%------------------------------------------------------------------------------
% 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/0023_Ground_Reduction_on_LV/prob_00077_003022__14408608_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1518 ( 553 unt; 242 typ;   0 def)
%            Number of atoms       : 3620 (1008 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 14557 ( 304   ~;  44   |; 241   &;12381   @)
%                                         (   0 <=>;1587  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :   29 (  28 usr)
%            Number of type conns  :  624 ( 624   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  217 ( 214 usr;  23 con; 0-5 aty)
%            Number of variables   : 4181 ( 507   ^;3569   !; 105   ?;4181   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:19:58.567
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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    member7030746907725424910_f_nat: produc2676906022994773093_f_nat > set_Pr516226060960033221_f_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    member4316649861536941904rm_f_v: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
    member5935997662887750672od_f_f: product_prod_f_f > set_Product_prod_f_f > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    member_term_f_v: term_f_v > set_term_f_v > $o ).

thf(sy_c_member_001tf__f,type,
    member_f: f > set_f > $o ).

thf(sy_v__092_060F_062,type,
    f2: set_Pr5245412377734449720_f_nat ).

thf(sy_v__092_060R_062,type,
    r: set_Pr8827868859434726151rm_f_v ).

thf(sy_v_c,type,
    c: f ).

thf(sy_v_d,type,
    d: f ).

thf(sy_v_s,type,
    s: term_f_v ).

thf(sy_v_t,type,
    t: term_f_v ).

% Relevant facts (1275)
thf(fact_0_diff,axiom,
    c != d ).

% diff
thf(fact_1_fresh_I1_J,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ f2 ) ).

% fresh(1)
thf(fact_2_fresh_I2_J,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ).

% fresh(2)
thf(fact_3__092_060open_062_092_060And_062t_As_A_092_060R_062_O_A_Is_M_At_J_A_092_060in_062_A_Isrstep_A_092_060H_062_A_092_060R_062_J_092_060_094sup_062_K_A_092_060Longrightarrow_062_A_Is_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062c_M_At_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062c_J_A_092_060in_062_A_Isrstep_A_092_060H_062_A_092_060R_062_J_092_060_094sup_062_K_092_060close_062,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ R ) ) ) )
     => ( member4316649861536941904rm_f_v
        @ ( produc2510790566102465375rm_f_v
          @ ( subst_7901567856578221233_f_v_v @ S
            @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) )
          @ ( subst_7901567856578221233_f_v_v @ T
            @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) ) )
        @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ R ) ) ) ) ) ).

% \<open>\<And>t s \<R>. (s, t) \<in> (srstep \<H> \<R>)\<^sup>* \<Longrightarrow> (s \<cdot> \<sigma>\<^sub>c, t \<cdot> \<sigma>\<^sub>c) \<in> (srstep \<H> \<R>)\<^sup>*\<close>
thf(fact_4_open__terms__two__const__lv__axioms,axiom,
    ground9003829210744935429lv_f_v @ r @ f2 @ c @ d ).

% open_terms_two_const_lv_axioms
thf(fact_5_lift,axiom,
    member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ s @ t ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( conver3099152742603637891rm_f_v @ r ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ r ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ r ) ) ) ).

% lift
thf(fact_6_assms,axiom,
    member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ s @ t ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ f2 @ ( conver3099152742603637891rm_f_v @ r ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ r ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ f2 @ r ) ) ) ).

% assms
thf(fact_7__092_060open_062_092_060And_062t_As_O_A_Is_M_At_J_A_092_060in_062_Asrsteps__with__root__step_A_092_060H_062_A_092_060R_062_A_092_060Longrightarrow_062_A_Is_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062c_M_At_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062d_J_A_092_060in_062_A_Isrstep_A_092_060H_062_A_092_060R_062_J_092_060_094sup_062_K_092_060close_062,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ r ) )
     => ( member4316649861536941904rm_f_v
        @ ( produc2510790566102465375rm_f_v
          @ ( subst_7901567856578221233_f_v_v @ S
            @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) )
          @ ( subst_7901567856578221233_f_v_v @ T
            @ ^ [X: v] : ( fun_f_v @ d @ nil_term_f_v ) ) )
        @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ r ) ) ) ) ) ).

% \<open>\<And>t s. (s, t) \<in> srsteps_with_root_step \<H> \<R> \<Longrightarrow> (s \<cdot> \<sigma>\<^sub>c, t \<cdot> \<sigma>\<^sub>d) \<in> (srstep \<H> \<R>)\<^sup>*\<close>
thf(fact_8__092_060open_062_092_060And_062t_As_O_A_Is_M_At_J_A_092_060in_062_Asrsteps__with__root__step_A_092_060H_062_A_I_092_060R_062_092_060inverse_062_J_A_092_060Longrightarrow_062_A_Is_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062c_M_At_A_092_060cdot_062_A_092_060sigma_062_092_060_094sub_062d_J_A_092_060in_062_A_Isrstep_A_092_060H_062_A_I_092_060R_062_092_060inverse_062_J_J_092_060_094sup_062_K_092_060close_062,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( conver3099152742603637891rm_f_v @ r ) ) )
     => ( member4316649861536941904rm_f_v
        @ ( produc2510790566102465375rm_f_v
          @ ( subst_7901567856578221233_f_v_v @ S
            @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) )
          @ ( subst_7901567856578221233_f_v_v @ T
            @ ^ [X: v] : ( fun_f_v @ d @ nil_term_f_v ) ) )
        @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ) ) ) ).

% \<open>\<And>t s. (s, t) \<in> srsteps_with_root_step \<H> (\<R>\<inverse>) \<Longrightarrow> (s \<cdot> \<sigma>\<^sub>c, t \<cdot> \<sigma>\<^sub>d) \<in> (srstep \<H> (\<R>\<inverse>))\<^sup>*\<close>
thf(fact_9_srsteps__eqD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) ) ) ).

% srsteps_eqD
thf(fact_10_lv__inv,axiom,
    rewrit2771924964435581537_f_v_f @ ( conver3099152742603637891rm_f_v @ r ) ).

% lv_inv
thf(fact_11_lv,axiom,
    rewrit2771924964435581537_f_v_f @ r ).

% lv
thf(fact_12_rstep__substI,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( rstep_f_v @ R ) ) ) ).

% rstep_substI
thf(fact_13_sig__mono,axiom,
    ord_le8976984241387448984_f_nat @ f2 @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) ).

% sig_mono
thf(fact_14_fresh__sym__d__inv,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ).

% fresh_sym_d_inv
thf(fact_15_fresh__sym__c__inv,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ).

% fresh_sym_c_inv
thf(fact_16_rtrancl__idemp__self__comp,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
      = ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_17_rstep__ruleI,axiom,
    ! [L: term_f_v,R3: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ R )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ ( rstep_f_v @ R ) ) ) ).

% rstep_ruleI
thf(fact_18_converse__iff,axiom,
    ! [A: nat,B: f,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A @ B ) @ ( converse_f_nat @ R3 ) )
      = ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ A ) @ R3 ) ) ).

% converse_iff
thf(fact_19_converse__iff,axiom,
    ! [A: f,B: nat,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( converse_nat_f @ R3 ) )
      = ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ B @ A ) @ R3 ) ) ).

% converse_iff
thf(fact_20_converse__iff,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( conver3099152742603637891rm_f_v @ R3 ) )
      = ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R3 ) ) ).

% converse_iff
thf(fact_21_fresh__sym__d,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ r ) ) ).

% fresh_sym_d
thf(fact_22_fresh__sym__c,axiom,
    ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ r ) ) ).

% fresh_sym_c
thf(fact_23_srstep__converse__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
      = ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% srstep_converse_dist
thf(fact_24_r__into__rtrancl,axiom,
    ! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P @ R3 )
     => ( member4316649861536941904rm_f_v @ P @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).

% r_into_rtrancl
thf(fact_25_rtrancl__idemp,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) )
      = ( transi3028809021613410336rm_f_v @ R3 ) ) ).

% rtrancl_idemp
thf(fact_26_converse__converse,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) )
      = R3 ) ).

% converse_converse
thf(fact_27_converse__inject,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ( conver3099152742603637891rm_f_v @ R3 )
        = ( conver3099152742603637891rm_f_v @ S ) )
      = ( R3 = S ) ) ).

% converse_inject
thf(fact_28_sig,axiom,
    ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ r ) @ f2 ).

% sig
thf(fact_29_sig__inv,axiom,
    ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ r ) ) @ f2 ).

% sig_inv
thf(fact_30_trancl__rtrancl__absorb,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) )
      = ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% trancl_rtrancl_absorb
thf(fact_31_rtrancl__trancl__absorb,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3208867851621559554rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) )
      = ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% rtrancl_trancl_absorb
thf(fact_32_converse__mono,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( conver3099152742603637891rm_f_v @ S ) )
      = ( ord_le4559761987009501863rm_f_v @ R3 @ S ) ) ).

% converse_mono
thf(fact_33_converse__mono,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8280083912720879166_nat_f @ ( converse_f_nat @ R3 ) @ ( converse_f_nat @ S ) )
      = ( ord_le8976984241387448984_f_nat @ R3 @ S ) ) ).

% converse_mono
thf(fact_34_converse__mono,axiom,
    ! [R3: set_Pr4548512049067879902_nat_f,S: set_Pr4548512049067879902_nat_f] :
      ( ( ord_le8976984241387448984_f_nat @ ( converse_nat_f @ R3 ) @ ( converse_nat_f @ S ) )
      = ( ord_le8280083912720879166_nat_f @ R3 @ S ) ) ).

% converse_mono
thf(fact_35_relcomp__distrib2,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) @ R2 )
      = ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ ( relcom2934258169684288428rm_f_v @ T2 @ R2 ) ) ) ).

% relcomp_distrib2
thf(fact_36_relcomp__distrib,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ R2 @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) )
      = ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ T2 ) ) ) ).

% relcomp_distrib
thf(fact_37_open__terms__two__const__lv_Olv,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( rewrit2771924964435581537_f_v_f @ R ) ) ).

% open_terms_two_const_lv.lv
thf(fact_38_open__terms__two__const__lv_Osig,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F ) ) ).

% open_terms_two_const_lv.sig
thf(fact_39_open__terms__two__const__lv_Odiff,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( C != D ) ) ).

% open_terms_two_const_lv.diff
thf(fact_40_funas__rel__converse,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ R ) ) @ F ) ) ).

% funas_rel_converse
thf(fact_41_trancl__mono,axiom,
    ! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
       => ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_mono
thf(fact_42_r__into__trancl_H,axiom,
    ! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P @ R3 )
     => ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% r_into_trancl'
thf(fact_43_open__terms__two__const__lv_Osig__inv,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ R ) ) @ F ) ) ).

% open_terms_two_const_lv.sig_inv
thf(fact_44_rtrancl__Un__subset,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) ) ).

% rtrancl_Un_subset
thf(fact_45_converse__rtrancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R3: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A2: term_f_v,B2: term_f_v,Aa: term_f_v,Ba: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R3 )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A2 @ B2 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_46_converse__rtrancl__induct2,axiom,
    ! [Ax: f,Ay: nat,Bx: f,By: nat,R3: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A2: f,B2: nat,Aa: f,Ba: nat] :
              ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A2 @ B2 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R3 )
             => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Aa @ Ba ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A2 @ B2 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_47_converse__rtranclE2,axiom,
    ! [Xa: term_f_v,Xb: term_f_v,Za: term_f_v,Zb: term_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa @ Xb ) @ ( produc2510790566102465375rm_f_v @ Za @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
     => ( ( ( produc2510790566102465375rm_f_v @ Xa @ Xb )
         != ( produc2510790566102465375rm_f_v @ Za @ Zb ) )
       => ~ ! [A2: term_f_v,B2: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa @ Xb ) @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) @ R3 )
             => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( produc2510790566102465375rm_f_v @ Za @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_48_converse__rtranclE2,axiom,
    ! [Xa: f,Xb: nat,Za: f,Zb: nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Xa @ Xb ) @ ( product_Pair_f_nat @ Za @ Zb ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
     => ( ( ( product_Pair_f_nat @ Xa @ Xb )
         != ( product_Pair_f_nat @ Za @ Zb ) )
       => ~ ! [A2: f,B2: nat] :
              ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Xa @ Xb ) @ ( product_Pair_f_nat @ A2 @ B2 ) ) @ R3 )
             => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A2 @ B2 ) @ ( product_Pair_f_nat @ Za @ Zb ) ) @ ( transi8903367562120364223_f_nat @ R3 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_49_rtrancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R3: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A2: term_f_v,B2: term_f_v,Aa: term_f_v,Ba: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A2 @ B2 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_50_rtrancl__induct2,axiom,
    ! [Ax: f,Ay: nat,Bx: f,By: nat,R3: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A2: f,B2: nat,Aa: f,Ba: nat] :
              ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A2 @ B2 ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
             => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A2 @ B2 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A2 @ B2 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_51_open__terms__two__const__lv__def,axiom,
    ( ground9003829210744935429lv_f_v
    = ( ^ [R4: set_Pr8827868859434726151rm_f_v,F2: set_Pr5245412377734449720_f_nat,C2: f,D2: f] :
          ( ( rewrit2771924964435581537_f_v_f @ R4 )
          & ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R4 ) @ F2 )
          & ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C2 @ zero_zero_nat ) @ F2 )
          & ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D2 @ zero_zero_nat ) @ F2 )
          & ( C2 != D2 ) ) ) ) ).

% open_terms_two_const_lv_def
thf(fact_52_mem__Collect__eq,axiom,
    ! [A: produc8027700073345061287rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ A @ ( collec7769483783292507282rm_f_v @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
    ! [A: product_prod_f_nat,P2: product_prod_f_nat > $o] :
      ( ( member2678038196313615385_f_nat @ A @ ( collec1417984143148964183_f_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
    ! [A: term_f_v,P2: term_f_v > $o] :
      ( ( member_term_f_v @ A @ ( collect_term_f_v @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( collec7769483783292507282rm_f_v
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( collec1417984143148964183_f_nat
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A3: set_term_f_v] :
      ( ( collect_term_f_v
        @ ^ [X: term_f_v] : ( member_term_f_v @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_58_Collect__cong,axiom,
    ! [P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ! [X2: term_f_v] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_term_f_v @ P2 )
        = ( collect_term_f_v @ Q ) ) ) ).

% Collect_cong
thf(fact_59_open__terms__two__const__lv_Ointro,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ F )
         => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F )
           => ( ( C != D )
             => ( ground9003829210744935429lv_f_v @ R @ F @ C @ D ) ) ) ) ) ) ).

% open_terms_two_const_lv.intro
thf(fact_60_subrelI,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: term_f_v,Y: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ S ) )
     => ( ord_le4559761987009501863rm_f_v @ R3 @ S ) ) ).

% subrelI
thf(fact_61_subrelI,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: f,Y: nat] :
          ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y ) @ R3 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y ) @ S ) )
     => ( ord_le8976984241387448984_f_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_62_rtrancl__subset__rtrancl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S ) )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_63_rtrancl__subset,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R2 @ S2 )
     => ( ( ord_le4559761987009501863rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R2 ) )
       => ( ( transi3028809021613410336rm_f_v @ S2 )
          = ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ).

% rtrancl_subset
thf(fact_64_rtrancl__mono,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ S ) ) ) ).

% rtrancl_mono
thf(fact_65_relcomp__mono,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v,S3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R5 @ R3 )
     => ( ( ord_le4559761987009501863rm_f_v @ S3 @ S )
       => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R5 @ S3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_66_relcomp__mono,axiom,
    ! [R5: set_Product_prod_f_f,R3: set_Product_prod_f_f,S3: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le7147000192471188839od_f_f @ R5 @ R3 )
     => ( ( ord_le8976984241387448984_f_nat @ S3 @ S )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ R5 @ S3 ) @ ( relcomp_f_f_nat @ R3 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_67_relcomp__mono,axiom,
    ! [R5: set_Pr5245412377734449720_f_nat,R3: set_Pr5245412377734449720_f_nat,S3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R5 @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ S3 @ S )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R5 @ S3 ) @ ( relcomp_f_nat_nat @ R3 @ S ) ) ) ) ).

% relcomp_mono
thf(fact_68_sig__step__rsteps__dist,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( sig_step_f_v @ F @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) )
        = ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% sig_step_rsteps_dist
thf(fact_69_converse__subset__swap,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ S ) )
      = ( ord_le4559761987009501863rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ S ) ) ).

% converse_subset_swap
thf(fact_70_converse__subset__swap,axiom,
    ! [R3: set_Pr4548512049067879902_nat_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8280083912720879166_nat_f @ R3 @ ( converse_f_nat @ S ) )
      = ( ord_le8976984241387448984_f_nat @ ( converse_nat_f @ R3 ) @ S ) ) ).

% converse_subset_swap
thf(fact_71_converse__subset__swap,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,S: set_Pr4548512049067879902_nat_f] :
      ( ( ord_le8976984241387448984_f_nat @ R3 @ ( converse_nat_f @ S ) )
      = ( ord_le8280083912720879166_nat_f @ ( converse_f_nat @ R3 ) @ S ) ) ).

% converse_subset_swap
thf(fact_72_open__terms__two__const__lv_Olv__inv,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( rewrit2771924964435581537_f_v_f @ ( conver3099152742603637891rm_f_v @ R ) ) ) ).

% open_terms_two_const_lv.lv_inv
thf(fact_73_trancl__unfold,axiom,
    ( transi3208867851621559554rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ R6 @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ R6 ) @ R6 ) ) ) ) ).

% trancl_unfold
thf(fact_74_open__terms__two__const__lv_Ofresh__sym__d,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) ) ) ).

% open_terms_two_const_lv.fresh_sym_d
thf(fact_75_open__terms__two__const__lv_Ofresh__sym__c,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) ) ) ).

% open_terms_two_const_lv.fresh_sym_c
thf(fact_76_converse__trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [Y: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ B ) @ R3 )
           => ( P2 @ Y ) )
       => ( ! [Y: term_f_v,Z: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ R3 )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
               => ( ( P2 @ Z )
                 => ( P2 @ Y ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_77_trancl__trans__induct,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [X2: term_f_v,Y: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 )
           => ( P2 @ X2 @ Y ) )
       => ( ! [X2: term_f_v,Y: term_f_v,Z: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ( ( P2 @ X2 @ Y )
               => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
                 => ( ( P2 @ Y @ Z )
                   => ( P2 @ X2 @ Z ) ) ) ) )
         => ( P2 @ X3 @ Y2 ) ) ) ) ).

% trancl_trans_induct
thf(fact_78_trancl__into__trancl2,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_79_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R3 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_80_irrefl__trancl__rD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ! [X2: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ X2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ R3 )
       => ( X3 != Y2 ) ) ) ).

% irrefl_trancl_rD
thf(fact_81_converse__tranclE,axiom,
    ! [X3: term_f_v,Z2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R3 )
       => ~ ! [Y: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R3 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ) ).

% converse_tranclE
thf(fact_82_r__r__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R2 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% r_r_into_trancl
thf(fact_83_trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [Y: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y ) @ R3 )
           => ( P2 @ Y ) )
       => ( ! [Y: term_f_v,Z: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ R3 )
               => ( ( P2 @ Y )
                 => ( P2 @ Z ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_84_trancl__trans,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_trans
thf(fact_85_tranclE,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
       => ~ ! [C3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C3 @ B ) @ R3 ) ) ) ) ).

