040790rm_a_b @ C5 @ T5 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T5 ) @ ( rrstep_a_b @ R ) ) ) ) ) ).

% rstep.simps
thf(fact_177_rstep_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rstep_a_b @ R ) )
     => ~ ! [S3: term_a_b,T4: term_a_b,C6: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C6 @ S3 ) )
           => ( ( A2
                = ( subter2376574525758040790rm_a_b @ C6 @ T4 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T4 ) @ ( rrstep_a_b @ R ) ) ) ) ) ).

% rstep.cases
thf(fact_178_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_179_steps__map,axiom,
    ! [P: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F2: set_term_a_b > term_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr4386577575007340137rm_a_b,T: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b,S4: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
      ( ! [T4: set_term_a_b,U2: set_term_a_b,R7: set_Pr7151917219565317845rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R6 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R6 ) @ ( relcom3320975603336820193rm_a_b @ S4 @ ( transi6904761420441344775rm_a_b @ R6 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_180_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > set_term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr7151917219565317845rm_a_b,T: term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S4 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_181_steps__map,axiom,
    ! [P: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F2: set_term_a_b > set_term_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr7151917219565317845rm_a_b,T: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b,S4: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
      ( ! [T4: set_term_a_b,U2: set_term_a_b,R7: set_Pr7151917219565317845rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R6 ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R6 ) @ ( relcom3320975603336820193rm_a_b @ S4 @ ( transi6904761420441344775rm_a_b @ R6 ) ) ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S4 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_182_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > produc357393685978478089rm_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b,T: term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S4 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_183_steps__map,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F2: produc357393685978478089rm_a_b > term_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b,T: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
      ( ! [T4: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R7: set_Pr2972776593051762503rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R6 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R6 ) @ ( relcom6159724393391289352rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ R6 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_184_steps__map,axiom,
    ! [P: set_Pr4386577575007340137rm_a_b > $o,Q: set_Pr647378613824986951rm_a_b > $o,F2: set_Pr4386577575007340137rm_a_b > term_a_b,G: set_Pr647378613824986951rm_a_b > set_Pr4386577575007340137rm_a_b,T: set_Pr4386577575007340137rm_a_b,R6: set_Pr647378613824986951rm_a_b,S4: set_Pr647378613824986951rm_a_b,U: set_Pr4386577575007340137rm_a_b] :
      ( ! [T4: set_Pr4386577575007340137rm_a_b,U2: set_Pr4386577575007340137rm_a_b,R7: set_Pr647378613824986951rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ ( transi8812752159401800684rm_a_b @ R6 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ R6 ) @ ( relcom8520403917366345832rm_a_b @ S4 @ ( transi8812752159401800684rm_a_b @ R6 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_185_steps__map,axiom,
    ! [P: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F2: set_term_a_b > produc357393685978478089rm_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr2972776593051762503rm_a_b,T: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b,S4: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
      ( ! [T4: set_term_a_b,U2: set_term_a_b,R7: set_Pr7151917219565317845rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R6 ) )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R6 ) @ ( relcom3320975603336820193rm_a_b @ S4 @ ( transi6904761420441344775rm_a_b @ R6 ) ) ) )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S4 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_186_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > set_Pr4386577575007340137rm_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr647378613824986951rm_a_b,T: term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
               => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi8812752159401800684rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) )
               => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ ( G @ R6 ) ) @ ( relcom8520403917366345832rm_a_b @ ( G @ S4 ) @ ( transi8812752159401800684rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_187_steps__map,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F2: produc357393685978478089rm_a_b > set_term_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr7151917219565317845rm_a_b,T: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
      ( ! [T4: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R7: set_Pr2972776593051762503rm_a_b] :
          ( ( P @ T4 )
         => ( ( Q @ R7 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U2 ) @ R7 )
             => ( ( P @ U2 )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T4 ) @ ( F2 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R6 )
         => ( ( Q @ S4 )
           => ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R6 ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) )
              & ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R6 ) @ ( relcom6159724393391289352rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ R6 ) ) ) )
               => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S4 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R6 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_188_adapt__vars__ctxt,axiom,
    ! [C4: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_a2780309480206857766_a_b_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) )
      = ( subter2376574525758040790rm_a_b @ ( term_a1758706544955394649_a_b_b @ C4 ) @ ( term_a2780309480206857766_a_b_b @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_189_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4802440762858034464rm_a_b,R6: set_Pr6510905305361987970rm_a_b] :
      ( ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ S4 @ R6 ) ) @ S4 )
      = ( relcom4000041261975091669rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ R6 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_190_rtrancl__O__shift,axiom,
    ! [S4: set_Pr6510905305361987970rm_a_b,R6: set_Pr4802440762858034464rm_a_b] :
      ( ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ S4 @ R6 ) ) @ S4 )
      = ( relcom3777457926417599474rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ R6 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_191_rtrancl__O__shift,axiom,
    ! [S4: set_Pr2972776593051762503rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ S4 @ R6 ) ) @ S4 )
      = ( relcom6159724393391289352rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ R6 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_192_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) ) @ S4 )
      = ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_193_relcomp3__I,axiom,
    ! [T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,B5: set_Pr2972776593051762503rm_a_b,V3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ A4 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ B5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ V3 ) @ B5 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ V3 ) @ ( relcom6159724393391289352rm_a_b @ B5 @ ( relcom6159724393391289352rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_194_relcomp3__I,axiom,
    ! [T: set_term_a_b,U: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,S: set_term_a_b,B5: set_Pr7151917219565317845rm_a_b,V3: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ A4 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ T ) @ B5 )
       => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U @ V3 ) @ B5 )
         => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ V3 ) @ ( relcom3320975603336820193rm_a_b @ B5 @ ( relcom3320975603336820193rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_195_relcomp3__I,axiom,
    ! [T: set_Pr4386577575007340137rm_a_b,U: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,S: set_Pr4386577575007340137rm_a_b,B5: set_Pr647378613824986951rm_a_b,V3: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ A4 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ T ) @ B5 )
       => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U @ V3 ) @ B5 )
         => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ V3 ) @ ( relcom8520403917366345832rm_a_b @ B5 @ ( relcom8520403917366345832rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_196_relcomp3__I,axiom,
    ! [T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V3 ) @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V3 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_197_relcompEpair,axiom,
    ! [A: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
     => ~ ! [B4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ R2 )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_198_relcompEpair,axiom,
    ! [A: set_term_a_b,C: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( relcom3320975603336820193rm_a_b @ R2 @ S ) )
     => ~ ! [B4: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B4 ) @ R2 )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_199_relcompEpair,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( relcom8520403917366345832rm_a_b @ R2 @ S ) )
     => ~ ! [B4: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B4 ) @ R2 )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_200_relcompEpair,axiom,
    ! [A: term_a_b,C: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
     => ~ ! [B4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_201_relcompE,axiom,
    ! [Xz: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ Xz @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
     => ~ ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
            ( ( Xz
              = ( produc1763473618796451543rm_a_b @ X3 @ Z3 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_202_relcompE,axiom,
    ! [Xz: produc9001763001631046773rm_a_b,R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ Xz @ ( relcom3320975603336820193rm_a_b @ R2 @ S ) )
     => ~ ! [X3: set_term_a_b,Y4: set_term_a_b,Z3: set_term_a_b] :
            ( ( Xz
              = ( produc8517751122659631533rm_a_b @ X3 @ Z3 ) )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_203_relcompE,axiom,
    ! [Xz: produc3421996462152255079rm_a_b,R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ Xz @ ( relcom8520403917366345832rm_a_b @ R2 @ S ) )
     => ~ ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
            ( ( Xz
              = ( produc3977138704197412567rm_a_b @ X3 @ Z3 ) )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ R2 )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_204_relcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
     => ~ ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_205_relcomp_OrelcompI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ S )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_206_relcomp_OrelcompI,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ S )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( relcom3320975603336820193rm_a_b @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_207_relcomp_OrelcompI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ S )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( relcom8520403917366345832rm_a_b @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_208_relcomp_OrelcompI,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_209_relcomp_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
      = ( ? [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_210_relcomp_Osimps,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( relcom3320975603336820193rm_a_b @ R2 @ S ) )
      = ( ? [A3: set_term_a_b,B2: set_term_a_b,C3: set_term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ R2 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B2 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_211_relcomp_Osimps,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( relcom8520403917366345832rm_a_b @ R2 @ S ) )
      = ( ? [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ R2 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B2 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_212_relcomp_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
      = ( ? [A3: term_a_b,B2: term_a_b,C3: term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_213_relcomp_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
     => ~ ! [B4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B4 ) @ R2 )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ A2 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_214_relcomp_Ocases,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( relcom3320975603336820193rm_a_b @ R2 @ S ) )
     => ~ ! [B4: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B4 ) @ R2 )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ A2 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_215_relcomp_Ocases,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( relcom8520403917366345832rm_a_b @ R2 @ S ) )
     => ~ ! [B4: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B4 ) @ R2 )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ A2 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_216_relcomp_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
     => ~ ! [B4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B4 ) @ R2 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A2 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_217_O__assoc,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S4 ) @ T6 )
      = ( relcom370159955682700863rm_a_b @ R6 @ ( relcom370159955682700863rm_a_b @ S4 @ T6 ) ) ) ).

% O_assoc
thf(fact_218_meetI,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra7998755932702363853rm_a_b @ A4 ) ) ) ) ).

% meetI
thf(fact_219_meetI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra818859418854869414rm_a_b @ A4 ) ) ) ) ).

% meetI
thf(fact_220_meetI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) ) ) ) ).

% meetI
thf(fact_221_meetI,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) ) ) ) ).

% meetI
thf(fact_222_map__gterm__of__term,axiom,
    ! [T: term_a_b,F2: a > a,G: b > b] :
      ( ( term_ground_a_b @ T )
     => ( ( ground3594116404241818627rm_a_b @ ( map_term_a_a_b_b @ F2 @ G @ T ) )
        = ( ground_map_gterm_a_a @ F2 @ ( ground3594116404241818627rm_a_b @ T ) ) ) ) ).

% map_gterm_of_term
thf(fact_223_meetD,axiom,
    ! [B: set_term_a_b,C: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra7998755932702363853rm_a_b @ A4 ) )
     => ? [A6: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
          & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ C ) @ ( transi6904761420441344775rm_a_b @ A4 ) ) ) ) ).

% meetD
thf(fact_224_meetD,axiom,
    ! [B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra818859418854869414rm_a_b @ A4 ) )
     => ? [A6: set_Pr4386577575007340137rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
          & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ C ) @ ( transi8812752159401800684rm_a_b @ A4 ) ) ) ) ).

% meetD
thf(fact_225_meetD,axiom,
    ! [B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) )
     => ? [A6: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).

% meetD
thf(fact_226_meetD,axiom,
    ! [B: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
     => ? [A6: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% meetD
thf(fact_227_meetE,axiom,
    ! [B: set_term_a_b,C: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra7998755932702363853rm_a_b @ A4 ) )
     => ~ ! [A6: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ C ) @ ( transi6904761420441344775rm_a_b @ A4 ) ) ) ) ).

% meetE
thf(fact_228_meetE,axiom,
    ! [B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra818859418854869414rm_a_b @ A4 ) )
     => ~ ! [A6: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ C ) @ ( transi8812752159401800684rm_a_b @ A4 ) ) ) ) ).

% meetE
thf(fact_229_meetE,axiom,
    ! [B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) )
     => ~ ! [A6: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).

% meetE
thf(fact_230_meetE,axiom,
    ! [B: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
     => ~ ! [A6: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% meetE
thf(fact_231_same__fstI,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b,Y5: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R6: produc357393685978478089rm_a_b > set_Pr2972776593051762503rm_a_b] :
      ( ( P @ X )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) )
       => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y5 ) @ ( produc1763473618796451543rm_a_b @ X @ Y ) ) @ ( same_f7764617898092871340rm_a_b @ P @ R6 ) ) ) ) ).

% same_fstI
thf(fact_232_same__fstI,axiom,
    ! [P: set_term_a_b > $o,X: set_term_a_b,Y5: set_term_a_b,Y: set_term_a_b,R6: set_term_a_b > set_Pr7151917219565317845rm_a_b] :
      ( ( P @ X )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) )
       => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y5 ) @ ( produc8517751122659631533rm_a_b @ X @ Y ) ) @ ( same_f2062142611643789826rm_a_b @ P @ R6 ) ) ) ) ).

% same_fstI
thf(fact_233_same__fstI,axiom,
    ! [P: set_Pr4386577575007340137rm_a_b > $o,X: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b > set_Pr647378613824986951rm_a_b] :
      ( ( P @ X )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) )
       => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y5 ) @ ( produc3977138704197412567rm_a_b @ X @ Y ) ) @ ( same_f1934838165808735916rm_a_b @ P @ R6 ) ) ) ) ).

% same_fstI
thf(fact_234_same__fstI,axiom,
    ! [P: term_a_b > $o,X: term_a_b,Y5: term_a_b,Y: term_a_b,R6: term_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( P @ X )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y5 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P @ R6 ) ) ) ) ).

% same_fstI
thf(fact_235_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C4: subterm_and_ctxt_a_b,T: term_a_b,Mu: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) @ Mu )
      = ( subter2376574525758040790rm_a_b @ ( subter4086027020339385547_a_b_b @ C4 @ Mu ) @ ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_236_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C4: subter6147397892252016653rm_a_b,T: term_a7872195449889686814rm_a_b,Mu: produc357393685978478089rm_a_b > term_a_b] :
      ( ( subst_4153759000330037794_a_b_b @ ( subter1074907186990661472rm_a_b @ C4 @ T ) @ Mu )
      = ( subter2376574525758040790rm_a_b @ ( subter7224274371358213697_a_b_b @ C4 @ Mu ) @ ( subst_4153759000330037794_a_b_b @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_237_ground__ctxt__apply,axiom,
    ! [C4: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_ground_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) )
      = ( ( term_ground_ctxt_a_b @ C4 )
        & ( term_ground_a_b @ T ) ) ) ).

% ground_ctxt_apply
thf(fact_238_restrict__subst__domain__empty,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( restri22458263168500592in_b_a @ bot_bot_set_b @ Sigma )
      = var_b_a ) ).

% restrict_subst_domain_empty
thf(fact_239_normalizability__I_H,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra4462713708631513506rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( abstra4462713708631513506rm_a_b @ A4 ) ) ) ) ).

% normalizability_I'
thf(fact_240_normalizability__I_H,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra845589504903352593rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( abstra845589504903352593rm_a_b @ A4 ) ) ) ) ).

% normalizability_I'
thf(fact_241_normalizability__I_H,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra1496011390211842225rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( abstra1496011390211842225rm_a_b @ A4 ) ) ) ) ).

% normalizability_I'
thf(fact_242_normalizability__I_H,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).

% normalizability_I'
thf(fact_243_relcomp__empty2,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ R6 @ bot_bo197521221353338581rm_a_b )
      = bot_bo197521221353338581rm_a_b ) ).

% relcomp_empty2
thf(fact_244_relcomp__empty1,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ bot_bo197521221353338581rm_a_b @ R6 )
      = bot_bo197521221353338581rm_a_b ) ).

% relcomp_empty1
thf(fact_245_map__term__of__gterm,axiom,
    ! [F2: a > a,G: b > b,T: ground_gterm_a] :
      ( ( map_term_a_a_b_b @ F2 @ G @ ( ground5786051259813020157rm_a_b @ T ) )
      = ( ground5786051259813020157rm_a_b @ ( ground_map_gterm_a_a @ F2 @ T ) ) ) ).

% map_term_of_gterm
thf(fact_246_UNF__onI,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ! [A6: produc4523971788519308903rm_a_b,B4: produc4523971788519308903rm_a_b,C2: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ A6 @ A4 )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ B4 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
           => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ C2 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra3562400838740635925rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_247_UNF__onI,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b] :
      ( ! [A6: b,B4: b,C2: b] :
          ( ( member_b @ A6 @ A4 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B4 ) @ ( abstra380364156431640999lity_b @ R2 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ C2 ) @ ( abstra380364156431640999lity_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstract_UNF_on_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_248_UNF__onI,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ! [A6: produc9001763001631046773rm_a_b,B4: produc9001763001631046773rm_a_b,C2: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ A6 @ A4 )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ B4 ) @ ( abstra1586503138784356125rm_a_b @ R2 ) )
           => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ C2 ) @ ( abstra1586503138784356125rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra556754771559176739rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_249_UNF__onI,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ! [A6: produc3421996462152255079rm_a_b,B4: produc3421996462152255079rm_a_b,C2: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ A6 @ A4 )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ B4 ) @ ( abstra6891539728454233359rm_a_b @ R2 ) )
           => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ C2 ) @ ( abstra6891539728454233359rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra3798689227379738901rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_250_UNF__onI,axiom,
    ! [A4: set_set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ! [A6: set_term_a_b,B4: set_term_a_b,C2: set_term_a_b] :
          ( ( member_set_term_a_b @ A6 @ A4 )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ C2 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra2699617484662362908rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_251_UNF__onI,axiom,
    ! [A4: set_se1467346398288745161rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ A6 @ A4 )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ C2 ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra80904564981652503rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_252_UNF__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A4 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra1303798319023342519rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_253_UNF__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A6 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
             => ( B4 = C2 ) ) ) )
     => ( abstra3619855735679963878rm_a_b @ R2 @ A4 ) ) ).

% UNF_onI
thf(fact_254_WN__onI,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ! [A6: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ A6 @ A4 )
         => ? [B6: produc4523971788519308903rm_a_b] : ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ B6 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) )
     => ( abstra1431869762417195701rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_255_WN__onI,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b] :
      ( ! [A6: b] :
          ( ( member_b @ A6 @ A4 )
         => ? [B6: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B6 ) @ ( abstra380364156431640999lity_b @ R2 ) ) )
     => ( abstract_WN_on_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_256_WN__onI,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ! [A6: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ A6 @ A4 )
         => ? [B6: produc9001763001631046773rm_a_b] : ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ B6 ) @ ( abstra1586503138784356125rm_a_b @ R2 ) ) )
     => ( abstra8897730914534552515rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_257_WN__onI,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ! [A6: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ A6 @ A4 )
         => ? [B6: produc3421996462152255079rm_a_b] : ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ B6 ) @ ( abstra6891539728454233359rm_a_b @ R2 ) ) )
     => ( abstra7570851922252731061rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_258_WN__onI,axiom,
    ! [A4: set_set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ! [A6: set_term_a_b] :
          ( ( member_set_term_a_b @ A6 @ A4 )
         => ? [B6: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B6 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) ) )
     => ( abstra2544905187720589692rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_259_WN__onI,axiom,
    ! [A4: set_se1467346398288745161rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ A6 @ A4 )
         => ? [B6: set_Pr4386577575007340137rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B6 ) @ ( abstra845589504903352593rm_a_b @ R2 ) ) )
     => ( abstra2171967793693783479rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_260_WN__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A4 )
         => ? [B6: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B6 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) )
     => ( abstra2721873542170697047rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_261_WN__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b] :
          ( ( member_term_a_b @ A6 @ A4 )
         => ? [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
     => ( abstra4644735066934440262rm_a_b @ R2 @ A4 ) ) ).

% WN_onI
thf(fact_262_normalizability__I,axiom,
    ! [A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( transi5332651807426350186rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ B @ ( abstra1925539948312754931rm_a_b @ A4 ) )
       => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra7490947582319486223rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_263_normalizability__I,axiom,
    ! [A: b,B: b,A4: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ A4 ) )
     => ( ( member_b @ B @ ( abstract_NF_b @ A4 ) )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_264_normalizability__I,axiom,
    ! [A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( transi2165682467528973816rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ B @ ( abstra5673374384174597889rm_a_b @ A4 ) )
       => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( abstra1586503138784356125rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_265_normalizability__I,axiom,
    ! [A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( transi5504100148209286250rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ B @ ( abstra8119681912743043827rm_a_b @ A4 ) )
       => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( abstra6891539728454233359rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_266_normalizability__I,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( ( member_set_term_a_b @ B @ ( abstra5309567699948295358rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra4462713708631513506rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_267_normalizability__I,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( ( member8823642877475723282rm_a_b @ B @ ( abstra6901259896266686709rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra845589504903352593rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_268_normalizability__I,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_269_normalizability__I,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).

% normalizability_I
thf(fact_270_UNF__onD,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
      ( ( abstra3562400838740635925rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_271_UNF__onD,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b,B: b,C: b] :
      ( ( abstract_UNF_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ R2 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ ( abstra380364156431640999lity_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_272_UNF__onD,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,C: produc9001763001631046773rm_a_b] :
      ( ( abstra556754771559176739rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( abstra1586503138784356125rm_a_b @ R2 ) )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ C ) @ ( abstra1586503138784356125rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_273_UNF__onD,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,C: produc3421996462152255079rm_a_b] :
      ( ( abstra3798689227379738901rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( abstra6891539728454233359rm_a_b @ R2 ) )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ C ) @ ( abstra6891539728454233359rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_274_UNF__onD,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C: set_term_a_b] :
      ( ( abstra2699617484662362908rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_275_UNF__onD,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra80904564981652503rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_276_UNF__onD,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_277_UNF__onD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_278_UNF__onE,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
      ( ( abstra3562400838740635925rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
           => ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_279_UNF__onE,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b,B: b,C: b] :
      ( ( abstract_UNF_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ R2 ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ ( abstra380364156431640999lity_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_280_UNF__onE,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,C: produc9001763001631046773rm_a_b] :
      ( ( abstra556754771559176739rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( abstra1586503138784356125rm_a_b @ R2 ) )
           => ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ C ) @ ( abstra1586503138784356125rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_281_UNF__onE,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,C: produc3421996462152255079rm_a_b] :
      ( ( abstra3798689227379738901rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( abstra6891539728454233359rm_a_b @ R2 ) )
           => ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ C ) @ ( abstra6891539728454233359rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_282_UNF__onE,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C: set_term_a_b] :
      ( ( abstra2699617484662362908rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( abstra4462713708631513506rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_283_UNF__onE,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra80904564981652503rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( abstra845589504903352593rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_284_UNF__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_285_UNF__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ( B != C )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_286_UNF__on__def,axiom,
    ( abstra2699617484662362908rm_a_b
    = ( ^ [R4: set_Pr7151917219565317845rm_a_b,A7: set_set_term_a_b] :
        ! [X2: set_term_a_b] :
          ( ( member_set_term_a_b @ X2 @ A7 )
         => ! [B2: set_term_a_b,C3: set_term_a_b] :
              ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ B2 ) @ ( abstra4462713708631513506rm_a_b @ R4 ) )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ C3 ) @ ( abstra4462713708631513506rm_a_b @ R4 ) ) )
             => ( B2 = C3 ) ) ) ) ) ).

% UNF_on_def
thf(fact_287_UNF__on__def,axiom,
    ( abstra80904564981652503rm_a_b
    = ( ^ [R4: set_Pr647378613824986951rm_a_b,A7: set_se1467346398288745161rm_a_b] :
        ! [X2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ X2 @ A7 )
         => ! [B2: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
              ( ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X2 @ B2 ) @ ( abstra845589504903352593rm_a_b @ R4 ) )
                & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X2 @ C3 ) @ ( abstra845589504903352593rm_a_b @ R4 ) ) )
             => ( B2 = C3 ) ) ) ) ) ).

% UNF_on_def
thf(fact_288_UNF__on__def,axiom,
    ( abstra1303798319023342519rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
        ! [X2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X2 @ A7 )
         => ! [B2: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
              ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ C3 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) ) )
             => ( B2 = C3 ) ) ) ) ) ).

% UNF_on_def
thf(fact_289_UNF__on__def,axiom,
    ( abstra3619855735679963878rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [B2: term_a_b,C3: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
             => ( B2 = C3 ) ) ) ) ) ).

% UNF_on_def
thf(fact_290_ground__ctxt__substI,axiom,
    ! [C4: subter7314335377364111083rm_a_b,Sigma: produc4523971788519308903rm_a_b > term_a_b] :
      ( ! [X3: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ X3 @ ( subter7904028378834178391rm_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter5283163014667465571_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_291_ground__ctxt__substI,axiom,
    ! [C4: subter2881564190832396198rm_a_b,Sigma: term_a_b > term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ ( subter7619234649594586916rm_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter6362074857403937054_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_292_ground__ctxt__substI,axiom,
    ! [C4: subterm_and_ctxt_a_b,Sigma: b > term_a_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( subter6041326469635110383xt_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter4086027020339385547_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_293_ground__ctxt__substI,axiom,
    ! [C4: subter44646029943400569rm_a_b,Sigma: produc9001763001631046773rm_a_b > term_a_b] :
      ( ! [X3: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ X3 @ ( subter625065731738019685rm_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter7509023468883661525_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_294_ground__ctxt__substI,axiom,
    ! [C4: subter5071808320638160619rm_a_b,Sigma: produc3421996462152255079rm_a_b > term_a_b] :
      ( ! [X3: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ X3 @ ( subter2009212160888183127rm_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter8166909670413405027_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_295_ground__ctxt__substI,axiom,
    ! [C4: subter6147397892252016653rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ ( subter7674038538283327737rm_a_b @ C4 ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_ctxt_a_b @ ( subter7224274371358213697_a_b_b @ C4 @ Sigma ) ) ) ).

% ground_ctxt_substI
thf(fact_296_range__vars__Var,axiom,
    ( ( range_vars_b_a @ var_b_a )
    = bot_bot_set_b ) ).

% range_vars_Var
thf(fact_297_WN__onD,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b] :
      ( ( abstra1431869762417195701rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ? [B4: produc4523971788519308903rm_a_b] : ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B4 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_298_WN__onD,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b] :
      ( ( abstract_WN_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ? [B4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B4 ) @ ( abstra380364156431640999lity_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_299_WN__onD,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b] :
      ( ( abstra8897730914534552515rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ? [B4: produc9001763001631046773rm_a_b] : ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B4 ) @ ( abstra1586503138784356125rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_300_WN__onD,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b] :
      ( ( abstra7570851922252731061rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ? [B4: produc3421996462152255079rm_a_b] : ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B4 ) @ ( abstra6891539728454233359rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_301_WN__onD,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b] :
      ( ( abstra2544905187720589692rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ? [B4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B4 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_302_WN__onD,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2171967793693783479rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ? [B4: set_Pr4386577575007340137rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B4 ) @ ( abstra845589504903352593rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_303_WN__onD,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ? [B4: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_304_WN__onD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b] :
      ( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).

% WN_onD
thf(fact_305_NF__I,axiom,
    ! [A: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b] :
      ( ! [B4: produc4523971788519308903rm_a_b] :
          ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B4 ) @ A4 )
     => ( member8417600551952982416rm_a_b @ A @ ( abstra1925539948312754931rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_306_NF__I,axiom,
    ! [A: b,A4: set_Product_prod_b_b] :
      ( ! [B4: b] :
          ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B4 ) @ A4 )
     => ( member_b @ A @ ( abstract_NF_b @ A4 ) ) ) ).

% NF_I
thf(fact_307_NF__I,axiom,
    ! [A: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b] :
      ( ! [B4: produc9001763001631046773rm_a_b] :
          ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B4 ) @ A4 )
     => ( member8888773126115863070rm_a_b @ A @ ( abstra5673374384174597889rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_308_NF__I,axiom,
    ! [A: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b] :
      ( ! [B4: produc3421996462152255079rm_a_b] :
          ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B4 ) @ A4 )
     => ( member1375104729371992464rm_a_b @ A @ ( abstra8119681912743043827rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_309_NF__I,axiom,
    ! [A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ! [B4: set_term_a_b] :
          ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B4 ) @ A4 )
     => ( member_set_term_a_b @ A @ ( abstra5309567699948295358rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_310_NF__I,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ! [B4: set_Pr4386577575007340137rm_a_b] :
          ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B4 ) @ A4 )
     => ( member8823642877475723282rm_a_b @ A @ ( abstra6901259896266686709rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_311_NF__I,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ! [B4: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ A4 )
     => ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_312_NF__I,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ! [B4: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ A4 )
     => ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) ) ) ).

% NF_I
thf(fact_313_NF__E,axiom,
    ! [A: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b,B: produc4523971788519308903rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ A @ ( abstra1925539948312754931rm_a_b @ A4 ) )
     => ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_314_NF__E,axiom,
    ! [A: b,A4: set_Product_prod_b_b,B: b] :
      ( ( member_b @ A @ ( abstract_NF_b @ A4 ) )
     => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_315_NF__E,axiom,
    ! [A: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b,B: produc9001763001631046773rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ A @ ( abstra5673374384174597889rm_a_b @ A4 ) )
     => ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_316_NF__E,axiom,
    ! [A: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b,B: produc3421996462152255079rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ A @ ( abstra8119681912743043827rm_a_b @ A4 ) )
     => ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_317_NF__E,axiom,
    ! [A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,B: set_term_a_b] :
      ( ( member_set_term_a_b @ A @ ( abstra5309567699948295358rm_a_b @ A4 ) )
     => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_318_NF__E,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ A @ ( abstra6901259896266686709rm_a_b @ A4 ) )
     => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_319_NF__E,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
     => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_320_NF__E,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A4 ) ) ).

% NF_E
thf(fact_321_NF__no__step,axiom,
    ! [A: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ A @ ( abstra1925539948312754931rm_a_b @ A4 ) )
     => ! [B6: produc4523971788519308903rm_a_b] :
          ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_322_NF__no__step,axiom,
    ! [A: b,A4: set_Product_prod_b_b] :
      ( ( member_b @ A @ ( abstract_NF_b @ A4 ) )
     => ! [B6: b] :
          ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_323_NF__no__step,axiom,
    ! [A: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ A @ ( abstra5673374384174597889rm_a_b @ A4 ) )
     => ! [B6: produc9001763001631046773rm_a_b] :
          ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_324_NF__no__step,axiom,
    ! [A: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ A @ ( abstra8119681912743043827rm_a_b @ A4 ) )
     => ! [B6: produc3421996462152255079rm_a_b] :
          ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_325_NF__no__step,axiom,
    ! [A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member_set_term_a_b @ A @ ( abstra5309567699948295358rm_a_b @ A4 ) )
     => ! [B6: set_term_a_b] :
          ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_326_NF__no__step,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ A @ ( abstra6901259896266686709rm_a_b @ A4 ) )
     => ! [B6: set_Pr4386577575007340137rm_a_b] :
          ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_327_NF__no__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
     => ! [B6: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_328_NF__no__step,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
     => ! [B6: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B6 ) @ A4 ) ) ).

% NF_no_step
thf(fact_329_NF__iff__no__step,axiom,
    ! [A: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ A @ ( abstra1925539948312754931rm_a_b @ A4 ) )
      = ( ! [B2: produc4523971788519308903rm_a_b] :
            ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_330_NF__iff__no__step,axiom,
    ! [A: b,A4: set_Product_prod_b_b] :
      ( ( member_b @ A @ ( abstract_NF_b @ A4 ) )
      = ( ! [B2: b] :
            ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_331_NF__iff__no__step,axiom,
    ! [A: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ A @ ( abstra5673374384174597889rm_a_b @ A4 ) )
      = ( ! [B2: produc9001763001631046773rm_a_b] :
            ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_332_NF__iff__no__step,axiom,
    ! [A: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ A @ ( abstra8119681912743043827rm_a_b @ A4 ) )
      = ( ! [B2: produc3421996462152255079rm_a_b] :
            ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_333_NF__iff__no__step,axiom,
    ! [A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member_set_term_a_b @ A @ ( abstra5309567699948295358rm_a_b @ A4 ) )
      = ( ! [B2: set_term_a_b] :
            ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_334_NF__iff__no__step,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ A @ ( abstra6901259896266686709rm_a_b @ A4 ) )
      = ( ! [B2: set_Pr4386577575007340137rm_a_b] :
            ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_335_NF__iff__no__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
      = ( ! [B2: produc357393685978478089rm_a_b] :
            ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_336_NF__iff__no__step,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
      = ( ! [B2: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B2 ) @ A4 ) ) ) ).

% NF_iff_no_step
thf(fact_337_ctxt__subst__eq,axiom,
    ! [C4: subter6147397892252016653rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b,Tau: produc357393685978478089rm_a_b > term_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ ( subter7674038538283327737rm_a_b @ C4 ) )
         => ( ( Sigma @ X3 )
            = ( Tau @ X3 ) ) )
     => ( ( subter7224274371358213697_a_b_b @ C4 @ Sigma )
        = ( subter7224274371358213697_a_b_b @ C4 @ Tau ) ) ) ).

% ctxt_subst_eq
thf(fact_338_mem__range__varsI,axiom,
    ! [Sigma: b > term_a_b,X: b,Y: b] :
      ( ( ( Sigma @ X )
        = ( var_b_a @ Y ) )
     => ( ( X != Y )
       => ( member_b @ Y @ ( range_vars_b_a @ Sigma ) ) ) ) ).

% mem_range_varsI
thf(fact_339_NF__not__suc,axiom,
    ! [X: produc4523971788519308903rm_a_b,Y: produc4523971788519308903rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ X @ ( abstra1925539948312754931rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_340_NF__not__suc,axiom,
    ! [X: b,Y: b,R2: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ R2 ) )
     => ( ( member_b @ X @ ( abstract_NF_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_341_NF__not__suc,axiom,
    ! [X: produc9001763001631046773rm_a_b,Y: produc9001763001631046773rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ X @ ( abstra5673374384174597889rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_342_NF__not__suc,axiom,
    ! [X: produc3421996462152255079rm_a_b,Y: produc3421996462152255079rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ X @ ( abstra8119681912743043827rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_343_NF__not__suc,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( member_set_term_a_b @ X @ ( abstra5309567699948295358rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_344_NF__not__suc,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( member8823642877475723282rm_a_b @ X @ ( abstra6901259896266686709rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_345_NF__not__suc,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_346_NF__not__suc,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_347_normalizability__E,axiom,
    ! [A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra7490947582319486223rm_a_b @ A4 ) )
     => ~ ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( transi5332651807426350186rm_a_b @ A4 ) )
         => ~ ( member8417600551952982416rm_a_b @ B @ ( abstra1925539948312754931rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_348_normalizability__E,axiom,
    ! [A: b,B: b,A4: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ A4 ) )
     => ~ ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ A4 ) )
         => ~ ( member_b @ B @ ( abstract_NF_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_349_normalizability__E,axiom,
    ! [A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( abstra1586503138784356125rm_a_b @ A4 ) )
     => ~ ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( transi2165682467528973816rm_a_b @ A4 ) )
         => ~ ( member8888773126115863070rm_a_b @ B @ ( abstra5673374384174597889rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_350_normalizability__E,axiom,
    ! [A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( abstra6891539728454233359rm_a_b @ A4 ) )
     => ~ ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( transi5504100148209286250rm_a_b @ A4 ) )
         => ~ ( member1375104729371992464rm_a_b @ B @ ( abstra8119681912743043827rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_351_normalizability__E,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra4462713708631513506rm_a_b @ A4 ) )
     => ~ ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
         => ~ ( member_set_term_a_b @ B @ ( abstra5309567699948295358rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_352_normalizability__E,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra845589504903352593rm_a_b @ A4 ) )
     => ~ ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
         => ~ ( member8823642877475723282rm_a_b @ B @ ( abstra6901259896266686709rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_353_normalizability__E,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A4 ) )
     => ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
         => ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_354_normalizability__E,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
         => ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) ) ) ) ).

% normalizability_E
thf(fact_355_WN__on__def,axiom,
    ( abstra2544905187720589692rm_a_b
    = ( ^ [R4: set_Pr7151917219565317845rm_a_b,A7: set_set_term_a_b] :
        ! [X2: set_term_a_b] :
          ( ( member_set_term_a_b @ X2 @ A7 )
         => ? [B2: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ B2 ) @ ( abstra4462713708631513506rm_a_b @ R4 ) ) ) ) ) ).

% WN_on_def
thf(fact_356_WN__on__def,axiom,
    ( abstra2171967793693783479rm_a_b
    = ( ^ [R4: set_Pr647378613824986951rm_a_b,A7: set_se1467346398288745161rm_a_b] :
        ! [X2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ X2 @ A7 )
         => ? [B2: set_Pr4386577575007340137rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X2 @ B2 ) @ ( abstra845589504903352593rm_a_b @ R4 ) ) ) ) ) ).

% WN_on_def
thf(fact_357_WN__on__def,axiom,
    ( abstra2721873542170697047rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
        ! [X2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X2 @ A7 )
         => ? [B2: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) ) ) ) ) ).

% WN_on_def
thf(fact_358_WN__on__def,axiom,
    ( abstra4644735066934440262rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ? [B2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) ) ) ) ).

% WN_on_def
thf(fact_359_WN__onE,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b] :
      ( ( abstra1431869762417195701rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ~ ! [B4: produc4523971788519308903rm_a_b] :
              ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B4 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_360_WN__onE,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b] :
      ( ( abstract_WN_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ~ ! [B4: b] :
              ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B4 ) @ ( abstra380364156431640999lity_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_361_WN__onE,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b] :
      ( ( abstra8897730914534552515rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ~ ! [B4: produc9001763001631046773rm_a_b] :
              ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B4 ) @ ( abstra1586503138784356125rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_362_WN__onE,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b] :
      ( ( abstra7570851922252731061rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ~ ! [B4: produc3421996462152255079rm_a_b] :
              ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B4 ) @ ( abstra6891539728454233359rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_363_WN__onE,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b] :
      ( ( abstra2544905187720589692rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ~ ! [B4: set_term_a_b] :
              ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B4 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_364_WN__onE,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2171967793693783479rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ~ ! [B4: set_Pr4386577575007340137rm_a_b] :
              ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B4 ) @ ( abstra845589504903352593rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_365_WN__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ~ ! [B4: produc357393685978478089rm_a_b] :
              ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_366_WN__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b] :
      ( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ~ ! [B4: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).

% WN_onE
thf(fact_367_NF__join__imp__reach,axiom,
    ! [Y: produc4523971788519308903rm_a_b,A4: set_Pr6848273517546626247rm_a_b,X: produc4523971788519308903rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ Y @ ( abstra1925539948312754931rm_a_b @ A4 ) )
     => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( abstra4037915826850753057rm_a_b @ A4 ) )
       => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( transi5332651807426350186rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_368_NF__join__imp__reach,axiom,
    ! [Y: b,A4: set_Product_prod_b_b,X: b] :
      ( ( member_b @ Y @ ( abstract_NF_b @ A4 ) )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( abstract_join_b @ A4 ) )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_369_NF__join__imp__reach,axiom,
    ! [Y: produc9001763001631046773rm_a_b,A4: set_Pr7061646551094443591rm_a_b,X: produc9001763001631046773rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ Y @ ( abstra5673374384174597889rm_a_b @ A4 ) )
     => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( abstra180165145518022959rm_a_b @ A4 ) )
       => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( transi2165682467528973816rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_370_NF__join__imp__reach,axiom,
    ! [Y: produc3421996462152255079rm_a_b,A4: set_Pr4284312964791606471rm_a_b,X: produc3421996462152255079rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ Y @ ( abstra8119681912743043827rm_a_b @ A4 ) )
     => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( abstra2994028023477150241rm_a_b @ A4 ) )
       => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( transi5504100148209286250rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_371_NF__join__imp__reach,axiom,
    ! [Y: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,X: set_term_a_b] :
      ( ( member_set_term_a_b @ Y @ ( abstra5309567699948295358rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_372_NF__join__imp__reach,axiom,
    ! [Y: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ Y @ ( abstra6901259896266686709rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_373_NF__join__imp__reach,axiom,
    ! [Y: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_374_NF__join__imp__reach,axiom,
    ! [Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_375_Id__on__empty,axiom,
    ( ( id_on_term_a_b @ bot_bot_set_term_a_b )
    = bot_bo197521221353338581rm_a_b ) ).

% Id_on_empty
thf(fact_376_Id__on__empty,axiom,
    ( ( id_on_115818524817812774rm_a_b @ bot_bo197521221353338581rm_a_b )
    = bot_bo5040873672931234227rm_a_b ) ).

% Id_on_empty
thf(fact_377_Id__on__empty,axiom,
    ( ( id_on_b @ bot_bot_set_b )
    = bot_bo2792761326896053555od_b_b ) ).

% Id_on_empty
thf(fact_378_meet__def,axiom,
    ( abstra8296971427885378886rm_a_b
    = ( ^ [A7: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ A7 ) ) @ ( transi2615809358984392588rm_a_b @ A7 ) ) ) ) ).

% meet_def
thf(fact_379_meet__def,axiom,
    ( abstra1093061187967292567rm_a_b
    = ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A7 ) ) @ ( transi7742714808557438673rm_a_b @ A7 ) ) ) ) ).

% meet_def
thf(fact_380_some__NF__UNF,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,X: produc4523971788519308903rm_a_b,Y: produc4523971788519308903rm_a_b] :
      ( ( abstra3562400838740635925rm_a_b @ R2 @ top_to1302229308806327447rm_a_b )
     => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
       => ( ( member8417600551952982416rm_a_b @ Y @ ( abstra1925539948312754931rm_a_b @ R2 ) )
         => ( Y
            = ( abstra5064197162818443967rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_381_some__NF__UNF,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,X: produc9001763001631046773rm_a_b,Y: produc9001763001631046773rm_a_b] :
      ( ( abstra556754771559176739rm_a_b @ R2 @ top_to1488846792128062501rm_a_b )
     => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
       => ( ( member8888773126115863070rm_a_b @ Y @ ( abstra5673374384174597889rm_a_b @ R2 ) )
         => ( Y
            = ( abstra4685688745131889869rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_382_some__NF__UNF,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,X: produc3421996462152255079rm_a_b,Y: produc3421996462152255079rm_a_b] :
      ( ( abstra3798689227379738901rm_a_b @ R2 @ top_to1603475325181578391rm_a_b )
     => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
       => ( ( member1375104729371992464rm_a_b @ Y @ ( abstra8119681912743043827rm_a_b @ R2 ) )
         => ( Y
            = ( abstra8963693433810084543rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_383_some__NF__UNF,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ( abstra2699617484662362908rm_a_b @ R2 @ top_to5470816188456996400rm_a_b )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
       => ( ( member_set_term_a_b @ Y @ ( abstra5309567699948295358rm_a_b @ R2 ) )
         => ( Y
            = ( abstra4354321892380618738rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_384_some__NF__UNF,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra80904564981652503rm_a_b @ R2 @ top_to8981750078293510169rm_a_b )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
       => ( ( member8823642877475723282rm_a_b @ Y @ ( abstra6901259896266686709rm_a_b @ R2 ) )
         => ( Y
            = ( abstra6563230700747095745rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_385_some__NF__UNF,axiom,
    ! [R2: set_Product_prod_a_a,X: a,Y: a] :
      ( ( abstract_UNF_on_a @ R2 @ top_top_set_a )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R2 ) )
       => ( ( member_a @ Y @ ( abstract_NF_a @ R2 ) )
         => ( Y
            = ( abstract_some_NF_a @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_386_some__NF__UNF,axiom,
    ! [R2: set_Product_prod_b_b,X: b,Y: b] :
      ( ( abstract_UNF_on_b @ R2 @ top_top_set_b )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ R2 ) )
       => ( ( member_b @ Y @ ( abstract_NF_b @ R2 ) )
         => ( Y
            = ( abstract_some_NF_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_387_some__NF__UNF,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
         => ( Y
            = ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_388_some__NF__UNF,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
         => ( Y
            = ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_389_non__strict__into__strict,axiom,
    ! [NS: set_Pr7151917219565317845rm_a_b,S4: set_Pr7151917219565317845rm_a_b,S: set_term_a_b,T: set_term_a_b] :
      ( ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ T ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ NS ) @ S4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_390_non__strict__into__strict,axiom,
    ! [NS: set_Pr647378613824986951rm_a_b,S4: set_Pr647378613824986951rm_a_b,S: set_Pr4386577575007340137rm_a_b,T: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le6958925370334017255rm_a_b @ ( relcom8520403917366345832rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ T ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ NS ) @ S4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_391_non__strict__into__strict,axiom,
    ! [NS: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ NS ) @ S4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_392_non__strict__into__strict,axiom,
    ! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_393_non__strict__into__strict,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_394_conversionI_H,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra6253361338556788316rm_a_b @ A4 ) ) ) ).

% conversionI'
thf(fact_395_conversionI_H,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra5398323551124918487rm_a_b @ A4 ) ) ) ).

% conversionI'
thf(fact_396_conversionI_H,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ).

% conversionI'
thf(fact_397_conversionI_H,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).

% conversionI'
thf(fact_398_srsteps__eq__subst__relcomp__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) )
     => ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ).

% srsteps_eq_subst_relcomp_closed
thf(fact_399_NF__empty,axiom,
    ( ( abstract_NF_term_a_b @ bot_bo197521221353338581rm_a_b )
    = top_top_set_term_a_b ) ).

% NF_empty
thf(fact_400_NF__empty,axiom,
    ( ( abstra1902783451776770197rm_a_b @ bot_bo5040873672931234227rm_a_b )
    = top_to1314267278846557113rm_a_b ) ).

% NF_empty
thf(fact_401_NF__empty,axiom,
    ( ( abstract_NF_a @ bot_bo3357376287454694259od_a_a )
    = top_top_set_a ) ).

% NF_empty
thf(fact_402_NF__empty,axiom,
    ( ( abstract_NF_b @ bot_bo2792761326896053555od_b_b )
    = top_top_set_b ) ).

% NF_empty
thf(fact_403_ssubst__Pair__rhs,axiom,
    ! [R2: produc357393685978478089rm_a_b,S: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,S6: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S ) @ R6 )
     => ( ( S6 = S )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S6 ) @ R6 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_404_ssubst__Pair__rhs,axiom,
    ! [R2: set_term_a_b,S: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b,S6: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ R2 @ S ) @ R6 )
     => ( ( S6 = S )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ R2 @ S6 ) @ R6 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_405_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b,R6: set_Pr647378613824986951rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ R2 @ S ) @ R6 )
     => ( ( S6 = S )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ R2 @ S6 ) @ R6 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_406_ssubst__Pair__rhs,axiom,
    ! [R2: term_a_b,S: term_a_b,R6: set_Pr4386577575007340137rm_a_b,S6: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S ) @ R6 )
     => ( ( S6 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S6 ) @ R6 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_407_converse__UNIV,axiom,
    ( ( conver1575690562132235771rm_a_b @ top_to1302229308806327447rm_a_b )
    = top_to1302229308806327447rm_a_b ) ).

% converse_UNIV
thf(fact_408_converse__UNIV,axiom,
    ( ( conver7608559976937845093rm_a_b @ top_to1314267278846557113rm_a_b )
    = top_to1314267278846557113rm_a_b ) ).

% converse_UNIV
thf(fact_409_converse__inject,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ( conver7608559976937845093rm_a_b @ R2 )
        = ( conver7608559976937845093rm_a_b @ S ) )
      = ( R2 = S ) ) ).

% converse_inject
thf(fact_410_converse__inject,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ( conver1575690562132235771rm_a_b @ R2 )
        = ( conver1575690562132235771rm_a_b @ S ) )
      = ( R2 = S ) ) ).

% converse_inject
thf(fact_411_converse__converse,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
      = R2 ) ).

% converse_converse
thf(fact_412_converse__converse,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( conver1575690562132235771rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
      = R2 ) ).

% converse_converse
thf(fact_413_conversion__conversion__idemp,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra1130568886744591910rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
      = ( abstra1130568886744591910rm_a_b @ A4 ) ) ).

% conversion_conversion_idemp
thf(fact_414_conversion__conversion__idemp,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra6125656530262095991rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
      = ( abstra6125656530262095991rm_a_b @ A4 ) ) ).

% conversion_conversion_idemp
thf(fact_415_converse__iff,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( conver1904685531124697809rm_a_b @ R2 ) )
      = ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ R2 ) ) ).

% converse_iff
thf(fact_416_converse__iff,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( conver7531182934069937659rm_a_b @ R2 ) )
      = ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ R2 ) ) ).

% converse_iff
thf(fact_417_converse__iff,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
      = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R2 ) ) ).

% converse_iff
thf(fact_418_converse__iff,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ).

% converse_iff
thf(fact_419_converse__mono,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( conver1575690562132235771rm_a_b @ S ) )
      = ( ord_le2593958535671617255rm_a_b @ R2 @ S ) ) ).

% converse_mono
thf(fact_420_converse__mono,axiom,
    ! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R2 ) @ ( converse_a_nat @ S ) )
      = ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).

% converse_mono
thf(fact_421_converse__mono,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( conver7608559976937845093rm_a_b @ S ) )
      = ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).

% converse_mono
thf(fact_422_converse__mono,axiom,
    ! [R2: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R2 ) @ ( converse_nat_a @ S ) )
      = ( ord_le7924913712489149241_nat_a @ R2 @ S ) ) ).

% converse_mono
thf(fact_423_Id__onI,axiom,
    ! [A: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ A @ A4 )
     => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ A ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_424_Id__onI,axiom,
    ! [A: b,A4: set_b] :
      ( ( member_b @ A @ A4 )
     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( id_on_b @ A4 ) ) ) ).

% Id_onI
thf(fact_425_Id__onI,axiom,
    ! [A: produc9001763001631046773rm_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ A @ A4 )
     => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ A ) @ ( id_on_458948420642081682rm_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_426_Id__onI,axiom,
    ! [A: produc3421996462152255079rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ A @ A4 )
     => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ A ) @ ( id_on_9131758789178798340rm_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_427_Id__onI,axiom,
    ! [A: set_term_a_b,A4: set_set_term_a_b] :
      ( ( member_set_term_a_b @ A @ A4 )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ A ) @ ( id_on_set_term_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_428_Id__onI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_se1467346398288745161rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ A @ A4 )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ A ) @ ( id_on_9064245942984952198rm_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_429_Id__onI,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A4 )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( id_on_115818524817812774rm_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_430_Id__onI,axiom,
    ! [A: term_a_b,A4: set_term_a_b] :
      ( ( member_term_a_b @ A @ A4 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( id_on_term_a_b @ A4 ) ) ) ).

% Id_onI
thf(fact_431_converse__empty,axiom,
    ( ( conver1575690562132235771rm_a_b @ bot_bo5040873672931234227rm_a_b )
    = bot_bo5040873672931234227rm_a_b ) ).

% converse_empty
thf(fact_432_converse__empty,axiom,
    ( ( conver7608559976937845093rm_a_b @ bot_bo197521221353338581rm_a_b )
    = bot_bo197521221353338581rm_a_b ) ).

% converse_empty
thf(fact_433_conversion__rtrancl,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
      = ( abstra6125656530262095991rm_a_b @ A4 ) ) ).