% tranclE
thf(fact_86_trancl_Or__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% trancl.r_into_trancl
thf(fact_87_trancl_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
      = ( ? [A4: term_f_v,B3: term_f_v] :
            ( ( A1 = A4 )
            & ( A22 = B3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ R3 ) )
        | ? [A4: term_f_v,B3: term_f_v,C2: term_f_v] :
            ( ( A1 = A4 )
            & ( A22 = C2 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C2 ) @ R3 ) ) ) ) ).

% trancl.simps
thf(fact_88_trancl_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ R3 )
       => ~ ! [B2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ A22 ) @ R3 ) ) ) ) ).

% trancl.cases
thf(fact_89_trancl__converse,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3208867851621559554rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) )
      = ( conver3099152742603637891rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% trancl_converse
thf(fact_90_rtrancl__Un__rtrancl,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
      = ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) ) ).

% rtrancl_Un_rtrancl
thf(fact_91_in__rtrancl__UnI,axiom,
    ! [X3: produc8027700073345061287rm_f_v,R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ R2 ) )
        | ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
     => ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) ) ) ).

% in_rtrancl_UnI
thf(fact_92_converse__Un,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S ) )
      = ( sup_su1994854688125285083rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( conver3099152742603637891rm_f_v @ S ) ) ) ).

% converse_Un
thf(fact_93_open__terms__two__const__lv_Ofresh__sym__d__inv,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% open_terms_two_const_lv.fresh_sym_d_inv
thf(fact_94_open__terms__two__const__lv_Ofresh__sym__c__inv,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% open_terms_two_const_lv.fresh_sym_c_inv
thf(fact_95_open__terms__two__const__lv_Osig__mono,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ord_le8976984241387448984_f_nat @ F @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F ) ) ) ) ).

% open_terms_two_const_lv.sig_mono
thf(fact_96_open__terms__two__const__lv_Ofresh_I1_J,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ F ) ) ).

% open_terms_two_const_lv.fresh(1)
thf(fact_97_open__terms__two__const__lv_Ofresh_I2_J,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F ) ) ).

% open_terms_two_const_lv.fresh(2)
thf(fact_98_trancl__rtrancl__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_99_rtrancl__trancl__trancl,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_100_rtrancl__into__trancl2,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_101_rtrancl__into__trancl1,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R3 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_102_rtrancl__eq__or__trancl,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
      = ( ( X3 = Y2 )
        | ( ( X3 != Y2 )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_103_trancl__into__rtrancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).

% trancl_into_rtrancl
thf(fact_104_tranclD2,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ? [Z: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y2 ) @ R2 ) ) ) ).

% tranclD2
thf(fact_105_rtranclD,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ) ).

% rtranclD
thf(fact_106_tranclD,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ? [Z: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z ) @ R2 )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ).

% tranclD
thf(fact_107_trancl__converseI,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( conver3099152742603637891rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ) ).

% trancl_converseI
thf(fact_108_trancl__converseD,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( conver3099152742603637891rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_converseD
thf(fact_109_rtrancl__Un__separator__converseE,axiom,
    ! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q ) ) )
     => ( ! [X2: term_f_v,Y: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X2 ) @ Q )
             => ( Y = X2 ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_110_rtrancl__Un__separatorE,axiom,
    ! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q ) ) )
     => ( ! [X2: term_f_v,Y: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ X2 ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ Q )
             => ( X2 = Y ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_111_trancl__unfold__left,axiom,
    ( transi3208867851621559554rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ R6 @ ( transi3028809021613410336rm_f_v @ R6 ) ) ) ) ).

% trancl_unfold_left
thf(fact_112_O__assoc,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ T2 )
      = ( relcom2934258169684288428rm_f_v @ R2 @ ( relcom2934258169684288428rm_f_v @ S2 @ T2 ) ) ) ).

% O_assoc
thf(fact_113_rstep__smycl__dist,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( rstep_f_v @ R ) @ ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R ) ) )
      = ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% rstep_smycl_dist
thf(fact_114_sig__step__symcl__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ R ) @ ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ R ) ) )
      = ( sig_step_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% sig_step_symcl_dist
thf(fact_115_pred__equals__eq2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_116_pred__equals__eq2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_117_srstep__symcl__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
      = ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ) ).

% srstep_symcl_dist
thf(fact_118_converse__rtrancl__into__rtrancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_119_converse__rtrancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( P2 @ B )
       => ( ! [Y: term_f_v,Z: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ R3 )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
               => ( ( P2 @ Z )
                 => ( P2 @ Y ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_120_converse__rtranclE,axiom,
    ! [X3: term_f_v,Z2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( X3 != Z2 )
       => ~ ! [Y: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R3 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_121_rtrancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( P2 @ A )
       => ( ! [Y: term_f_v,Z: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ R3 )
               => ( ( P2 @ Y )
                 => ( P2 @ Z ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_122_rtrancl__trans,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ).

% rtrancl_trans
thf(fact_123_rtranclE,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( A != B )
       => ~ ! [Y: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ B ) @ R3 ) ) ) ) ).

% rtranclE
thf(fact_124_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R3 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_125_rtrancl_Ortrancl__refl,axiom,
    ! [A: term_f_v,R3: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_126_rtrancl_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
      = ( ? [A4: term_f_v] :
            ( ( A1 = A4 )
            & ( A22 = A4 ) )
        | ? [A4: term_f_v,B3: term_f_v,C2: term_f_v] :
            ( ( A1 = A4 )
            & ( A22 = C2 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C2 ) @ R3 ) ) ) ) ).

% rtrancl.simps
thf(fact_127_rtrancl_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( A22 != A1 )
       => ~ ! [B2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ A22 ) @ R3 ) ) ) ) ).

% rtrancl.cases
thf(fact_128_relcompEpair,axiom,
    ! [A: term_f_v,C: term_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) )
     => ~ ! [B2: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B2 ) @ R3 )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_129_relcompEpair,axiom,
    ! [A: f,C: nat,R3: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C ) @ ( relcomp_f_f_nat @ R3 @ S ) )
     => ~ ! [B2: f] :
            ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B2 ) @ R3 )
           => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_130_relcompEpair,axiom,
    ! [A: f,C: nat,R3: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C ) @ ( relcomp_f_nat_nat @ R3 @ S ) )
     => ~ ! [B2: nat] :
            ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B2 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_131_relcompE,axiom,
    ! [Xz: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Xz @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) )
     => ~ ! [X2: term_f_v,Y: term_f_v,Z: term_f_v] :
            ( ( Xz
              = ( produc2510790566102465375rm_f_v @ X2 @ Z ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_132_relcompE,axiom,
    ! [Xz: product_prod_f_nat,R3: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ Xz @ ( relcomp_f_f_nat @ R3 @ S ) )
     => ~ ! [X2: f,Y: f,Z: nat] :
            ( ( Xz
              = ( product_Pair_f_nat @ X2 @ Z ) )
           => ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y ) @ R3 )
             => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_133_relcompE,axiom,
    ! [Xz: product_prod_f_nat,R3: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2678038196313615385_f_nat @ Xz @ ( relcomp_f_nat_nat @ R3 @ S ) )
     => ~ ! [X2: f,Y: nat,Z: nat] :
            ( ( Xz
              = ( product_Pair_f_nat @ X2 @ Z ) )
           => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y ) @ R3 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_134_relcomp_OrelcompI,axiom,
    ! [A: f,B: f,R3: set_Product_prod_f_f,C: nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R3 )
     => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ C ) @ S )
       => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C ) @ ( relcomp_f_f_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_135_relcomp_OrelcompI,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ S )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_136_relcomp_OrelcompI,axiom,
    ! [A: f,B: nat,R3: set_Pr5245412377734449720_f_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S )
       => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C ) @ ( relcomp_f_nat_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_137_relcomp_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) )
      = ( ? [A4: term_f_v,B3: term_f_v,C2: term_f_v] :
            ( ( A1 = A4 )
            & ( A22 = C2 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ R3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_138_relcomp_Osimps,axiom,
    ! [A1: f,A22: nat,R3: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_f_nat @ R3 @ S ) )
      = ( ? [A4: f,B3: f,C2: nat] :
            ( ( A1 = A4 )
            & ( A22 = C2 )
            & ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A4 @ B3 ) @ R3 )
            & ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_139_relcomp_Osimps,axiom,
    ! [A1: f,A22: nat,R3: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_nat_nat @ R3 @ S ) )
      = ( ? [A4: f,B3: nat,C2: nat] :
            ( ( A1 = A4 )
            & ( A22 = C2 )
            & ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A4 @ B3 ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_140_relcomp_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) )
     => ~ ! [B2: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B2 ) @ R3 )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_141_relcomp_Ocases,axiom,
    ! [A1: f,A22: nat,R3: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_f_nat @ R3 @ S ) )
     => ~ ! [B2: f] :
            ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ B2 ) @ R3 )
           => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B2 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_142_relcomp_Ocases,axiom,
    ! [A1: f,A22: nat,R3: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_nat_nat @ R3 @ S ) )
     => ~ ! [B2: nat] :
            ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ B2 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_143_converseI,axiom,
    ! [A: nat,B: f,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A @ B ) @ R3 )
     => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ A ) @ ( converse_nat_f @ R3 ) ) ) ).

% converseI
thf(fact_144_converseI,axiom,
    ! [A: f,B: nat,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ R3 )
     => ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ B @ A ) @ ( converse_f_nat @ R3 ) ) ) ).

% converseI
thf(fact_145_converseI,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ).

% converseI
thf(fact_146_converseE,axiom,
    ! [Yx: product_prod_nat_f,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member8962352073626239679_nat_f @ Yx @ ( converse_f_nat @ R3 ) )
     => ~ ! [X2: f,Y: nat] :
            ( ( Yx
              = ( product_Pair_nat_f @ Y @ X2 ) )
           => ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y ) @ R3 ) ) ) ).

% converseE
thf(fact_147_converseE,axiom,
    ! [Yx: product_prod_f_nat,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member2678038196313615385_f_nat @ Yx @ ( converse_nat_f @ R3 ) )
     => ~ ! [X2: nat,Y: f] :
            ( ( Yx
              = ( product_Pair_f_nat @ Y @ X2 ) )
           => ~ ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X2 @ Y ) @ R3 ) ) ) ).

% converseE
thf(fact_148_converseE,axiom,
    ! [Yx: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Yx @ ( conver3099152742603637891rm_f_v @ R3 ) )
     => ~ ! [X2: term_f_v,Y: term_f_v] :
            ( ( Yx
              = ( produc2510790566102465375rm_f_v @ Y @ X2 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 ) ) ) ).

% converseE
thf(fact_149_converseD,axiom,
    ! [A: nat,B: f,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A @ B ) @ ( converse_f_nat @ R3 ) )
     => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ A ) @ R3 ) ) ).

% converseD
thf(fact_150_converseD,axiom,
    ! [A: f,B: nat,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( converse_nat_f @ R3 ) )
     => ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ B @ A ) @ R3 ) ) ).

% converseD
thf(fact_151_converseD,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( conver3099152742603637891rm_f_v @ R3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R3 ) ) ).

% converseD
thf(fact_152_converse_Osimps,axiom,
    ! [A1: nat,A22: f,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A1 @ A22 ) @ ( converse_f_nat @ R3 ) )
      = ( ? [A4: f,B3: nat] :
            ( ( A1 = B3 )
            & ( A22 = A4 )
            & ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A4 @ B3 ) @ R3 ) ) ) ) ).

% converse.simps
thf(fact_153_converse_Osimps,axiom,
    ! [A1: f,A22: nat,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( converse_nat_f @ R3 ) )
      = ( ? [A4: nat,B3: f] :
            ( ( A1 = B3 )
            & ( A22 = A4 )
            & ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A4 @ B3 ) @ R3 ) ) ) ) ).

% converse.simps
thf(fact_154_converse_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( conver3099152742603637891rm_f_v @ R3 ) )
      = ( ? [A4: term_f_v,B3: term_f_v] :
            ( ( A1 = B3 )
            & ( A22 = A4 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ R3 ) ) ) ) ).

% converse.simps
thf(fact_155_converse_Ocases,axiom,
    ! [A1: nat,A22: f,R3: set_Pr5245412377734449720_f_nat] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A1 @ A22 ) @ ( converse_f_nat @ R3 ) )
     => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A22 @ A1 ) @ R3 ) ) ).

% converse.cases
thf(fact_156_converse_Ocases,axiom,
    ! [A1: f,A22: nat,R3: set_Pr4548512049067879902_nat_f] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( converse_nat_f @ R3 ) )
     => ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ A22 @ A1 ) @ R3 ) ) ).

% converse.cases
thf(fact_157_converse_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( conver3099152742603637891rm_f_v @ R3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A22 @ A1 ) @ R3 ) ) ).

% converse.cases
thf(fact_158_r__comp__rtrancl__eq,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ R3 ) )
      = ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ R3 ) ) ).

% r_comp_rtrancl_eq
thf(fact_159_rtrancl__converse,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) )
      = ( conver3099152742603637891rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).

% rtrancl_converse
thf(fact_160_converse__relcomp,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) )
      = ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ S ) @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ).

% converse_relcomp
thf(fact_161_rstep__converse__dist,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R ) )
      = ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ).

% rstep_converse_dist
thf(fact_162_rtrancl__converseI,axiom,
    ! [Y2: term_f_v,X3: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ X3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ) ).

% rtrancl_converseI
thf(fact_163_rtrancl__converseD,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ X3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).

% rtrancl_converseD
thf(fact_164_sig__step__converse__rstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
      = ( sig_step_f_v @ F @ ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R ) ) ) ) ).

% sig_step_converse_rstep
thf(fact_165_well__subst_I1_J,axiom,
    ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( fun_f_v @ c @ nil_term_f_v ) ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) ).

% well_subst(1)
thf(fact_166_well__subst_I2_J,axiom,
    ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( fun_f_v @ d @ nil_term_f_v ) ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) ).

% well_subst(2)
thf(fact_167_comp__rtrancl__trancl,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ S2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) @ S2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ S2 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_168_srsteps__with__root__step__srstepsD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% srsteps_with_root_step_srstepsD
thf(fact_169_srsteps__with__root__step__sresteps__eqD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% srsteps_with_root_step_sresteps_eqD
thf(fact_170_symcl__idemp,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) @ ( conver3099152742603637891rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) )
      = ( sup_su1994854688125285083rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ).

% symcl_idemp
thf(fact_171_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_172_Un__insert__right,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( sup_su770888850772321124_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
      = ( insert5008453031520946354_f_nat @ A @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_173_Un__insert__right,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
      = ( insert5456475782666790775rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_174_Un__insert__left,axiom,
    ! [A: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat] :
      ( ( sup_su770888850772321124_f_nat @ ( insert5008453031520946354_f_nat @ A @ B4 ) @ C4 )
      = ( insert5008453031520946354_f_nat @ A @ ( sup_su770888850772321124_f_nat @ B4 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_175_Un__insert__left,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B4 ) @ C4 )
      = ( insert5456475782666790775rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_176_Un__subset__iff,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ C4 )
      = ( ( ord_le4559761987009501863rm_f_v @ A3 @ C4 )
        & ( ord_le4559761987009501863rm_f_v @ B4 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_177_Un__subset__iff,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) @ C4 )
      = ( ( ord_le8976984241387448984_f_nat @ A3 @ C4 )
        & ( ord_le8976984241387448984_f_nat @ B4 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_178_insert__subset,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A3 ) @ B4 )
      = ( ( member4316649861536941904rm_f_v @ X3 @ B4 )
        & ( ord_le4559761987009501863rm_f_v @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_179_insert__subset,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ X3 @ A3 ) @ B4 )
      = ( ( member2678038196313615385_f_nat @ X3 @ B4 )
        & ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_180_subset__antisym,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( ord_le8976984241387448984_f_nat @ B4 @ A3 )
       => ( A3 = B4 ) ) ) ).

% subset_antisym
thf(fact_181_subsetI,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
         => ( member4316649861536941904rm_f_v @ X2 @ B4 ) )
     => ( ord_le4559761987009501863rm_f_v @ A3 @ B4 ) ) ).

% subsetI
thf(fact_182_subsetI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ X2 @ A3 )
         => ( member2678038196313615385_f_nat @ X2 @ B4 ) )
     => ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ).

% subsetI
thf(fact_183_insertCI,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( ~ ( member4316649861536941904rm_f_v @ A @ B4 )
       => ( A = B ) )
     => ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ B4 ) ) ) ).

% insertCI
thf(fact_184_insertCI,axiom,
    ! [A: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
      ( ( ~ ( member2678038196313615385_f_nat @ A @ B4 )
       => ( A = B ) )
     => ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_185_insert__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ A3 ) )
      = ( ( A = B )
        | ( member4316649861536941904rm_f_v @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_186_insert__iff,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ A3 ) )
      = ( ( A = B )
        | ( member2678038196313615385_f_nat @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_187_insert__absorb2,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( insert5008453031520946354_f_nat @ X3 @ ( insert5008453031520946354_f_nat @ X3 @ A3 ) )
      = ( insert5008453031520946354_f_nat @ X3 @ A3 ) ) ).

% insert_absorb2
thf(fact_188_Un__iff,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) )
      = ( ( member2678038196313615385_f_nat @ C @ A3 )
        | ( member2678038196313615385_f_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_189_Un__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
      = ( ( member4316649861536941904rm_f_v @ C @ A3 )
        | ( member4316649861536941904rm_f_v @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_190_UnCI,axiom,
    ! [C: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( ~ ( member2678038196313615385_f_nat @ C @ B4 )
       => ( member2678038196313615385_f_nat @ C @ A3 ) )
     => ( member2678038196313615385_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ) ).

% UnCI
thf(fact_191_UnCI,axiom,
    ! [C: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ~ ( member4316649861536941904rm_f_v @ C @ B4 )
       => ( member4316649861536941904rm_f_v @ C @ A3 ) )
     => ( member4316649861536941904rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ) ).

% UnCI
thf(fact_192_sig__stepI,axiom,
    ! [S: term_f_v,F: set_Pr5245412377734449720_f_nat,T: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ R )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ R ) ) ) ) ) ).

% sig_stepI
thf(fact_193_rstep__srstepI,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ).

% rstep_srstepI
thf(fact_194_rsteps__eq__srsteps__eqI,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% rsteps_eq_srsteps_eqI
thf(fact_195_rsteps__srstepsI,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% rsteps_srstepsI
thf(fact_196_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ S4 ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ S4 ) ) ) )
             => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) ) ) ) ) ) ) ).

% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_197_trancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R3: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
     => ( ! [A2: term_f_v,B2: term_f_v] :
            ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) @ R3 )
           => ( P2 @ A2 @ B2 ) )
       => ( ! [A2: term_f_v,B2: term_f_v,Aa: term_f_v,Ba: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A2 @ B2 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_198_trancl__induct2,axiom,
    ! [Ax: f,Ay: nat,Bx: f,By: nat,R3: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi3759436115690491041_f_nat @ R3 ) )
     => ( ! [A2: f,B2: nat] :
            ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A2 @ B2 ) ) @ R3 )
           => ( P2 @ A2 @ B2 ) )
       => ( ! [A2: f,B2: nat,Aa: f,Ba: nat] :
              ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A2 @ B2 ) ) @ ( transi3759436115690491041_f_nat @ R3 ) )
             => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A2 @ B2 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A2 @ B2 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_199_sig__step__mono2,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,L2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le4559761987009501863rm_f_v @ R @ L2 )
     => ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ R ) @ ( sig_step_f_v @ F @ L2 ) ) ) ).

% sig_step_mono2
thf(fact_200_sig__step__mono,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ R ) @ ( sig_step_f_v @ G @ R ) ) ) ).

% sig_step_mono
thf(fact_201_srsteps__with__root__step__sig__mono,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ R ) @ ( srstep3130564645414158319ep_f_v @ G @ R ) ) ) ).

% srsteps_with_root_step_sig_mono
thf(fact_202_sig__stepE,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ R ) )
     => ~ ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ R )
         => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
           => ~ ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ) ).

% sig_stepE
thf(fact_203_Collect__mono__iff,axiom,
    ! [P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ( ord_le947722214152454265rm_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) )
      = ( ! [X: term_f_v] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_204_Collect__mono__iff,axiom,
    ! [P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P2 ) @ ( collec1417984143148964183_f_nat @ Q ) )
      = ( ! [X: product_prod_f_nat] :
            ( ( P2 @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_205_set__eq__subset,axiom,
    ( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A5 @ B5 )
          & ( ord_le8976984241387448984_f_nat @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_206_subset__trans,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( ord_le8976984241387448984_f_nat @ B4 @ C4 )
       => ( ord_le8976984241387448984_f_nat @ A3 @ C4 ) ) ) ).

% subset_trans
thf(fact_207_Collect__mono,axiom,
    ! [P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ! [X2: term_f_v] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le947722214152454265rm_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) ) ) ).

% Collect_mono
thf(fact_208_Collect__mono,axiom,
    ! [P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ! [X2: product_prod_f_nat] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P2 ) @ ( collec1417984143148964183_f_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_209_subset__refl,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A3 @ A3 ) ).

% subset_refl
thf(fact_210_subset__iff,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
        ! [T3: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ T3 @ A5 )
         => ( member4316649861536941904rm_f_v @ T3 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_211_subset__iff,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
        ! [T3: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ T3 @ A5 )
         => ( member2678038196313615385_f_nat @ T3 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_212_equalityD2,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( A3 = B4 )
     => ( ord_le8976984241387448984_f_nat @ B4 @ A3 ) ) ).

% equalityD2
thf(fact_213_equalityD1,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( A3 = B4 )
     => ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ).

% equalityD1
thf(fact_214_subset__eq,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
        ! [X: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ X @ A5 )
         => ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_215_subset__eq,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
        ! [X: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ X @ A5 )
         => ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_216_equalityE,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( A3 = B4 )
     => ~ ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
         => ~ ( ord_le8976984241387448984_f_nat @ B4 @ A3 ) ) ) ).

% equalityE
thf(fact_217_subsetD,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ B4 )
     => ( ( member4316649861536941904rm_f_v @ C @ A3 )
       => ( member4316649861536941904rm_f_v @ C @ B4 ) ) ) ).

% subsetD
thf(fact_218_subsetD,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,C: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( member2678038196313615385_f_nat @ C @ A3 )
       => ( member2678038196313615385_f_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_219_in__mono,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ B4 )
     => ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
       => ( member4316649861536941904rm_f_v @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_220_in__mono,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( member2678038196313615385_f_nat @ X3 @ A3 )
       => ( member2678038196313615385_f_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_221_insertE,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ A3 ) )
     => ( ( A != B )
       => ( member4316649861536941904rm_f_v @ A @ A3 ) ) ) ).

% insertE
thf(fact_222_insertE,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ A3 ) )
     => ( ( A != B )
       => ( member2678038196313615385_f_nat @ A @ A3 ) ) ) ).

% insertE
thf(fact_223_insertI1,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ A @ B4 ) ) ).

% insertI1
thf(fact_224_insertI1,axiom,
    ! [A: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat] : ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_225_insertI2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ B4 )
     => ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ B4 ) ) ) ).

% insertI2
thf(fact_226_insertI2,axiom,
    ! [A: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ B4 )
     => ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_227_Set_Oset__insert,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
     => ~ ! [B6: set_Pr8827868859434726151rm_f_v] :
            ( ( A3
              = ( insert5456475782666790775rm_f_v @ X3 @ B6 ) )
           => ( member4316649861536941904rm_f_v @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_228_Set_Oset__insert,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ X3 @ A3 )
     => ~ ! [B6: set_Pr5245412377734449720_f_nat] :
            ( ( A3
              = ( insert5008453031520946354_f_nat @ X3 @ B6 ) )
           => ( member2678038196313615385_f_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_229_insert__ident,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ X3 @ A3 )
     => ( ~ ( member4316649861536941904rm_f_v @ X3 @ B4 )
       => ( ( ( insert5456475782666790775rm_f_v @ X3 @ A3 )
            = ( insert5456475782666790775rm_f_v @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_230_insert__ident,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ X3 @ A3 )
     => ( ~ ( member2678038196313615385_f_nat @ X3 @ B4 )
       => ( ( ( insert5008453031520946354_f_nat @ X3 @ A3 )
            = ( insert5008453031520946354_f_nat @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_231_insert__absorb,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ( insert5456475782666790775rm_f_v @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_232_insert__absorb,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ( insert5008453031520946354_f_nat @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_233_insert__eq__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ~ ( member4316649861536941904rm_f_v @ B @ B4 )
       => ( ( ( insert5456475782666790775rm_f_v @ A @ A3 )
            = ( insert5456475782666790775rm_f_v @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_Pr8827868859434726151rm_f_v] :
                  ( ( A3
                    = ( insert5456475782666790775rm_f_v @ B @ C5 ) )
                  & ~ ( member4316649861536941904rm_f_v @ B @ C5 )
                  & ( B4
                    = ( insert5456475782666790775rm_f_v @ A @ C5 ) )
                  & ~ ( member4316649861536941904rm_f_v @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_234_insert__eq__iff,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ~ ( member2678038196313615385_f_nat @ B @ B4 )
       => ( ( ( insert5008453031520946354_f_nat @ A @ A3 )
            = ( insert5008453031520946354_f_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_Pr5245412377734449720_f_nat] :
                  ( ( A3
                    = ( insert5008453031520946354_f_nat @ B @ C5 ) )
                  & ~ ( member2678038196313615385_f_nat @ B @ C5 )
                  & ( B4
                    = ( insert5008453031520946354_f_nat @ A @ C5 ) )
                  & ~ ( member2678038196313615385_f_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_235_insert__commute,axiom,
    ! [X3: product_prod_f_nat,Y2: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( insert5008453031520946354_f_nat @ X3 @ ( insert5008453031520946354_f_nat @ Y2 @ A3 ) )
      = ( insert5008453031520946354_f_nat @ Y2 @ ( insert5008453031520946354_f_nat @ X3 @ A3 ) ) ) ).

% insert_commute
thf(fact_236_mk__disjoint__insert,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ A3 )
     => ? [B6: set_Pr8827868859434726151rm_f_v] :
          ( ( A3
            = ( insert5456475782666790775rm_f_v @ A @ B6 ) )
          & ~ ( member4316649861536941904rm_f_v @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_237_mk__disjoint__insert,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ A3 )
     => ? [B6: set_Pr5245412377734449720_f_nat] :
          ( ( A3
            = ( insert5008453031520946354_f_nat @ A @ B6 ) )
          & ~ ( member2678038196313615385_f_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_238_Un__left__commute,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) )
      = ( sup_su1994854688125285083rm_f_v @ B4 @ ( sup_su1994854688125285083rm_f_v @ A3 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_239_Un__left__absorb,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
      = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ).

% Un_left_absorb
thf(fact_240_Un__commute,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_241_Un__absorb,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_242_Un__assoc,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ C4 )
      = ( sup_su1994854688125285083rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) ) ) ).

% Un_assoc
thf(fact_243_ball__Un,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( ! [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
           => ( P2 @ X ) ) )
      = ( ! [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ A3 )
           => ( P2 @ X ) )
        & ! [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ B4 )
           => ( P2 @ X ) ) ) ) ).

% ball_Un
thf(fact_244_bex__Un,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( ? [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
            & ( P2 @ X ) ) )
      = ( ? [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ A3 )
            & ( P2 @ X ) )
        | ? [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ B4 )
            & ( P2 @ X ) ) ) ) ).

% bex_Un
thf(fact_245_UnI2,axiom,
    ! [C: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ B4 )
     => ( member2678038196313615385_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ) ).

% UnI2
thf(fact_246_UnI2,axiom,
    ! [C: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ B4 )
     => ( member4316649861536941904rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ) ).

% UnI2
thf(fact_247_UnI1,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ A3 )
     => ( member2678038196313615385_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ) ).

% UnI1
thf(fact_248_UnI1,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ A3 )
     => ( member4316649861536941904rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ) ).

% UnI1
thf(fact_249_UnE,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) )
     => ( ~ ( member2678038196313615385_f_nat @ C @ A3 )
       => ( member2678038196313615385_f_nat @ C @ B4 ) ) ) ).

% UnE
thf(fact_250_UnE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ C @ A3 )
       => ( member4316649861536941904rm_f_v @ C @ B4 ) ) ) ).

% UnE
thf(fact_251_srstep__monp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ G @ ( rstep_f_v @ R ) ) ) ) ).

% srstep_monp
thf(fact_252_sigstep__trancl__funas_I2_J,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) )
     => ( ( S != T )
       => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).

% sigstep_trancl_funas(2)
thf(fact_253_sigstep__trancl__funas_I1_J,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) )
     => ( ( S != T )
       => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F ) ) ) ).

% sigstep_trancl_funas(1)
thf(fact_254_srstepD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).

% srstepD
thf(fact_255_srsteps__eq__monp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ R ) ) ) ) ) ).

% srsteps_eq_monp
thf(fact_256_srsteps__monp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ R ) ) ) ) ) ).

% srsteps_monp
thf(fact_257_Collect__subset,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ord_le4559761987009501863rm_f_v
      @ ( collec7769483783292507282rm_f_v
        @ ^ [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ A3 )
            & ( P2 @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_258_Collect__subset,axiom,
    ! [A3: set_term_f_v,P2: term_f_v > $o] :
      ( ord_le947722214152454265rm_f_v
      @ ( collect_term_f_v
        @ ^ [X: term_f_v] :
            ( ( member_term_f_v @ X @ A3 )
            & ( P2 @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_259_Collect__subset,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,P2: product_prod_f_nat > $o] :
      ( ord_le8976984241387448984_f_nat
      @ ( collec1417984143148964183_f_nat
        @ ^ [X: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X @ A3 )
            & ( P2 @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_260_insert__compr,axiom,
    ( insert5456475782666790775rm_f_v
    = ( ^ [A4: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [X: produc8027700073345061287rm_f_v] :
              ( ( X = A4 )
              | ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_261_insert__compr,axiom,
    ( insert5008453031520946354_f_nat
    = ( ^ [A4: product_prod_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( X = A4 )
              | ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_262_insert__compr,axiom,
    ( insert_term_f_v
    = ( ^ [A4: term_f_v,B5: set_term_f_v] :
          ( collect_term_f_v
          @ ^ [X: term_f_v] :
              ( ( X = A4 )
              | ( member_term_f_v @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_263_insert__Collect,axiom,
    ! [A: product_prod_f_nat,P2: product_prod_f_nat > $o] :
      ( ( insert5008453031520946354_f_nat @ A @ ( collec1417984143148964183_f_nat @ P2 ) )
      = ( collec1417984143148964183_f_nat
        @ ^ [U: product_prod_f_nat] :
            ( ( U != A )
           => ( P2 @ U ) ) ) ) ).

% insert_Collect
thf(fact_264_insert__Collect,axiom,
    ! [A: term_f_v,P2: term_f_v > $o] :
      ( ( insert_term_f_v @ A @ ( collect_term_f_v @ P2 ) )
      = ( collect_term_f_v
        @ ^ [U: term_f_v] :
            ( ( U != A )
           => ( P2 @ U ) ) ) ) ).

% insert_Collect
thf(fact_265_Un__def,axiom,
    ( sup_su770888850772321124_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X @ A5 )
              | ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_266_Un__def,axiom,
    ( sup_sup_set_term_f_v
    = ( ^ [A5: set_term_f_v,B5: set_term_f_v] :
          ( collect_term_f_v
          @ ^ [X: term_f_v] :
              ( ( member_term_f_v @ X @ A5 )
              | ( member_term_f_v @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_267_Un__def,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [X: produc8027700073345061287rm_f_v] :
              ( ( member4316649861536941904rm_f_v @ X @ A5 )
              | ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_268_Collect__disj__eq,axiom,
    ! [P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ( collect_term_f_v
        @ ^ [X: term_f_v] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_term_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_269_Collect__disj__eq,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
      ( ( collec7769483783292507282rm_f_v
        @ ^ [X: produc8027700073345061287rm_f_v] :
            ( ( P2 @ X )
            | ( Q @ X ) ) )
      = ( sup_su1994854688125285083rm_f_v @ ( collec7769483783292507282rm_f_v @ P2 ) @ ( collec7769483783292507282rm_f_v @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_270_srstep__subst__closed,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
     => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% srstep_subst_closed
thf(fact_271_srstepsD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).

% srstepsD
thf(fact_272_srsteps__eq__subst__closed,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
     => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ).

% srsteps_eq_subst_closed
thf(fact_273_srsteps__subst__closed,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
     => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ).

% srsteps_subst_closed
thf(fact_274_insert__mono,axiom,
    ! [C4: set_Pr5245412377734449720_f_nat,D3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C4 @ D3 )
     => ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ A @ C4 ) @ ( insert5008453031520946354_f_nat @ A @ D3 ) ) ) ).

% insert_mono
thf(fact_275_subset__insert,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ X3 @ A3 )
     => ( ( ord_le4559761987009501863rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ X3 @ B4 ) )
        = ( ord_le4559761987009501863rm_f_v @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_276_subset__insert,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ X3 @ A3 )
     => ( ( ord_le8976984241387448984_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ X3 @ B4 ) )
        = ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_277_subset__insertI,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ B4 @ ( insert5008453031520946354_f_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_278_subset__insertI2,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ord_le8976984241387448984_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_279_srsteps__eq__subst__relcomp__closed,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) )
     => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) ) ) ) ).

% srsteps_eq_subst_relcomp_closed
thf(fact_280_subset__Un__eq,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( ( sup_su1994854688125285083rm_f_v @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_281_subset__Un__eq,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( ( sup_su770888850772321124_f_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_282_subset__UnE,axiom,
    ! [C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C4 @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ~ ! [A6: set_Pr8827868859434726151rm_f_v] :
            ( ( ord_le4559761987009501863rm_f_v @ A6 @ A3 )
           => ! [B7: set_Pr8827868859434726151rm_f_v] :
                ( ( ord_le4559761987009501863rm_f_v @ B7 @ B4 )
               => ( C4
                 != ( sup_su1994854688125285083rm_f_v @ A6 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_283_subset__UnE,axiom,
    ! [C4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C4 @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) )
     => ~ ! [A6: set_Pr5245412377734449720_f_nat] :
            ( ( ord_le8976984241387448984_f_nat @ A6 @ A3 )
           => ! [B7: set_Pr5245412377734449720_f_nat] :
                ( ( ord_le8976984241387448984_f_nat @ B7 @ B4 )
               => ( C4
                 != ( sup_su770888850772321124_f_nat @ A6 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_284_Un__absorb2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B4 @ A3 )
     => ( ( sup_su1994854688125285083rm_f_v @ A3 @ B4 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_285_Un__absorb2,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B4 @ A3 )
     => ( ( sup_su770888850772321124_f_nat @ A3 @ B4 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_286_Un__absorb1,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ B4 )
     => ( ( sup_su1994854688125285083rm_f_v @ A3 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_287_Un__absorb1,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( sup_su770888850772321124_f_nat @ A3 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_288_Un__upper2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ B4 @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ).

% Un_upper2
thf(fact_289_Un__upper2,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ B4 @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ).

% Un_upper2
thf(fact_290_Un__upper1,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) ).

% Un_upper1
thf(fact_291_Un__upper1,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A3 @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) ) ).

% Un_upper1
thf(fact_292_Un__least,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ C4 )
     => ( ( ord_le4559761987009501863rm_f_v @ B4 @ C4 )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ C4 ) ) ) ).

% Un_least
thf(fact_293_Un__least,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ C4 )
     => ( ( ord_le8976984241387448984_f_nat @ B4 @ C4 )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) @ C4 ) ) ) ).

% Un_least
thf(fact_294_Un__mono,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,D3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ C4 )
     => ( ( ord_le4559761987009501863rm_f_v @ B4 @ D3 )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ ( sup_su1994854688125285083rm_f_v @ C4 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_295_Un__mono,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,D3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ C4 )
     => ( ( ord_le8976984241387448984_f_nat @ B4 @ D3 )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A3 @ B4 ) @ ( sup_su770888850772321124_f_nat @ C4 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_296_relcomp3__I,axiom,
    ! [T: nat,U2: f,A3: set_Pr4548512049067879902_nat_f,S: f,B4: set_Pr5245412377734449720_f_nat,V: nat] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ T @ U2 ) @ A3 )
     => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ B4 )
       => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ U2 @ V ) @ B4 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ V ) @ ( relcomp_f_nat_nat @ B4 @ ( relcomp_nat_f_nat @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_297_relcomp3__I,axiom,
    ! [T: term_f_v,U2: term_f_v,A3: set_Pr8827868859434726151rm_f_v,S: term_f_v,B4: set_Pr8827868859434726151rm_f_v,V: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U2 ) @ A3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ B4 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U2 @ V ) @ B4 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ V ) @ ( relcom2934258169684288428rm_f_v @ B4 @ ( relcom2934258169684288428rm_f_v @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_298_relcomp3__I,axiom,
    ! [T: f,U2: nat,A3: set_Pr5245412377734449720_f_nat,S: nat,B4: set_Pr4548512049067879902_nat_f,V: f] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ T @ U2 ) @ A3 )
     => ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ S @ T ) @ B4 )
       => ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ U2 @ V ) @ B4 )
         => ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ S @ V ) @ ( relcomp_nat_f_f @ B4 @ ( relcomp_f_nat_f @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_299_open__terms__two__const__lv_Owell__subst_I2_J,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( fun_f_v @ D @ nil_term_f_v ) ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F ) ) ) ) ).

% open_terms_two_const_lv.well_subst(2)
thf(fact_300_open__terms__two__const__lv_Owell__subst_I1_J,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( fun_f_v @ C @ nil_term_f_v ) ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F ) ) ) ) ).

% open_terms_two_const_lv.well_subst(1)
thf(fact_301_rtrancl__O__shift,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) ) @ S2 )
      = ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_302_O__mono1,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,R7: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R2 @ R7 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ R7 ) ) ) ).