% conversion_rtrancl
thf(fact_434_conversion__rtrancl,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
      = ( abstra1130568886744591910rm_a_b @ A4 ) ) ).

% conversion_rtrancl
thf(fact_435_conversion__O__conversion,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
      = ( abstra1130568886744591910rm_a_b @ A4 ) ) ).

% conversion_O_conversion
thf(fact_436_conversion__O__conversion,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) )
      = ( abstra6125656530262095991rm_a_b @ A4 ) ) ).

% conversion_O_conversion
thf(fact_437_conversion__converse,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
      = ( abstra1130568886744591910rm_a_b @ A4 ) ) ).

% conversion_converse
thf(fact_438_conversion__converse,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( conver1575690562132235771rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
      = ( abstra6125656530262095991rm_a_b @ A4 ) ) ).

% conversion_converse
thf(fact_439_converse__Id__on,axiom,
    ! [A4: set_term_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( id_on_term_a_b @ A4 ) )
      = ( id_on_term_a_b @ A4 ) ) ).

% converse_Id_on
thf(fact_440_converse__Id__on,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( conver1575690562132235771rm_a_b @ ( id_on_115818524817812774rm_a_b @ A4 ) )
      = ( id_on_115818524817812774rm_a_b @ A4 ) ) ).

% converse_Id_on
thf(fact_441_funas__term__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_funas_term_a_b @ ( term_a2780309480206857766_a_b_b @ T ) )
      = ( term_funas_term_a_b @ T ) ) ).

% funas_term_adapt_vars
thf(fact_442_joinI__right,axiom,
    ! [B: set_term_a_b,A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ).

% joinI_right
thf(fact_443_joinI__right,axiom,
    ! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ).

% joinI_right
thf(fact_444_joinI__right,axiom,
    ! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).

% joinI_right
thf(fact_445_joinI__right,axiom,
    ! [B: term_a_b,A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).

% joinI_right
thf(fact_446_joinI__left,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ).

% joinI_left
thf(fact_447_joinI__left,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ).

% joinI_left
thf(fact_448_joinI__left,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).

% joinI_left
thf(fact_449_joinI__left,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).

% joinI_left
thf(fact_450_joinI,axiom,
    ! [A: set_term_a_b,C: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,B: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ) ).

% joinI
thf(fact_451_joinI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ) ).

% joinI
thf(fact_452_joinI,axiom,
    ! [A: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).

% joinI
thf(fact_453_joinI,axiom,
    ! [A: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).

% joinI
thf(fact_454_sig__stepI,axiom,
    ! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R ) ) ) ) ) ).

% sig_stepI
thf(fact_455_NF__anti__mono,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
     => ( ord_le118470702582115849rm_a_b @ ( abstra1902783451776770197rm_a_b @ B5 ) @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ).

% NF_anti_mono
thf(fact_456_NF__anti__mono,axiom,
    ! [A4: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat] :
      ( ( ord_le1018141841545957851_a_nat @ A4 @ B5 )
     => ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B5 ) @ ( abstra2685462378305987313_a_nat @ A4 ) ) ) ).

% NF_anti_mono
thf(fact_457_NF__anti__mono,axiom,
    ! [A4: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A4 @ B5 )
     => ( ord_less_eq_set_b @ ( abstract_NF_b @ B5 ) @ ( abstract_NF_b @ A4 ) ) ) ).

% NF_anti_mono
thf(fact_458_NF__anti__mono,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
     => ( ord_le2705286416250468010rm_a_b @ ( abstract_NF_term_a_b @ B5 ) @ ( abstract_NF_term_a_b @ A4 ) ) ) ).

% NF_anti_mono
thf(fact_459_join__mono,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
     => ( ord_le2593958535671617255rm_a_b @ ( abstra4889682940153293507rm_a_b @ R2 ) @ ( abstra4889682940153293507rm_a_b @ S ) ) ) ).

% join_mono
thf(fact_460_join__mono,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
     => ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra4096080454567261402rm_a_b @ S ) ) ) ).

% join_mono
thf(fact_461_converse__subset__swap,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ S ) )
      = ( ord_le2593958535671617255rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ S ) ) ).

% converse_subset_swap
thf(fact_462_converse__subset__swap,axiom,
    ! [R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le7924913712489149241_nat_a @ R2 @ ( converse_a_nat @ S ) )
      = ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R2 ) @ S ) ) ).

% converse_subset_swap
thf(fact_463_converse__subset__swap,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ S ) )
      = ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ S ) ) ).

% converse_subset_swap
thf(fact_464_converse__subset__swap,axiom,
    ! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ R2 @ ( converse_nat_a @ S ) )
      = ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R2 ) @ S ) ) ).

% converse_subset_swap
thf(fact_465_conversion__mono,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
     => ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) @ ( abstra1130568886744591910rm_a_b @ B5 ) ) ) ).

% conversion_mono
thf(fact_466_conversion__mono,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
     => ( ord_le2593958535671617255rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) @ ( abstra6125656530262095991rm_a_b @ B5 ) ) ) ).

% conversion_mono
thf(fact_467_join__imp__conversion,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra1130568886744591910rm_a_b @ R2 ) ) ).

% join_imp_conversion
thf(fact_468_join__imp__conversion,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( abstra4889682940153293507rm_a_b @ R2 ) @ ( abstra6125656530262095991rm_a_b @ R2 ) ) ).

% join_imp_conversion
thf(fact_469_sig__step__mono,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R ) @ ( sig_step_a_b @ G2 @ R ) ) ) ).

% sig_step_mono
thf(fact_470_srsteps__with__root__step__sig__mono,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ R ) @ ( srstep7844470518422762656ep_a_b @ G2 @ R ) ) ) ).

% srsteps_with_root_step_sig_mono
thf(fact_471_meet__imp__conversion,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R2 ) @ ( abstra1130568886744591910rm_a_b @ R2 ) ) ).

% meet_imp_conversion
thf(fact_472_meet__imp__conversion,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( abstra8296971427885378886rm_a_b @ R2 ) @ ( abstra6125656530262095991rm_a_b @ R2 ) ) ).

% meet_imp_conversion
thf(fact_473_sig__stepE,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R ) )
     => ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R )
         => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
           => ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ) ).

% sig_stepE
thf(fact_474_converse_Ocases,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( conver1904685531124697809rm_a_b @ R2 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A2 @ A1 ) @ R2 ) ) ).

% converse.cases
thf(fact_475_converse_Ocases,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( conver7531182934069937659rm_a_b @ R2 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A2 @ A1 ) @ R2 ) ) ).

% converse.cases
thf(fact_476_converse_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A1 ) @ R2 ) ) ).

% converse.cases
thf(fact_477_converse_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A1 ) @ R2 ) ) ).

% converse.cases
thf(fact_478_converse_Osimps,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( conver1904685531124697809rm_a_b @ R2 ) )
      = ( ? [A3: set_term_a_b,B2: set_term_a_b] :
            ( ( A1 = B2 )
            & ( A2 = A3 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% converse.simps
thf(fact_479_converse_Osimps,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( conver7531182934069937659rm_a_b @ R2 ) )
      = ( ? [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = B2 )
            & ( A2 = A3 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% converse.simps
thf(fact_480_converse_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
      = ( ? [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
            ( ( A1 = B2 )
            & ( A2 = A3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% converse.simps
thf(fact_481_converse_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
      = ( ? [A3: term_a_b,B2: term_a_b] :
            ( ( A1 = B2 )
            & ( A2 = A3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% converse.simps
thf(fact_482_converseD,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( conver1904685531124697809rm_a_b @ R2 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ R2 ) ) ).

% converseD
thf(fact_483_converseD,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( conver7531182934069937659rm_a_b @ R2 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ R2 ) ) ).

% converseD
thf(fact_484_converseD,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R2 ) ) ).

% converseD
thf(fact_485_converseD,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ).

% converseD
thf(fact_486_converseE,axiom,
    ! [Yx: produc9001763001631046773rm_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ Yx @ ( conver1904685531124697809rm_a_b @ R2 ) )
     => ~ ! [X3: set_term_a_b,Y4: set_term_a_b] :
            ( ( Yx
              = ( produc8517751122659631533rm_a_b @ Y4 @ X3 ) )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 ) ) ) ).

% converseE
thf(fact_487_converseE,axiom,
    ! [Yx: produc3421996462152255079rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ Yx @ ( conver7531182934069937659rm_a_b @ R2 ) )
     => ~ ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
            ( ( Yx
              = ( produc3977138704197412567rm_a_b @ Y4 @ X3 ) )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ R2 ) ) ) ).

% converseE
thf(fact_488_converseE,axiom,
    ! [Yx: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ Yx @ ( conver1575690562132235771rm_a_b @ R2 ) )
     => ~ ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
            ( ( Yx
              = ( produc1763473618796451543rm_a_b @ Y4 @ X3 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 ) ) ) ).

% converseE
thf(fact_489_converseE,axiom,
    ! [Yx: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R2 ) )
     => ~ ! [X3: term_a_b,Y4: term_a_b] :
            ( ( Yx
              = ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 ) ) ) ).

% converseE
thf(fact_490_converseI,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ ( conver1904685531124697809rm_a_b @ R2 ) ) ) ).

% converseI
thf(fact_491_converseI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ ( conver7531182934069937659rm_a_b @ R2 ) ) ) ).

% converseI
thf(fact_492_converseI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).

% converseI
thf(fact_493_converseI,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).

% converseI
thf(fact_494_join__def,axiom,
    ( abstra4889682940153293507rm_a_b
    = ( ^ [A7: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ A7 ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ A7 ) ) ) ) ) ).

% join_def
thf(fact_495_join__def,axiom,
    ( abstra4096080454567261402rm_a_b
    = ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A7 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A7 ) ) ) ) ) ).

% join_def
thf(fact_496_rtrancl__converse,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
      = ( conver1575690562132235771rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).

% rtrancl_converse
thf(fact_497_rtrancl__converse,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
      = ( conver7608559976937845093rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% rtrancl_converse
thf(fact_498_converse__relcomp,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
      = ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ S ) @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).

% converse_relcomp
thf(fact_499_converse__relcomp,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( conver1575690562132235771rm_a_b @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
      = ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ S ) @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).

% converse_relcomp
thf(fact_500_funas__term_Osimps_I1_J,axiom,
    ! [X: b] :
      ( ( term_funas_term_a_b @ ( var_b_a @ X ) )
      = bot_bo9049108969261143879_a_nat ) ).

% funas_term.simps(1)
thf(fact_501_srstep__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ).

% srstep_monp
thf(fact_502_rstep__converse__dist,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) )
      = ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).

% rstep_converse_dist
thf(fact_503_subrelI,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ S ) )
     => ( ord_le2593958535671617255rm_a_b @ R2 @ S ) ) ).

% subrelI
thf(fact_504_subrelI,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ! [X3: set_term_a_b,Y4: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ S ) )
     => ( ord_le2663425029446080117rm_a_b @ R2 @ S ) ) ).

% subrelI
thf(fact_505_subrelI,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ S ) )
     => ( ord_le6958925370334017255rm_a_b @ R2 @ S ) ) ).

% subrelI
thf(fact_506_subrelI,axiom,
    ! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: a,Y4: nat] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ R2 )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ S ) )
     => ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_507_subrelI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ S ) )
     => ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).

% subrelI
thf(fact_508_srrstep__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) @ ( sig_step_a_b @ G2 @ ( rrstep_a_b @ R ) ) ) ) ).

% srrstep_monp
thf(fact_509_rtrancl__subset__rtrancl,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2 @ ( transi2615809358984392588rm_a_b @ S ) )
     => ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_510_rtrancl__subset__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ S ) )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_511_rtrancl__subset,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R6 @ S4 )
     => ( ( ord_le2593958535671617255rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ R6 ) )
       => ( ( transi2615809358984392588rm_a_b @ S4 )
          = ( transi2615809358984392588rm_a_b @ R6 ) ) ) ) ).

% rtrancl_subset
thf(fact_512_rtrancl__subset,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R6 @ S4 )
     => ( ( ord_le118470702582115849rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R6 ) )
       => ( ( transi7742714808557438673rm_a_b @ S4 )
          = ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ).

% rtrancl_subset
thf(fact_513_rtrancl__mono,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
     => ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).

% rtrancl_mono
thf(fact_514_rtrancl__mono,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).

% rtrancl_mono
thf(fact_515_O__mono2,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R6 @ R8 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ T6 ) @ ( relcom370159955682700863rm_a_b @ R8 @ T6 ) ) ) ).

% O_mono2
thf(fact_516_O__mono2,axiom,
    ! [R6: set_Pr4934435412358123699_a_nat,R8: set_Pr4934435412358123699_a_nat,T6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R6 @ R8 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R6 @ T6 ) @ ( relcomp_a_nat_nat @ R8 @ T6 ) ) ) ).

% O_mono2
thf(fact_517_O__mono1,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R6 @ R8 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ R8 ) ) ) ).

% O_mono1
thf(fact_518_O__mono1,axiom,
    ! [R6: set_Pr4934435412358123699_a_nat,R8: set_Pr4934435412358123699_a_nat,S4: set_Product_prod_a_a] :
      ( ( ord_le8666007276011122963_a_nat @ R6 @ R8 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S4 @ R6 ) @ ( relcomp_a_a_nat @ S4 @ R8 ) ) ) ).

% O_mono1
thf(fact_519_relcomp__mono,axiom,
    ! [R9: set_Product_prod_a_a,R2: set_Product_prod_a_a,S6: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le746702958409616551od_a_a @ R9 @ R2 )
     => ( ( ord_le8666007276011122963_a_nat @ S6 @ S )
       => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R9 @ S6 ) @ ( relcomp_a_a_nat @ R2 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_520_relcomp__mono,axiom,
    ! [R9: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R9 @ R2 )
     => ( ( ord_le118470702582115849rm_a_b @ S6 @ S )
       => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R9 @ S6 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_521_relcomp__mono,axiom,
    ! [R9: set_Pr4934435412358123699_a_nat,R2: set_Pr4934435412358123699_a_nat,S6: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R9 @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ S6 @ S )
       => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R9 @ S6 ) @ ( relcomp_a_nat_nat @ R2 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_522_join__sym,axiom,
    ! [S: set_term_a_b,T: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ T ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ S ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ).

% join_sym
thf(fact_523_join__sym,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,T: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ T ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ S ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ).

% join_sym
thf(fact_524_join__sym,axiom,
    ! [S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ S ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).

% join_sym
thf(fact_525_join__sym,axiom,
    ! [S: term_a_b,T: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).

% join_sym
thf(fact_526_sig__step__mono2,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,L4: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le118470702582115849rm_a_b @ R @ L4 )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R ) @ ( sig_step_a_b @ F @ L4 ) ) ) ).

% sig_step_mono2
thf(fact_527_conversion__refl,axiom,
    ! [A: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ A ) @ ( abstra6253361338556788316rm_a_b @ A4 ) ) ).

% conversion_refl
thf(fact_528_conversion__refl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ A ) @ ( abstra5398323551124918487rm_a_b @ A4 ) ) ).

% conversion_refl
thf(fact_529_conversion__refl,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ).

% conversion_refl
thf(fact_530_conversion__refl,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ).

% conversion_refl
thf(fact_531_conversion__inv,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( abstra6253361338556788316rm_a_b @ R6 ) )
      = ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ X ) @ ( abstra6253361338556788316rm_a_b @ R6 ) ) ) ).

% conversion_inv
thf(fact_532_conversion__inv,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R6: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( abstra5398323551124918487rm_a_b @ R6 ) )
      = ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ X ) @ ( abstra5398323551124918487rm_a_b @ R6 ) ) ) ).

% conversion_inv
thf(fact_533_conversion__inv,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra6125656530262095991rm_a_b @ R6 ) )
      = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( abstra6125656530262095991rm_a_b @ R6 ) ) ) ).

% conversion_inv
thf(fact_534_conversion__inv,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R6 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R6 ) ) ) ).

% conversion_inv
thf(fact_535_Id__on__iff,axiom,
    ! [X: produc4523971788519308903rm_a_b,Y: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member8417600551952982416rm_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_536_Id__on__iff,axiom,
    ! [X: b,Y: b,A4: set_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( id_on_b @ A4 ) )
      = ( ( X = Y )
        & ( member_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_537_Id__on__iff,axiom,
    ! [X: produc9001763001631046773rm_a_b,Y: produc9001763001631046773rm_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( id_on_458948420642081682rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member8888773126115863070rm_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_538_Id__on__iff,axiom,
    ! [X: produc3421996462152255079rm_a_b,Y: produc3421996462152255079rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( id_on_9131758789178798340rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member1375104729371992464rm_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_539_Id__on__iff,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,A4: set_set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( id_on_set_term_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member_set_term_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_540_Id__on__iff,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,A4: set_se1467346398288745161rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( id_on_9064245942984952198rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member8823642877475723282rm_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_541_Id__on__iff,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( id_on_115818524817812774rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_542_Id__on__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( id_on_term_a_b @ A4 ) )
      = ( ( X = Y )
        & ( member_term_a_b @ X @ A4 ) ) ) ).

% Id_on_iff
thf(fact_543_Id__on__eqI,axiom,
    ! [A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( A = B )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_544_Id__on__eqI,axiom,
    ! [A: b,B: b,A4: set_b] :
      ( ( A = B )
     => ( ( member_b @ A @ A4 )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( id_on_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_545_Id__on__eqI,axiom,
    ! [A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( A = B )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( id_on_458948420642081682rm_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_546_Id__on__eqI,axiom,
    ! [A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( A = B )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( id_on_9131758789178798340rm_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_547_Id__on__eqI,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_set_term_a_b] :
      ( ( A = B )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( id_on_set_term_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_548_Id__on__eqI,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_se1467346398288745161rm_a_b] :
      ( ( A = B )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( id_on_9064245942984952198rm_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_549_Id__on__eqI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( id_on_115818524817812774rm_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_550_Id__on__eqI,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_term_a_b] :
      ( ( A = B )
     => ( ( member_term_a_b @ A @ A4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( id_on_term_a_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_551_Id__onE,axiom,
    ! [C: produc2957090337049255143rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ C @ ( id_on_4065744930832257284rm_a_b @ A4 ) )
     => ~ ! [X3: produc4523971788519308903rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ X3 @ A4 )
           => ( C
             != ( produc2499998789532207959rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_552_Id__onE,axiom,
    ! [C: product_prod_b_b,A4: set_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( id_on_b @ A4 ) )
     => ~ ! [X3: b] :
            ( ( member_b @ X3 @ A4 )
           => ( C
             != ( product_Pair_b_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_553_Id__onE,axiom,
    ! [C: produc3601453998762642791rm_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ C @ ( id_on_458948420642081682rm_a_b @ A4 ) )
     => ~ ! [X3: produc9001763001631046773rm_a_b] :
            ( ( member8888773126115863070rm_a_b @ X3 @ A4 )
           => ( C
             != ( produc1448243995722353623rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_554_Id__onE,axiom,
    ! [C: produc8791377122869980391rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ C @ ( id_on_9131758789178798340rm_a_b @ A4 ) )
     => ~ ! [X3: produc3421996462152255079rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ X3 @ A4 )
           => ( C
             != ( produc765457364750699351rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_555_Id__onE,axiom,
    ! [C: produc9001763001631046773rm_a_b,A4: set_set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ C @ ( id_on_set_term_a_b @ A4 ) )
     => ~ ! [X3: set_term_a_b] :
            ( ( member_set_term_a_b @ X3 @ A4 )
           => ( C
             != ( produc8517751122659631533rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_556_Id__onE,axiom,
    ! [C: produc3421996462152255079rm_a_b,A4: set_se1467346398288745161rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ C @ ( id_on_9064245942984952198rm_a_b @ A4 ) )
     => ~ ! [X3: set_Pr4386577575007340137rm_a_b] :
            ( ( member8823642877475723282rm_a_b @ X3 @ A4 )
           => ( C
             != ( produc3977138704197412567rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_557_Id__onE,axiom,
    ! [C: produc4523971788519308903rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ C @ ( id_on_115818524817812774rm_a_b @ A4 ) )
     => ~ ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
           => ( C
             != ( produc1763473618796451543rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_558_Id__onE,axiom,
    ! [C: produc357393685978478089rm_a_b,A4: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ C @ ( id_on_term_a_b @ A4 ) )
     => ~ ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
           => ( C
             != ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_559_srstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srstepD
thf(fact_560_srrstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srrstepD
thf(fact_561_srsteps__eq__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ) ).

% srsteps_eq_monp
thf(fact_562_rtrancl__converseI,axiom,
    ! [Y: set_term_a_b,X: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ X ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ ( conver1904685531124697809rm_a_b @ R2 ) ) ) ) ).

% rtrancl_converseI
thf(fact_563_rtrancl__converseI,axiom,
    ! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ X ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ ( conver7531182934069937659rm_a_b @ R2 ) ) ) ) ).

% rtrancl_converseI
thf(fact_564_rtrancl__converseI,axiom,
    ! [Y: produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ) ).

% rtrancl_converseI
thf(fact_565_rtrancl__converseI,axiom,
    ! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).

% rtrancl_converseI
thf(fact_566_rtrancl__converseD,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ ( conver1904685531124697809rm_a_b @ R2 ) ) )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ X ) @ ( transi6904761420441344775rm_a_b @ R2 ) ) ) ).

% rtrancl_converseD
thf(fact_567_rtrancl__converseD,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ ( conver7531182934069937659rm_a_b @ R2 ) ) )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ X ) @ ( transi8812752159401800684rm_a_b @ R2 ) ) ) ).

% rtrancl_converseD
thf(fact_568_rtrancl__converseD,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).

% rtrancl_converseD
thf(fact_569_rtrancl__converseD,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% rtrancl_converseD
thf(fact_570_peak__iff,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,B5: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ ( conver1904685531124697809rm_a_b @ A4 ) @ B5 ) )
      = ( ? [U3: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U3 @ X ) @ A4 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).

% peak_iff
thf(fact_571_peak__iff,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,B5: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ ( conver7531182934069937659rm_a_b @ A4 ) @ B5 ) )
      = ( ? [U3: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U3 @ X ) @ A4 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).

% peak_iff
thf(fact_572_peak__iff,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A4 ) @ B5 ) )
      = ( ? [U3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ X ) @ A4 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).

% peak_iff
thf(fact_573_peak__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ B5 ) )
      = ( ? [U3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).

% peak_iff
thf(fact_574_srstep__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
     => ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).

% srstep_subst_closed
thf(fact_575_sig__step__converse__rstep,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
      = ( sig_step_a_b @ F @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) ) ) ) ).

% sig_step_converse_rstep
thf(fact_576_srstep__converse__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
      = ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% srstep_converse_dist
thf(fact_577_srrstep__converse__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) )
      = ( sig_step_a_b @ F @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% srrstep_converse_dist
thf(fact_578_rtrancl__O__push,axiom,
    ! [S4: set_Pr2972776593051762503rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ S4 @ R6 ) @ ( relcom6159724393391289352rm_a_b @ R6 @ ( transi2615809358984392588rm_a_b @ S4 ) ) )
     => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S4 ) @ R6 ) @ ( relcom6159724393391289352rm_a_b @ R6 @ ( transi2615809358984392588rm_a_b @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_579_rtrancl__O__push,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_580_compat__tr__compat,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) @ S4 ) ) ).

% compat_tr_compat
thf(fact_581_compat__tr__compat,axiom,
    ! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) @ S4 ) ) ).

% compat_tr_compat
thf(fact_582_rtrancl__join__join,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ) ).

% rtrancl_join_join
thf(fact_583_rtrancl__join__join,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ) ).

% rtrancl_join_join
thf(fact_584_rtrancl__join__join,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).

% rtrancl_join_join
thf(fact_585_rtrancl__join__join,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).

% rtrancl_join_join
thf(fact_586_join__rtrancl__join,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ C @ B ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( abstra2139004588645393680rm_a_b @ A4 ) ) ) ) ).

% join_rtrancl_join
thf(fact_587_join__rtrancl__join,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ C @ B ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( abstra6273572973012767523rm_a_b @ A4 ) ) ) ) ).

% join_rtrancl_join
thf(fact_588_join__rtrancl__join,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).

% join_rtrancl_join
thf(fact_589_join__rtrancl__join,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).

% join_rtrancl_join
thf(fact_590_joinE,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
     => ~ ! [C2: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi6904761420441344775rm_a_b @ A4 ) ) ) ) ).

% joinE
thf(fact_591_joinE,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
     => ~ ! [C2: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi8812752159401800684rm_a_b @ A4 ) ) ) ) ).

% joinE
thf(fact_592_joinE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
     => ~ ! [C2: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).

% joinE
thf(fact_593_joinE,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
     => ~ ! [C2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% joinE
thf(fact_594_joinD,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,A4: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra2139004588645393680rm_a_b @ A4 ) )
     => ? [C2: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi6904761420441344775rm_a_b @ A4 ) )
          & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi6904761420441344775rm_a_b @ A4 ) ) ) ) ).

% joinD
thf(fact_595_joinD,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A4: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra6273572973012767523rm_a_b @ A4 ) )
     => ? [C2: set_Pr4386577575007340137rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi8812752159401800684rm_a_b @ A4 ) )
          & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi8812752159401800684rm_a_b @ A4 ) ) ) ) ).

% joinD
thf(fact_596_joinD,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
     => ? [C2: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).

% joinD
thf(fact_597_joinD,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
     => ? [C2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% joinD
thf(fact_598_rrstep__rstep__mono,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R ) @ ( rstep_a_b @ R ) ) ).

% rrstep_rstep_mono
thf(fact_599_srsteps__eq__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
     => ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).

% srsteps_eq_subst_closed
thf(fact_600_join__NF__imp__eq,axiom,
    ! [X: produc4523971788519308903rm_a_b,Y: produc4523971788519308903rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( abstra4037915826850753057rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ X @ ( abstra1925539948312754931rm_a_b @ R2 ) )
       => ( ( member8417600551952982416rm_a_b @ Y @ ( abstra1925539948312754931rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_601_join__NF__imp__eq,axiom,
    ! [X: b,Y: b,R2: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( abstract_join_b @ R2 ) )
     => ( ( member_b @ X @ ( abstract_NF_b @ R2 ) )
       => ( ( member_b @ Y @ ( abstract_NF_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_602_join__NF__imp__eq,axiom,
    ! [X: produc9001763001631046773rm_a_b,Y: produc9001763001631046773rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ X @ Y ) @ ( abstra180165145518022959rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ X @ ( abstra5673374384174597889rm_a_b @ R2 ) )
       => ( ( member8888773126115863070rm_a_b @ Y @ ( abstra5673374384174597889rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_603_join__NF__imp__eq,axiom,
    ! [X: produc3421996462152255079rm_a_b,Y: produc3421996462152255079rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ X @ Y ) @ ( abstra2994028023477150241rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ X @ ( abstra8119681912743043827rm_a_b @ R2 ) )
       => ( ( member1375104729371992464rm_a_b @ Y @ ( abstra8119681912743043827rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_604_join__NF__imp__eq,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( abstra2139004588645393680rm_a_b @ R2 ) )
     => ( ( member_set_term_a_b @ X @ ( abstra5309567699948295358rm_a_b @ R2 ) )
       => ( ( member_set_term_a_b @ Y @ ( abstra5309567699948295358rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_605_join__NF__imp__eq,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( abstra6273572973012767523rm_a_b @ R2 ) )
     => ( ( member8823642877475723282rm_a_b @ X @ ( abstra6901259896266686709rm_a_b @ R2 ) )
       => ( ( member8823642877475723282rm_a_b @ Y @ ( abstra6901259896266686709rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_606_join__NF__imp__eq,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_607_join__NF__imp__eq,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
     => ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_608_commute__def,axiom,
    ( abstra5145658890658669097rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,S2: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) ) @ ( transi2615809358984392588rm_a_b @ S2 ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S2 ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) ) ) ) ) ) ).

% commute_def
thf(fact_609_commute__def,axiom,
    ( abstra455195320138238452rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) ) @ ( transi7742714808557438673rm_a_b @ S2 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) ) ) ) ) ) ).

% commute_def
thf(fact_610_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ( abstra7536183037145338092rm_a_b @ R2 @ top_to5470816188456996400rm_a_b )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( abstra6253361338556788316rm_a_b @ R2 ) )
        = ( ! [U3: set_term_a_b,V4: set_term_a_b] :
              ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ U3 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ V4 ) @ ( abstra4462713708631513506rm_a_b @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_611_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4222541640063877703rm_a_b @ R2 @ top_to8981750078293510169rm_a_b )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( abstra5398323551124918487rm_a_b @ R2 ) )
        = ( ! [U3: set_Pr4386577575007340137rm_a_b,V4: set_Pr4386577575007340137rm_a_b] :
              ( ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ U3 ) @ ( abstra845589504903352593rm_a_b @ R2 ) )
                & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ V4 ) @ ( abstra845589504903352593rm_a_b @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_612_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Product_prod_a_a,X: a,Y: a] :
      ( ( abstra6495048823662926812e_on_a @ R2 @ top_top_set_a )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( abstra3918519255145857644sion_a @ R2 ) )
        = ( ! [U3: a,V4: a] :
              ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ U3 ) @ ( abstra380364156431640998lity_a @ R2 ) )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ V4 ) @ ( abstra380364156431640998lity_a @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_613_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Product_prod_b_b,X: b,Y: b] :
      ( ( abstra6495048823662926813e_on_b @ R2 @ top_top_set_b )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( abstra3918519255145857645sion_b @ R2 ) )
        = ( ! [U3: b,V4: b] :
              ( ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ U3 ) @ ( abstra380364156431640999lity_b @ R2 ) )
                & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ V4 ) @ ( abstra380364156431640999lity_b @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_614_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( abstra5399326240126609895rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra6125656530262095991rm_a_b @ R2 ) )
        = ( ! [U3: produc357393685978478089rm_a_b,V4: produc357393685978478089rm_a_b] :
              ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ U3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ V4 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_615_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( abstra7162307272211697846rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
        = ( ! [U3: term_a_b,V4: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
             => ( U3 = V4 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_616_diamond__imp__semi__confluence,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra7467186385360731951rm_a_b @ R2 )
     => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ).

% diamond_imp_semi_confluence
thf(fact_617_diamond__imp__semi__confluence,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8151087076625346926rm_a_b @ R2 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).

% diamond_imp_semi_confluence
thf(fact_618_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S5 ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
         => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S5 ) ) ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ) ) ) ).

% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_619_diamond__I,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ R2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) )
     => ( abstra7467186385360731951rm_a_b @ R2 ) ) ).

% diamond_I
thf(fact_620_diamond__I,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
     => ( abstra8151087076625346926rm_a_b @ R2 ) ) ).

% diamond_I
thf(fact_621_kleene__induct,axiom,
    ! [A4: set_Pr4802440762858034464rm_a_b,X4: set_Pr4802440762858034464rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le7334406955775145344rm_a_b @ A4 @ X4 )
     => ( ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ B5 @ X4 ) @ X4 )
       => ( ( ord_le7334406955775145344rm_a_b @ ( relcom4000041261975091669rm_a_b @ X4 @ C4 ) @ X4 )
         => ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom4000041261975091669rm_a_b @ A4 @ ( transi2615809358984392588rm_a_b @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_622_kleene__induct,axiom,
    ! [A4: set_Pr6510905305361987970rm_a_b,X4: set_Pr6510905305361987970rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le9042871498279098850rm_a_b @ A4 @ X4 )
     => ( ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ B5 @ X4 ) @ X4 )
       => ( ( ord_le9042871498279098850rm_a_b @ ( relcom3777457926417599474rm_a_b @ X4 @ C4 ) @ X4 )
         => ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ B5 ) @ ( relcom3777457926417599474rm_a_b @ A4 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_623_kleene__induct,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,X4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A4 @ X4 )
     => ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ B5 @ X4 ) @ X4 )
       => ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ X4 @ C4 ) @ X4 )
         => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ B5 ) @ ( relcom6159724393391289352rm_a_b @ A4 @ ( transi2615809358984392588rm_a_b @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_624_kleene__induct,axiom,
    ! [A4: set_Pr4934435412358123699_a_nat,X4: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A4 @ X4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X4 ) @ X4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X4 @ C4 ) @ X4 )
         => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A4 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_625_kleene__induct,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A4 @ X4 )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X4 ) @ X4 )
       => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ C4 ) @ X4 )
         => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A4 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_626_rsteps__eq__srsteps__eqI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).

% rsteps_eq_srsteps_eqI
thf(fact_627_WCR__alt__def,axiom,
    ! [A4: set_Product_prod_a_a] :
      ( ( abstract_WCR_on_a @ A4 @ top_top_set_a )
      = ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( converse_a_a @ A4 ) @ A4 ) @ ( abstract_join_a @ A4 ) ) ) ).

% WCR_alt_def
thf(fact_628_WCR__alt__def,axiom,
    ! [A4: set_Product_prod_b_b] :
      ( ( abstract_WCR_on_b @ A4 @ top_top_set_b )
      = ( ord_le182087997850975847od_b_b @ ( relcomp_b_b_b @ ( converse_b_b @ A4 ) @ A4 ) @ ( abstract_join_b @ A4 ) ) ) ).

% WCR_alt_def
thf(fact_629_WCR__alt__def,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
      = ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A4 ) @ A4 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).

% WCR_alt_def
thf(fact_630_WCR__alt__def,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ A4 @ top_top_set_term_a_b )
      = ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ A4 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).

% WCR_alt_def
thf(fact_631_WCR__onI,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ! [A6: produc4523971788519308903rm_a_b,B4: produc4523971788519308903rm_a_b,C2: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ A6 @ A4 )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B4 @ C2 ) @ ( abstra4037915826850753057rm_a_b @ R2 ) ) ) ) )
     => ( abstra4159178081913940476rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_632_WCR__onI,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b] :
      ( ! [A6: b,B4: b,C2: b] :
          ( ( member_b @ A6 @ A4 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B4 ) @ R2 )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ C2 ) @ R2 )
             => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ C2 ) @ ( abstract_join_b @ R2 ) ) ) ) )
     => ( abstract_WCR_on_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_633_WCR__onI,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ! [A6: produc9001763001631046773rm_a_b,B4: produc9001763001631046773rm_a_b,C2: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ A6 @ A4 )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ B4 @ C2 ) @ ( abstra180165145518022959rm_a_b @ R2 ) ) ) ) )
     => ( abstra445168549219233930rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_634_WCR__onI,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ! [A6: produc3421996462152255079rm_a_b,B4: produc3421996462152255079rm_a_b,C2: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ A6 @ A4 )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ B4 @ C2 ) @ ( abstra2994028023477150241rm_a_b @ R2 ) ) ) ) )
     => ( abstra2712031032269736444rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_635_WCR__onI,axiom,
    ! [A4: set_set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ! [A6: set_term_a_b,B4: set_term_a_b,C2: set_term_a_b] :
          ( ( member_set_term_a_b @ A6 @ A4 )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C2 ) @ ( abstra2139004588645393680rm_a_b @ R2 ) ) ) ) )
     => ( abstra2457992723331821045rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_636_WCR__onI,axiom,
    ! [A4: set_se1467346398288745161rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ A6 @ A4 )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C2 ) @ ( abstra6273572973012767523rm_a_b @ R2 ) ) ) ) )
     => ( abstra6757280596944162430rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_637_WCR__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A4 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
     => ( abstra875157153669124638rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_638_WCR__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A6 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ R2 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C2 ) @ R2 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
     => ( abstra5026849107050860991rm_a_b @ R2 @ A4 ) ) ).

% WCR_onI
thf(fact_639_rstep__srstepI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).

% rstep_srstepI
thf(fact_640_funas__rel__converse,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R ) ) @ F ) ) ).

% funas_rel_converse
thf(fact_641_WCR__on__def,axiom,
    ( abstra2457992723331821045rm_a_b
    = ( ^ [R4: set_Pr7151917219565317845rm_a_b,A7: set_set_term_a_b] :
        ! [X2: set_term_a_b] :
          ( ( member_set_term_a_b @ X2 @ A7 )
         => ! [B2: set_term_a_b,C3: set_term_a_b] :
              ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ B2 ) @ R4 )
                & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ C3 ) @ R4 ) )
             => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B2 @ C3 ) @ ( abstra2139004588645393680rm_a_b @ R4 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_642_WCR__on__def,axiom,
    ( abstra6757280596944162430rm_a_b
    = ( ^ [R4: set_Pr647378613824986951rm_a_b,A7: set_se1467346398288745161rm_a_b] :
        ! [X2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ X2 @ A7 )
         => ! [B2: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
              ( ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X2 @ B2 ) @ R4 )
                & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X2 @ C3 ) @ R4 ) )
             => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B2 @ C3 ) @ ( abstra6273572973012767523rm_a_b @ R4 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_643_WCR__on__def,axiom,
    ( abstra875157153669124638rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
        ! [X2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X2 @ A7 )
         => ! [B2: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
              ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B2 ) @ R4 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ C3 ) @ R4 ) )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R4 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_644_WCR__on__def,axiom,
    ( abstra5026849107050860991rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [B2: term_a_b,C3: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B2 ) @ R4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ R4 ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_645_WCR__onE,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
      ( ( abstra4159178081913940476rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ( ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B @ C ) @ ( abstra4037915826850753057rm_a_b @ R2 ) )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ R2 )
           => ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_646_WCR__onE,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b,B: b,C: b] :
      ( ( abstract_WCR_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R2 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_647_WCR__onE,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,C: produc9001763001631046773rm_a_b] :
      ( ( abstra445168549219233930rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ( ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ B @ C ) @ ( abstra180165145518022959rm_a_b @ R2 ) )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ R2 )
           => ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_648_WCR__onE,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,C: produc3421996462152255079rm_a_b] :
      ( ( abstra2712031032269736444rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ( ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ B @ C ) @ ( abstra2994028023477150241rm_a_b @ R2 ) )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ R2 )
           => ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_649_WCR__onE,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C: set_term_a_b] :
      ( ( abstra2457992723331821045rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra2139004588645393680rm_a_b @ R2 ) )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_650_WCR__onE,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra6757280596944162430rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra6273572973012767523rm_a_b @ R2 ) )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_651_WCR__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_652_WCR__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ R2 ) ) ) ) ) ).

% WCR_onE
thf(fact_653_WCR__onD,axiom,
    ! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
      ( ( abstra4159178081913940476rm_a_b @ R2 @ A4 )
     => ( ( member8417600551952982416rm_a_b @ A @ A4 )
       => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ R2 )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C ) @ R2 )
           => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B @ C ) @ ( abstra4037915826850753057rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_654_WCR__onD,axiom,
    ! [R2: set_Product_prod_b_b,A4: set_b,A: b,B: b,C: b] :
      ( ( abstract_WCR_on_b @ R2 @ A4 )
     => ( ( member_b @ A @ A4 )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ R2 )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_655_WCR__onD,axiom,
    ! [R2: set_Pr7061646551094443591rm_a_b,A4: set_Pr7151917219565317845rm_a_b,A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b,C: produc9001763001631046773rm_a_b] :
      ( ( abstra445168549219233930rm_a_b @ R2 @ A4 )
     => ( ( member8888773126115863070rm_a_b @ A @ A4 )
       => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ R2 )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ C ) @ R2 )
           => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ B @ C ) @ ( abstra180165145518022959rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_656_WCR__onD,axiom,
    ! [R2: set_Pr4284312964791606471rm_a_b,A4: set_Pr647378613824986951rm_a_b,A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b,C: produc3421996462152255079rm_a_b] :
      ( ( abstra2712031032269736444rm_a_b @ R2 @ A4 )
     => ( ( member1375104729371992464rm_a_b @ A @ A4 )
       => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ R2 )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ C ) @ R2 )
           => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ B @ C ) @ ( abstra2994028023477150241rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_657_WCR__onD,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C: set_term_a_b] :
      ( ( abstra2457992723331821045rm_a_b @ R2 @ A4 )
     => ( ( member_set_term_a_b @ A @ A4 )
       => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ R2 )
           => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( abstra2139004588645393680rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_658_WCR__onD,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,A4: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra6757280596944162430rm_a_b @ R2 @ A4 )
     => ( ( member8823642877475723282rm_a_b @ A @ A4 )
       => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ R2 )
           => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( abstra6273572973012767523rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_659_WCR__onD,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ R2 )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_660_WCR__onD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ R2 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_661_diamond__E,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra7467186385360731951rm_a_b @ R2 )
     => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ R2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ) ).

% diamond_E
thf(fact_662_diamond__E,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8151087076625346926rm_a_b @ R2 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).

% diamond_E
thf(fact_663_diamond__def,axiom,
    ( abstra7467186385360731951rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) @ R4 ) @ ( relcom6159724393391289352rm_a_b @ R4 @ ( conver1575690562132235771rm_a_b @ R4 ) ) ) ) ) ).

% diamond_def
thf(fact_664_diamond__def,axiom,
    ( abstra8151087076625346926rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( conver7608559976937845093rm_a_b @ R4 ) ) ) ) ) ).

% diamond_def
thf(fact_665_srsteps__eq__ctxt__closed,axiom,
    ! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).

% srsteps_eq_ctxt_closed
thf(fact_666_UNC__E,axiom,
    ! [A4: set_Pr6848273517546626247rm_a_b,A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b] :
      ( ( abstra5923572653484115479rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member8417600551952982416rm_a_b @ A @ ( abstra1925539948312754931rm_a_b @ A4 ) )
         => ( ( member8417600551952982416rm_a_b @ B @ ( abstra1925539948312754931rm_a_b @ A4 ) )
           => ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra3453776217580455381rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_667_UNC__E,axiom,
    ! [A4: set_Product_prod_b_b,A: b,B: b] :
      ( ( abstract_UNC_b @ A4 )
     => ( ( A != B )
       => ( ( member_b @ A @ ( abstract_NF_b @ A4 ) )
         => ( ( member_b @ B @ ( abstract_NF_b @ A4 ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra3918519255145857645sion_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_668_UNC__E,axiom,
    ! [A4: set_Pr7061646551094443591rm_a_b,A: produc9001763001631046773rm_a_b,B: produc9001763001631046773rm_a_b] :
      ( ( abstra4736007659589303333rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member8888773126115863070rm_a_b @ A @ ( abstra5673374384174597889rm_a_b @ A4 ) )
         => ( ( member8888773126115863070rm_a_b @ B @ ( abstra5673374384174597889rm_a_b @ A4 ) )
           => ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A @ B ) @ ( abstra2517255444688506595rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_669_UNC__E,axiom,
    ! [A4: set_Pr4284312964791606471rm_a_b,A: produc3421996462152255079rm_a_b,B: produc3421996462152255079rm_a_b] :
      ( ( abstra9132637070351371799rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member1375104729371992464rm_a_b @ A @ ( abstra8119681912743043827rm_a_b @ A4 ) )
         => ( ( member1375104729371992464rm_a_b @ B @ ( abstra8119681912743043827rm_a_b @ A4 ) )
           => ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A @ B ) @ ( abstra3908750571075493333rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_670_UNC__E,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b,A: set_term_a_b,B: set_term_a_b] :
      ( ( abstra425847635066497690rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member_set_term_a_b @ A @ ( abstra5309567699948295358rm_a_b @ A4 ) )
         => ( ( member_set_term_a_b @ B @ ( abstra5309567699948295358rm_a_b @ A4 ) )
           => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( abstra6253361338556788316rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_671_UNC__E,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra5358366364211414553rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member8823642877475723282rm_a_b @ A @ ( abstra6901259896266686709rm_a_b @ A4 ) )
         => ( ( member8823642877475723282rm_a_b @ B @ ( abstra6901259896266686709rm_a_b @ A4 ) )
           => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( abstra5398323551124918487rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_672_UNC__E,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( abstra8480497335821202873rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
         => ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_673_UNC__E,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4649771476585718372rm_a_b @ A4 )
     => ( ( A != B )
       => ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
         => ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ) ) ) ).

% UNC_E
thf(fact_674_UNC__I,axiom,
    ! [A4: set_Pr6848273517546626247rm_a_b] :
      ( ! [A6: produc4523971788519308903rm_a_b,B4: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ A6 @ ( abstra1925539948312754931rm_a_b @ A4 ) )
         => ( ( member8417600551952982416rm_a_b @ B4 @ ( abstra1925539948312754931rm_a_b @ A4 ) )
           => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ B4 ) @ ( abstra3453776217580455381rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra5923572653484115479rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_675_UNC__I,axiom,
    ! [A4: set_Product_prod_b_b] :
      ( ! [A6: b,B4: b] :
          ( ( member_b @ A6 @ ( abstract_NF_b @ A4 ) )
         => ( ( member_b @ B4 @ ( abstract_NF_b @ A4 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B4 ) @ ( abstra3918519255145857645sion_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstract_UNC_b @ A4 ) ) ).

% UNC_I
thf(fact_676_UNC__I,axiom,
    ! [A4: set_Pr7061646551094443591rm_a_b] :
      ( ! [A6: produc9001763001631046773rm_a_b,B4: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ A6 @ ( abstra5673374384174597889rm_a_b @ A4 ) )
         => ( ( member8888773126115863070rm_a_b @ B4 @ ( abstra5673374384174597889rm_a_b @ A4 ) )
           => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ B4 ) @ ( abstra2517255444688506595rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra4736007659589303333rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_677_UNC__I,axiom,
    ! [A4: set_Pr4284312964791606471rm_a_b] :
      ( ! [A6: produc3421996462152255079rm_a_b,B4: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ A6 @ ( abstra8119681912743043827rm_a_b @ A4 ) )
         => ( ( member1375104729371992464rm_a_b @ B4 @ ( abstra8119681912743043827rm_a_b @ A4 ) )
           => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ B4 ) @ ( abstra3908750571075493333rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra9132637070351371799rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_678_UNC__I,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b] :
      ( ! [A6: set_term_a_b,B4: set_term_a_b] :
          ( ( member_set_term_a_b @ A6 @ ( abstra5309567699948295358rm_a_b @ A4 ) )
         => ( ( member_set_term_a_b @ B4 @ ( abstra5309567699948295358rm_a_b @ A4 ) )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( abstra6253361338556788316rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra425847635066497690rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_679_UNC__I,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ A6 @ ( abstra6901259896266686709rm_a_b @ A4 ) )
         => ( ( member8823642877475723282rm_a_b @ B4 @ ( abstra6901259896266686709rm_a_b @ A4 ) )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( abstra5398323551124918487rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra5358366364211414553rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_680_UNC__I,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
         => ( ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra8480497335821202873rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_681_UNC__I,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b] :
          ( ( member_term_a_b @ A6 @ ( abstract_NF_term_a_b @ A4 ) )
         => ( ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A4 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
             => ( A6 = B4 ) ) ) )
     => ( abstra4649771476585718372rm_a_b @ A4 ) ) ).

% UNC_I
thf(fact_682_UNC__def,axiom,
    ( abstra5923572653484115479rm_a_b
    = ( ^ [A7: set_Pr6848273517546626247rm_a_b] :
        ! [A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b] :
          ( ( ( member8417600551952982416rm_a_b @ A3 @ ( abstra1925539948312754931rm_a_b @ A7 ) )
            & ( member8417600551952982416rm_a_b @ B2 @ ( abstra1925539948312754931rm_a_b @ A7 ) )
            & ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ ( abstra3453776217580455381rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_683_UNC__def,axiom,
    ( abstract_UNC_b
    = ( ^ [A7: set_Product_prod_b_b] :
        ! [A3: b,B2: b] :
          ( ( ( member_b @ A3 @ ( abstract_NF_b @ A7 ) )
            & ( member_b @ B2 @ ( abstract_NF_b @ A7 ) )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A3 @ B2 ) @ ( abstra3918519255145857645sion_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_684_UNC__def,axiom,
    ( abstra4736007659589303333rm_a_b
    = ( ^ [A7: set_Pr7061646551094443591rm_a_b] :
        ! [A3: produc9001763001631046773rm_a_b,B2: produc9001763001631046773rm_a_b] :
          ( ( ( member8888773126115863070rm_a_b @ A3 @ ( abstra5673374384174597889rm_a_b @ A7 ) )
            & ( member8888773126115863070rm_a_b @ B2 @ ( abstra5673374384174597889rm_a_b @ A7 ) )
            & ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A3 @ B2 ) @ ( abstra2517255444688506595rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_685_UNC__def,axiom,
    ( abstra9132637070351371799rm_a_b
    = ( ^ [A7: set_Pr4284312964791606471rm_a_b] :
        ! [A3: produc3421996462152255079rm_a_b,B2: produc3421996462152255079rm_a_b] :
          ( ( ( member1375104729371992464rm_a_b @ A3 @ ( abstra8119681912743043827rm_a_b @ A7 ) )
            & ( member1375104729371992464rm_a_b @ B2 @ ( abstra8119681912743043827rm_a_b @ A7 ) )
            & ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A3 @ B2 ) @ ( abstra3908750571075493333rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_686_UNC__def,axiom,
    ( abstra425847635066497690rm_a_b
    = ( ^ [A7: set_Pr7151917219565317845rm_a_b] :
        ! [A3: set_term_a_b,B2: set_term_a_b] :
          ( ( ( member_set_term_a_b @ A3 @ ( abstra5309567699948295358rm_a_b @ A7 ) )
            & ( member_set_term_a_b @ B2 @ ( abstra5309567699948295358rm_a_b @ A7 ) )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ ( abstra6253361338556788316rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_687_UNC__def,axiom,
    ( abstra5358366364211414553rm_a_b
    = ( ^ [A7: set_Pr647378613824986951rm_a_b] :
        ! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
          ( ( ( member8823642877475723282rm_a_b @ A3 @ ( abstra6901259896266686709rm_a_b @ A7 ) )
            & ( member8823642877475723282rm_a_b @ B2 @ ( abstra6901259896266686709rm_a_b @ A7 ) )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ ( abstra5398323551124918487rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_688_UNC__def,axiom,
    ( abstra8480497335821202873rm_a_b
    = ( ^ [A7: set_Pr2972776593051762503rm_a_b] :
        ! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
          ( ( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A7 ) )
            & ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A7 ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_689_UNC__def,axiom,
    ( abstra4649771476585718372rm_a_b
    = ( ^ [A7: set_Pr4386577575007340137rm_a_b] :
        ! [A3: term_a_b,B2: term_a_b] :
          ( ( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A7 ) )
            & ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A7 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A7 ) ) )
         => ( A3 = B2 ) ) ) ) ).

% UNC_def
thf(fact_690_semi__confluence__imp__CR,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( converse_a_a @ R2 ) @ ( transitive_rtrancl_a @ R2 ) ) @ ( abstract_join_a @ R2 ) )
     => ( abstract_CR_on_a @ R2 @ top_top_set_a ) ) ).