% O_mono1
thf(fact_303_O__mono1,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,R7: set_Pr5245412377734449720_f_nat,S2: set_Product_prod_f_f] :
      ( ( ord_le8976984241387448984_f_nat @ R2 @ R7 )
     => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ S2 @ R2 ) @ ( relcomp_f_f_nat @ S2 @ R7 ) ) ) ).

% O_mono1
thf(fact_304_O__mono2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,R7: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R2 @ R7 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ T2 ) @ ( relcom2934258169684288428rm_f_v @ R7 @ T2 ) ) ) ).

% O_mono2
thf(fact_305_O__mono2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,R7: set_Pr5245412377734449720_f_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R2 @ R7 )
     => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R2 @ T2 ) @ ( relcomp_f_nat_nat @ R7 @ T2 ) ) ) ).

% O_mono2
thf(fact_306_trancl__mono__set,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ).

% trancl_mono_set
thf(fact_307_symcl__Un,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ ( conver3099152742603637891rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) ) )
      = ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ ( conver3099152742603637891rm_f_v @ A3 ) ) @ ( sup_su1994854688125285083rm_f_v @ B4 @ ( conver3099152742603637891rm_f_v @ B4 ) ) ) ) ).

% symcl_Un
thf(fact_308_symcl__converse,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ ( conver3099152742603637891rm_f_v @ A3 ) ) )
      = ( sup_su1994854688125285083rm_f_v @ A3 @ ( conver3099152742603637891rm_f_v @ A3 ) ) ) ).

% symcl_converse
thf(fact_309_comp__rrstep__rel_H__sig__mono,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ R ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) ) ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ G @ R ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ S4 ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ R ) ) ) @ ( srstep3130564645414158319ep_f_v @ G @ S4 ) ) ) ) ) ).

% comp_rrstep_rel'_sig_mono
thf(fact_310_insert__def,axiom,
    ( insert5008453031520946354_f_nat
    = ( ^ [A4: product_prod_f_nat] :
          ( sup_su770888850772321124_f_nat
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_311_insert__def,axiom,
    ( insert_term_f_v
    = ( ^ [A4: term_f_v] :
          ( sup_sup_set_term_f_v
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_312_insert__def,axiom,
    ( insert5456475782666790775rm_f_v
    = ( ^ [A4: produc8027700073345061287rm_f_v] :
          ( sup_su1994854688125285083rm_f_v
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_313_steps__map,axiom,
    ! [P2: term_f_v > $o,Q: set_Pr8827868859434726151rm_f_v > $o,F3: term_f_v > term_f_v,G2: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,U2: term_f_v] :
      ( ! [T4: term_f_v,U3: term_f_v,R8: set_Pr8827868859434726151rm_f_v] :
          ( ( P2 @ T4 )
         => ( ( Q @ R8 )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ U3 ) @ R8 )
             => ( ( P2 @ U3 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F3 @ T4 ) @ ( F3 @ U3 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q @ R2 )
         => ( ( Q @ S2 )
           => ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U2 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F3 @ T ) @ ( F3 @ U2 ) ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R2 ) ) ) )
              & ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U2 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F3 @ T ) @ ( F3 @ U2 ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( G2 @ R2 ) ) @ ( relcom2934258169684288428rm_f_v @ ( G2 @ S2 ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R2 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_314_peak__iff,axiom,
    ! [X3: nat,Y2: nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( relcomp_nat_f_nat @ ( converse_f_nat @ A3 ) @ B4 ) )
      = ( ? [U: f] :
            ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ U @ X3 ) @ A3 )
            & ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ U @ Y2 ) @ B4 ) ) ) ) ).

% peak_iff
thf(fact_315_peak__iff,axiom,
    ! [X3: f,Y2: nat,A3: set_Product_prod_f_f,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X3 @ Y2 ) @ ( relcomp_f_f_nat @ ( converse_f_f @ A3 ) @ B4 ) )
      = ( ? [U: f] :
            ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ U @ X3 ) @ A3 )
            & ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ U @ Y2 ) @ B4 ) ) ) ) ).

% peak_iff
thf(fact_316_peak__iff,axiom,
    ! [X3: term_f_v,Y2: term_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ A3 ) @ B4 ) )
      = ( ? [U: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ X3 ) @ A3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ Y2 ) @ B4 ) ) ) ) ).

% peak_iff
thf(fact_317_rtrancl__O__push,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ R2 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3028809021613410336rm_f_v @ S2 ) ) ) ) ).

% rtrancl_O_push
thf(fact_318_compat__tr__compat,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ NS ) @ S2 ) @ S2 ) ) ).

% compat_tr_compat
thf(fact_319_compat__tr__compat,axiom,
    ! [NS: set_Product_prod_f_f,S2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ NS @ S2 ) @ S2 )
     => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ NS ) @ S2 ) @ S2 ) ) ).

% compat_tr_compat
thf(fact_320_O__rtrancl__O__O,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ R2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) ) @ S2 ) )
      = ( transi3208867851621559554rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) ) ) ).

% O_rtrancl_O_O
thf(fact_321_rtrancl__comp__trancl__conv,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ R3 )
      = ( transi3208867851621559554rm_f_v @ R3 ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_322_comp__trancl,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ S2 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3208867851621559554rm_f_v @ S2 ) ) @ ( transi3208867851621559554rm_f_v @ S2 ) ) ) ).

% comp_trancl
thf(fact_323_rtrancl__union__subset__rtrancl__union__trancl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ ( transi3208867851621559554rm_f_v @ S ) ) )
      = ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S ) ) ) ).

% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_324_trancl__union__right,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ ( transi3208867851621559554rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S @ R3 ) ) ) ).

% trancl_union_right
thf(fact_325_non__strict__into__strict,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ NS ) @ S2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ S2 ) ) ) ).

% non_strict_into_strict
thf(fact_326_non__strict__into__strict,axiom,
    ! [NS: set_Product_prod_f_f,S2: set_Pr5245412377734449720_f_nat,S: f,T: nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ NS @ S2 ) @ S2 )
     => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ NS ) @ S2 ) )
       => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ S2 ) ) ) ).

% non_strict_into_strict
thf(fact_327_firstStep,axiom,
    ! [L3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ( L3
        = ( sup_su1994854688125285083rm_f_v @ S2 @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ L3 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
          | ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ L3 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_328_first__step,axiom,
    ! [C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v,Z2: term_f_v] :
      ( ( C4
        = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ C4 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ B4 )
         => ? [Y: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ B4 ) ) ) ) ) ).

% first_step
thf(fact_329_first__step__O,axiom,
    ! [C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ( C4
        = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ C4 ) @ B4 ) )
       => ? [Y: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ B4 ) ) ) ) ).

% first_step_O
thf(fact_330_right__comp__S,axiom,
    ! [X3: term_f_v,Y2: term_f_v,S2: set_Pr8827868859434726151rm_f_v,NS: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_331_rtrancl__U__push,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
     => ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S2 @ R2 ) )
        = ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) ) ) ).

% rtrancl_U_push
thf(fact_332_compatible__conv,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) ) )
        = ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).

% compatible_conv
thf(fact_333_lv__srsteps__with__root__step__idep__subst,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,Sigma: v > term_f_v,Tau: v > term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
       => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
         => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Tau @ X2 ) ) @ F )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Tau ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) ) ) ) ).

% lv_srsteps_with_root_step_idep_subst
thf(fact_334_const__subst__eq__ground__eq,axiom,
    ! [S: term_f_v,C: f,T: term_f_v,D: f] :
      ( ( ( subst_7901567856578221233_f_v_v @ S
          @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
        = ( subst_7901567856578221233_f_v_v @ T
          @ ^ [X: v] : ( fun_f_v @ D @ nil_term_f_v ) ) )
     => ( ( C != D )
       => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( term_funas_term_f_v @ T ) )
         => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ ( term_funas_term_f_v @ S ) )
           => ( S = T ) ) ) ) ) ).

% const_subst_eq_ground_eq
thf(fact_335_sup_Obounded__iff,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ B @ C ) @ A )
      = ( ( ord_le4559761987009501863rm_f_v @ B @ A )
        & ( ord_le4559761987009501863rm_f_v @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_336_sup_Obounded__iff,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C ) @ A )
      = ( ( ord_le8976984241387448984_f_nat @ B @ A )
        & ( ord_le8976984241387448984_f_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_337_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_338_le__sup__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ Z2 )
      = ( ( ord_le4559761987009501863rm_f_v @ X3 @ Z2 )
        & ( ord_le4559761987009501863rm_f_v @ Y2 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_339_le__sup__iff,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) @ Z2 )
      = ( ( ord_le8976984241387448984_f_nat @ X3 @ Z2 )
        & ( ord_le8976984241387448984_f_nat @ Y2 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_340_le__sup__iff,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ Z2 )
      = ( ( ord_less_eq_nat @ X3 @ Z2 )
        & ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_341_CR__Aux,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) )
         => ( rewrit4152091719787942777dp_f_v @ F @ R @ R @ S5 @ T4 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) )
       => ( rewrit4152091719787942777dp_f_v @ F @ R @ R @ S @ T ) ) ) ).

% CR_Aux
thf(fact_342_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_343_commute__def,axiom,
    ( abstra4964661570048985923rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R6 ) ) @ ( transi3028809021613410336rm_f_v @ S6 ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S6 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R6 ) ) ) ) ) ) ).

% commute_def
thf(fact_344_remove__const__lv__mondaic__steps,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,C: f,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
       => ( ( rewrit3916899077429870368adic_f @ F )
         => ( ( member4316649861536941904rm_f_v
              @ ( produc2510790566102465375rm_f_v
                @ ( subst_7901567856578221233_f_v_v @ S
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
                @ ( subst_7901567856578221233_f_v_v @ T
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) ) )
              @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% remove_const_lv_mondaic_steps
thf(fact_345_remove__const__lv__mondaic__steps__lhs,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,C: f,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
       => ( ( rewrit3916899077429870368adic_f @ F )
         => ( ( member4316649861536941904rm_f_v
              @ ( produc2510790566102465375rm_f_v
                @ ( subst_7901567856578221233_f_v_v @ S
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
                @ T )
              @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% remove_const_lv_mondaic_steps_lhs
thf(fact_346_sup_Oidem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A @ A )
      = A ) ).

% sup.idem
thf(fact_347_sup__idem,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ X3 )
      = X3 ) ).

% sup_idem
thf(fact_348_sup_Oleft__idem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ A @ B ) )
      = ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ).

% sup.left_idem
thf(fact_349_sup__left__idem,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) )
      = ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% sup_left_idem
thf(fact_350_sup_Oright__idem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A @ B ) @ B )
      = ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ).

% sup.right_idem
thf(fact_351_sup__Un__eq,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( sup_su7253025400623347233_nat_o
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ R2 )
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ S2 ) )
      = ( ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ ( sup_su770888850772321124_f_nat @ R2 @ S2 ) ) ) ) ).

% sup_Un_eq
thf(fact_352_sup__Un__eq,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su755325427420126146_f_v_o
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ R2 )
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ S2 ) )
      = ( ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) ) ) ).

% sup_Un_eq
thf(fact_353_sup__set__def,axiom,
    ( sup_su770888850772321124_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( collec1417984143148964183_f_nat
          @ ( sup_su7253025400623347233_nat_o
            @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A5 )
            @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_354_sup__set__def,axiom,
    ( sup_sup_set_term_f_v
    = ( ^ [A5: set_term_f_v,B5: set_term_f_v] :
          ( collect_term_f_v
          @ ( sup_sup_term_f_v_o
            @ ^ [X: term_f_v] : ( member_term_f_v @ X @ A5 )
            @ ^ [X: term_f_v] : ( member_term_f_v @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_355_sup__set__def,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ( sup_su755325427420126146_f_v_o
            @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A5 )
            @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_356_pred__subset__eq,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4597002773593529078_f_v_o
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ R2 )
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ S2 ) )
      = ( ord_le4559761987009501863rm_f_v @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_357_pred__subset__eq,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le3748900571519617005_nat_o
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ R2 )
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ S2 ) )
      = ( ord_le8976984241387448984_f_nat @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_358_less__eq__set__def,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( ord_le4597002773593529078_f_v_o
          @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A5 )
          @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_359_less__eq__set__def,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( ord_le3748900571519617005_nat_o
          @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A5 )
          @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_360_sup__Un__eq2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( sup_sup_f_nat_o
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R2 )
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ ( sup_su770888850772321124_f_nat @ R2 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_361_sup__Un__eq2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su8963822174138643348_f_v_o
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( sup_su1994854688125285083rm_f_v @ R2 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_362_pred__subset__eq2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le194975665355383648_f_v_o
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ S2 ) )
      = ( ord_le4559761987009501863rm_f_v @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_363_pred__subset__eq2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_f_nat_o
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R2 )
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ S2 ) )
      = ( ord_le8976984241387448984_f_nat @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_364_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_365_inf__sup__aci_I8_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) )
      = ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_366_inf__sup__aci_I7_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) )
      = ( sup_su1994854688125285083rm_f_v @ Y2 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_367_inf__sup__aci_I6_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ Z2 )
      = ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_368_inf__sup__aci_I5_J,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ Y3 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_369_sup_Oassoc,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A @ B ) @ C )
      = ( sup_su1994854688125285083rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ B @ C ) ) ) ).

% sup.assoc
thf(fact_370_sup__assoc,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ Z2 )
      = ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_371_sup_Ocommute,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [A4: set_Pr8827868859434726151rm_f_v,B3: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ B3 @ A4 ) ) ) ).

% sup.commute
thf(fact_372_sup__commute,axiom,
    ( sup_su1994854688125285083rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ Y3 @ X ) ) ) ).

% sup_commute
thf(fact_373_sup_Oleft__commute,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ B @ ( sup_su1994854688125285083rm_f_v @ A @ C ) )
      = ( sup_su1994854688125285083rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_374_sup__left__commute,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) )
      = ( sup_su1994854688125285083rm_f_v @ Y2 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_375_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_376_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ Y2 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_377_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y2 @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_378_inf__sup__ord_I4_J,axiom,
    ! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_379_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_380_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_381_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_382_le__supE,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A @ B ) @ X3 )
     => ~ ( ( ord_le4559761987009501863rm_f_v @ A @ X3 )
         => ~ ( ord_le4559761987009501863rm_f_v @ B @ X3 ) ) ) ).

% le_supE
thf(fact_383_le__supE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_le8976984241387448984_f_nat @ A @ X3 )
         => ~ ( ord_le8976984241387448984_f_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_384_le__supE,axiom,
    ! [A: nat,B: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_nat @ A @ X3 )
         => ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_385_le__supI,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ X3 )
     => ( ( ord_le4559761987009501863rm_f_v @ B @ X3 )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_386_le__supI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ X3 )
     => ( ( ord_le8976984241387448984_f_nat @ B @ X3 )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_387_le__supI,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ( ord_less_eq_nat @ B @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_388_sup__ge1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% sup_ge1
thf(fact_389_sup__ge1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) ) ).

% sup_ge1
thf(fact_390_sup__ge1,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% sup_ge1
thf(fact_391_sup__ge2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ Y2 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) ) ).

% sup_ge2
thf(fact_392_sup__ge2,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y2 @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) ) ).

% sup_ge2
thf(fact_393_sup__ge2,axiom,
    ! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% sup_ge2
thf(fact_394_le__supI1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ A )
     => ( ord_le4559761987009501863rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ).

% le_supI1
thf(fact_395_le__supI1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ A )
     => ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_396_le__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_397_le__supI2,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ B )
     => ( ord_le4559761987009501863rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ).

% le_supI2
thf(fact_398_le__supI2,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ B )
     => ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_399_le__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X3 @ B )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_400_sup_Omono,axiom,
    ! [C: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,D: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C @ A )
     => ( ( ord_le4559761987009501863rm_f_v @ D @ B )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ C @ D ) @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_401_sup_Omono,axiom,
    ! [C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C @ A )
     => ( ( ord_le8976984241387448984_f_nat @ D @ B )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ C @ D ) @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_402_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_403_sup__mono,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,D: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ C )
     => ( ( ord_le4559761987009501863rm_f_v @ B @ D )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A @ B ) @ ( sup_su1994854688125285083rm_f_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_404_sup__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ( ord_le8976984241387448984_f_nat @ B @ D )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ ( sup_su770888850772321124_f_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_405_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_406_sup__least,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ Y2 @ X3 )
     => ( ( ord_le4559761987009501863rm_f_v @ Z2 @ X3 )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_407_sup__least,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y2 @ X3 )
     => ( ( ord_le8976984241387448984_f_nat @ Z2 @ X3 )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ Y2 @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_408_sup__least,axiom,
    ! [Y2: nat,X3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( ord_less_eq_nat @ Z2 @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_409_le__iff__sup,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] :
          ( ( sup_su1994854688125285083rm_f_v @ X @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_410_le__iff__sup,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
          ( ( sup_su770888850772321124_f_nat @ X @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_411_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y3: nat] :
          ( ( sup_sup_nat @ X @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_412_sup_OorderE,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ A )
     => ( A
        = ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ).

% sup.orderE
thf(fact_413_sup_OorderE,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( A
        = ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_414_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_415_sup_OorderI,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( A
        = ( sup_su1994854688125285083rm_f_v @ A @ B ) )
     => ( ord_le4559761987009501863rm_f_v @ B @ A ) ) ).

% sup.orderI
thf(fact_416_sup_OorderI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( A
        = ( sup_su770888850772321124_f_nat @ A @ B ) )
     => ( ord_le8976984241387448984_f_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_417_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_418_sup__unique,axiom,
    ! [F3: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ X2 @ ( F3 @ X2 @ Y ) )
     => ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ Y @ ( F3 @ X2 @ Y ) )
       => ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v,Z: set_Pr8827868859434726151rm_f_v] :
              ( ( ord_le4559761987009501863rm_f_v @ Y @ X2 )
             => ( ( ord_le4559761987009501863rm_f_v @ Z @ X2 )
               => ( ord_le4559761987009501863rm_f_v @ ( F3 @ Y @ Z ) @ X2 ) ) )
         => ( ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_419_sup__unique,axiom,
    ! [F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X2 @ ( F3 @ X2 @ Y ) )
     => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y @ ( F3 @ X2 @ Y ) )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ Y @ X2 )
             => ( ( ord_le8976984241387448984_f_nat @ Z @ X2 )
               => ( ord_le8976984241387448984_f_nat @ ( F3 @ Y @ Z ) @ X2 ) ) )
         => ( ( sup_su770888850772321124_f_nat @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_420_sup__unique,axiom,
    ! [F3: nat > nat > nat,X3: nat,Y2: nat] :
      ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( F3 @ X2 @ Y ) )
     => ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F3 @ X2 @ Y ) )
       => ( ! [X2: nat,Y: nat,Z: nat] :
              ( ( ord_less_eq_nat @ Y @ X2 )
             => ( ( ord_less_eq_nat @ Z @ X2 )
               => ( ord_less_eq_nat @ ( F3 @ Y @ Z ) @ X2 ) ) )
         => ( ( sup_sup_nat @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_421_sup_Oabsorb1,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ A )
     => ( ( sup_su1994854688125285083rm_f_v @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_422_sup_Oabsorb1,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( sup_su770888850772321124_f_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_423_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_424_sup_Oabsorb2,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ B )
     => ( ( sup_su1994854688125285083rm_f_v @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_425_sup_Oabsorb2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( sup_su770888850772321124_f_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_426_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_427_sup__absorb1,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ Y2 @ X3 )
     => ( ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_428_sup__absorb1,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y2 @ X3 )
     => ( ( sup_su770888850772321124_f_nat @ X3 @ Y2 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_429_sup__absorb1,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( sup_sup_nat @ X3 @ Y2 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_430_sup__absorb2,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ Y2 )
     => ( ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_431_sup__absorb2,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
     => ( ( sup_su770888850772321124_f_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_432_sup__absorb2,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( sup_sup_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_433_sup_OboundedE,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ B @ C ) @ A )
     => ~ ( ( ord_le4559761987009501863rm_f_v @ B @ A )
         => ~ ( ord_le4559761987009501863rm_f_v @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_434_sup_OboundedE,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C ) @ A )
     => ~ ( ( ord_le8976984241387448984_f_nat @ B @ A )
         => ~ ( ord_le8976984241387448984_f_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_435_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_436_sup_OboundedI,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ A )
     => ( ( ord_le4559761987009501863rm_f_v @ C @ A )
       => ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_437_sup_OboundedI,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( ord_le8976984241387448984_f_nat @ C @ A )
       => ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_438_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_439_sup_Oorder__iff,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [B3: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
          ( A4
          = ( sup_su1994854688125285083rm_f_v @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_440_sup_Oorder__iff,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [B3: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] :
          ( A4
          = ( sup_su770888850772321124_f_nat @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_441_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_442_sup_Ocobounded1,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ A @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ).