% semi_confluence_imp_CR
thf(fact_691_semi__confluence__imp__CR,axiom,
    ! [R2: set_Product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ ( relcomp_b_b_b @ ( converse_b_b @ R2 ) @ ( transitive_rtrancl_b @ R2 ) ) @ ( abstract_join_b @ R2 ) )
     => ( abstract_CR_on_b @ R2 @ top_top_set_b ) ) ).

% semi_confluence_imp_CR
thf(fact_692_semi__confluence__imp__CR,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
     => ( abstra7359694574113289279rm_a_b @ R2 @ top_to1314267278846557113rm_a_b ) ) ).

% semi_confluence_imp_CR
thf(fact_693_semi__confluence__imp__CR,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
     => ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).

% semi_confluence_imp_CR
thf(fact_694_rsteps__srstepsI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).

% rsteps_srstepsI
thf(fact_695_srstep__ctxt__closed,axiom,
    ! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).

% srstep_ctxt_closed
thf(fact_696_trancl__rtrancl__absorb,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( transi9211502839322181930rm_a_b @ R6 ) )
      = ( transi2615809358984392588rm_a_b @ R6 ) ) ).

% trancl_rtrancl_absorb
thf(fact_697_trancl__rtrancl__absorb,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) )
      = ( transi7742714808557438673rm_a_b @ R6 ) ) ).

% trancl_rtrancl_absorb
thf(fact_698_rtrancl__trancl__absorb,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b] :
      ( ( transi9211502839322181930rm_a_b @ ( transi2615809358984392588rm_a_b @ R6 ) )
      = ( transi2615809358984392588rm_a_b @ R6 ) ) ).

% rtrancl_trancl_absorb
thf(fact_699_rtrancl__trancl__absorb,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = ( transi7742714808557438673rm_a_b @ R6 ) ) ).

% rtrancl_trancl_absorb
thf(fact_700_trancl__empty,axiom,
    ( ( transi7922773638565587891rm_a_b @ bot_bo197521221353338581rm_a_b )
    = bot_bo197521221353338581rm_a_b ) ).

% trancl_empty
thf(fact_701_trancl__empty,axiom,
    ( ( transi9211502839322181930rm_a_b @ bot_bo5040873672931234227rm_a_b )
    = bot_bo5040873672931234227rm_a_b ) ).

% trancl_empty
thf(fact_702_trancl__supteq,axiom,
    ( ( transi7922773638565587891rm_a_b @ subter523971068842742411eq_a_b )
    = subter523971068842742411eq_a_b ) ).

% trancl_supteq
thf(fact_703_semi__complete__onI,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
     => ( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
       => ( abstra5399326240126609895rm_a_b @ R2 @ A4 ) ) ) ).

% semi_complete_onI
thf(fact_704_semi__complete__onI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
     => ( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
       => ( abstra7162307272211697846rm_a_b @ R2 @ A4 ) ) ) ).

% semi_complete_onI
thf(fact_705_CR__onI,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
      ( ! [A6: produc4523971788519308903rm_a_b,B4: produc4523971788519308903rm_a_b,C2: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ A6 @ A4 )
         => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ B4 ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
           => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A6 @ C2 ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
             => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B4 @ C2 ) @ ( abstra4037915826850753057rm_a_b @ R2 ) ) ) ) )
     => ( abstra1401220437545577885rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_706_CR__onI,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b] :
      ( ! [A6: b,B4: b,C2: b] :
          ( ( member_b @ A6 @ A4 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ B4 ) @ ( transitive_rtrancl_b @ R2 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A6 @ C2 ) @ ( transitive_rtrancl_b @ R2 ) )
             => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ C2 ) @ ( abstract_join_b @ R2 ) ) ) ) )
     => ( abstract_CR_on_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_707_CR__onI,axiom,
    ! [A4: set_Pr7151917219565317845rm_a_b,R2: set_Pr7061646551094443591rm_a_b] :
      ( ! [A6: produc9001763001631046773rm_a_b,B4: produc9001763001631046773rm_a_b,C2: produc9001763001631046773rm_a_b] :
          ( ( member8888773126115863070rm_a_b @ A6 @ A4 )
         => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ B4 ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
           => ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ A6 @ C2 ) @ ( transi2165682467528973816rm_a_b @ R2 ) )
             => ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ B4 @ C2 ) @ ( abstra180165145518022959rm_a_b @ R2 ) ) ) ) )
     => ( abstra3276740543777786539rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_708_CR__onI,axiom,
    ! [A4: set_Pr647378613824986951rm_a_b,R2: set_Pr4284312964791606471rm_a_b] :
      ( ! [A6: produc3421996462152255079rm_a_b,B4: produc3421996462152255079rm_a_b,C2: produc3421996462152255079rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ A6 @ A4 )
         => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ B4 ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
           => ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ A6 @ C2 ) @ ( transi5504100148209286250rm_a_b @ R2 ) )
             => ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ B4 @ C2 ) @ ( abstra2994028023477150241rm_a_b @ R2 ) ) ) ) )
     => ( abstra3183407928357012893rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_709_CR__onI,axiom,
    ! [A4: set_set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ! [A6: set_term_a_b,B4: set_term_a_b,C2: set_term_a_b] :
          ( ( member_set_term_a_b @ A6 @ A4 )
         => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ B4 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
           => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A6 @ C2 ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
             => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C2 ) @ ( abstra2139004588645393680rm_a_b @ R2 ) ) ) ) )
     => ( abstra3512238261578967188rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_710_CR__onI,axiom,
    ! [A4: set_se1467346398288745161rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ! [A6: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
          ( ( member8823642877475723282rm_a_b @ A6 @ A4 )
         => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ B4 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
           => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A6 @ C2 ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
             => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C2 ) @ ( abstra6273572973012767523rm_a_b @ R2 ) ) ) ) )
     => ( abstra1879145755980819615rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_711_CR__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A4 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
     => ( abstra7359694574113289279rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_712_CR__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A6 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R2 @ A4 ) ) ).

% CR_onI
thf(fact_713_r__into__trancl_H,axiom,
    ! [P2: produc9001763001631046773rm_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ P2 @ R2 )
     => ( member8888773126115863070rm_a_b @ P2 @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ).

% r_into_trancl'
thf(fact_714_r__into__trancl_H,axiom,
    ! [P2: produc3421996462152255079rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ P2 @ R2 )
     => ( member1375104729371992464rm_a_b @ P2 @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ).

% r_into_trancl'
thf(fact_715_r__into__trancl_H,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ R2 )
     => ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).

% r_into_trancl'
thf(fact_716_r__into__trancl_H,axiom,
    ! [P2: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ R2 )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).

% r_into_trancl'
thf(fact_717_CR__imp__UNC,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( abstract_CR_on_a @ R2 @ top_top_set_a )
     => ( abstract_UNC_a @ R2 ) ) ).

% CR_imp_UNC
thf(fact_718_CR__imp__UNC,axiom,
    ! [R2: set_Product_prod_b_b] :
      ( ( abstract_CR_on_b @ R2 @ top_top_set_b )
     => ( abstract_UNC_b @ R2 ) ) ).

% CR_imp_UNC
thf(fact_719_CR__imp__UNC,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( abstra4649771476585718372rm_a_b @ R2 ) ) ).

% CR_imp_UNC
thf(fact_720_CR__imp__UNC,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
     => ( abstra8480497335821202873rm_a_b @ R2 ) ) ).

% CR_imp_UNC
thf(fact_721_converse__trancl__induct,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,P: set_term_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ! [Y4: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: set_term_a_b,Z3: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z3 @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_722_converse__trancl__induct,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,P: set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ! [Y4: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z3 @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_723_converse__trancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ! [Y4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z3 @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_724_converse__trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ! [Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_725_trancl__trans__induct,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,P: set_term_a_b > set_term_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ! [X3: set_term_a_b,Y4: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: set_term_a_b,Y4: set_term_a_b,Z3: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
                 => ( ( P @ Y4 @ Z3 )
                   => ( P @ X3 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_726_trancl__trans__induct,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,P: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
                 => ( ( P @ Y4 @ Z3 )
                   => ( P @ X3 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_727_trancl__trans__induct,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
                 => ( ( P @ Y4 @ Z3 )
                   => ( P @ X3 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_728_trancl__trans__induct,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ! [X3: term_a_b,Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
                 => ( ( P @ Y4 @ Z3 )
                   => ( P @ X3 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_729_trancl__into__trancl2,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_730_trancl__into__trancl2,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_731_trancl__into__trancl2,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_732_trancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_733_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ R2 )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_734_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ R2 )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_735_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ R2 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_736_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_737_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ! [X3: set_term_a_b] :
          ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ X3 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_738_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: set_Pr4386577575007340137rm_a_b] :
          ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ X3 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_739_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ X3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_740_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_741_converse__tranclE,axiom,
    ! [X: set_term_a_b,Z2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ R2 )
       => ~ ! [Y4: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_742_converse__tranclE,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ R2 )
       => ~ ! [Y4: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_743_converse__tranclE,axiom,
    ! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
       => ~ ! [Y4: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_744_converse__tranclE,axiom,
    ! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
       => ~ ! [Y4: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R2 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_745_r__r__into__trancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R6: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R6 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ R6 )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi2170614581841672169rm_a_b @ R6 ) ) ) ) ).

% r_r_into_trancl
thf(fact_746_r__r__into__trancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R6: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R6 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ R6 )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi6883191930764840330rm_a_b @ R6 ) ) ) ) ).

% r_r_into_trancl
thf(fact_747_r__r__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R6 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ R6 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R6 ) ) ) ) ).

% r_r_into_trancl
thf(fact_748_r__r__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R6: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R6 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R6 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R6 ) ) ) ) ).

% r_r_into_trancl
thf(fact_749_trancl__induct,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,P: set_term_a_b > $o] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ! [Y4: set_term_a_b] :
            ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: set_term_a_b,Z3: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y4 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
             => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_750_trancl__induct,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,P: set_Pr4386577575007340137rm_a_b > $o] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ! [Y4: set_Pr4386577575007340137rm_a_b] :
            ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y4 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
             => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_751_trancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ! [Y4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y4 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_752_trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ! [Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_753_trancl__trans,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,Z2: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_754_trancl__trans,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_755_trancl__trans,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_756_trancl__trans,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_757_tranclE,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
       => ~ ! [C2: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ C2 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_758_tranclE,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
       => ~ ! [C2: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ C2 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_759_tranclE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
       => ~ ! [C2: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C2 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_760_tranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
       => ~ ! [C2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_761_trancl_Or__into__trancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_762_trancl_Or__into__trancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_763_trancl_Or__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_764_trancl_Or__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_765_trancl_Osimps,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
      = ( ? [A3: set_term_a_b,B2: set_term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = B2 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ R2 ) )
        | ? [A3: set_term_a_b,B2: set_term_a_b,C3: set_term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
            & ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_766_trancl_Osimps,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
      = ( ? [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = B2 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ R2 ) )
        | ? [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
            & ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_767_trancl_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
      = ( ? [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = B2 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 ) )
        | ? [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_768_trancl_Osimps,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
      = ( ? [A3: term_a_b,B2: term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = B2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) )
        | ? [A3: term_a_b,B2: term_a_b,C3: term_a_b] :
            ( ( A1 = A3 )
            & ( A2 = C3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_769_trancl_Ocases,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ R2 )
       => ~ ! [B4: set_term_a_b] :
              ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B4 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
             => ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_770_trancl_Ocases,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ R2 )
       => ~ ! [B4: set_Pr4386577575007340137rm_a_b] :
              ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B4 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
             => ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_771_trancl_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ R2 )
       => ~ ! [B4: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B4 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_772_trancl_Ocases,axiom,
    ! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ R2 )
       => ~ ! [B4: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A2 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_773_trancl__map,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_774_trancl__map,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,F2: set_term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ! [X3: set_term_a_b,Y4: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_775_trancl__map,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > set_term_a_b,S: set_Pr7151917219565317845rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2170614581841672169rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_776_trancl__map,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,F2: set_term_a_b > set_term_a_b,S: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ! [X3: set_term_a_b,Y4: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2170614581841672169rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_777_trancl__map,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_778_trancl__map,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,F2: produc357393685978478089rm_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_779_trancl__map,axiom,
    ! [R2: set_Pr647378613824986951rm_a_b,F2: set_Pr4386577575007340137rm_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
          ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_780_trancl__map,axiom,
    ! [R2: set_Pr7151917219565317845rm_a_b,F2: set_term_a_b > produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
      ( ! [X3: set_term_a_b,Y4: set_term_a_b] :
          ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_781_trancl__map,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > set_Pr4386577575007340137rm_a_b,S: set_Pr647378613824986951rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6883191930764840330rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_782_trancl__map,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,F2: produc357393685978478089rm_a_b > set_term_a_b,S: set_Pr7151917219565317845rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2170614581841672169rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_783_trancl__mono__set,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
     => ( ord_le2593958535671617255rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ).

% trancl_mono_set
thf(fact_784_trancl__mono__set,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
     => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).

% trancl_mono_set
thf(fact_785_trancl__mono,axiom,
    ! [P2: produc9001763001631046773rm_a_b,R2: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ P2 @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ( ord_le2663425029446080117rm_a_b @ R2 @ S )
       => ( member8888773126115863070rm_a_b @ P2 @ ( transi2170614581841672169rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_786_trancl__mono,axiom,
    ! [P2: produc3421996462152255079rm_a_b,R2: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ P2 @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ( ord_le6958925370334017255rm_a_b @ R2 @ S )
       => ( member1375104729371992464rm_a_b @ P2 @ ( transi6883191930764840330rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_787_trancl__mono,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
       => ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_788_trancl__mono,axiom,
    ! [P2: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
       => ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_789_trancl__converse,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
      = ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).

% trancl_converse
thf(fact_790_trancl__converse,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
      = ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).

% trancl_converse
thf(fact_791_trancl__rtrancl__trancl,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_792_trancl__rtrancl__trancl,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_793_trancl__rtrancl__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_794_trancl__rtrancl__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_795_rtrancl__trancl__trancl,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,Z2: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_796_rtrancl__trancl__trancl,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_797_rtrancl__trancl__trancl,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_798_rtrancl__trancl__trancl,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_799_rtrancl__into__trancl2,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,R2: set_Pr7151917219565317845rm_a_b,C: set_term_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
     => ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C ) @ ( transi6904761420441344775rm_a_b @ R2 ) )
       => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C ) @ ( transi2170614581841672169rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_800_rtrancl__into__trancl2,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R2: set_Pr647378613824986951rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R2 )
     => ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C ) @ ( transi8812752159401800684rm_a_b @ R2 ) )
       => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C ) @ ( transi6883191930764840330rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_801_rtrancl__into__trancl2,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_802_rtrancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_803_rtrancl__into__trancl1,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ R2 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_804_rtrancl__into__trancl1,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_805_rtrancl__eq__or__trancl,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R6 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_806_trancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% trancl_into_rtrancl
thf(fact_807_tranclD2,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R6 ) )
     => ? [Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ R6 ) ) ) ).

% tranclD2
thf(fact_808_rtranclD,axiom,
    ! [A: term_a_b,B: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R6 ) ) ) ) ) ).

% rtranclD
thf(fact_809_tranclD,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R6 ) )
     => ? [Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R6 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ).

% tranclD
thf(fact_810_trancl__converseD,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% trancl_converseD
thf(fact_811_trancl__converseI,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).

% trancl_converseI
thf(fact_812_trancl__unfold__left,axiom,
    ( transi7922773638565587891rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).

% trancl_unfold_left
thf(fact_813_rtrancl__comp__trancl__conv,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 )
      = ( transi7922773638565587891rm_a_b @ R2 ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_814_NF__no__trancl__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
     => ! [B6: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B6 ) @ ( transi9211502839322181930rm_a_b @ A4 ) ) ) ).

% NF_no_trancl_step
thf(fact_815_NF__no__trancl__step,axiom,
    ! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
     => ! [B6: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B6 ) @ ( transi7922773638565587891rm_a_b @ A4 ) ) ) ).

% NF_no_trancl_step
thf(fact_816_srsteps__ctxt__closed,axiom,
    ! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).

% srsteps_ctxt_closed
thf(fact_817_CR__on__def,axiom,
    ( abstra8448919418672941150rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [B2: term_a_b,C3: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).

% CR_on_def
thf(fact_818_CR__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ) ).

% CR_onE
thf(fact_819_CR__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ) ).

% CR_onE
thf(fact_820_CR__onD,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).

% CR_onD
thf(fact_821_CR__onD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).

% CR_onD
thf(fact_822_sig__step__rsteps__dist,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
        = ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).

% sig_step_rsteps_dist
thf(fact_823_srsteps__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G2 )
     => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ) ).

% srsteps_monp
thf(fact_824_CR__divergence__imp__join,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).

% CR_divergence_imp_join
thf(fact_825_partially__localize__CR,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
      = ( ! [X2: term_a_b,Y3: term_a_b,Z4: term_a_b] :
            ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R2 )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).

% partially_localize_CR
thf(fact_826_CR__join__right__I,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).

% CR_join_right_I
thf(fact_827_CR__join__left__I,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).

% CR_join_left_I
thf(fact_828_srstepsD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srstepsD
thf(fact_829_diamond__imp__CR_H,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8151087076625346926rm_a_b @ S )
     => ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
       => ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b ) ) ) ) ).

% diamond_imp_CR'
thf(fact_830_srsteps__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
     => ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).

% srsteps_subst_closed
thf(fact_831_CR__NF__conv,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
     => ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R2 ) )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R2 ) )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ).

% CR_NF_conv
thf(fact_832_CR__NF__conv,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ).

% CR_NF_conv
thf(fact_833_sig__steps__join__ctxt__closed,axiom,
    ! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).

% sig_steps_join_ctxt_closed
thf(fact_834_rstep__trancl__sig__step__l,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V3: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V3 @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_l
thf(fact_835_rstep__trancl__sig__step__r,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V3: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V3 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_r
thf(fact_836_restrict__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S4: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R2 @ S4 ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% restrict_rtrancl
thf(fact_837_rstep__term__to__sig__l,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V3: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V3 @ S ) @ T ) @ ( rstep_a_b @ R ) ) ) ) ) ).

% rstep_term_to_sig_l
thf(fact_838_rstep__term__to__sig__r,axiom,
    ! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V3: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V3 @ T ) ) @ ( rstep_a_b @ R ) ) ) ) ) ).

% rstep_term_to_sig_r
thf(fact_839_trancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
     => ( ! [A6: term_a_b,B4: term_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) @ R2 )
           => ( P @ A6 @ B4 ) )
       => ( ! [A6: term_a_b,B4: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A6 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_840_CR__on__singletonI,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [B4: term_a_b,C2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% CR_on_singletonI
thf(fact_841_funas__gterm__gterm__of__term,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( ground_funas_gterm_a @ ( ground3594116404241818627rm_a_b @ T ) )
        = ( term_funas_term_a_b @ T ) ) ) ).

% funas_gterm_gterm_of_term
thf(fact_842_irrefl__tranclI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
        = bot_bo197521221353338581rm_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).

% irrefl_tranclI
thf(fact_843_conversionI,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).

% conversionI
thf(fact_844_rtrancl__Un__rtrancl,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
      = ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) ) ) ).

% rtrancl_Un_rtrancl
thf(fact_845_in__rtrancl__UnI,axiom,
    ! [X: produc357393685978478089rm_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R6 ) )
        | ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) ) ) ) ).

% in_rtrancl_UnI
thf(fact_846_rtrancl__Un__separator__converseE,axiom,
    ! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
     => ( ! [X3: term_a_b,Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ Q )
             => ( Y4 = X3 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_847_rtrancl__Un__separatorE,axiom,
    ! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
     => ( ! [X3: term_a_b,Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ Q )
             => ( X3 = Y4 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_848_rtrancl__Un__subset,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) ) ) ).

% rtrancl_Un_subset
thf(fact_849_rtrancl__union__subset__rtrancl__union__trancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( transi7922773638565587891rm_a_b @ S ) ) )
      = ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ S ) ) ) ).

% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_850_rstep__smycl__dist,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) ) )
      = ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% rstep_smycl_dist
thf(fact_851_sig__step__symcl__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ R ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ R ) ) )
      = ( sig_step_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% sig_step_symcl_dist
thf(fact_852_right__comp__S,axiom,
    ! [X: term_a_b,Y: term_a_b,S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_853_first__step__O,axiom,
    ! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( C4
        = ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C4 ) @ B5 ) )
       => ? [Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B5 ) ) ) ) ).

% first_step_O
thf(fact_854_first__step,axiom,
    ! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( C4
        = ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C4 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B5 )
         => ? [Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B5 ) ) ) ) ) ).

% first_step
thf(fact_855_firstStep,axiom,
    ! [L5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( L5
        = ( sup_su6776935440552674877rm_a_b @ S4 @ R6 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_856_rtrancl__U__push,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S4 @ R6 ) )
        = ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) ) ).

% rtrancl_U_push
thf(fact_857_compatible__conv,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) ) )
        = ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% compatible_conv
thf(fact_858_rtrancl__Un2__separatorE,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( ( relcom370159955682700863rm_a_b @ B5 @ A4 )
        = bot_bo197521221353338581rm_a_b )
     => ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
        = ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) ) ) ) ).

% rtrancl_Un2_separatorE
thf(fact_859_conversion__def,axiom,
    ( abstra1130568886744591910rm_a_b
    = ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A7 @ ( conver7608559976937845093rm_a_b @ A7 ) ) ) ) ) ).

% conversion_def
thf(fact_860_srstep__symcl__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
      = ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ) ).

% srstep_symcl_dist
thf(fact_861_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_862_conversionE,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) ) ) ).

% conversionE
thf(fact_863_sig__step__conversion__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
      = ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ) ) ).

% sig_step_conversion_dist
thf(fact_864_comp__rtrancl__trancl,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) ) @ S4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_865_quasi__commute__def,axiom,
    ( abstra4317384144805385751rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S2 ) ) ) ) ) ) ).

% quasi_commute_def
thf(fact_866_strongly__confluentI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 )
           => ? [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) )
     => ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).

% strongly_confluentI
thf(fact_867_sig__step__rsteps__eq__dist,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
     => ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) ) @ id_term_a_b )
        = ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).

% sig_step_rsteps_eq_dist
thf(fact_868_IdI,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).

% IdI
thf(fact_869_pair__in__Id__conv,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_870_rtrancl__reflcl__absorb,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ id_term_a_b )
      = ( transi7742714808557438673rm_a_b @ R6 ) ) ).

% rtrancl_reflcl_absorb
thf(fact_871_rtrancl__reflcl,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ id_term_a_b ) )
      = ( transi7742714808557438673rm_a_b @ R6 ) ) ).

% rtrancl_reflcl
thf(fact_872_rtrancl__empty,axiom,
    ( ( transi7742714808557438673rm_a_b @ bot_bo197521221353338581rm_a_b )
    = id_term_a_b ) ).

% rtrancl_empty
thf(fact_873_trancl__reflcl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
      = ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% trancl_reflcl
thf(fact_874_IdE,axiom,
    ! [P2: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
     => ~ ! [X3: term_a_b] :
            ( P2
           != ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_875_rtrancl__unfold,axiom,
    ( transi7742714808557438673rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ id_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ R4 ) ) ) ) ).

% rtrancl_unfold
thf(fact_876_rtrancl__trancl__reflcl,axiom,
    ( transi7742714808557438673rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ id_term_a_b ) ) ) ).

% rtrancl_trancl_reflcl
thf(fact_877_rtrancl__Int__subset,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ id_term_a_b @ S )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ S ) @ R2 ) @ S )
       => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ S ) ) ) ).

% rtrancl_Int_subset
thf(fact_878_strongly__confluent__on__def,axiom,
    ( abstra5664096641628173427rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [Y3: term_a_b,Z4: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z4 ) @ R4 ) )
             => ? [U3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_def
thf(fact_879_strongly__confluent__on__E11,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( abstra837640964651051114rm_a_b @ R2 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ X @ A4 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
           => ? [U2: produc357393685978478089rm_a_b] :
                ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_880_strongly__confluent__on__E11,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra5664096641628173427rm_a_b @ R2 @ A4 )
     => ( ( member_term_a_b @ X @ A4 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
           => ? [U2: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_881_IdD,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
     => ( A = B ) ) ).

% IdD
thf(fact_882_strongly__confluent__E1n,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b,N: nat] :
      ( ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
         => ? [U2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ).

% strongly_confluent_E1n
thf(fact_883_some__NF,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
        & ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).

% some_NF
thf(fact_884_some__NF,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
        & ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).

% some_NF
thf(fact_885_relpow__image,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R9: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
      ( ! [S3: term_a_b,T4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S3 ) @ ( F2 @ T4 ) ) @ R9 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S ) @ ( F2 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R9 ) ) ) ) ).

% relpow_image
thf(fact_886_relpow__Suc__D2_H,axiom,
    ! [N: nat,R6: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y6: term_a_b,Z5: term_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y6 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z5 ) @ R6 ) )
     => ? [W: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ W ) @ R6 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) ) ) ) ).

% relpow_Suc_D2'
thf(fact_887_SN__on__irrefl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
     => ! [X5: term_a_b] :
          ( ( member_term_a_b @ X5 @ A4 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ X5 ) @ R2 ) ) ) ).

% SN_on_irrefl
thf(fact_888_SN__on__induct,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( abstra2398554102055911763rm_a_b @ R6 @ A4 )
     => ( ( member5869715511025134514rm_a_b @ S @ A4 )
       => ( ! [T4: produc357393685978478089rm_a_b] :
              ( ! [U4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U4 ) @ R6 )
                 => ( P @ U4 ) )
             => ( P @ T4 ) )
         => ( P @ S ) ) ) ) ).