% sup.cobounded1
thf(fact_443_sup_Ocobounded1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_444_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_445_sup_Ocobounded2,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ B @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ).

% sup.cobounded2
thf(fact_446_sup_Ocobounded2,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ B @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_447_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_448_sup_Oabsorb__iff1,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [B3: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
          ( ( sup_su1994854688125285083rm_f_v @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_449_sup_Oabsorb__iff1,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [B3: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] :
          ( ( sup_su770888850772321124_f_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_450_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_451_sup_Oabsorb__iff2,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A4: set_Pr8827868859434726151rm_f_v,B3: set_Pr8827868859434726151rm_f_v] :
          ( ( sup_su1994854688125285083rm_f_v @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_452_sup_Oabsorb__iff2,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
          ( ( sup_su770888850772321124_f_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_453_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( sup_sup_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_454_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C @ A )
     => ( ord_le4559761987009501863rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_455_sup_OcoboundedI1,axiom,
    ! [C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C @ A )
     => ( ord_le8976984241387448984_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_456_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_457_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C @ B )
     => ( ord_le4559761987009501863rm_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_458_sup_OcoboundedI2,axiom,
    ! [C: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C @ B )
     => ( ord_le8976984241387448984_f_nat @ C @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_459_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_460_commute__to__rrstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( abstra4964661570048985923rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) ) ) )
       => ( rewrit4152091719787942777dp_f_v @ F @ R @ S4 @ S @ T ) ) ) ).

% commute_to_rrstep
thf(fact_461_commute__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) ) ) )
         => ( rewrit4152091719787942777dp_f_v @ F @ R @ S4 @ S5 @ T4 ) )
     => ( abstra4964661570048985923rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) ).

% commute_rrstep_intro
thf(fact_462_remove__const__lv__mondaic__step__rhs,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,C: f,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
       => ( ( rewrit3916899077429870368adic_f @ F )
         => ( ( member4316649861536941904rm_f_v
              @ ( produc2510790566102465375rm_f_v @ S
                @ ( subst_7901567856578221233_f_v_v @ T
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) ) )
              @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ).

% remove_const_lv_mondaic_step_rhs
thf(fact_463_remove__const__lv__mondaic__step__lhs,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,C: f,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
       => ( ( rewrit3916899077429870368adic_f @ F )
         => ( ( member4316649861536941904rm_f_v
              @ ( produc2510790566102465375rm_f_v
                @ ( subst_7901567856578221233_f_v_v @ S
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
                @ T )
              @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ).

% remove_const_lv_mondaic_step_lhs
thf(fact_464_commute__redp__def,axiom,
    ( rewrit4152091719787942777dp_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v,S7: set_Pr8827868859434726151rm_f_v,S6: term_f_v,T3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S6 @ T3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ S7 ) ) ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) ) ) ) ) ) ).

% commute_redp_def
thf(fact_465_remove__const__lv__mondaic__steps__rhs,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,C: f,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
       => ( ( rewrit3916899077429870368adic_f @ F )
         => ( ( member4316649861536941904rm_f_v
              @ ( produc2510790566102465375rm_f_v @ S
                @ ( subst_7901567856578221233_f_v_v @ T
                  @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) ) )
              @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% remove_const_lv_mondaic_steps_rhs
thf(fact_466_const__replace__closed__remove__subst__lhs,axiom,
    ! [C: f,U4: set_Pr8827868859434726151rm_f_v,T: term_f_v,S: term_f_v] :
      ( ( replac3904333054701138457ed_f_v @ C @ U4 )
     => ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( term_funas_term_f_v @ T ) )
       => ( ( member4316649861536941904rm_f_v
            @ ( produc2510790566102465375rm_f_v
              @ ( subst_7901567856578221233_f_v_v @ S
                @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
              @ T )
            @ U4 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ U4 ) ) ) ) ).

% const_replace_closed_remove_subst_lhs
thf(fact_467_llrg__funas__term__steps__pres,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) )
       => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ ( sup_su770888850772321124_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ ( term_funas_term_f_v @ S ) ) ) ) ) ).

% llrg_funas_term_steps_pres
thf(fact_468_quasi__commute__def,axiom,
    ( abstra8826850394716133222rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S6 @ R6 ) @ ( relcom2934258169684288428rm_f_v @ R6 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R6 @ S6 ) ) ) ) ) ) ).

% quasi_commute_def
thf(fact_469_diamond__I,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) )
     => ( abstra3437181289681318589rm_f_v @ R3 ) ) ).

% diamond_I
thf(fact_470_kleene__trancl__induct,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,X4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ X4 )
     => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ B4 @ X4 ) @ X4 )
       => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ X4 @ C4 ) @ X4 )
         => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ B4 ) @ ( relcom2934258169684288428rm_f_v @ A3 @ ( transi3208867851621559554rm_f_v @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_471_kleene__trancl__induct,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,X4: set_Pr5245412377734449720_f_nat,B4: set_Product_prod_f_f,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ X4 )
     => ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ B4 @ X4 ) @ X4 )
       => ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ X4 @ C4 ) @ X4 )
         => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_trancl_f @ B4 ) @ ( relcomp_f_nat_nat @ A3 @ ( transi6264000038957366511cl_nat @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_472_kleene__induct,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,X4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ X4 )
     => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ B4 @ X4 ) @ X4 )
       => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ X4 @ C4 ) @ X4 )
         => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ B4 ) @ ( relcom2934258169684288428rm_f_v @ A3 @ ( transi3028809021613410336rm_f_v @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_473_kleene__induct,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,X4: set_Pr5245412377734449720_f_nat,B4: set_Product_prod_f_f,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ X4 )
     => ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ B4 @ X4 ) @ X4 )
       => ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ X4 @ C4 ) @ X4 )
         => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ B4 ) @ ( relcomp_f_nat_nat @ A3 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X4 ) ) ) ) ).

% kleene_induct
thf(fact_474_llrg__funas__term__step__pres,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
       => ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ ( sup_su770888850772321124_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ ( term_funas_term_f_v @ S ) ) ) ) ) ).

% llrg_funas_term_step_pres
thf(fact_475_const__replace__closed__rtrancl,axiom,
    ! [C: f,U4: set_Pr8827868859434726151rm_f_v] :
      ( ( replac3904333054701138457ed_f_v @ C @ U4 )
     => ( replac3904333054701138457ed_f_v @ C @ ( transi3028809021613410336rm_f_v @ U4 ) ) ) ).

% const_replace_closed_rtrancl
thf(fact_476_const__replace__closed__trancl,axiom,
    ! [C: f,U4: set_Pr8827868859434726151rm_f_v] :
      ( ( replac3904333054701138457ed_f_v @ C @ U4 )
     => ( replac3904333054701138457ed_f_v @ C @ ( transi3208867851621559554rm_f_v @ U4 ) ) ) ).

% const_replace_closed_trancl
thf(fact_477_const__replace__closed__relcomp,axiom,
    ! [C: f,U4: set_Pr8827868859434726151rm_f_v,V2: set_Pr8827868859434726151rm_f_v] :
      ( ( replac3904333054701138457ed_f_v @ C @ U4 )
     => ( ( replac3904333054701138457ed_f_v @ C @ V2 )
       => ( replac3904333054701138457ed_f_v @ C @ ( relcom2934258169684288428rm_f_v @ U4 @ V2 ) ) ) ) ).

% const_replace_closed_relcomp
thf(fact_478_qc__imp__qc__trancl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra8826850394716133222rm_f_v @ R3 @ S )
     => ( abstra8826850394716133222rm_f_v @ R3 @ ( transi3208867851621559554rm_f_v @ S ) ) ) ).

% qc_imp_qc_trancl
thf(fact_479_trancl__map,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,F3: term_f_v > term_f_v,S: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ! [X2: term_f_v,Y: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F3 @ X2 ) @ ( F3 @ Y ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_480_diamond__E,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra3437181289681318589rm_f_v @ R3 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) ) ).

% diamond_E
thf(fact_481_diamond__def,axiom,
    ( abstra3437181289681318589rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R6 ) @ R6 ) @ ( relcom2934258169684288428rm_f_v @ R6 @ ( conver3099152742603637891rm_f_v @ R6 ) ) ) ) ) ).

% diamond_def
thf(fact_482_rstep__trancl__sig__step__l,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( terms_3805575757503158827ig_f_v @ F @ V @ S ) @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_l
thf(fact_483_rstep__trancl__sig__step__r,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ ( terms_3805575757503158827ig_f_v @ F @ V @ T ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_r
thf(fact_484_order__refl,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_485_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_486_dual__order_Orefl,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ A ) ).

% dual_order.refl
thf(fact_487_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_488_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_489_le__cases3,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_490_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
    = ( ^ [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
          & ( ord_le8976984241387448984_f_nat @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_491_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [X: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_492_ord__eq__le__trans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( A = B )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_493_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_494_ord__le__eq__trans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( B = C )
       => ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_495_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_496_order__antisym,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
     => ( ( ord_le8976984241387448984_f_nat @ Y2 @ X3 )
       => ( X3 = Y2 ) ) ) ).

% order_antisym
thf(fact_497_order__antisym,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X3 )
       => ( X3 = Y2 ) ) ) ).

% order_antisym
thf(fact_498_order_Otrans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_499_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_500_order__trans,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
     => ( ( ord_le8976984241387448984_f_nat @ Y2 @ Z2 )
       => ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_501_order__trans,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_502_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
         => ( P2 @ A2 @ B2 ) )
     => ( ! [A2: nat,B2: nat] :
            ( ( P2 @ B2 @ A2 )
           => ( P2 @ A2 @ B2 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_503_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ B3 @ A4 )
          & ( ord_le8976984241387448984_f_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_504_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_505_dual__order_Oantisym,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( ord_le8976984241387448984_f_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_506_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_507_dual__order_Otrans,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( ord_le8976984241387448984_f_nat @ C @ B )
       => ( ord_le8976984241387448984_f_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_508_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_509_antisym,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_510_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_511_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A4 @ B3 )
          & ( ord_le8976984241387448984_f_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_512_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_513_order__subst1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_514_order__subst1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F3: nat > set_Pr5245412377734449720_f_nat,B: nat,C: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_515_order__subst1,axiom,
    ! [A: nat,F3: set_Pr5245412377734449720_f_nat > nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_516_order__subst1,axiom,
    ! [A: nat,F3: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_517_order__subst2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ ( F3 @ B ) @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_518_order__subst2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > nat,C: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F3 @ B ) @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_519_order__subst2,axiom,
    ! [A: nat,B: nat,F3: nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ ( F3 @ B ) @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_520_order__subst2,axiom,
    ! [A: nat,B: nat,F3: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F3 @ B ) @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_521_order__eq__refl,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ( X3 = Y2 )
     => ( ord_le8976984241387448984_f_nat @ X3 @ Y2 ) ) ).

% order_eq_refl
thf(fact_522_order__eq__refl,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( X3 = Y2 )
     => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).

% order_eq_refl
thf(fact_523_linorder__linear,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_linear
thf(fact_524_ord__eq__le__subst,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( A
        = ( F3 @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_525_ord__eq__le__subst,axiom,
    ! [A: nat,F3: set_Pr5245412377734449720_f_nat > nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( A
        = ( F3 @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_526_ord__eq__le__subst,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F3: nat > set_Pr5245412377734449720_f_nat,B: nat,C: nat] :
      ( ( A
        = ( F3 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_527_ord__eq__le__subst,axiom,
    ! [A: nat,F3: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F3 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_528_ord__le__eq__subst,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ( F3 @ B )
          = C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_529_ord__le__eq__subst,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > nat,C: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ( F3 @ B )
          = C )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_530_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F3: nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F3 @ B )
          = C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_531_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F3: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F3 @ B )
          = C )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_532_linorder__le__cases,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_le_cases
thf(fact_533_order__antisym__conv,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y2 @ X3 )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_534_order__antisym__conv,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_535_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,K: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( A3
        = ( sup_su1994854688125285083rm_f_v @ K @ A ) )
     => ( ( sup_su1994854688125285083rm_f_v @ A3 @ B )
        = ( sup_su1994854688125285083rm_f_v @ K @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_536_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,K: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( B4
        = ( sup_su1994854688125285083rm_f_v @ K @ B ) )
     => ( ( sup_su1994854688125285083rm_f_v @ A @ B4 )
        = ( sup_su1994854688125285083rm_f_v @ K @ ( sup_su1994854688125285083rm_f_v @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_537_rstep__term__to__sig__l,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( terms_3805575757503158827ig_f_v @ F @ V @ S ) @ T ) @ ( rstep_f_v @ R ) ) ) ) ) ).

% rstep_term_to_sig_l
thf(fact_538_rstep__term__to__sig__r,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ ( terms_3805575757503158827ig_f_v @ F @ V @ T ) ) @ ( rstep_f_v @ R ) ) ) ) ) ).

% rstep_term_to_sig_r
thf(fact_539_term__to__sig__subst__sig,axiom,
    ! [T: term_f_v,F: set_Pr5245412377734449720_f_nat,V: v,Sigma: v > term_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
     => ( ( terms_3805575757503158827ig_f_v @ F @ V @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) )
        = ( subst_7901567856578221233_f_v_v @ T
          @ ^ [X: v] : ( terms_3805575757503158827ig_f_v @ F @ V @ ( Sigma @ X ) ) ) ) ) ).

% term_to_sig_subst_sig
thf(fact_540_term_Oinject_I2_J,axiom,
    ! [X21: f,X22: list_term_f_v,Y21: f,Y22: list_term_f_v] :
      ( ( ( fun_f_v @ X21 @ X22 )
        = ( fun_f_v @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% term.inject(2)
thf(fact_541_srsteps__with__root__step__to__grsteps,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ f2 @ r ) )
     => ( member4316649861536941904rm_f_v
        @ ( produc2510790566102465375rm_f_v
          @ ( subst_7901567856578221233_f_v_v @ S
            @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) )
          @ ( subst_7901567856578221233_f_v_v @ T
            @ ^ [X: v] : ( fun_f_v @ d @ nil_term_f_v ) ) )
        @ ( transi3028809021613410336rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ r ) )
            @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
              @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ).

% srsteps_with_root_step_to_grsteps
thf(fact_542_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_543_inf_Oidem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A @ A )
      = A ) ).

% inf.idem
thf(fact_544_inf__idem,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ X3 )
      = X3 ) ).

% inf_idem
thf(fact_545_inf_Oleft__idem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A @ B ) )
      = ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ).

% inf.left_idem
thf(fact_546_inf__left__idem,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) )
      = ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) ) ).

% inf_left_idem
thf(fact_547_inf_Oright__idem,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ B )
      = ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ).

% inf.right_idem
thf(fact_548_inf__right__idem,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ Y2 )
      = ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) ) ).

% inf_right_idem
thf(fact_549_IntI,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ A3 )
     => ( ( member2678038196313615385_f_nat @ C @ B4 )
       => ( member2678038196313615385_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_550_IntI,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ A3 )
     => ( ( member4316649861536941904rm_f_v @ C @ B4 )
       => ( member4316649861536941904rm_f_v @ C @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_551_Int__iff,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) )
      = ( ( member2678038196313615385_f_nat @ C @ A3 )
        & ( member2678038196313615385_f_nat @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_552_Int__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
      = ( ( member4316649861536941904rm_f_v @ C @ A3 )
        & ( member4316649861536941904rm_f_v @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_553_inf_Obounded__iff,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B @ C ) )
      = ( ( ord_le4559761987009501863rm_f_v @ A @ B )
        & ( ord_le4559761987009501863rm_f_v @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_554_inf_Obounded__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C ) )
      = ( ( ord_le8976984241387448984_f_nat @ A @ B )
        & ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_555_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_556_le__inf__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
      = ( ( ord_le4559761987009501863rm_f_v @ X3 @ Y2 )
        & ( ord_le4559761987009501863rm_f_v @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_557_le__inf__iff,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y2 @ Z2 ) )
      = ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
        & ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_558_le__inf__iff,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) )
      = ( ( ord_less_eq_nat @ X3 @ Y2 )
        & ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_559_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_560_sup__inf__absorb,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) )
      = X3 ) ).

% sup_inf_absorb
thf(fact_561_inf__sup__absorb,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) )
      = X3 ) ).

% inf_sup_absorb
thf(fact_562_Int__subset__iff,axiom,
    ! [C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C4 @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
      = ( ( ord_le4559761987009501863rm_f_v @ C4 @ A3 )
        & ( ord_le4559761987009501863rm_f_v @ C4 @ B4 ) ) ) ).

% Int_subset_iff
thf(fact_563_Int__subset__iff,axiom,
    ! [C4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C4 @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) )
      = ( ( ord_le8976984241387448984_f_nat @ C4 @ A3 )
        & ( ord_le8976984241387448984_f_nat @ C4 @ B4 ) ) ) ).

% Int_subset_iff
thf(fact_564_Int__insert__left__if0,axiom,
    ! [A: product_prod_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ A @ C4 )
     => ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ B4 ) @ C4 )
        = ( inf_in4124010953442585290_f_nat @ B4 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_565_Int__insert__left__if0,axiom,
    ! [A: produc8027700073345061287rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ A @ C4 )
     => ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B4 ) @ C4 )
        = ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_566_Int__insert__left__if1,axiom,
    ! [A: product_prod_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ C4 )
     => ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ B4 ) @ C4 )
        = ( insert5008453031520946354_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B4 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_567_Int__insert__left__if1,axiom,
    ! [A: produc8027700073345061287rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ C4 )
     => ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B4 ) @ C4 )
        = ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_568_insert__inter__insert,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ A3 ) @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
      = ( insert5008453031520946354_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_569_insert__inter__insert,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ A3 ) @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
      = ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_570_Int__insert__right__if0,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ( inf_in4124010953442585290_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
        = ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_571_Int__insert__right__if0,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
        = ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_572_Int__insert__right__if1,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ( inf_in4124010953442585290_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
        = ( insert5008453031520946354_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_573_Int__insert__right__if1,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
        = ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_574_Un__Int__eq_I1_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) @ S2 )
      = S2 ) ).