% SN_on_induct
thf(fact_889_SN__on__induct,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R6 @ A4 )
     => ( ( member_term_a_b @ S @ A4 )
       => ( ! [T4: term_a_b] :
              ( ! [U4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ R6 )
                 => ( P @ U4 ) )
             => ( P @ T4 ) )
         => ( P @ S ) ) ) ) ).

% SN_on_induct
thf(fact_890_rtrancl__imp__relpow,axiom,
    ! [P2: produc357393685978478089rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R6 ) )
     => ? [N2: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R6 ) ) ) ).

% rtrancl_imp_relpow
thf(fact_891_relpow__imp__rtrancl,axiom,
    ! [P2: produc357393685978478089rm_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ).

% relpow_imp_rtrancl
thf(fact_892_rtrancl__power,axiom,
    ! [P2: produc357393685978478089rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = ( ? [N3: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R6 ) ) ) ) ).

% rtrancl_power
thf(fact_893_SN__induct__rule,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ top_top_set_term_a_b )
     => ( ! [A6: term_a_b] :
            ( ! [B6: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ R2 )
               => ( P @ B6 ) )
           => ( P @ A6 ) )
       => ( P @ A ) ) ) ).

% SN_induct_rule
thf(fact_894_SN__imp__minimal,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
     => ! [Q2: set_Pr4386577575007340137rm_a_b] :
          ( ? [X5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X5 @ Q2 )
         => ? [X3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
              & ! [Y6: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y6 ) @ A4 )
                 => ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_895_SN__imp__minimal,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
     => ! [Q2: set_term_a_b] :
          ( ? [X5: term_a_b] : ( member_term_a_b @ X5 @ Q2 )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ Q2 )
              & ! [Y6: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ A4 )
                 => ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_896_refl__not__SN,axiom,
    ! [X: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R6 )
     => ~ ( abstra4720023341729745482rm_a_b @ R6 @ top_top_set_term_a_b ) ) ).

% refl_not_SN
thf(fact_897_rtrancl__len__E,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ~ ! [N2: nat] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N2 @ R2 ) ) ) ).

% rtrancl_len_E
thf(fact_898_rtrancl__r__diff__Id,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
      = ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% rtrancl_r_diff_Id
thf(fact_899_SN__on__induct_H,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R6 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
     => ( ! [T4: term_a_b] :
            ( ! [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ R6 )
               => ( P @ U4 ) )
           => ( P @ T4 ) )
       => ( P @ S ) ) ) ).

% SN_on_induct'
thf(fact_900_step__reflects__SN__on,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 )
         => ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) )
     => ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% step_reflects_SN_on
thf(fact_901_SN__on__imp__on__minimal,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ! [Q2: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ Q2 )
         => ? [X3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
              & ! [Y6: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y6 ) @ R2 )
                 => ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_902_SN__on__imp__on__minimal,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ! [Q2: set_term_a_b] :
          ( ( member_term_a_b @ X @ Q2 )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ Q2 )
              & ! [Y6: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ R2 )
                 => ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_903_step__preserves__SN__on,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
       => ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).

% step_preserves_SN_on
thf(fact_904_SN__on__induct__acc__style,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,A: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R6 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
     => ( ! [X3: term_a_b] :
            ( ( abstra4720023341729745482rm_a_b @ R6 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
           => ( ! [Y6: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ R6 )
                 => ( P @ Y6 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% SN_on_induct_acc_style
thf(fact_905_lower__set__imp__not__SN__on,axiom,
    ! [S: produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ S @ X4 )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ X4 )
           => ? [Xa2: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ Xa2 @ X4 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R6 ) ) )
       => ~ ( abstra2398554102055911763rm_a_b @ R6 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_906_lower__set__imp__not__SN__on,axiom,
    ! [S: term_a_b,X4: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ S @ X4 )
     => ( ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ X4 )
           => ? [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ X4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R6 ) ) )
       => ~ ( abstra4720023341729745482rm_a_b @ R6 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_907_SN__on__all__reducts__SN__on__conv,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
      = ( ! [B2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B2 ) @ R2 )
           => ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ) ).

% SN_on_all_reducts_SN_on_conv
thf(fact_908_SN__on__O__push,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( ( abstra4720023341729745482rm_a_b @ R6 @ X4 )
       => ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ X4 ) ) ) ).

% SN_on_O_push
thf(fact_909_NF__imp__SN__on,axiom,
    ! [A: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ R6 ) )
     => ( abstra2398554102055911763rm_a_b @ R6 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ).

% NF_imp_SN_on
thf(fact_910_rtrancl__diff__decomp,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ) ).

% rtrancl_diff_decomp
thf(fact_911_steps__reflect__SN__on,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
      ( ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
       => ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ) ).

% steps_reflect_SN_on
thf(fact_912_steps__preserve__SN__on,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
     => ( ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
       => ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).

% steps_preserve_SN_on
thf(fact_913_compatible__SN,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
       => ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).

% compatible_SN
thf(fact_914_SN__on__imp__normalizability,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
     => ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ).

% SN_on_imp_normalizability
thf(fact_915_SN__reaches__NF,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ? [Y4: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
          & ( member5869715511025134514rm_a_b @ Y4 @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).

% SN_reaches_NF
thf(fact_916_SN__reaches__NF,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ? [Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
          & ( member_term_a_b @ Y4 @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).

% SN_reaches_NF
thf(fact_917_compatible__SN_H,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
       => ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).

% compatible_SN'
thf(fact_918_qc__SN__relto__iff,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R2 ) ) ) )
     => ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) @ top_top_set_term_a_b )
        = ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).

% qc_SN_relto_iff
thf(fact_919_the__NF,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( abstra2242782261322123075rm_a_b @ A4 @ A ) ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).

% the_NF
thf(fact_920_the__NF__conv,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
         => ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
            = ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).

% the_NF_conv
thf(fact_921_the__NF__step,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A4 )
         => ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
            = ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).

% the_NF_step
thf(fact_922_the__NF__NF,axiom,
    ! [A4: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
     => ( ( abstra7359694574113289279rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
       => ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A4 @ A ) @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).

% the_NF_NF
thf(fact_923_the__NF__steps,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
         => ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
            = ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).

% the_NF_steps
thf(fact_924_Field__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) )
      = ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) ) ) ).

% Field_insert
thf(fact_925_acyclic__insert,axiom,
    ! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
      = ( ( transi5314701259734769157rm_a_b @ R2 )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% acyclic_insert
thf(fact_926_FieldI2,axiom,
    ! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R6 )
     => ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R6 ) ) ) ).

% FieldI2
thf(fact_927_FieldI2,axiom,
    ! [I: term_a_b,J: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R6 )
     => ( member_term_a_b @ J @ ( field_term_a_b @ R6 ) ) ) ).

% FieldI2
thf(fact_928_FieldI1,axiom,
    ! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R6 )
     => ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R6 ) ) ) ).

% FieldI1
thf(fact_929_FieldI1,axiom,
    ! [I: term_a_b,J: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R6 )
     => ( member_term_a_b @ I @ ( field_term_a_b @ R6 ) ) ) ).

% FieldI1
thf(fact_930_acyclic__def,axiom,
    ( transi5314701259734769157rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).

% acyclic_def
thf(fact_931_acyclicI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
     => ( transi5314701259734769157rm_a_b @ R2 ) ) ).

% acyclicI
thf(fact_932_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
       => ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_933_refl__on__singleton,axiom,
    ! [X: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).

% refl_on_singleton
thf(fact_934_Total__subset__Id,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( total_on_term_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( ord_le118470702582115849rm_a_b @ R2 @ id_term_a_b )
       => ( ( R2 = bot_bo197521221353338581rm_a_b )
          | ? [A6: term_a_b] :
              ( R2
              = ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).

% Total_subset_Id
thf(fact_935_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b,I: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R2 ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_936_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,J: term_a_b,I: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J @ I ) @ R2 )
          | ( J = I ) )
       => ( ( member_term_a_b @ I @ ( field_term_a_b @ R2 ) )
         => ( ( member_term_a_b @ J @ ( field_term_a_b @ R2 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_937_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ! [X5: term_a_b] :
          ( ( member_term_a_b @ X5 @ ( field_term_a_b @ R2 ) )
         => ! [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Xa2 ) @ R2 )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X5 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_938_refl__on__domain,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
       => ( ( member5869715511025134514rm_a_b @ A @ A4 )
          & ( member5869715511025134514rm_a_b @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_939_refl__on__domain,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( refl_on_term_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
       => ( ( member_term_a_b @ A @ A4 )
          & ( member_term_a_b @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_940_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ! [X3: term_a_b] :
            ( ! [Y6: term_a_b] :
                ( ( ( Y6 != X3 )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 ) )
               => ( P @ Y6 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_941_refl__onD2,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
       => ( member5869715511025134514rm_a_b @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_942_refl__onD2,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( member_term_a_b @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_943_refl__onD1,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
       => ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).

% refl_onD1
thf(fact_944_refl__onD1,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( member_term_a_b @ X @ A4 ) ) ) ).

% refl_onD1
thf(fact_945_refl__onD,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_946_refl__onD,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ A4 @ R2 )
     => ( ( member_term_a_b @ A @ A4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_947_total__on__def,axiom,
    ( total_on_term_a_b
    = ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [Y3: term_a_b] :
              ( ( member_term_a_b @ Y3 @ A7 )
             => ( ( X2 != Y3 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R4 )
                  | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_948_total__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ Y4 @ A4 )
           => ( ( X3 != Y4 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
                | ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) ) )
     => ( total_4010809703584253837rm_a_b @ A4 @ R2 ) ) ).

% total_onI
thf(fact_949_total__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member_term_a_b @ X3 @ A4 )
         => ( ( member_term_a_b @ Y4 @ A4 )
           => ( ( X3 != Y4 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) ) )
     => ( total_on_term_a_b @ A4 @ R2 ) ) ).

% total_onI
thf(fact_950_reflD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_951_reflI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
     => ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% reflI
thf(fact_952_well__order__induct__imp,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ! [Y6: produc357393685978478089rm_a_b] :
                ( ( ( Y6 != X3 )
                  & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ X3 ) @ R2 ) )
               => ( ( member5869715511025134514rm_a_b @ Y6 @ ( field_6884932134483023318rm_a_b @ R2 ) )
                 => ( P @ Y6 ) ) )
           => ( ( member5869715511025134514rm_a_b @ X3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
             => ( P @ X3 ) ) )
       => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_953_well__order__induct__imp,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ! [X3: term_a_b] :
            ( ! [Y6: term_a_b] :
                ( ( ( Y6 != X3 )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 ) )
               => ( ( member_term_a_b @ Y6 @ ( field_term_a_b @ R2 ) )
                 => ( P @ Y6 ) ) )
           => ( ( member_term_a_b @ X3 @ ( field_term_a_b @ R2 ) )
             => ( P @ X3 ) ) )
       => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_954_refl__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% refl_rtrancl
thf(fact_955_totalI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( X3 != Y4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
            | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) ) )
     => ( total_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% totalI
thf(fact_956_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
       => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_957_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ B5 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_958_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) @ B ) @ R2 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_959_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ A @ B5 )
         => ( ! [B4: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ B4 @ B5 )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ R2 ) )
           => ( A
              = ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_960_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ A @ B5 )
         => ( ! [B4: term_a_b] :
                ( ( member_term_a_b @ B4 @ B5 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 ) )
           => ( A
              = ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_961_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_962_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_963_wo__rel_Omax2__def,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
         => ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
            = B ) )
        & ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
         => ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_964_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B )
              = A )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_965_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
              = A )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_966_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B )
              = B )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_967_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
              = B )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_968_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_969_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_970_linear__order__on__singleton,axiom,
    ! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).

% linear_order_on_singleton
thf(fact_971_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
            = ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_972_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
            = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_973_Range__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( range_8688746797885257759rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) )
      = ( insert_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R2 ) ) ) ).

% Range_insert
thf(fact_974_Range__rtrancl,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( range_8688746797885257759rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = top_top_set_term_a_b ) ).

% Range_rtrancl
thf(fact_975_term_Oinj__map,axiom,
    ! [F1: a > a,F22: b > b] :
      ( ( inj_on_a_a @ F1 @ top_top_set_a )
     => ( ( inj_on_b_b @ F22 @ top_top_set_b )
       => ( inj_on7375953189722882843rm_a_b @ ( map_term_a_a_b_b @ F1 @ F22 ) @ top_top_set_term_a_b ) ) ) ).

% term.inj_map
thf(fact_976_Range__iff,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R2 ) )
      = ( ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_977_RangeE,axiom,
    ! [B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R2 ) )
     => ~ ! [A6: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_978_Range_Ointros,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( member_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R2 ) ) ) ).

% Range.intros
thf(fact_979_Range_Osimps,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R2 ) )
      = ( ? [A3: term_a_b,B2: term_a_b] :
            ( ( A = B2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_980_Range_Ocases,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R2 ) )
     => ~ ! [A6: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_981_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
        = ( ! [A7: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R2 ) )
             => ( ( A7 != bot_bot_set_term_a_b )
               => ? [X2: term_a_b] :
                    ( ( member_term_a_b @ X2 @ A7 )
                    & ! [Y3: term_a_b] :
                        ( ( member_term_a_b @ Y3 @ A7 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_982_apfst__conv,axiom,
    ! [F2: term_a_b > term_a_b,X: term_a_b,Y: term_a_b] :
      ( ( produc4217944709291808634rm_a_b @ F2 @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
      = ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_983_apsnd__conv,axiom,
    ! [F2: term_a_b > term_a_b,X: term_a_b,Y: term_a_b] :
      ( ( produc6963505556121627836rm_a_b @ F2 @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
      = ( produc7020197800436672577rm_a_b @ X @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_984_minimal__imp__wf,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [Q3: set_Pr4386577575007340137rm_a_b] :
          ( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q3 )
         => ? [X5: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X5 @ Q3 )
              & ! [Y4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Y4 ) @ R2 )
                 => ~ ( member5869715511025134514rm_a_b @ Y4 @ Q3 ) ) ) )
     => ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).

% minimal_imp_wf
thf(fact_985_minimal__imp__wf,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [Q3: set_term_a_b] :
          ( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q3 )
         => ? [X5: term_a_b] :
              ( ( member_term_a_b @ X5 @ Q3 )
              & ! [Y4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y4 ) @ R2 )
                 => ~ ( member_term_a_b @ Y4 @ Q3 ) ) ) )
     => ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).

% minimal_imp_wf
thf(fact_986_gterm__of__term__inj,axiom,
    ! [S4: set_term_a_b] :
      ( ! [T4: term_a_b] :
          ( ( member_term_a_b @ T4 @ S4 )
         => ( term_ground_a_b @ T4 ) )
     => ( inj_on1763302225069724520term_a @ ground3594116404241818627rm_a_b @ S4 ) ) ).

% gterm_of_term_inj
thf(fact_987_wf__eq__minimal2,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [A7: set_term_a_b] :
          ( ( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R4 ) )
            & ( A7 != bot_bot_set_term_a_b ) )
         => ? [X2: term_a_b] :
              ( ( member_term_a_b @ X2 @ A7 )
              & ! [Y3: term_a_b] :
                  ( ( member_term_a_b @ Y3 @ A7 )
                 => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_988_wf__insert,axiom,
    ! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
      = ( ( wf_term_a_b @ R2 )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% wf_insert
thf(fact_989_qc__wf__relto__iff,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) ) @ R6 ) )
     => ( ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) )
        = ( wf_term_a_b @ R6 ) ) ) ).

% qc_wf_relto_iff
thf(fact_990_wf__def,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [P3: term_a_b > $o] :
          ( ! [X2: term_a_b] :
              ( ! [Y3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 )
                 => ( P3 @ Y3 ) )
             => ( P3 @ X2 ) )
         => ! [X6: term_a_b] : ( P3 @ X6 ) ) ) ) ).

% wf_def
thf(fact_991_wfE__min,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R6 )
     => ( ( member5869715511025134514rm_a_b @ X @ Q )
       => ~ ! [Z3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z3 @ Q )
             => ~ ! [Y6: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z3 ) @ R6 )
                   => ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_992_wfE__min,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
      ( ( wf_term_a_b @ R6 )
     => ( ( member_term_a_b @ X @ Q )
       => ~ ! [Z3: term_a_b] :
              ( ( member_term_a_b @ Z3 @ Q )
             => ~ ! [Y6: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z3 ) @ R6 )
                   => ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_993_wfI__min,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Q3: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ Q3 )
         => ? [Xa2: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Xa2 @ Q3 )
              & ! [Y4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Xa2 ) @ R6 )
                 => ~ ( member5869715511025134514rm_a_b @ Y4 @ Q3 ) ) ) )
     => ( wf_Pro2335863617654816626rm_a_b @ R6 ) ) ).

% wfI_min
thf(fact_994_wfI__min,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Q3: set_term_a_b] :
          ( ( member_term_a_b @ X3 @ Q3 )
         => ? [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ Q3 )
              & ! [Y4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Xa2 ) @ R6 )
                 => ~ ( member_term_a_b @ Y4 @ Q3 ) ) ) )
     => ( wf_term_a_b @ R6 ) ) ).

% wfI_min
thf(fact_995_wfUNIVI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [P4: term_a_b > $o,X3: term_a_b] :
          ( ! [Xa2: term_a_b] :
              ( ! [Y4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Xa2 ) @ R2 )
                 => ( P4 @ Y4 ) )
             => ( P4 @ Xa2 ) )
         => ( P4 @ X3 ) )
     => ( wf_term_a_b @ R2 ) ) ).

% wfUNIVI
thf(fact_996_wf__asym,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R2 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R2 ) ) ) ).

% wf_asym
thf(fact_997_wf__induct,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ( ! [X3: term_a_b] :
            ( ! [Y6: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 )
               => ( P @ Y6 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_998_wf__irrefl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).

% wf_irrefl
thf(fact_999_wf__not__sym,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R2 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R2 ) ) ) ).

% wf_not_sym
thf(fact_1000_wf__not__refl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).

% wf_not_refl
thf(fact_1001_wf__eq__minimal,axiom,
    ( wf_Pro2335863617654816626rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b] :
        ! [Q4: set_Pr4386577575007340137rm_a_b] :
          ( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ Q4 )
         => ? [X2: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X2 @ Q4 )
              & ! [Y3: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R4 )
                 => ~ ( member5869715511025134514rm_a_b @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1002_wf__eq__minimal,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [Q4: set_term_a_b] :
          ( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ Q4 )
         => ? [X2: term_a_b] :
              ( ( member_term_a_b @ X2 @ Q4 )
              & ! [Y3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 )
                 => ~ ( member_term_a_b @ Y3 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1003_wf__induct__rule,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ( ! [X3: term_a_b] :
            ( ! [Y6: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 )
               => ( P @ Y6 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_1004_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ( wf_term_a_b @ S )
     => ( ! [X3: term_a_b,Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) @ S ) )
       => ( wf_term_a_b @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1005_wfE__min_H,axiom,
    ! [R6: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R6 )
     => ( ( Q != bot_bo197521221353338581rm_a_b )
       => ~ ! [Z3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z3 @ Q )
             => ~ ! [Y6: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z3 ) @ R6 )
                   => ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_1006_wfE__min_H,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
      ( ( wf_term_a_b @ R6 )
     => ( ( Q != bot_bot_set_term_a_b )
       => ~ ! [Z3: term_a_b] :
              ( ( member_term_a_b @ Z3 @ Q )
             => ~ ! [Y6: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z3 ) @ R6 )
                   => ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_1007_in__inv__image,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( inv_im3004224898869642620rm_a_b @ R2 @ F2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_1008_term_Omap__disc__iff,axiom,
    ! [F1: a > a,F22: b > b,A: term_a_b] :
      ( ( is_Var_a_b @ ( map_term_a_a_b_b @ F1 @ F22 @ A ) )
      = ( is_Var_a_b @ A ) ) ).

% term.map_disc_iff
thf(fact_1009_ground__vars__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( vars_term_a_b @ T )
        = bot_bot_set_b ) ) ).

% ground_vars_term_empty
thf(fact_1010_ground__substI,axiom,
    ! [S: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ ( vars_t1977812157029556855rm_a_b @ S ) )
         => ( term_ground_a_b @ ( Sigma @ X3 ) ) )
     => ( term_ground_a_b @ ( subst_4153759000330037794_a_b_b @ S @ Sigma ) ) ) ).

% ground_substI
thf(fact_1011_vars__term__empty__ground,axiom,
    ! [S: term_a_b] :
      ( ( ( vars_term_a_b @ S )
        = bot_bot_set_b )
     => ( term_ground_a_b @ S ) ) ).

% vars_term_empty_ground
thf(fact_1012_ground__substD,axiom,
    ! [L: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b,X: produc357393685978478089rm_a_b] :
      ( ( term_ground_a_b @ ( subst_4153759000330037794_a_b_b @ L @ Sigma ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( vars_t1977812157029556855rm_a_b @ L ) )
       => ( term_ground_a_b @ ( Sigma @ X ) ) ) ) ).

% ground_substD
thf(fact_1013_supteq__imp__vars__term__subset,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ ( vars_term_a_b @ S ) ) ) ).

% supteq_imp_vars_term_subset
thf(fact_1014_vars__term__supteq,axiom,
    ! [X: b,T: term_a_b] :
      ( ( member_b @ X @ ( vars_term_a_b @ T ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( var_b_a @ X ) ) @ subter523971068842742411eq_a_b ) ) ).

% vars_term_supteq
thf(fact_1015_trans__ctxt__sig__imp__all__ctxt__closed,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ! [T4: term_a_b] :
            ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T4 ) @ F4 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ T4 ) @ R2 ) )
       => ( ! [C6: subterm_and_ctxt_a_b,S3: term_a_b,T4: term_a_b] :
              ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C6 ) @ F4 )
             => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S3 ) @ F4 )
               => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T4 ) @ F4 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T4 ) @ R2 )
                   => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C6 @ S3 ) @ ( subter2376574525758040790rm_a_b @ C6 @ T4 ) ) @ R2 ) ) ) ) )
         => ( terms_5226143800768910156ed_a_b @ F4 @ R2 ) ) ) ) ).

% trans_ctxt_sig_imp_all_ctxt_closed
thf(fact_1016_in__lex__prod,axiom,
    ! [A: term_a_b,B: term_a_b,A5: term_a_b,B3: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R2 @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1017_trans__on__def,axiom,
    ( trans_on_term_a_b
    = ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [Y3: term_a_b] :
              ( ( member_term_a_b @ Y3 @ A7 )
             => ! [Z4: term_a_b] :
                  ( ( member_term_a_b @ Z4 @ A7 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R4 )
                   => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R4 )
                     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z4 ) @ R4 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_1018_trans__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ Y4 @ A4 )
           => ( ( member5869715511025134514rm_a_b @ Z3 @ A4 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R2 )
                 => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z3 ) @ R2 ) ) ) ) ) )
     => ( trans_5404704112166290345rm_a_b @ A4 @ R2 ) ) ).

% trans_onI
thf(fact_1019_trans__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A4 )
         => ( ( member_term_a_b @ Y4 @ A4 )
           => ( ( member_term_a_b @ Z3 @ A4 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
                 => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 ) ) ) ) ) )
     => ( trans_on_term_a_b @ A4 @ R2 ) ) ).

% trans_onI
thf(fact_1020_trans__onD,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( trans_5404704112166290345rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ X @ A4 )
       => ( ( member5869715511025134514rm_a_b @ Y @ A4 )
         => ( ( member5869715511025134514rm_a_b @ Z2 @ A4 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ R2 )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1021_trans__onD,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ A4 @ R2 )
     => ( ( member_term_a_b @ X @ A4 )
       => ( ( member_term_a_b @ Y @ A4 )
         => ( ( member_term_a_b @ Z2 @ A4 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R2 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1022_transD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1023_transE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1024_transI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R2 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 ) ) )
     => ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% transI
thf(fact_1025_trans__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] : ( trans_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).

% trans_rtrancl
thf(fact_1026_relcomp3__transI,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,V3: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ B5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V3 ) @ B5 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V3 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) ) ) ) ) ) ).

% relcomp3_transI
thf(fact_1027_trans__refl__imp__rtrancl__id,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 )
       => ( ( transi7742714808557438673rm_a_b @ R2 )
          = R2 ) ) ) ).

% trans_refl_imp_rtrancl_id
thf(fact_1028_trans__singleton,axiom,
    ! [A: term_a_b] : ( trans_on_term_a_b @ top_top_set_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ bot_bo197521221353338581rm_a_b ) ) ).

% trans_singleton
thf(fact_1029_under__incr,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
       => ( ord_le2705286416250468010rm_a_b @ ( order_under_term_a_b @ R2 @ A ) @ ( order_under_term_a_b @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1030_antisym__on__def,axiom,
    ( antisym_on_term_a_b
    = ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [Y3: term_a_b] :
              ( ( member_term_a_b @ Y3 @ A7 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 )
                 => ( X2 = Y3 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1031_antisym__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ Y4 @ A4 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisy5041284635856308844rm_a_b @ A4 @ R2 ) ) ).

% antisym_onI
thf(fact_1032_antisym__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member_term_a_b @ X3 @ A4 )
         => ( ( member_term_a_b @ Y4 @ A4 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisym_on_term_a_b @ A4 @ R2 ) ) ).