% Un_Int_eq(1)
thf(fact_575_Un__Int__eq_I2_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_576_Un__Int__eq_I3_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ S2 @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) )
      = S2 ) ).

% Un_Int_eq(3)
thf(fact_577_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ T2 @ ( sup_su1994854688125285083rm_f_v @ S2 @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_578_Int__Un__eq_I1_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ S2 @ T2 ) @ S2 )
      = S2 ) ).

% Int_Un_eq(1)
thf(fact_579_Int__Un__eq_I2_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ S2 @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_580_Int__Un__eq_I3_J,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,T2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ S2 @ ( inf_in6135112614888068085rm_f_v @ S2 @ T2 ) )
      = S2 ) ).

% Int_Un_eq(3)
thf(fact_581_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ T2 @ ( inf_in6135112614888068085rm_f_v @ S2 @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_582_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,K: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( A3
        = ( inf_in6135112614888068085rm_f_v @ K @ A ) )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ B )
        = ( inf_in6135112614888068085rm_f_v @ K @ ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_583_boolean__algebra__cancel_Oinf2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,K: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( B4
        = ( inf_in6135112614888068085rm_f_v @ K @ B ) )
     => ( ( inf_in6135112614888068085rm_f_v @ A @ B4 )
        = ( inf_in6135112614888068085rm_f_v @ K @ ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_584_inf__sup__aci_I4_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) )
      = ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) ) ).

% inf_sup_aci(4)
thf(fact_585_inf__sup__aci_I3_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
      = ( inf_in6135112614888068085rm_f_v @ Y2 @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) ) ).

% inf_sup_aci(3)
thf(fact_586_inf__sup__aci_I2_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ Z2 )
      = ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) ) ) ).

% inf_sup_aci(2)
thf(fact_587_inf__sup__aci_I1_J,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] : ( inf_in6135112614888068085rm_f_v @ Y3 @ X ) ) ) ).

% inf_sup_aci(1)
thf(fact_588_inf_Oassoc,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ C )
      = ( inf_in6135112614888068085rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B @ C ) ) ) ).

% inf.assoc
thf(fact_589_inf__assoc,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ Z2 )
      = ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) ) ) ).

% inf_assoc
thf(fact_590_inf_Ocommute,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [A4: set_Pr8827868859434726151rm_f_v,B3: set_Pr8827868859434726151rm_f_v] : ( inf_in6135112614888068085rm_f_v @ B3 @ A4 ) ) ) ).

% inf.commute
thf(fact_591_inf__commute,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] : ( inf_in6135112614888068085rm_f_v @ Y3 @ X ) ) ) ).

% inf_commute
thf(fact_592_inf_Oleft__commute,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ B @ ( inf_in6135112614888068085rm_f_v @ A @ C ) )
      = ( inf_in6135112614888068085rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_593_inf__left__commute,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
      = ( inf_in6135112614888068085rm_f_v @ Y2 @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) ) ).

% inf_left_commute
thf(fact_594_Restr__converse,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( conver3099152742603637891rm_f_v
        @ ( inf_in6135112614888068085rm_f_v @ R
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : A3 ) ) )
      = ( inf_in6135112614888068085rm_f_v @ ( conver3099152742603637891rm_f_v @ R )
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : A3 ) ) ) ).

% Restr_converse
thf(fact_595_sig__step__grstep__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
        @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
          @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) )
      = ( sig_step_f_v @ F
        @ ( inf_in6135112614888068085rm_f_v @ ( rstep_f_v @ R )
          @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
            @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ).

% sig_step_grstep_dist
thf(fact_596_ground__subst__apply,axiom,
    ! [T: term_f_v,Sigma: v > term_f_v] :
      ( ( term_ground_f_v @ T )
     => ( ( subst_7901567856578221233_f_v_v @ T @ Sigma )
        = T ) ) ).

% ground_subst_apply
thf(fact_597_IntE,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) )
     => ~ ( ( member2678038196313615385_f_nat @ C @ A3 )
         => ~ ( member2678038196313615385_f_nat @ C @ B4 ) ) ) ).

% IntE
thf(fact_598_IntE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
     => ~ ( ( member4316649861536941904rm_f_v @ C @ A3 )
         => ~ ( member4316649861536941904rm_f_v @ C @ B4 ) ) ) ).

% IntE
thf(fact_599_IntD1,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) )
     => ( member2678038196313615385_f_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_600_IntD1,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
     => ( member4316649861536941904rm_f_v @ C @ A3 ) ) ).

% IntD1
thf(fact_601_IntD2,axiom,
    ! [C: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) )
     => ( member2678038196313615385_f_nat @ C @ B4 ) ) ).

% IntD2
thf(fact_602_IntD2,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
     => ( member4316649861536941904rm_f_v @ C @ B4 ) ) ).

% IntD2
thf(fact_603_Int__def,axiom,
    ( inf_in4124010953442585290_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X @ A5 )
              & ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_604_Int__def,axiom,
    ( inf_inf_set_term_f_v
    = ( ^ [A5: set_term_f_v,B5: set_term_f_v] :
          ( collect_term_f_v
          @ ^ [X: term_f_v] :
              ( ( member_term_f_v @ X @ A5 )
              & ( member_term_f_v @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_605_Int__def,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [X: produc8027700073345061287rm_f_v] :
              ( ( member4316649861536941904rm_f_v @ X @ A5 )
              & ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_606_Int__assoc,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ C4 )
      = ( inf_in6135112614888068085rm_f_v @ A3 @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) ) ).

% Int_assoc
thf(fact_607_Int__absorb,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_608_Int__Collect,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,P2: product_prod_f_nat > $o] :
      ( ( member2678038196313615385_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A3 @ ( collec1417984143148964183_f_nat @ P2 ) ) )
      = ( ( member2678038196313615385_f_nat @ X3 @ A3 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_609_Int__Collect,axiom,
    ! [X3: term_f_v,A3: set_term_f_v,P2: term_f_v > $o] :
      ( ( member_term_f_v @ X3 @ ( inf_inf_set_term_f_v @ A3 @ ( collect_term_f_v @ P2 ) ) )
      = ( ( member_term_f_v @ X3 @ A3 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_610_Int__Collect,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ A3 @ ( collec7769483783292507282rm_f_v @ P2 ) ) )
      = ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_611_Int__commute,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] : ( inf_in6135112614888068085rm_f_v @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_612_Collect__conj__eq,axiom,
    ! [P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ( collect_term_f_v
        @ ^ [X: term_f_v] :
            ( ( P2 @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_term_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_613_Collect__conj__eq,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
      ( ( collec7769483783292507282rm_f_v
        @ ^ [X: produc8027700073345061287rm_f_v] :
            ( ( P2 @ X )
            & ( Q @ X ) ) )
      = ( inf_in6135112614888068085rm_f_v @ ( collec7769483783292507282rm_f_v @ P2 ) @ ( collec7769483783292507282rm_f_v @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_614_Int__left__absorb,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) )
      = ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ).

% Int_left_absorb
thf(fact_615_Int__left__commute,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) )
      = ( inf_in6135112614888068085rm_f_v @ B4 @ ( inf_in6135112614888068085rm_f_v @ A3 @ C4 ) ) ) ).

% Int_left_commute
thf(fact_616_Restr__mono,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ord_le4559761987009501863rm_f_v
      @ ( inf_in6135112614888068085rm_f_v @ R3
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : A3 ) )
      @ R3 ) ).

% Restr_mono
thf(fact_617_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_618_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_619_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_620_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_621_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_622_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_623_inf__le1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ X3 ) ).

% inf_le1
thf(fact_624_inf__le1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_le1
thf(fact_625_inf__le1,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_le1
thf(fact_626_inf__le2,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_627_inf__le2,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_628_inf__le2,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_629_le__infE,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ A @ B ) )
     => ~ ( ( ord_le4559761987009501863rm_f_v @ X3 @ A )
         => ~ ( ord_le4559761987009501863rm_f_v @ X3 @ B ) ) ) ).

% le_infE
thf(fact_630_le__infE,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A @ B ) )
     => ~ ( ( ord_le8976984241387448984_f_nat @ X3 @ A )
         => ~ ( ord_le8976984241387448984_f_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_631_le__infE,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X3 @ A )
         => ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_632_le__infI,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ A )
     => ( ( ord_le4559761987009501863rm_f_v @ X3 @ B )
       => ( ord_le4559761987009501863rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ) ) ).

% le_infI
thf(fact_633_le__infI,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ A )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ B )
       => ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_634_le__infI,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ( ord_less_eq_nat @ X3 @ B )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_635_inf__mono,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,D: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ C )
     => ( ( ord_le4559761987009501863rm_f_v @ B @ D )
       => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ ( inf_in6135112614888068085rm_f_v @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_636_inf__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ( ord_le8976984241387448984_f_nat @ B @ D )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ ( inf_in4124010953442585290_f_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_637_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_638_le__infI1,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ X3 )
     => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_639_le__infI1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ X3 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_640_le__infI1,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_641_le__infI2,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ X3 )
     => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_642_le__infI2,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ X3 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_643_le__infI2,axiom,
    ! [B: nat,X3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_644_inf_OorderE,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ B )
     => ( A
        = ( inf_in6135112614888068085rm_f_v @ A @ B ) ) ) ).

% inf.orderE
thf(fact_645_inf_OorderE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( A
        = ( inf_in4124010953442585290_f_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_646_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_647_inf_OorderI,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( A
        = ( inf_in6135112614888068085rm_f_v @ A @ B ) )
     => ( ord_le4559761987009501863rm_f_v @ A @ B ) ) ).

% inf.orderI
thf(fact_648_inf_OorderI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( A
        = ( inf_in4124010953442585290_f_nat @ A @ B ) )
     => ( ord_le8976984241387448984_f_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_649_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_650_inf__unique,axiom,
    ! [F3: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( F3 @ X2 @ Y ) @ X2 )
     => ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( F3 @ X2 @ Y ) @ Y )
       => ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v,Z: set_Pr8827868859434726151rm_f_v] :
              ( ( ord_le4559761987009501863rm_f_v @ X2 @ Y )
             => ( ( ord_le4559761987009501863rm_f_v @ X2 @ Z )
               => ( ord_le4559761987009501863rm_f_v @ X2 @ ( F3 @ Y @ Z ) ) ) )
         => ( ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_651_inf__unique,axiom,
    ! [F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 @ Y ) @ X2 )
     => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 @ Y ) @ Y )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y )
             => ( ( ord_le8976984241387448984_f_nat @ X2 @ Z )
               => ( ord_le8976984241387448984_f_nat @ X2 @ ( F3 @ Y @ Z ) ) ) )
         => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_652_inf__unique,axiom,
    ! [F3: nat > nat > nat,X3: nat,Y2: nat] :
      ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F3 @ X2 @ Y ) @ X2 )
     => ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F3 @ X2 @ Y ) @ Y )
       => ( ! [X2: nat,Y: nat,Z: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ( ord_less_eq_nat @ X2 @ Z )
               => ( ord_less_eq_nat @ X2 @ ( F3 @ Y @ Z ) ) ) )
         => ( ( inf_inf_nat @ X3 @ Y2 )
            = ( F3 @ X3 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_653_le__iff__inf,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [X: set_Pr8827868859434726151rm_f_v,Y3: set_Pr8827868859434726151rm_f_v] :
          ( ( inf_in6135112614888068085rm_f_v @ X @ Y3 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_654_le__iff__inf,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ X @ Y3 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_655_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y3: nat] :
          ( ( inf_inf_nat @ X @ Y3 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_656_inf_Oabsorb1,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ B )
     => ( ( inf_in6135112614888068085rm_f_v @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_657_inf_Oabsorb1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( inf_in4124010953442585290_f_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_658_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_659_inf_Oabsorb2,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ A )
     => ( ( inf_in6135112614888068085rm_f_v @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_660_inf_Oabsorb2,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( inf_in4124010953442585290_f_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_661_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_662_inf__absorb1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ Y2 )
     => ( ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_663_inf__absorb1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
     => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y2 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_664_inf__absorb1,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( inf_inf_nat @ X3 @ Y2 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_665_inf__absorb2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ Y2 @ X3 )
     => ( ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_666_inf__absorb2,axiom,
    ! [Y2: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y2 @ X3 )
     => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_667_inf__absorb2,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( inf_inf_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_668_inf_OboundedE,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B @ C ) )
     => ~ ( ( ord_le4559761987009501863rm_f_v @ A @ B )
         => ~ ( ord_le4559761987009501863rm_f_v @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_669_inf_OboundedE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C ) )
     => ~ ( ( ord_le8976984241387448984_f_nat @ A @ B )
         => ~ ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_670_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_671_inf_OboundedI,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ B )
     => ( ( ord_le4559761987009501863rm_f_v @ A @ C )
       => ( ord_le4559761987009501863rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_672_inf_OboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ A @ C )
       => ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_673_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_674_inf__greatest,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ X3 @ Y2 )
     => ( ( ord_le4559761987009501863rm_f_v @ X3 @ Z2 )
       => ( ord_le4559761987009501863rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_675_inf__greatest,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y2 )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ Z2 )
       => ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y2 @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_676_inf__greatest,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ X3 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_677_inf_Oorder__iff,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A4: set_Pr8827868859434726151rm_f_v,B3: set_Pr8827868859434726151rm_f_v] :
          ( A4
          = ( inf_in6135112614888068085rm_f_v @ A4 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_678_inf_Oorder__iff,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
          ( A4
          = ( inf_in4124010953442585290_f_nat @ A4 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_679_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_680_inf_Ocobounded1,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_681_inf_Ocobounded1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_682_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_683_inf_Ocobounded2,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_684_inf_Ocobounded2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_685_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_686_inf_Oabsorb__iff1,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [A4: set_Pr8827868859434726151rm_f_v,B3: set_Pr8827868859434726151rm_f_v] :
          ( ( inf_in6135112614888068085rm_f_v @ A4 @ B3 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_687_inf_Oabsorb__iff1,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_688_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( inf_inf_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_689_inf_Oabsorb__iff2,axiom,
    ( ord_le4559761987009501863rm_f_v
    = ( ^ [B3: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
          ( ( inf_in6135112614888068085rm_f_v @ A4 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_690_inf_Oabsorb__iff2,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [B3: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_691_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_692_inf_OcoboundedI1,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A @ C )
     => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_693_inf_OcoboundedI1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_694_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_695_inf_OcoboundedI2,axiom,
    ! [B: set_Pr8827868859434726151rm_f_v,C: set_Pr8827868859434726151rm_f_v,A: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B @ C )
     => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_696_inf_OcoboundedI2,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ C )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_697_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_698_rtrancl__Restr,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 )
        @ ( transi3028809021613410336rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ R3
            @ ( produc6667182599969081488rm_f_v @ A3
              @ ^ [Uu: term_f_v] : A3 ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).

% rtrancl_Restr
thf(fact_699_distrib__imp1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v,Z: set_Pr8827868859434726151rm_f_v] :
          ( ( inf_in6135112614888068085rm_f_v @ X2 @ ( sup_su1994854688125285083rm_f_v @ Y @ Z ) )
          = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X2 @ Y ) @ ( inf_in6135112614888068085rm_f_v @ X2 @ Z ) ) )
     => ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
        = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ) ).

% distrib_imp1
thf(fact_700_distrib__imp2,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: set_Pr8827868859434726151rm_f_v,Y: set_Pr8827868859434726151rm_f_v,Z: set_Pr8827868859434726151rm_f_v] :
          ( ( sup_su1994854688125285083rm_f_v @ X2 @ ( inf_in6135112614888068085rm_f_v @ Y @ Z ) )
          = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X2 @ Y ) @ ( sup_su1994854688125285083rm_f_v @ X2 @ Z ) ) )
     => ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) )
        = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) ) ) ).

% distrib_imp2
thf(fact_701_inf__sup__distrib1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) ) ).

% inf_sup_distrib1
thf(fact_702_inf__sup__distrib2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) @ X3 )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ Y2 @ X3 ) @ ( inf_in6135112614888068085rm_f_v @ Z2 @ X3 ) ) ) ).

% inf_sup_distrib2
thf(fact_703_sup__inf__distrib1,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ).

% sup_inf_distrib1
thf(fact_704_sup__inf__distrib2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) @ X3 )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ Y2 @ X3 ) @ ( sup_su1994854688125285083rm_f_v @ Z2 @ X3 ) ) ) ).

% sup_inf_distrib2
thf(fact_705_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_706_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_707_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) @ X3 )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ Y2 @ X3 ) @ ( inf_in6135112614888068085rm_f_v @ Z2 @ X3 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_708_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) @ X3 )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ Y2 @ X3 ) @ ( sup_su1994854688125285083rm_f_v @ Z2 @ X3 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_709_Int__Collect__mono,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,P2: term_f_v > $o,Q: term_f_v > $o] :
      ( ( ord_le947722214152454265rm_f_v @ A3 @ B4 )
     => ( ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ A3 )
           => ( ( P2 @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le947722214152454265rm_f_v @ ( inf_inf_set_term_f_v @ A3 @ ( collect_term_f_v @ P2 ) ) @ ( inf_inf_set_term_f_v @ B4 @ ( collect_term_f_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_710_Int__Collect__mono,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ B4 )
     => ( ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
           => ( ( P2 @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ ( collec7769483783292507282rm_f_v @ P2 ) ) @ ( inf_in6135112614888068085rm_f_v @ B4 @ ( collec7769483783292507282rm_f_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_711_Int__Collect__mono,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A3 )
           => ( ( P2 @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A3 @ ( collec1417984143148964183_f_nat @ P2 ) ) @ ( inf_in4124010953442585290_f_nat @ B4 @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_712_Int__greatest,axiom,
    ! [C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ C4 @ A3 )
     => ( ( ord_le4559761987009501863rm_f_v @ C4 @ B4 )
       => ( ord_le4559761987009501863rm_f_v @ C4 @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% Int_greatest
thf(fact_713_Int__greatest,axiom,
    ! [C4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C4 @ A3 )
     => ( ( ord_le8976984241387448984_f_nat @ C4 @ B4 )
       => ( ord_le8976984241387448984_f_nat @ C4 @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% Int_greatest
thf(fact_714_Int__absorb2,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ B4 )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_715_Int__absorb2,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( inf_in4124010953442585290_f_nat @ A3 @ B4 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_716_Int__absorb1,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ B4 @ A3 )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = B4 ) ) ).

% Int_absorb1
thf(fact_717_Int__absorb1,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B4 @ A3 )
     => ( ( inf_in4124010953442585290_f_nat @ A3 @ B4 )
        = B4 ) ) ).