% antisym_onI
thf(fact_1033_antisym__onD,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( antisy5041284635856308844rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ X @ A4 )
       => ( ( member5869715511025134514rm_a_b @ Y @ A4 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1034_antisym__onD,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( antisym_on_term_a_b @ A4 @ R2 )
     => ( ( member_term_a_b @ X @ A4 )
       => ( ( member_term_a_b @ Y @ A4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1035_sym__on__def,axiom,
    ( sym_on_term_a_b
    = ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ! [Y3: term_a_b] :
              ( ( member_term_a_b @ Y3 @ A7 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R4 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).

% sym_on_def
thf(fact_1036_sym__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
         => ( ( member5869715511025134514rm_a_b @ Y4 @ A4 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R2 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) )
     => ( sym_on5489148433978082986rm_a_b @ A4 @ R2 ) ) ).

% sym_onI
thf(fact_1037_sym__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member_term_a_b @ X3 @ A4 )
         => ( ( member_term_a_b @ Y4 @ A4 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) )
     => ( sym_on_term_a_b @ A4 @ R2 ) ) ).

% sym_onI
thf(fact_1038_sym__onD,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( sym_on5489148433978082986rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ X @ A4 )
       => ( ( member5869715511025134514rm_a_b @ Y @ A4 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R2 ) ) ) ) ) ).

% sym_onD
thf(fact_1039_sym__onD,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( sym_on_term_a_b @ A4 @ R2 )
     => ( ( member_term_a_b @ X @ A4 )
       => ( ( member_term_a_b @ Y @ A4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) ) ) ) ) ).

% sym_onD
thf(fact_1040_symI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) )
     => ( sym_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% symI
thf(fact_1041_symE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
      ( ( sym_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ).

% symE
thf(fact_1042_symD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( sym_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) ) ) ).

% symD
thf(fact_1043_sym__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( sym_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( sym_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% sym_rtrancl
thf(fact_1044_antisymI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 )
           => ( X3 = Y4 ) ) )
     => ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% antisymI
thf(fact_1045_antisymD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1046_acyclic__impl__antisym__rtrancl,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( transi5314701259734769157rm_a_b @ R2 )
     => ( antisym_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% acyclic_impl_antisym_rtrancl
thf(fact_1047_underS__incr,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
         => ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R2 @ A ) @ ( order_1619770106104096206rm_a_b @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1048_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
        = ( ! [A7: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R2 ) )
             => ( ( A7 != bot_bot_set_term_a_b )
               => ? [X2: term_a_b] :
                    ( ( member_term_a_b @ X2 @ A7 )
                    & ! [Y3: term_a_b] :
                        ( ( member_term_a_b @ Y3 @ A7 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1049_underS__incl__iff,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
         => ( ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R2 @ A ) @ ( order_8163520290499871439rm_a_b @ R2 @ B ) )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1050_underS__incl__iff,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
         => ( ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R2 @ A ) @ ( order_1619770106104096206rm_a_b @ R2 @ B ) )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1051_well__order__on__domain,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_8637870607158948775rm_a_b @ A4 @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
       => ( ( member5869715511025134514rm_a_b @ A @ A4 )
          & ( member5869715511025134514rm_a_b @ B @ A4 ) ) ) ) ).

% well_order_on_domain
thf(fact_1052_well__order__on__domain,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
       => ( ( member_term_a_b @ A @ A4 )
          & ( member_term_a_b @ B @ A4 ) ) ) ) ).

% well_order_on_domain
thf(fact_1053_underS__I,axiom,
    ! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( I != J )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R6 )
       => ( member5869715511025134514rm_a_b @ I @ ( order_8163520290499871439rm_a_b @ R6 @ J ) ) ) ) ).

% underS_I
thf(fact_1054_underS__I,axiom,
    ! [I: term_a_b,J: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( I != J )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R6 )
       => ( member_term_a_b @ I @ ( order_1619770106104096206rm_a_b @ R6 @ J ) ) ) ) ).

% underS_I
thf(fact_1055_underS__E,axiom,
    ! [I: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ I @ ( order_8163520290499871439rm_a_b @ R6 @ J ) )
     => ( ( I != J )
        & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R6 ) ) ) ).

% underS_E
thf(fact_1056_underS__E,axiom,
    ! [I: term_a_b,R6: set_Pr4386577575007340137rm_a_b,J: term_a_b] :
      ( ( member_term_a_b @ I @ ( order_1619770106104096206rm_a_b @ R6 @ J ) )
     => ( ( I != J )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R6 ) ) ) ).

% underS_E
thf(fact_1057_bsqr__max2,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A1: term_a_b,A2: term_a_b,B1: term_a_b,B22: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( produc7020197800436672577rm_a_b @ B1 @ B22 ) ) @ ( bNF_We8522098035112346167rm_a_b @ R2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A1 @ A2 ) @ ( bNF_We4618759378340593734rm_a_b @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).

% bsqr_max2
thf(fact_1058_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R2 )
     => ( ( bNF_We8430659725849320639rm_a_b @ R2 @ A4 @ B )
        = ( ( member5869715511025134514rm_a_b @ B @ A4 )
          & ! [X2: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X2 @ A4 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ X2 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1059_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R2 )
     => ( ( bNF_We695392074589568990rm_a_b @ R2 @ A4 @ B )
        = ( ( member_term_a_b @ B @ A4 )
          & ! [X2: term_a_b] :
              ( ( member_term_a_b @ X2 @ A4 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ X2 ) @ R2 ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1060_trancl__steps__relpow,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
      ( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
       => ? [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M @ B ) ) ) ) ) ).

% trancl_steps_relpow
thf(fact_1061_relpow__refl__mono,axiom,
    ! [Rel: set_Pr4386577575007340137rm_a_b,M2: nat,N: nat,A: term_a_b,B: term_a_b] :
      ( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M2 @ Rel ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).

% relpow_refl_mono
thf(fact_1062_non__strict__ending,axiom,
    ! [T: nat > term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S4 ) @ S4 )
       => ( ( abstra4720023341729745482rm_a_b @ S4 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
         => ? [J2: nat] :
            ! [I3: nat] :
              ( ( ord_less_eq_nat @ J2 @ I3 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R6 @ S4 ) ) ) ) ) ) ).

% non_strict_ending
thf(fact_1063_mlex__leq,axiom,
    ! [F2: term_a_b > nat,X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R6 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F2 @ R6 ) ) ) ) ).

% mlex_leq
thf(fact_1064_Domain__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( domain6518119340328324150rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) )
      = ( insert_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) ) ) ).

% Domain_insert
thf(fact_1065_Domain__rtrancl,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b] :
      ( ( domain6518119340328324150rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = top_top_set_term_a_b ) ).

% Domain_rtrancl
thf(fact_1066_SN__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [F5: nat > produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
         => ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) )
     => ( abstra2398554102055911763rm_a_b @ R2 @ A4 ) ) ).

% SN_onI
thf(fact_1067_SN__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [F5: nat > term_a_b] :
          ( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
         => ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) )
     => ( abstra4720023341729745482rm_a_b @ R2 @ A4 ) ) ).

% SN_onI
thf(fact_1068_not__SN__onI,axiom,
    ! [F2: nat > produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( F2 @ zero_zero_nat ) @ X4 )
     => ( ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R6 )
       => ~ ( abstra2398554102055911763rm_a_b @ R6 @ X4 ) ) ) ).

% not_SN_onI
thf(fact_1069_not__SN__onI,axiom,
    ! [F2: nat > term_a_b,X4: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ ( F2 @ zero_zero_nat ) @ X4 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R6 )
       => ~ ( abstra4720023341729745482rm_a_b @ R6 @ X4 ) ) ) ).

% not_SN_onI
thf(fact_1070_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R8: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R8 @ R6 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R8 )
       => ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R6 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_1071_Domain_Ocases,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) )
     => ~ ! [B4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_1072_Domain_Osimps,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) )
      = ( ? [A3: term_a_b,B2: term_a_b] :
            ( ( A = A3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_1073_Domain_ODomainI,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_1074_DomainE,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) )
     => ~ ! [B4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_1075_Domain__iff,axiom,
    ! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R2 ) )
      = ( ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_1076_relpow__E2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X != Z2 ) )
       => ~ ! [Y4: term_a_b,M: nat] :
              ( ( N
                = ( suc @ M ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R6 )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ M @ R6 ) ) ) ) ) ) ).

% relpow_E2
thf(fact_1077_relpow__E,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X != Z2 ) )
       => ~ ! [Y4: term_a_b,M: nat] :
              ( ( N
                = ( suc @ M ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( compow4057154403645558940rm_a_b @ M @ R6 ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ R6 ) ) ) ) ) ).

% relpow_E
thf(fact_1078_SN__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
     => ~ ? [F6: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
            & ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).

% SN_onE
thf(fact_1079_SN__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
     => ~ ? [F6: nat > term_a_b] :
            ( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
            & ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).

% SN_onE
thf(fact_1080_SN__on__def,axiom,
    ( abstra2398554102055911763rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F7: nat > produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( F7 @ zero_zero_nat ) @ A7 )
              & ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ).

% SN_on_def
thf(fact_1081_SN__on__def,axiom,
    ( abstra4720023341729745482rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
          ~ ? [F7: nat > term_a_b] :
              ( ( member_term_a_b @ ( F7 @ zero_zero_nat ) @ A7 )
              & ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ).

% SN_on_def
thf(fact_1082_not__SN__onE,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
     => ~ ! [F5: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
           => ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).

% not_SN_onE
thf(fact_1083_not__SN__onE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ~ ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
     => ~ ! [F5: nat > term_a_b] :
            ( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
           => ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).

% not_SN_onE
thf(fact_1084_relpow__Suc__I2,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b,Z2: term_a_b,N: nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R6 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R6 ) ) ) ) ).

% relpow_Suc_I2
thf(fact_1085_relpow__Suc__E2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R6 ) )
     => ~ ! [Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R6 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) ) ) ) ).

% relpow_Suc_E2
thf(fact_1086_relpow__Suc__D2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R6 ) )
     => ? [Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R6 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) ) ) ) ).

% relpow_Suc_D2
thf(fact_1087_relpow__Suc__I,axiom,
    ! [X: term_a_b,Y: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R6 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R6 ) ) ) ) ).

% relpow_Suc_I
thf(fact_1088_relpow__Suc__E,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R6 ) )
     => ~ ! [Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ R6 ) ) ) ).

% relpow_Suc_E
thf(fact_1089_Not__Domain__rtrancl,axiom,
    ! [X: produc357393685978478089rm_a_b,R6: set_Pr2972776593051762503rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ ( domain7769946957726046796rm_a_b @ R6 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R6 ) )
        = ( X = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1090_Not__Domain__rtrancl,axiom,
    ! [X: term_a_b,R6: set_Pr4386577575007340137rm_a_b,Y: term_a_b] :
      ( ~ ( member_term_a_b @ X @ ( domain6518119340328324150rm_a_b @ R6 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
        = ( X = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1091_relpow__0__E,axiom,
    ! [X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R6 ) )
     => ( X = Y ) ) ).

% relpow_0_E
thf(fact_1092_relpow__0__I,axiom,
    ! [X: term_a_b,R6: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R6 ) ) ).

% relpow_0_I
thf(fact_1093_subterm_Olift__Suc__mono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_mono_le
thf(fact_1094_subterm_Olift__Suc__antimono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N ) @ ( F2 @ N4 ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_antimono_le
thf(fact_1095_relpow__Suc__E2_H,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A4 ) )
     => ~ ! [Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ A4 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).

% relpow_Suc_E2'
thf(fact_1096_chain__imp__not__SN__on,axiom,
    ! [F2: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R2 )
     => ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ ( F2 @ I ) @ bot_bot_set_term_a_b ) ) ) ).

% chain_imp_not_SN_on
thf(fact_1097_chain__imp__rtrancl,axiom,
    ! [S4: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ ( suc @ I2 ) ) ) @ R2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I ) @ ( S4 @ J ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).

% chain_imp_rtrancl
thf(fact_1098_wf__no__infinite__down__chainE,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
      ( ( wf_term_a_b @ R2 )
     => ~ ! [K: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ K ) ) @ ( F2 @ K ) ) @ R2 ) ) ).

% wf_no_infinite_down_chainE
thf(fact_1099_wf__iff__no__infinite__down__chain,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F7: nat > term_a_b] :
            ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ ( suc @ I4 ) ) @ ( F7 @ I4 ) ) @ R4 ) ) ) ).

% wf_iff_no_infinite_down_chain
thf(fact_1100_chain__Un__SN__on__imp__first__step,axiom,
    ! [T: nat > term_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ S4 ) )
     => ( ( abstra4720023341729745482rm_a_b @ S4 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
       => ? [I2: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ R6 )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ S4 )
                  & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ R6 ) ) ) ) ) ) ).

% chain_Un_SN_on_imp_first_step
thf(fact_1101_mlex__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,F2: term_a_b > nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F2 @ R6 ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R6 ) ) ) ) ).

% mlex_iff
thf(fact_1102_mlex__less,axiom,
    ! [F2: term_a_b > nat,X: term_a_b,Y: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F2 @ R6 ) ) ) ).

% mlex_less
thf(fact_1103_chain__imp__trancl,axiom,
    ! [S4: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ ( suc @ I2 ) ) ) @ R2 )
     => ( ( ord_less_nat @ I @ J )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I ) @ ( S4 @ J ) ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% chain_imp_trancl
thf(fact_1104_wf__bounded__measure,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F2: term_a_b > nat] :
      ( ! [A6: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A6 ) @ R2 )
         => ( ( ord_less_eq_nat @ ( Ub @ B4 ) @ ( Ub @ A6 ) )
            & ( ord_less_eq_nat @ ( F2 @ B4 ) @ ( Ub @ A6 ) )
            & ( ord_less_nat @ ( F2 @ A6 ) @ ( F2 @ B4 ) ) ) )
     => ( wf_term_a_b @ R2 ) ) ).

% wf_bounded_measure
thf(fact_1105_trancl__power,axiom,
    ! [P2: produc357393685978478089rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R6 ) )
      = ( ? [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
            & ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R6 ) ) ) ) ) ).

% trancl_power
thf(fact_1106_rtrancl__fun__conv,axiom,
    ! [S: term_a_b,T: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
      = ( ? [F7: nat > term_a_b,N3: nat] :
            ( ( ( F7 @ zero_zero_nat )
              = S )
            & ( ( F7 @ N3 )
              = T )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N3 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R6 ) ) ) ) ) ).

% rtrancl_fun_conv
thf(fact_1107_rtrancl__imp__seq,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ? [F5: nat > term_a_b,N2: nat] :
          ( ( ( F5 @ zero_zero_nat )
            = X )
          & ( ( F5 @ N2 )
            = Y )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N2 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ) ).

% rtrancl_imp_seq
thf(fact_1108_relpow__fun__conv,axiom,
    ! [A: term_a_b,B: term_a_b,N: nat,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) )
      = ( ? [F7: nat > term_a_b] :
            ( ( ( F7 @ zero_zero_nat )
              = A )
            & ( ( F7 @ N )
              = B )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R6 ) ) ) ) ) ).

% relpow_fun_conv
thf(fact_1109_relpow__seq,axiom,
    ! [X: term_a_b,Y: term_a_b,N: nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
     => ? [F5: nat > term_a_b] :
          ( ( ( F5 @ zero_zero_nat )
            = X )
          & ( ( F5 @ N )
            = Y )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ) ).

% relpow_seq
thf(fact_1110_in__measure,axiom,
    ! [X: term_a_b,Y: term_a_b,F2: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measure_term_a_b @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1111_max__ext_Omax__extI,axiom,
    ! [X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( finite7663359492169029298rm_a_b @ X4 )
     => ( ( finite7663359492169029298rm_a_b @ Y7 )
       => ( ( Y7 != bot_bo197521221353338581rm_a_b )
         => ( ! [X3: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ X3 @ X4 )
               => ? [Xa2: produc357393685978478089rm_a_b] :
                    ( ( member5869715511025134514rm_a_b @ Xa2 @ Y7 )
                    & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R6 ) ) )
           => ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X4 @ Y7 ) @ ( max_ex2405036269911770436rm_a_b @ R6 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1112_max__ext_Omax__extI,axiom,
    ! [X4: set_term_a_b,Y7: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( finite1334135622471826475rm_a_b @ X4 )
     => ( ( finite1334135622471826475rm_a_b @ Y7 )
       => ( ( Y7 != bot_bot_set_term_a_b )
         => ( ! [X3: term_a_b] :
                ( ( member_term_a_b @ X3 @ X4 )
               => ? [Xa2: term_a_b] :
                    ( ( member_term_a_b @ Xa2 @ Y7 )
                    & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R6 ) ) )
           => ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X4 @ Y7 ) @ ( max_ext_term_a_b @ R6 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1113_subterm_Ofinite__has__minimal2,axiom,
    ! [A4: set_term_a_b,A: term_a_b] :
      ( ( finite1334135622471826475rm_a_b @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ? [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ subter523971068842742411eq_a_b )
            & ! [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ A4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ subter523971068842742411eq_a_b )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% subterm.finite_has_minimal2
thf(fact_1114_subterm_Ofinite__has__maximal2,axiom,
    ! [A4: set_term_a_b,A: term_a_b] :
      ( ( finite1334135622471826475rm_a_b @ A4 )
     => ( ( member_term_a_b @ A @ A4 )
       => ? [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ A ) @ subter523971068842742411eq_a_b )
            & ! [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ A4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X3 ) @ subter523971068842742411eq_a_b )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% subterm.finite_has_maximal2
thf(fact_1115_subterm_Ofinite__has__maximal,axiom,
    ! [A4: set_term_a_b] :
      ( ( finite1334135622471826475rm_a_b @ A4 )
     => ( ( A4 != bot_bot_set_term_a_b )
       => ? [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
            & ! [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ A4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X3 ) @ subter523971068842742411eq_a_b )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% subterm.finite_has_maximal
thf(fact_1116_subterm_Ofinite__has__minimal,axiom,
    ! [A4: set_term_a_b] :
      ( ( finite1334135622471826475rm_a_b @ A4 )
     => ( ( A4 != bot_bot_set_term_a_b )
       => ? [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
            & ! [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ A4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ subter523971068842742411eq_a_b )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% subterm.finite_has_minimal
thf(fact_1117_max__ext_Ocases,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( max_ex2405036269911770436rm_a_b @ R6 ) )
     => ~ ( ( finite7663359492169029298rm_a_b @ A1 )
         => ( ( finite7663359492169029298rm_a_b @ A2 )
           => ( ( A2 != bot_bo197521221353338581rm_a_b )
             => ~ ! [X5: produc357393685978478089rm_a_b] :
                    ( ( member5869715511025134514rm_a_b @ X5 @ A1 )
                   => ? [Xa3: produc357393685978478089rm_a_b] :
                        ( ( member5869715511025134514rm_a_b @ Xa3 @ A2 )
                        & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Xa3 ) @ R6 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1118_max__ext_Ocases,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( max_ext_term_a_b @ R6 ) )
     => ~ ( ( finite1334135622471826475rm_a_b @ A1 )
         => ( ( finite1334135622471826475rm_a_b @ A2 )
           => ( ( A2 != bot_bot_set_term_a_b )
             => ~ ! [X5: term_a_b] :
                    ( ( member_term_a_b @ X5 @ A1 )
                   => ? [Xa3: term_a_b] :
                        ( ( member_term_a_b @ Xa3 @ A2 )
                        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Xa3 ) @ R6 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1119_max__ext_Osimps,axiom,
    ! [A1: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,R6: set_Pr2972776593051762503rm_a_b] :
      ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A2 ) @ ( max_ex2405036269911770436rm_a_b @ R6 ) )
      = ( ( finite7663359492169029298rm_a_b @ A1 )
        & ( finite7663359492169029298rm_a_b @ A2 )
        & ( A2 != bot_bo197521221353338581rm_a_b )
        & ! [X2: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X2 @ A1 )
           => ? [Y3: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ Y3 @ A2 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R6 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1120_max__ext_Osimps,axiom,
    ! [A1: set_term_a_b,A2: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
      ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A2 ) @ ( max_ext_term_a_b @ R6 ) )
      = ( ( finite1334135622471826475rm_a_b @ A1 )
        & ( finite1334135622471826475rm_a_b @ A2 )
        & ( A2 != bot_bot_set_term_a_b )
        & ! [X2: term_a_b] :
            ( ( member_term_a_b @ X2 @ A1 )
           => ? [Y3: term_a_b] :
                ( ( member_term_a_b @ Y3 @ A2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1121_cons__chain,axiom,
    ! [X: term_a_b,S4: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( S4 @ zero_zero_nat ) ) @ R2 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ ( suc @ I2 ) ) ) @ R2 )
       => ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( case_nat_term_a_b @ X @ S4 @ I3 ) @ ( case_nat_term_a_b @ X @ S4 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).

% cons_chain
thf(fact_1122_relpow__union__cases,axiom,
    ! [A: term_a_b,D: term_a_b,N: nat,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ D ) @ ( compow4057154403645558940rm_a_b @ N @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ D ) @ ( compow4057154403645558940rm_a_b @ N @ B5 ) )
        | ? [B4: term_a_b,C2: term_a_b,K: nat,M: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ ( compow4057154403645558940rm_a_b @ K @ B5 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ A4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ D ) @ ( compow4057154403645558940rm_a_b @ M @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) )
            & ( N
              = ( suc @ ( plus_plus_nat @ K @ M ) ) ) ) ) ) ).

% relpow_union_cases
thf(fact_1123_chain__imp__relpow,axiom,
    ! [S4: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ ( suc @ I2 ) ) ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I ) @ ( S4 @ ( plus_plus_nat @ I @ J ) ) ) @ ( compow4057154403645558940rm_a_b @ J @ R2 ) ) ) ).

% chain_imp_relpow
thf(fact_1124_term_Omap__sel_I1_J,axiom,
    ! [A: term_a_b,F1: a > a,F22: b > b] :
      ( ( is_Var_a_b @ A )
     => ( ( the_Var_a_b @ ( map_term_a_a_b_b @ F1 @ F22 @ A ) )
        = ( F22 @ ( the_Var_a_b @ A ) ) ) ) ).

% term.map_sel(1)
thf(fact_1125_irrefl__on__def,axiom,
    ( irrefl_on_term_a_b
    = ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X2: term_a_b] :
          ( ( member_term_a_b @ X2 @ A7 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ R4 ) ) ) ) ).

% irrefl_on_def
thf(fact_1126_irrefl__onI,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A4 )
         => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ A6 ) @ R2 ) )
     => ( irrefl6620156600162922239rm_a_b @ A4 @ R2 ) ) ).

% irrefl_onI
thf(fact_1127_irrefl__onI,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b] :
          ( ( member_term_a_b @ A6 @ A4 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ R2 ) )
     => ( irrefl_on_term_a_b @ A4 @ R2 ) ) ).

% irrefl_onI
thf(fact_1128_irrefl__onD,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( irrefl6620156600162922239rm_a_b @ A4 @ R2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A4 )
       => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R2 ) ) ) ).

% irrefl_onD
thf(fact_1129_irrefl__onD,axiom,
    ! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( irrefl_on_term_a_b @ A4 @ R2 )
     => ( ( member_term_a_b @ A @ A4 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ) ).

% irrefl_onD
thf(fact_1130_irreflD,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 ) ) ).

% irreflD
thf(fact_1131_irreflI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ R2 )
     => ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).

% irreflI
thf(fact_1132_some__NF__WCR,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ( ( abstra875157153669124638rm_a_b @ R2
          @ ( collec99177395925924084rm_a_b
            @ ^ [X2: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
         => ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
           => ( Y
              = ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) ) ) ) ) ).

% some_NF_WCR
thf(fact_1133_some__NF__WCR,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ( ( abstra5026849107050860991rm_a_b @ R2
          @ ( collect_term_a_b
            @ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
         => ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
           => ( Y
              = ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) ) ) ) ) ).

% some_NF_WCR
thf(fact_1134_suptI,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ) ).

% suptI
thf(fact_1135_supt__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ subterm_and_supt_a_b ) ) ).

% supt_subst
thf(fact_1136_adapt__vars__gr__map__vars,axiom,
    ! [T: term_a_b,F2: b > b] :
      ( ( term_ground_a_b @ T )
     => ( ( map_term_a_a_b_b
          @ ^ [X2: a] : X2
          @ F2
          @ T )
        = ( term_a2780309480206857766_a_b_b @ T ) ) ) ).

% adapt_vars_gr_map_vars
thf(fact_1137_supt__supteq__not__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b ) ) ) ).

% supt_supteq_not_supteq
thf(fact_1138_supteq__supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supteq_supt_trans
thf(fact_1139_supt__supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_supteq_trans
thf(fact_1140_supteq__supt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
        | ( S = T ) ) ) ).

% supteq_supt_conv
thf(fact_1141_supt__supteq__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ( S != T ) ) ) ).

% supt_supteq_conv
thf(fact_1142_supteq__not__supt,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( S = T ) ) ) ).

% supteq_not_supt
thf(fact_1143_supt__imp__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% supt_imp_supteq
thf(fact_1144_suptE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( S != T ) ) ) ).

% suptE
thf(fact_1145_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.not_eq_order_implies_strict
thf(fact_1146_subterm_Odual__order_Ostrict__implies__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.dual_order.strict_implies_order
thf(fact_1147_subterm_Odual__order_Ostrict__iff__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.dual_order.strict_iff_order
thf(fact_1148_subterm_Odual__order_Oorder__iff__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.dual_order.order_iff_strict
thf(fact_1149_subterm_Oorder_Ostrict__implies__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.order.strict_implies_order
thf(fact_1150_subterm_Odual__order_Ostrict__iff__not,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.strict_iff_not
thf(fact_1151_subterm_Odual__order_Ostrict__trans2,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans2
thf(fact_1152_subterm_Odual__order_Ostrict__trans1,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans1
thf(fact_1153_subterm_Oorder_Ostrict__iff__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.order.strict_iff_order
thf(fact_1154_subterm_Oorder_Oorder__iff__strict,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.order.order_iff_strict
thf(fact_1155_subterm_Oorder_Ostrict__iff__not,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.strict_iff_not
thf(fact_1156_subterm_Oorder_Ostrict__trans2,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans2
thf(fact_1157_subterm_Oorder_Ostrict__trans1,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans1
thf(fact_1158_subterm_Ole__imp__less__or__eq,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
        | ( X = Y ) ) ) ).

% subterm.le_imp_less_or_eq
thf(fact_1159_subterm_Oless__le__not__le,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.less_le_not_le
thf(fact_1160_subterm_Oless__le__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_le_trans
thf(fact_1161_subterm_Ole__less__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_less_trans
thf(fact_1162_subterm_Oantisym__conv2,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b ) )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv2
thf(fact_1163_subterm_Oantisym__conv1,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv1
thf(fact_1164_subterm_Ole__neq__trans,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( A != B )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_neq_trans
thf(fact_1165_subterm_Oless__imp__le,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.less_imp_le
thf(fact_1166_subterm_Onless__le,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        | ( A = B ) ) ) ).

% subterm.nless_le
thf(fact_1167_subterm_Oless__le,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
        & ( X != Y ) ) ) ).

% subterm.less_le
thf(fact_1168_subterm_Ole__less,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
        | ( X = Y ) ) ) ).

% subterm.le_less
thf(fact_1169_subterm_OleD,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b ) ) ).

% subterm.leD
thf(fact_1170_supt__var,axiom,
    ! [X: b,U: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X ) @ U ) @ subterm_and_supt_a_b ) ).

% supt_var
thf(fact_1171_Var__supt,axiom,
    ! [X: b,T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X ) @ T ) @ subterm_and_supt_a_b ) ).

% Var_supt
thf(fact_1172_under__def,axiom,
    ( order_under_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
          ( collect_term_a_b
          @ ^ [B2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R4 ) ) ) ) ).

% under_def
thf(fact_1173_subterm_Oless__asym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym
thf(fact_1174_subterm_Oless__asym_H,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym'
thf(fact_1175_subterm_Oless__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_trans
thf(fact_1176_subterm_Oorder_Oasym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.order.asym
thf(fact_1177_subterm_Oless__irrefl,axiom,
    ! [X: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subterm_and_supt_a_b ) ).

% subterm.less_irrefl
thf(fact_1178_subterm_Oless__imp__neq,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( X != Y ) ) ).

% subterm.less_imp_neq
thf(fact_1179_subterm_Oless__not__sym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_not_sym
thf(fact_1180_subterm_Oless__imp__triv,axiom,
    ! [Y: term_a_b,X: term_a_b,P: $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b )
       => P ) ) ).

% subterm.less_imp_triv
thf(fact_1181_subterm_Odual__order_Oasym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ).

% subterm.dual_order.asym
thf(fact_1182_subterm_Oless__imp__not__eq,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( X != Y ) ) ).

% subterm.less_imp_not_eq
thf(fact_1183_subterm_Oless__imp__not__eq2,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( Y != X ) ) ).

% subterm.less_imp_not_eq2
thf(fact_1184_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subterm_and_supt_a_b ) ).

% subterm.dual_order.irrefl
thf(fact_1185_subterm_Oless__imp__not__less,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_imp_not_less
thf(fact_1186_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_1187_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_1188_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans
thf(fact_1189_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_1190_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_1191_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_1192_supt__neqD,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_1193_supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_trans
thf(fact_1194_supt__irrefl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_irrefl
thf(fact_1195_supt__not__sym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subterm_and_supt_a_b ) ) ).

% supt_not_sym
thf(fact_1196_supt__not__refl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_not_refl
thf(fact_1197_subterm__induct,axiom,
    ! [P: term_a_b > $o,T: term_a_b] :
      ( ! [T4: term_a_b] :
          ( ! [S7: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ S7 ) @ subterm_and_supt_a_b )
             => ( P @ S7 ) )
         => ( P @ T4 ) )
     => ( P @ T ) ) ).

% subterm_induct
thf(fact_1198_pred__equals__eq2,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ( ^ [X2: term_a_b,Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 ) )
        = ( ^ [X2: term_a_b,Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ S4 ) ) )
      = ( R6 = S4 ) ) ).

% pred_equals_eq2
thf(fact_1199_term_Omap__ident,axiom,
    ! [T: term_a_b] :
      ( ( map_term_a_a_b_b
        @ ^ [X2: a] : X2
        @ ^ [X2: b] : X2
        @ T )
      = T ) ).

% term.map_ident
thf(fact_1200_underS__def,axiom,
    ( order_1619770106104096206rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
          ( collect_term_a_b
          @ ^ [B2: term_a_b] :
              ( ( B2 != A3 )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R4 ) ) ) ) ) ).

% underS_def
thf(fact_1201_wf__finite__segments,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] :
      ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R2 )
     => ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
       => ( ! [X3: term_a_b] :
              ( finite1334135622471826475rm_a_b
              @ ( collect_term_a_b
                @ ^ [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R2 ) ) )
         => ( wf_term_a_b @ R2 ) ) ) ) ).

% wf_finite_segments
thf(fact_1202_subterm_Olift__Suc__mono__less__iff,axiom,
    ! [F2: nat > term_a_b,M2: nat,N: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ M2 ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% subterm.lift_Suc_mono_less_iff
thf(fact_1203_subterm_Olift__Suc__mono__less,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subterm_and_supt_a_b )
     => ( ( ord_less_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.lift_Suc_mono_less
thf(fact_1204_subterm_OacyclicI__order,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ A6 ) @ ( F2 @ B4 ) ) @ subterm_and_supt_a_b ) )
     => ( transi5314701259734769157rm_a_b @ R2 ) ) ).

% subterm.acyclicI_order
thf(fact_1205_SN__WCR__reaches__NF,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ( ( abstra875157153669124638rm_a_b @ R2
          @ ( collec99177395925924084rm_a_b
            @ ^ [X2: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) )
       => ? [X3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
            & ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R2 ) )
            & ! [Y6: produc357393685978478089rm_a_b] :
                ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
                  & ( member5869715511025134514rm_a_b @ Y6 @ ( abstra1902783451776770197rm_a_b @ R2 ) ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% SN_WCR_reaches_NF
thf(fact_1206_SN__WCR__reaches__NF,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ( ( abstra5026849107050860991rm_a_b @ R2
          @ ( collect_term_a_b
            @ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
       => ? [X3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
            & ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R2 ) )
            & ! [Y6: term_a_b] :
                ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
                  & ( member_term_a_b @ Y6 @ ( abstract_NF_term_a_b @ R2 ) ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% SN_WCR_reaches_NF
thf(fact_1207_subterm_OgreaterThanLessThan__empty,axiom,
    ! [K2: term_a_b,L: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ L ) @ subter523971068842742411eq_a_b )
     => ( ( set_gr2587711877945340532rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2
          @ L )
        = bot_bot_set_term_a_b ) ) ).

% subterm.greaterThanLessThan_empty
thf(fact_1208_subterm_OgreaterThanAtMost__empty,axiom,
    ! [K2: term_a_b,L: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ L ) @ subter523971068842742411eq_a_b )
     => ( ( set_gr5290542583680275920rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2
          @ L )
        = bot_bot_set_term_a_b ) ) ).

% subterm.greaterThanAtMost_empty
thf(fact_1209_subterm_OgreaterThanLessThan__iff,axiom,
    ! [I: term_a_b,L: term_a_b,U: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_gr2587711877945340532rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ L
          @ U ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ L ) @ subterm_and_supt_a_b )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ I ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.greaterThanLessThan_iff
thf(fact_1210_subterm_OgreaterThanAtMost__iff,axiom,
    ! [I: term_a_b,L: term_a_b,U: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_gr5290542583680275920rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ L
          @ U ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ L ) @ subterm_and_supt_a_b )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ I ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.greaterThanAtMost_iff
thf(fact_1211_subterm_OgreaterThanAtMost__empty__iff2,axiom,
    ! [K2: term_a_b,L: term_a_b] :
      ( ( bot_bot_set_term_a_b
        = ( set_gr5290542583680275920rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2
          @ L ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ K2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.greaterThanAtMost_empty_iff2
thf(fact_1212_subterm_OgreaterThanAtMost__empty__iff,axiom,
    ! [K2: term_a_b,L: term_a_b] :
      ( ( ( set_gr5290542583680275920rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2
          @ L )
        = bot_bot_set_term_a_b )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ K2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.greaterThanAtMost_empty_iff
thf(fact_1213_subterm_OatLeastLessThan__empty__iff2,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( bot_bot_set_term_a_b
        = ( set_at5694027715021433807rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ A
          @ B ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.atLeastLessThan_empty_iff2
thf(fact_1214_subterm_OatLeastLessThan__empty__iff,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( ( set_at5694027715021433807rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ A
          @ B )
        = bot_bot_set_term_a_b )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.atLeastLessThan_empty_iff
thf(fact_1215_subterm_OatLeastLessThan__iff,axiom,
    ! [I: term_a_b,L: term_a_b,U: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_at5694027715021433807rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ L
          @ U ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ L ) @ subter523971068842742411eq_a_b )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ I ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.atLeastLessThan_iff
thf(fact_1216_subterm_OatLeastLessThan__empty,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( set_at5694027715021433807rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ A
          @ B )
        = bot_bot_set_term_a_b ) ) ).

% subterm.atLeastLessThan_empty
thf(fact_1217_inj__on__convol__ident,axiom,
    ! [F2: term_a_b > term_a_b,X4: set_term_a_b] :
      ( inj_on7922359415145192386rm_a_b
      @ ^ [X2: term_a_b] : ( produc7020197800436672577rm_a_b @ X2 @ ( F2 @ X2 ) )
      @ X4 ) ).

% inj_on_convol_ident
thf(fact_1218_subterm_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( set_gr5290542583680275920rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B )
      = ( minus_4613853266328121507rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% subterm.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1219_subterm_OIcc__eq__Icc,axiom,
    ! [L: term_a_b,H: term_a_b,L6: term_a_b,H2: term_a_b] :
      ( ( ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L
          @ H )
        = ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L6
          @ H2 ) )
      = ( ( ( L = L6 )
          & ( H = H2 ) )
        | ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ H @ L ) @ subter523971068842742411eq_a_b )
          & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ H2 @ L6 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.Icc_eq_Icc
thf(fact_1220_subterm_OatLeastAtMost__iff,axiom,
    ! [I: term_a_b,L: term_a_b,U: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L
          @ U ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ L ) @ subter523971068842742411eq_a_b )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ I ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atLeastAtMost_iff
thf(fact_1221_subterm_OatLeastatMost__empty__iff2,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( bot_bot_set_term_a_b
        = ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atLeastatMost_empty_iff2
thf(fact_1222_subterm_OatLeastatMost__empty__iff,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        = bot_bot_set_term_a_b )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atLeastatMost_empty_iff
thf(fact_1223_subterm_OatLeastatMost__subset__iff,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b,D: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ C
          @ D ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        | ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ D @ B ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.atLeastatMost_subset_iff
thf(fact_1224_subterm_OatLeastatMost__empty,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        = bot_bot_set_term_a_b ) ) ).

% subterm.atLeastatMost_empty
thf(fact_1225_subterm_OatLeastAtMost__singleton__iff,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        = ( insert_term_a_b @ C @ bot_bot_set_term_a_b ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% subterm.atLeastAtMost_singleton_iff
thf(fact_1226_subterm_OatLeastAtMost__singleton,axiom,
    ! [A: term_a_b] :
      ( ( set_at1176928818165845163rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ A
        @ A )
      = ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ).

% subterm.atLeastAtMost_singleton
thf(fact_1227_subterm_OatLeastatMost__psubset__iff,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b,D: term_a_b] :
      ( ( ord_le7073768085308280478rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ C
          @ D ) )
      = ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
          | ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ D @ B ) @ subter523971068842742411eq_a_b )
            & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ D @ B ) @ subterm_and_supt_a_b ) ) ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ D @ C ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atLeastatMost_psubset_iff
thf(fact_1228_subterm_OatLeastAtMost__singleton_H,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A = B )
     => ( ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        = ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% subterm.atLeastAtMost_singleton'
thf(fact_1229_subterm_OatLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b,D: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        @ ( set_at5694027715021433807rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ C
          @ D ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ D @ B ) @ subterm_and_supt_a_b ) ) ) ) ).

% subterm.atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1230_subterm_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( set_at5694027715021433807rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B )
      = ( minus_4613853266328121507rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ A
          @ B )
        @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ).

% subterm.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1231_subterm_OIcc__subset__Iic__iff,axiom,
    ! [L: term_a_b,H: term_a_b,H2: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L
          @ H )
        @ ( set_atMost_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ H2 ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ H @ L ) @ subter523971068842742411eq_a_b )
        | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ H2 @ H ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.Icc_subset_Iic_iff
thf(fact_1232_subterm_OIcc__subset__Ici__iff,axiom,
    ! [L: term_a_b,H: term_a_b,L6: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b
        @ ( set_at1176928818165845163rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L
          @ H )
        @ ( set_atLeast_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L6 ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ H @ L ) @ subter523971068842742411eq_a_b )
        | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ L6 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.Icc_subset_Ici_iff
thf(fact_1233_subterm_OatLeast__iff,axiom,
    ! [I: term_a_b,K2: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_atLeast_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ K2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ K2 ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.atLeast_iff
thf(fact_1234_subterm_OatMost__iff,axiom,
    ! [I: term_a_b,K2: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_atMost_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ K2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ I ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.atMost_iff
thf(fact_1235_subterm_OatLeastAtMost__def,axiom,
    ! [L: term_a_b,U: term_a_b] :
      ( ( set_at1176928818165845163rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ L
        @ U )
      = ( inf_inf_set_term_a_b
        @ ( set_atLeast_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L )
        @ ( set_atMost_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ U ) ) ) ).

% subterm.atLeastAtMost_def
thf(fact_1236_subterm_OatLeast__def,axiom,
    ! [L: term_a_b] :
      ( ( set_atLeast_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ L )
      = ( collect_term_a_b
        @ ^ [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ L ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atLeast_def
thf(fact_1237_subterm_OatMost__def,axiom,
    ! [U: term_a_b] :
      ( ( set_atMost_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ U )
      = ( collect_term_a_b
        @ ^ [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ X2 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.atMost_def
thf(fact_1238_subterm_Onot__empty__eq__Ici__eq__empty,axiom,
    ! [L: term_a_b] :
      ( bot_bot_set_term_a_b
     != ( set_atLeast_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ L ) ) ).

% subterm.not_empty_eq_Ici_eq_empty
thf(fact_1239_subterm_Onot__empty__eq__Iic__eq__empty,axiom,
    ! [H: term_a_b] :
      ( bot_bot_set_term_a_b
     != ( set_atMost_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ H ) ) ).

% subterm.not_empty_eq_Iic_eq_empty
thf(fact_1240_subterm_OgreaterThanAtMost__def,axiom,
    ! [L: term_a_b,U: term_a_b] :
      ( ( set_gr5290542583680275920rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ L
        @ U )
      = ( inf_inf_set_term_a_b
        @ ( set_gr6755604162050517946rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ L )
        @ ( set_atMost_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ U ) ) ) ).

% subterm.greaterThanAtMost_def
thf(fact_1241_subterm_OatLeastLessThan__def,axiom,
    ! [L: term_a_b,U: term_a_b] :
      ( ( set_at5694027715021433807rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ L
        @ U )
      = ( inf_inf_set_term_a_b
        @ ( set_atLeast_term_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
          @ L )
        @ ( set_le311243489613833509rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ U ) ) ) ).

% subterm.atLeastLessThan_def
thf(fact_1242_subterm_OgreaterThan__iff,axiom,
    ! [I: term_a_b,K2: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_gr6755604162050517946rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ K2 ) @ subterm_and_supt_a_b ) ) ).

% subterm.greaterThan_iff
thf(fact_1243_subterm_OlessThan__iff,axiom,
    ! [I: term_a_b,K2: term_a_b] :
      ( ( member_term_a_b @ I
        @ ( set_le311243489613833509rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ K2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ I ) @ subterm_and_supt_a_b ) ) ).

% subterm.lessThan_iff
thf(fact_1244_subterm_OgreaterThanLessThan__def,axiom,
    ! [L: term_a_b,U: term_a_b] :
      ( ( set_gr2587711877945340532rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ L
        @ U )
      = ( inf_inf_set_term_a_b
        @ ( set_gr6755604162050517946rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ L )
        @ ( set_le311243489613833509rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ U ) ) ) ).

% subterm.greaterThanLessThan_def
thf(fact_1245_subterm_OgreaterThanLessThan__eq,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( set_gr2587711877945340532rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B )
      = ( inf_inf_set_term_a_b
        @ ( set_gr6755604162050517946rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ A )
        @ ( set_le311243489613833509rm_a_b
          @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
          @ B ) ) ) ).

% subterm.greaterThanLessThan_eq
thf(fact_1246_subterm_OgreaterThan__def,axiom,
    ! [L: term_a_b] :
      ( ( set_gr6755604162050517946rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ L )
      = ( collect_term_a_b
        @ ^ [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ L ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.greaterThan_def
thf(fact_1247_subterm_OlessThan__def,axiom,
    ! [U: term_a_b] :
      ( ( set_le311243489613833509rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ U )
      = ( collect_term_a_b
        @ ^ [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ X2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.lessThan_def
thf(fact_1248_subterm_OIio__Int__singleton,axiom,
    ! [K2: term_a_b,X: term_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ X ) @ subterm_and_supt_a_b )
       => ( ( inf_inf_set_term_a_b
            @ ( set_le311243489613833509rm_a_b
              @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
              @ K2 )
            @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
          = ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) ) )
      & ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ K2 @ X ) @ subterm_and_supt_a_b )
       => ( ( inf_inf_set_term_a_b
            @ ( set_le311243489613833509rm_a_b
              @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
              @ K2 )
            @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
          = bot_bot_set_term_a_b ) ) ) ).

% subterm.Iio_Int_singleton
thf(fact_1249_subterm_OIoi__le__Ico,axiom,
    ! [A: term_a_b] :
      ( ord_le2705286416250468010rm_a_b
      @ ( set_gr6755604162050517946rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A )
      @ ( set_atLeast_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ A ) ) ).

% subterm.Ioi_le_Ico
thf(fact_1250_gsrstep__conversion__dist,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra1130568886744591910rm_a_b
        @ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) )
          @ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
            @ ^ [Uu2: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
      = ( transi7742714808557438673rm_a_b
        @ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) )
          @ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
            @ ^ [Uu2: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ).

% gsrstep_conversion_dist
thf(fact_1251_subterm_Obdd__above__Ioo,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_gr2587711877945340532rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ioo
thf(fact_1252_SigmaI,axiom,
    ! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A4 )
     => ( ( member5869715511025134514rm_a_b @ B @ ( B5 @ A ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1253_SigmaI,axiom,
    ! [A: term_a_b,A4: set_term_a_b,B: term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member_term_a_b @ A @ A4 )
     => ( ( member_term_a_b @ B @ ( B5 @ A ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1254_mem__Sigma__iff,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A4 @ B5 ) )
      = ( ( member5869715511025134514rm_a_b @ A @ A4 )
        & ( member5869715511025134514rm_a_b @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1255_mem__Sigma__iff,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) )
      = ( ( member_term_a_b @ A @ A4 )
        & ( member_term_a_b @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1256_subterm_Obdd__above_OI,axiom,
    ! [A4: set_term_a_b,M3: term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M3 @ X3 ) @ subter523971068842742411eq_a_b ) )
     => ( condit416533379776037224rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ A4 ) ) ).

% subterm.bdd_above.I
thf(fact_1257_subterm_Obdd__above__empty,axiom,
    ( condit416533379776037224rm_a_b
    @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
    @ bot_bot_set_term_a_b ) ).

% subterm.bdd_above_empty
thf(fact_1258_subterm_Obdd__above__Icc,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_at1176928818165845163rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Icc
thf(fact_1259_subterm_Obdd__above__Iic,axiom,
    ! [B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_atMost_term_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ B ) ) ).

% subterm.bdd_above_Iic
thf(fact_1260_insert__Times__insert,axiom,
    ! [A: term_a_b,A4: set_term_a_b,B: term_a_b,B5: set_term_a_b] :
      ( ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A @ A4 )
        @ ^ [Uu2: term_a_b] : ( insert_term_a_b @ B @ B5 ) )
      = ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B )
        @ ( sup_su6776935440552674877rm_a_b
          @ ( produc1953217797448512882rm_a_b @ A4
            @ ^ [Uu2: term_a_b] : ( insert_term_a_b @ B @ B5 ) )
          @ ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A @ A4 )
            @ ^ [Uu2: term_a_b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1261_subterm_Obdd__above__Ico,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_at5694027715021433807rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ico
thf(fact_1262_subterm_Obdd__above__Ioc,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_gr5290542583680275920rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ioc
thf(fact_1263_subterm_Obdd__above__Iio,axiom,
    ! [B: term_a_b] :
      ( condit416533379776037224rm_a_b
      @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subter523971068842742411eq_a_b )
      @ ( set_le311243489613833509rm_a_b
        @ ^ [S2: term_a_b,T5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T5 @ S2 ) @ subterm_and_supt_a_b )
        @ B ) ) ).

% subterm.bdd_above_Iio
thf(fact_1264_rtrancl__Restr,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y )
        @ ( transi7742714808557438673rm_a_b
          @ ( inf_in1693821330460682071rm_a_b @ R2
            @ ( produc1953217797448512882rm_a_b @ A4
              @ ^ [Uu2: term_a_b] : A4 ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).

% rtrancl_Restr
thf(fact_1265_SigmaE2,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A4 @ B5 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ A @ A4 )
         => ~ ( member5869715511025134514rm_a_b @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1266_SigmaE2,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) )
     => ~ ( ( member_term_a_b @ A @ A4 )
         => ~ ( member_term_a_b @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1267_SigmaD2,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) )
     => ( member_term_a_b @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1268_SigmaD1,axiom,
    ! [A: term_a_b,B: term_a_b,A4: set_term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) )
     => ( member_term_a_b @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1269_SigmaE,axiom,
    ! [C: produc4523971788519308903rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ C @ ( produc5745020764710804872rm_a_b @ A4 @ B5 ) )
     => ~ ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
           => ! [Y4: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ Y4 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1270_SigmaE,axiom,
    ! [C: produc357393685978478089rm_a_b,A4: set_term_a_b,B5: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ C @ ( produc1953217797448512882rm_a_b @ A4 @ B5 ) )
     => ~ ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A4 )
           => ! [Y4: term_a_b] :
                ( ( member_term_a_b @ Y4 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1271_wfI,axiom,
    ! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R2
        @ ( produc5745020764710804872rm_a_b @ A4
          @ ^ [Uu2: produc357393685978478089rm_a_b] : B5 ) )
     => ( ! [X3: produc357393685978478089rm_a_b,P4: produc357393685978478089rm_a_b > $o] :
            ( ! [Xa2: produc357393685978478089rm_a_b] :
                ( ! [Y4: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Xa2 ) @ R2 )
                   => ( P4 @ Y4 ) )
               => ( P4 @ Xa2 ) )
           => ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
             => ( ( member5869715511025134514rm_a_b @ X3 @ B5 )
               => ( P4 @ X3 ) ) ) )
       => ( wf_Pro2335863617654816626rm_a_b @ R2 ) ) ) ).

% wfI
thf(fact_1272_wfI,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B5: set_term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2
        @ ( produc1953217797448512882rm_a_b @ A4
          @ ^ [Uu2: term_a_b] : B5 ) )
     => ( ! [X3: term_a_b,P4: term_a_b > $o] :
            ( ! [Xa2: term_a_b] :
                ( ! [Y4: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Xa2 ) @ R2 )
                   => ( P4 @ Y4 ) )
               => ( P4 @ Xa2 ) )
           => ( ( member_term_a_b @ X3 @ A4 )
             => ( ( member_term_a_b @ X3 @ B5 )
               => ( P4 @ X3 ) ) ) )
       => ( wf_term_a_b @ R2 ) ) ) ).

% wfI
thf(fact_1273_trancl__subset__Sigma__aux,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
     => ( ( ord_le2593958535671617255rm_a_b @ R2
          @ ( produc5745020764710804872rm_a_b @ A4
            @ ^ [Uu2: produc357393685978478089rm_a_b] : A4 ) )
       => ( ( A = B )
          | ( member5869715511025134514rm_a_b @ A @ A4 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1274_trancl__subset__Sigma__aux,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
     => ( ( ord_le118470702582115849rm_a_b @ R2
          @ ( produc1953217797448512882rm_a_b @ A4
            @ ^ [Uu2: term_a_b] : A4 ) )
       => ( ( A = B )
          | ( member_term_a_b @ A @ A4 ) ) ) ) ).

% trancl_subset_Sigma_aux

% Helper facts (3)
thf(help_If_3_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    term_ground_a_b @ s ).

%------------------------------------------------------------------------------