% Int_absorb1
thf(fact_718_Int__lower2,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ B4 ) ).

% Int_lower2
thf(fact_719_Int__lower2,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) @ B4 ) ).

% Int_lower2
thf(fact_720_Int__lower1,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ A3 ) ).

% Int_lower1
thf(fact_721_Int__lower1,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) @ A3 ) ).

% Int_lower1
thf(fact_722_Int__mono,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,D3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ C4 )
     => ( ( ord_le4559761987009501863rm_f_v @ B4 @ D3 )
       => ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ ( inf_in6135112614888068085rm_f_v @ C4 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_723_Int__mono,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,D3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ C4 )
     => ( ( ord_le8976984241387448984_f_nat @ B4 @ D3 )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) @ ( inf_in4124010953442585290_f_nat @ C4 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_724_Restr__trancl__mono__set,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ord_le4559761987009501863rm_f_v
      @ ( transi3208867851621559554rm_f_v
        @ ( inf_in6135112614888068085rm_f_v @ R3
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : A3 ) ) )
      @ ( produc6667182599969081488rm_f_v @ A3
        @ ^ [Uu: term_f_v] : A3 ) ) ).

% Restr_trancl_mono_set
thf(fact_725_Int__insert__left,axiom,
    ! [A: product_prod_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ( member2678038196313615385_f_nat @ A @ C4 )
       => ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ B4 ) @ C4 )
          = ( insert5008453031520946354_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B4 @ C4 ) ) ) )
      & ( ~ ( member2678038196313615385_f_nat @ A @ C4 )
       => ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ B4 ) @ C4 )
          = ( inf_in4124010953442585290_f_nat @ B4 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_726_Int__insert__left,axiom,
    ! [A: produc8027700073345061287rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( member4316649861536941904rm_f_v @ A @ C4 )
       => ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B4 ) @ C4 )
          = ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) ) )
      & ( ~ ( member4316649861536941904rm_f_v @ A @ C4 )
       => ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B4 ) @ C4 )
          = ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_727_Int__insert__right,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( inf_in4124010953442585290_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
          = ( insert5008453031520946354_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) )
      & ( ~ ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( inf_in4124010953442585290_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ B4 ) )
          = ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_728_Int__insert__right,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
          = ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) )
      & ( ~ ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ A @ B4 ) )
          = ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_729_Restr__smycl__dist,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,A7: set_term_f_v] :
      ( ( sup_su1994854688125285083rm_f_v
        @ ( inf_in6135112614888068085rm_f_v @ R
          @ ( produc6667182599969081488rm_f_v @ A7
            @ ^ [Uu: term_f_v] : A7 ) )
        @ ( conver3099152742603637891rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ R
            @ ( produc6667182599969081488rm_f_v @ A7
              @ ^ [Uu: term_f_v] : A7 ) ) ) )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) )
        @ ( produc6667182599969081488rm_f_v @ A7
          @ ^ [Uu: term_f_v] : A7 ) ) ) ).

% Restr_smycl_dist
thf(fact_730_Un__Int__distrib2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) @ A3 )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ B4 @ A3 ) @ ( sup_su1994854688125285083rm_f_v @ C4 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_731_Int__Un__distrib2,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) @ A3 )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ B4 @ A3 ) @ ( inf_in6135112614888068085rm_f_v @ C4 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_732_Un__Int__distrib,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ ( sup_su1994854688125285083rm_f_v @ A3 @ C4 ) ) ) ).

% Un_Int_distrib
thf(fact_733_Int__Un__distrib,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) )
      = ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ ( inf_in6135112614888068085rm_f_v @ A3 @ C4 ) ) ) ).

% Int_Un_distrib
thf(fact_734_Un__Int__crazy,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ ( inf_in6135112614888068085rm_f_v @ B4 @ C4 ) ) @ ( inf_in6135112614888068085rm_f_v @ C4 @ A3 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) ) @ ( sup_su1994854688125285083rm_f_v @ C4 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_735_converse__Int,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( inf_in6135112614888068085rm_f_v @ R3 @ S ) )
      = ( inf_in6135112614888068085rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( conver3099152742603637891rm_f_v @ S ) ) ) ).

% converse_Int
thf(fact_736_gsrsteps__eq__to__rsteps__eq,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( transi3028809021613410336rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
            @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
              @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) ) ) ).

% gsrsteps_eq_to_rsteps_eq
thf(fact_737_gsrsteps__eq__to__srsteps__eq,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( transi3028809021613410336rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
            @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
              @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% gsrsteps_eq_to_srsteps_eq
thf(fact_738_ground__srsteps__eq__gsrsteps__eq,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( term_ground_f_v @ S )
     => ( ( term_ground_f_v @ T )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
            @ ( transi3028809021613410336rm_f_v
              @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
                @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                  @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ) ) ).

% ground_srsteps_eq_gsrsteps_eq
thf(fact_739_gsrsteps__to__rsteps,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( transi3208867851621559554rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
            @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
              @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) ) ) ).

% gsrsteps_to_rsteps
thf(fact_740_gsrsteps__to__srsteps,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( transi3208867851621559554rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
            @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
              @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% gsrsteps_to_srsteps
thf(fact_741_ground__srsteps__gsrsteps,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( term_ground_f_v @ S )
     => ( ( term_ground_f_v @ T )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
            @ ( transi3208867851621559554rm_f_v
              @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
                @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                  @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ) ) ).

% ground_srsteps_gsrsteps
thf(fact_742_trancl__full__on,axiom,
    ! [X4: set_term_f_v] :
      ( ( transi3208867851621559554rm_f_v
        @ ( produc6667182599969081488rm_f_v @ X4
          @ ^ [Uu: term_f_v] : X4 ) )
      = ( produc6667182599969081488rm_f_v @ X4
        @ ^ [Uu: term_f_v] : X4 ) ) ).

% trancl_full_on
thf(fact_743_sig__step__def,axiom,
    ( sig_step_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v] :
          ( inf_in6135112614888068085rm_f_v @ R4
          @ ( produc6667182599969081488rm_f_v
            @ ( collect_term_f_v
              @ ^ [S6: term_f_v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S6 ) @ F2 ) )
            @ ^ [Uu: term_f_v] :
                ( collect_term_f_v
                @ ^ [S6: term_f_v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S6 ) @ F2 ) ) ) ) ) ) ).

% sig_step_def
thf(fact_744_srsteps__eq__relcomp__gsrsteps__relcomp,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) )
     => ( ( term_ground_f_v @ S )
       => ( ( term_ground_f_v @ T )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
            @ ( relcom2934258169684288428rm_f_v
              @ ( transi3028809021613410336rm_f_v
                @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
                  @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                    @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) )
              @ ( transi3028809021613410336rm_f_v
                @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) )
                  @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                    @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ) ) ) ).

% srsteps_eq_relcomp_gsrsteps_relcomp
thf(fact_745_gsrsteps__eq__relcomp__srsteps__relcompD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( relcom2934258169684288428rm_f_v
          @ ( transi3028809021613410336rm_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) )
          @ ( transi3028809021613410336rm_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) ) ) ).

% gsrsteps_eq_relcomp_srsteps_relcompD
thf(fact_746_gsrsteps__eq__relcomp__to__rsteps__relcomp,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T )
        @ ( relcom2934258169684288428rm_f_v
          @ ( transi3028809021613410336rm_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) )
          @ ( transi3028809021613410336rm_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ S4 ) ) ) ) ) ).

% gsrsteps_eq_relcomp_to_rsteps_relcomp
thf(fact_747_distrib__sup__le,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ ( inf_in6135112614888068085rm_f_v @ Y2 @ Z2 ) ) @ ( inf_in6135112614888068085rm_f_v @ ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) @ ( sup_su1994854688125285083rm_f_v @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_748_distrib__sup__le,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y2 @ Z2 ) ) @ ( inf_in4124010953442585290_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ Y2 ) @ ( sup_su770888850772321124_f_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_749_distrib__sup__le,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_750_distrib__inf__le,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v,Z2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) @ ( inf_in6135112614888068085rm_f_v @ X3 @ Z2 ) ) @ ( inf_in6135112614888068085rm_f_v @ X3 @ ( sup_su1994854688125285083rm_f_v @ Y2 @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_751_distrib__inf__le,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y2: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y2 ) @ ( inf_in4124010953442585290_f_nat @ X3 @ Z2 ) ) @ ( inf_in4124010953442585290_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ Y2 @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_752_distrib__inf__le,axiom,
    ! [X3: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y2 @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_753_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( sup_su1994854688125285083rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) @ C4 )
        = ( inf_in6135112614888068085rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) ) )
      = ( ord_le4559761987009501863rm_f_v @ C4 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_754_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat] :
      ( ( ( sup_su770888850772321124_f_nat @ ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) @ C4 )
        = ( inf_in4124010953442585290_f_nat @ A3 @ ( sup_su770888850772321124_f_nat @ B4 @ C4 ) ) )
      = ( ord_le8976984241387448984_f_nat @ C4 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_755_llrg__ground__rhs,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,L: term_f_v,R3: term_f_v] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ R )
       => ( term_ground_f_v @ R3 ) ) ) ).

% llrg_ground_rhs
thf(fact_756_relcomp__subset__Sigma,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,B4: set_term_f_v,S: set_Pr8827868859434726151rm_f_v,C4: set_term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : B4 ) )
     => ( ( ord_le4559761987009501863rm_f_v @ S
          @ ( produc6667182599969081488rm_f_v @ B4
            @ ^ [Uu: term_f_v] : C4 ) )
       => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S )
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : C4 ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_757_relcomp__subset__Sigma,axiom,
    ! [R3: set_Product_prod_f_f,A3: set_f,B4: set_f,S: set_Pr5245412377734449720_f_nat,C4: set_nat] :
      ( ( ord_le7147000192471188839od_f_f @ R3
        @ ( product_Sigma_f_f @ A3
          @ ^ [Uu: f] : B4 ) )
     => ( ( ord_le8976984241387448984_f_nat @ S
          @ ( product_Sigma_f_nat @ B4
            @ ^ [Uu: f] : C4 ) )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ R3 @ S )
          @ ( product_Sigma_f_nat @ A3
            @ ^ [Uu: f] : C4 ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_758_relcomp__subset__Sigma,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,A3: set_f,B4: set_nat,S: set_Pr1261947904930325089at_nat,C4: set_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R3
        @ ( product_Sigma_f_nat @ A3
          @ ^ [Uu: f] : B4 ) )
     => ( ( ord_le3146513528884898305at_nat @ S
          @ ( produc457027306803732586at_nat @ B4
            @ ^ [Uu: nat] : C4 ) )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R3 @ S )
          @ ( product_Sigma_f_nat @ A3
            @ ^ [Uu: f] : C4 ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_759_trancl__subset__Sigma,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : A3 ) )
     => ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 )
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : A3 ) ) ) ).

% trancl_subset_Sigma
thf(fact_760_trancl__subset__Sigma__aux,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
     => ( ( ord_le7518223742277120103rm_f_v @ R3
          @ ( produc7856489909025154568rm_f_v @ A3
            @ ^ [Uu: produc8027700073345061287rm_f_v] : A3 ) )
       => ( ( A = B )
          | ( member4316649861536941904rm_f_v @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_761_trancl__subset__Sigma__aux,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
     => ( ( ord_le8946695678602162533_f_nat @ R3
          @ ( produc7667213020494768334_f_nat @ A3
            @ ^ [Uu: product_prod_f_nat] : A3 ) )
       => ( ( A = B )
          | ( member2678038196313615385_f_nat @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_762_trancl__subset__Sigma__aux,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( ord_le4559761987009501863rm_f_v @ R3
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : A3 ) )
       => ( ( A = B )
          | ( member_term_f_v @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_763_trancl__Int__subset,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
     => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( inf_in6135112614888068085rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ S ) @ R3 ) @ S )
       => ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ S ) ) ) ).

% trancl_Int_subset
thf(fact_764_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_765_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_766_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_767_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_768_llrg__srsteps__with__root__step__ground,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
       => ( term_ground_f_v @ T ) ) ) ).

% llrg_srsteps_with_root_step_ground
thf(fact_769_subst__term__eqI,axiom,
    ! [Sigma: v > term_f_v,Tau: v > term_f_v] :
      ( ! [T4: term_f_v] :
          ( ( subst_7901567856578221233_f_v_v @ T4 @ Sigma )
          = ( subst_7901567856578221233_f_v_v @ T4 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_770_open__terms__two__const__lv_Osrsteps__with__root__step__to__grsteps,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,C: f,D: f,S: term_f_v,T: term_f_v] :
      ( ( ground9003829210744935429lv_f_v @ R @ F @ C @ D )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
       => ( member4316649861536941904rm_f_v
          @ ( produc2510790566102465375rm_f_v
            @ ( subst_7901567856578221233_f_v_v @ S
              @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
            @ ( subst_7901567856578221233_f_v_v @ T
              @ ^ [X: v] : ( fun_f_v @ D @ nil_term_f_v ) ) )
          @ ( transi3028809021613410336rm_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ D @ zero_zero_nat ) @ F ) ) @ ( rstep_f_v @ R ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ) ).

% open_terms_two_const_lv.srsteps_with_root_step_to_grsteps
thf(fact_771_llrg__srsteps__with__root__step__inv__ground,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) )
       => ( term_ground_f_v @ S ) ) ) ).

% llrg_srsteps_with_root_step_inv_ground
thf(fact_772_llrg__rsteps__pres__groundness,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( term_ground_f_v @ S )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
         => ( term_ground_f_v @ T ) ) ) ) ).

% llrg_rsteps_pres_groundness
thf(fact_773_llrg__monadic__rstep__pres__groundness,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( rewrit3916899077429870368adic_f @ F )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
         => ( term_ground_f_v @ T ) ) ) ) ).

% llrg_monadic_rstep_pres_groundness
thf(fact_774_llrg__monadic__rsteps__groundness,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( rewrit3916899077429870368adic_f @ F )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
         => ( term_ground_f_v @ T ) ) ) ) ).

% llrg_monadic_rsteps_groundness
thf(fact_775_insert__Times__insert,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( produc7667213020494768334_f_nat @ ( insert5008453031520946354_f_nat @ A @ A3 )
        @ ^ [Uu: product_prod_f_nat] : ( insert5008453031520946354_f_nat @ B @ B4 ) )
      = ( insert4764342813885385269_f_nat @ ( produc5861087217798509725_f_nat @ A @ B )
        @ ( sup_su8886893976626623897_f_nat
          @ ( produc7667213020494768334_f_nat @ A3
            @ ^ [Uu: product_prod_f_nat] : ( insert5008453031520946354_f_nat @ B @ B4 ) )
          @ ( produc7667213020494768334_f_nat @ ( insert5008453031520946354_f_nat @ A @ A3 )
            @ ^ [Uu: product_prod_f_nat] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_776_insert__Times__insert,axiom,
    ! [A: f,A3: set_f,B: nat,B4: set_nat] :
      ( ( product_Sigma_f_nat @ ( insert_f @ A @ A3 )
        @ ^ [Uu: f] : ( insert_nat @ B @ B4 ) )
      = ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ A @ B )
        @ ( sup_su770888850772321124_f_nat
          @ ( product_Sigma_f_nat @ A3
            @ ^ [Uu: f] : ( insert_nat @ B @ B4 ) )
          @ ( product_Sigma_f_nat @ ( insert_f @ A @ A3 )
            @ ^ [Uu: f] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_777_insert__Times__insert,axiom,
    ! [A: term_f_v,A3: set_term_f_v,B: term_f_v,B4: set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ ( insert_term_f_v @ A @ A3 )
        @ ^ [Uu: term_f_v] : ( insert_term_f_v @ B @ B4 ) )
      = ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B )
        @ ( sup_su1994854688125285083rm_f_v
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : ( insert_term_f_v @ B @ B4 ) )
          @ ( produc6667182599969081488rm_f_v @ ( insert_term_f_v @ A @ A3 )
            @ ^ [Uu: term_f_v] : B4 ) ) ) ) ).

% insert_Times_insert
thf(fact_778_mem__Sigma__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( produc7856489909025154568rm_f_v @ A3 @ B4 ) )
      = ( ( member4316649861536941904rm_f_v @ A @ A3 )
        & ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_779_mem__Sigma__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: product_prod_f_nat,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat] :
      ( ( member6881531084954517871_f_nat @ ( produc8402213614931650770_f_nat @ A @ B ) @ ( produc423937348911600481_f_nat @ A3 @ B4 ) )
      = ( ( member4316649861536941904rm_f_v @ A @ A3 )
        & ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_780_mem__Sigma__iff,axiom,
    ! [A: product_prod_f_nat,B: produc8027700073345061287rm_f_v,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr8827868859434726151rm_f_v] :
      ( ( member4222901287935865009rm_f_v @ ( produc4606116700006435276rm_f_v @ A @ B ) @ ( produc5851212470841160795rm_f_v @ A3 @ B4 ) )
      = ( ( member2678038196313615385_f_nat @ A @ A3 )
        & ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_781_mem__Sigma__iff,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( produc7667213020494768334_f_nat @ A3 @ B4 ) )
      = ( ( member2678038196313615385_f_nat @ A @ A3 )
        & ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_782_mem__Sigma__iff,axiom,
    ! [A: f,B: nat,A3: set_f,B4: f > set_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( product_Sigma_f_nat @ A3 @ B4 ) )
      = ( ( member_f @ A @ A3 )
        & ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_783_mem__Sigma__iff,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) )
      = ( ( member_term_f_v @ A @ A3 )
        & ( member_term_f_v @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_784_SigmaI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) )
       => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( produc7856489909025154568rm_f_v @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_785_SigmaI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B: product_prod_f_nat,B4: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat] :
      ( ( member4316649861536941904rm_f_v @ A @ A3 )
     => ( ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) )
       => ( member6881531084954517871_f_nat @ ( produc8402213614931650770_f_nat @ A @ B ) @ ( produc423937348911600481_f_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_786_SigmaI,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B: produc8027700073345061287rm_f_v,B4: product_prod_f_nat > set_Pr8827868859434726151rm_f_v] :
      ( ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) )
       => ( member4222901287935865009rm_f_v @ ( produc4606116700006435276rm_f_v @ A @ B ) @ ( produc5851212470841160795rm_f_v @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_787_SigmaI,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat,B4: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ A3 )
     => ( ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) )
       => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( produc7667213020494768334_f_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_788_SigmaI,axiom,
    ! [A: f,A3: set_f,B: nat,B4: f > set_nat] :
      ( ( member_f @ A @ A3 )
     => ( ( member_nat @ B @ ( B4 @ A ) )
       => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( product_Sigma_f_nat @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_789_SigmaI,axiom,
    ! [A: term_f_v,A3: set_term_f_v,B: term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member_term_f_v @ A @ A3 )
     => ( ( member_term_f_v @ B @ ( B4 @ A ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_790_fresh__const__subst__NF__pres,axiom,
    ! [C: f,R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,H: set_Pr5245412377734449720_f_nat] :
      ( ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ ( terms_7890395023448655851_f_v_v @ R ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( ( ord_le8976984241387448984_f_nat @ F @ H )
           => ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ C @ zero_zero_nat ) @ H )
             => ( ( member_term_f_v
                  @ ( subst_7901567856578221233_f_v_v @ S
                    @ ^ [X: v] : ( fun_f_v @ C @ nil_term_f_v ) )
                  @ ( abstract_NF_term_f_v
                    @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ H @ ( rstep_f_v @ R ) )
                      @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                        @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) )
               => ( member_term_f_v @ S @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ) ) ).

% fresh_const_subst_NF_pres
thf(fact_791_Times__Int__distrib1,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,C4: set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ ( inf_inf_set_term_f_v @ A3 @ B4 )
        @ ^ [Uu: term_f_v] : C4 )
      = ( inf_in6135112614888068085rm_f_v
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : C4 )
        @ ( produc6667182599969081488rm_f_v @ B4
          @ ^ [Uu: term_f_v] : C4 ) ) ) ).

% Times_Int_distrib1
thf(fact_792_old_Oprod_Oinject,axiom,
    ! [A: term_f_v,B: term_f_v,A8: term_f_v,B8: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A8 @ B8 ) )
      = ( ( A = A8 )
        & ( B = B8 ) ) ) ).

% old.prod.inject
thf(fact_793_old_Oprod_Oinject,axiom,
    ! [A: f,B: nat,A8: f,B8: nat] :
      ( ( ( product_Pair_f_nat @ A @ B )
        = ( product_Pair_f_nat @ A8 @ B8 ) )
      = ( ( A = A8 )
        & ( B = B8 ) ) ) ).

% old.prod.inject
thf(fact_794_prod_Oinject,axiom,
    ! [X1: term_f_v,X23: term_f_v,Y1: term_f_v,Y23: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ X1 @ X23 )
        = ( produc2510790566102465375rm_f_v @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_795_prod_Oinject,axiom,
    ! [X1: f,X23: nat,Y1: f,Y23: nat] :
      ( ( ( product_Pair_f_nat @ X1 @ X23 )
        = ( product_Pair_f_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_796_NF__I,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v] :
      ( ! [B2: produc8027700073345061287rm_f_v] :
          ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B2 ) @ A3 )
     => ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) ) ) ).

% NF_I
thf(fact_797_NF__I,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ! [B2: product_prod_f_nat] :
          ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B2 ) @ A3 )
     => ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) ) ) ).

% NF_I
thf(fact_798_NF__I,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ! [B2: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B2 ) @ A3 )
     => ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) ) ) ).

% NF_I
thf(fact_799_NF__iff__no__step,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) )
      = ( ! [B3: produc8027700073345061287rm_f_v] :
            ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ A3 ) ) ) ).

% NF_iff_no_step
thf(fact_800_NF__iff__no__step,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) )
      = ( ! [B3: product_prod_f_nat] :
            ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B3 ) @ A3 ) ) ) ).

% NF_iff_no_step
thf(fact_801_NF__iff__no__step,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) )
      = ( ! [B3: term_f_v] :
            ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ A3 ) ) ) ).

% NF_iff_no_step
thf(fact_802_NF__no__step,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) )
     => ! [B9: produc8027700073345061287rm_f_v] :
          ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B9 ) @ A3 ) ) ).

% NF_no_step
thf(fact_803_NF__no__step,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) )
     => ! [B9: product_prod_f_nat] :
          ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B9 ) @ A3 ) ) ).

% NF_no_step
thf(fact_804_NF__no__step,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) )
     => ! [B9: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B9 ) @ A3 ) ) ).

% NF_no_step
thf(fact_805_NF__E,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) )
     => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ A3 ) ) ).

% NF_E
thf(fact_806_NF__E,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat,B: product_prod_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) )
     => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ A3 ) ) ).

% NF_E
thf(fact_807_NF__E,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v,B: term_f_v] :
      ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ A3 ) ) ).

% NF_E
thf(fact_808_inf__set__def,axiom,
    ( inf_in4124010953442585290_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
          ( collec1417984143148964183_f_nat
          @ ( inf_in8255514135000801851_nat_o
            @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A5 )
            @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_809_inf__set__def,axiom,
    ( inf_inf_set_term_f_v
    = ( ^ [A5: set_term_f_v,B5: set_term_f_v] :
          ( collect_term_f_v
          @ ( inf_inf_term_f_v_o
            @ ^ [X: term_f_v] : ( member_term_f_v @ X @ A5 )
            @ ^ [X: term_f_v] : ( member_term_f_v @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_810_inf__set__def,axiom,
    ( inf_in6135112614888068085rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ( inf_in5475010204142724904_f_v_o
            @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A5 )
            @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_811_inf__Int__eq,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( inf_in8255514135000801851_nat_o
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ R2 )
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ S2 ) )
      = ( ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ ( inf_in4124010953442585290_f_nat @ R2 @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_812_inf__Int__eq,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in5475010204142724904_f_v_o
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ R2 )
        @ ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ S2 ) )
      = ( ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ ( inf_in6135112614888068085rm_f_v @ R2 @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_813_inf__Int__eq2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S2: set_Pr5245412377734449720_f_nat] :
      ( ( inf_inf_f_nat_o
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R2 )
        @ ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ ( inf_in4124010953442585290_f_nat @ R2 @ S2 ) ) ) ) ).

% inf_Int_eq2
thf(fact_814_inf__Int__eq2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in2796210053413908654_f_v_o
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( inf_in6135112614888068085rm_f_v @ R2 @ S2 ) ) ) ) ).

% inf_Int_eq2
thf(fact_815_NF__not__suc,axiom,
    ! [X3: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ X3 @ ( abstra349717802288577587rm_f_v @ R3 ) )
       => ( X3 = Y2 ) ) ) ).

% NF_not_suc
thf(fact_816_NF__not__suc,axiom,
    ! [X3: product_prod_f_nat,Y2: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
     => ( ( member2678038196313615385_f_nat @ X3 @ ( abstra8862684023087959798_f_nat @ R3 ) )
       => ( X3 = Y2 ) ) ) ).

% NF_not_suc
thf(fact_817_NF__not__suc,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
     => ( ( member_term_f_v @ X3 @ ( abstract_NF_term_f_v @ R3 ) )
       => ( X3 = Y2 ) ) ) ).

% NF_not_suc
thf(fact_818_NF__anti__mono,axiom,
    ! [A3: set_Pr516226060960033221_f_nat,B4: set_Pr516226060960033221_f_nat] :
      ( ( ord_le8946695678602162533_f_nat @ A3 @ B4 )
     => ( ord_le8976984241387448984_f_nat @ ( abstra8862684023087959798_f_nat @ B4 ) @ ( abstra8862684023087959798_f_nat @ A3 ) ) ) ).

% NF_anti_mono
thf(fact_819_NF__no__trancl__step,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) )
     => ! [B9: produc8027700073345061287rm_f_v] :
          ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B9 ) @ ( transi7658437189833989320rm_f_v @ A3 ) ) ) ).

% NF_no_trancl_step
thf(fact_820_NF__no__trancl__step,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) )
     => ! [B9: product_prod_f_nat] :
          ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B9 ) @ ( transi3759436115690491041_f_nat @ A3 ) ) ) ).

% NF_no_trancl_step
thf(fact_821_NF__no__trancl__step,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) )
     => ! [B9: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B9 ) @ ( transi3208867851621559554rm_f_v @ A3 ) ) ) ).

% NF_no_trancl_step
thf(fact_822_Pair__inject,axiom,
    ! [A: term_f_v,B: term_f_v,A8: term_f_v,B8: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A8 @ B8 ) )
     => ~ ( ( A = A8 )
         => ( B != B8 ) ) ) ).

% Pair_inject
thf(fact_823_Pair__inject,axiom,
    ! [A: f,B: nat,A8: f,B8: nat] :
      ( ( ( product_Pair_f_nat @ A @ B )
        = ( product_Pair_f_nat @ A8 @ B8 ) )
     => ~ ( ( A = A8 )
         => ( B != B8 ) ) ) ).

% Pair_inject
thf(fact_824_prod__cases,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,P: produc8027700073345061287rm_f_v] :
      ( ! [A2: term_f_v,B2: term_f_v] : ( P2 @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_825_prod__cases,axiom,
    ! [P2: product_prod_f_nat > $o,P: product_prod_f_nat] :
      ( ! [A2: f,B2: nat] : ( P2 @ ( product_Pair_f_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_826_surj__pair,axiom,
    ! [P: produc8027700073345061287rm_f_v] :
    ? [X2: term_f_v,Y: term_f_v] :
      ( P
      = ( produc2510790566102465375rm_f_v @ X2 @ Y ) ) ).

% surj_pair
thf(fact_827_surj__pair,axiom,
    ! [P: product_prod_f_nat] :
    ? [X2: f,Y: nat] :
      ( P
      = ( product_Pair_f_nat @ X2 @ Y ) ) ).

% surj_pair
thf(fact_828_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc8027700073345061287rm_f_v] :
      ~ ! [A2: term_f_v,B2: term_f_v] :
          ( Y2
         != ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_829_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_f_nat] :
      ~ ! [A2: f,B2: nat] :
          ( Y2
         != ( product_Pair_f_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_830_Sigma__cong,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,C4: term_f_v > set_term_f_v,D3: term_f_v > set_term_f_v] :
      ( ( A3 = B4 )
     => ( ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ B4 )
           => ( ( C4 @ X2 )
              = ( D3 @ X2 ) ) )
       => ( ( produc6667182599969081488rm_f_v @ A3 @ C4 )
          = ( produc6667182599969081488rm_f_v @ B4 @ D3 ) ) ) ) ).

% Sigma_cong
thf(fact_831_Times__eq__cancel2,axiom,
    ! [X3: term_f_v,C4: set_term_f_v,A3: set_term_f_v,B4: set_term_f_v] :
      ( ( member_term_f_v @ X3 @ C4 )
     => ( ( ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : C4 )
          = ( produc6667182599969081488rm_f_v @ B4
            @ ^ [Uu: term_f_v] : C4 ) )
        = ( A3 = B4 ) ) ) ).

% Times_eq_cancel2
thf(fact_832_ground__NF__srstep__gsrstep,axiom,
    ! [S: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( term_ground_f_v @ S )
     => ( ( member_term_f_v @ S @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
       => ( member_term_f_v @ S
          @ ( abstract_NF_term_f_v
            @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
              @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
                @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ) ) ).

% ground_NF_srstep_gsrstep
thf(fact_833_SigmaE,axiom,
    ! [C: produc3406425829894461415rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ C @ ( produc7856489909025154568rm_f_v @ A3 @ B4 ) )
     => ~ ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
           => ! [Y: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( produc3874942763110801239rm_f_v @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_834_SigmaE,axiom,
    ! [C: produc3492972538368621912_f_nat,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat] :
      ( ( member6881531084954517871_f_nat @ C @ ( produc423937348911600481_f_nat @ A3 @ B4 ) )
     => ~ ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
           => ! [Y: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( produc8402213614931650770_f_nat @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_835_SigmaE,axiom,
    ! [C: produc834342741349969050rm_f_v,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr8827868859434726151rm_f_v] :
      ( ( member4222901287935865009rm_f_v @ C @ ( produc5851212470841160795rm_f_v @ A3 @ B4 ) )
     => ~ ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A3 )
           => ! [Y: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( produc4606116700006435276rm_f_v @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_836_SigmaE,axiom,
    ! [C: produc2676906022994773093_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( member7030746907725424910_f_nat @ C @ ( produc7667213020494768334_f_nat @ A3 @ B4 ) )
     => ~ ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A3 )
           => ! [Y: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( produc5861087217798509725_f_nat @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_837_SigmaE,axiom,
    ! [C: product_prod_f_nat,A3: set_f,B4: f > set_nat] :
      ( ( member2678038196313615385_f_nat @ C @ ( product_Sigma_f_nat @ A3 @ B4 ) )
     => ~ ! [X2: f] :
            ( ( member_f @ X2 @ A3 )
           => ! [Y: nat] :
                ( ( member_nat @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( product_Pair_f_nat @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_838_SigmaE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) )
     => ~ ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ A3 )
           => ! [Y: term_f_v] :
                ( ( member_term_f_v @ Y @ ( B4 @ X2 ) )
               => ( C
                 != ( produc2510790566102465375rm_f_v @ X2 @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_839_SigmaD1,axiom,
    ! [A: f,B: nat,A3: set_f,B4: f > set_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( product_Sigma_f_nat @ A3 @ B4 ) )
     => ( member_f @ A @ A3 ) ) ).

% SigmaD1
thf(fact_840_SigmaD1,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) )
     => ( member_term_f_v @ A @ A3 ) ) ).

% SigmaD1
thf(fact_841_SigmaD2,axiom,
    ! [A: f,B: nat,A3: set_f,B4: f > set_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( product_Sigma_f_nat @ A3 @ B4 ) )
     => ( member_nat @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_842_SigmaD2,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) )
     => ( member_term_f_v @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_843_SigmaE2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( produc7856489909025154568rm_f_v @ A3 @ B4 ) )
     => ~ ( ( member4316649861536941904rm_f_v @ A @ A3 )
         => ~ ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_844_SigmaE2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: product_prod_f_nat,A3: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat] :
      ( ( member6881531084954517871_f_nat @ ( produc8402213614931650770_f_nat @ A @ B ) @ ( produc423937348911600481_f_nat @ A3 @ B4 ) )
     => ~ ( ( member4316649861536941904rm_f_v @ A @ A3 )
         => ~ ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_845_SigmaE2,axiom,
    ! [A: product_prod_f_nat,B: produc8027700073345061287rm_f_v,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr8827868859434726151rm_f_v] :
      ( ( member4222901287935865009rm_f_v @ ( produc4606116700006435276rm_f_v @ A @ B ) @ ( produc5851212470841160795rm_f_v @ A3 @ B4 ) )
     => ~ ( ( member2678038196313615385_f_nat @ A @ A3 )
         => ~ ( member4316649861536941904rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_846_SigmaE2,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( produc7667213020494768334_f_nat @ A3 @ B4 ) )
     => ~ ( ( member2678038196313615385_f_nat @ A @ A3 )
         => ~ ( member2678038196313615385_f_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_847_SigmaE2,axiom,
    ! [A: f,B: nat,A3: set_f,B4: f > set_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ ( product_Sigma_f_nat @ A3 @ B4 ) )
     => ~ ( ( member_f @ A @ A3 )
         => ~ ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_848_SigmaE2,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) )
     => ~ ( ( member_term_f_v @ A @ A3 )
         => ~ ( member_term_f_v @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_849_Sigma__mono,axiom,
    ! [A3: set_term_f_v,C4: set_term_f_v,B4: term_f_v > set_term_f_v,D3: term_f_v > set_term_f_v] :
      ( ( ord_le947722214152454265rm_f_v @ A3 @ C4 )
     => ( ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ A3 )
           => ( ord_le947722214152454265rm_f_v @ ( B4 @ X2 ) @ ( D3 @ X2 ) ) )
       => ( ord_le4559761987009501863rm_f_v @ ( produc6667182599969081488rm_f_v @ A3 @ B4 ) @ ( produc6667182599969081488rm_f_v @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_850_Sigma__mono,axiom,
    ! [A3: set_f,C4: set_f,B4: f > set_nat,D3: f > set_nat] :
      ( ( ord_less_eq_set_f @ A3 @ C4 )
     => ( ! [X2: f] :
            ( ( member_f @ X2 @ A3 )
           => ( ord_less_eq_set_nat @ ( B4 @ X2 ) @ ( D3 @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat @ ( product_Sigma_f_nat @ A3 @ B4 ) @ ( product_Sigma_f_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_851_Sigma__mono,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat,D3: produc8027700073345061287rm_f_v > set_Pr5245412377734449720_f_nat] :
      ( ( ord_le4559761987009501863rm_f_v @ A3 @ C4 )
     => ( ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
           => ( ord_le8976984241387448984_f_nat @ ( B4 @ X2 ) @ ( D3 @ X2 ) ) )
       => ( ord_le5954414151786787566_f_nat @ ( produc423937348911600481_f_nat @ A3 @ B4 ) @ ( produc423937348911600481_f_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_852_Sigma__mono,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: product_prod_f_nat > set_Pr5245412377734449720_f_nat,D3: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ C4 )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A3 )
           => ( ord_le8976984241387448984_f_nat @ ( B4 @ X2 ) @ ( D3 @ X2 ) ) )
       => ( ord_le8946695678602162533_f_nat @ ( produc7667213020494768334_f_nat @ A3 @ B4 ) @ ( produc7667213020494768334_f_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_853_Sigma__Un__distrib1,axiom,
    ! [I: set_term_f_v,J: set_term_f_v,C4: term_f_v > set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ ( sup_sup_set_term_f_v @ I @ J ) @ C4 )
      = ( sup_su1994854688125285083rm_f_v @ ( produc6667182599969081488rm_f_v @ I @ C4 ) @ ( produc6667182599969081488rm_f_v @ J @ C4 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_854_Times__subset__cancel2,axiom,
    ! [X3: term_f_v,C4: set_term_f_v,A3: set_term_f_v,B4: set_term_f_v] :
      ( ( member_term_f_v @ X3 @ C4 )
     => ( ( ord_le4559761987009501863rm_f_v
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : C4 )
          @ ( produc6667182599969081488rm_f_v @ B4
            @ ^ [Uu: term_f_v] : C4 ) )
        = ( ord_le947722214152454265rm_f_v @ A3 @ B4 ) ) ) ).

% Times_subset_cancel2
thf(fact_855_Times__subset__cancel2,axiom,
    ! [X3: produc8027700073345061287rm_f_v,C4: set_Pr8827868859434726151rm_f_v,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member4316649861536941904rm_f_v @ X3 @ C4 )
     => ( ( ord_le4772425595354554672rm_f_v
          @ ( produc5851212470841160795rm_f_v @ A3
            @ ^ [Uu: product_prod_f_nat] : C4 )
          @ ( produc5851212470841160795rm_f_v @ B4
            @ ^ [Uu: product_prod_f_nat] : C4 ) )
        = ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ) ).

% Times_subset_cancel2
thf(fact_856_Times__subset__cancel2,axiom,
    ! [X3: product_prod_f_nat,C4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( member2678038196313615385_f_nat @ X3 @ C4 )
     => ( ( ord_le8946695678602162533_f_nat
          @ ( produc7667213020494768334_f_nat @ A3
            @ ^ [Uu: product_prod_f_nat] : C4 )
          @ ( produc7667213020494768334_f_nat @ B4
            @ ^ [Uu: product_prod_f_nat] : C4 ) )
        = ( ord_le8976984241387448984_f_nat @ A3 @ B4 ) ) ) ).

% Times_subset_cancel2
thf(fact_857_Times__subset__cancel2,axiom,
    ! [X3: nat,C4: set_nat,A3: set_f,B4: set_f] :
      ( ( member_nat @ X3 @ C4 )
     => ( ( ord_le8976984241387448984_f_nat
          @ ( product_Sigma_f_nat @ A3
            @ ^ [Uu: f] : C4 )
          @ ( product_Sigma_f_nat @ B4
            @ ^ [Uu: f] : C4 ) )
        = ( ord_less_eq_set_f @ A3 @ B4 ) ) ) ).

% Times_subset_cancel2
thf(fact_858_Sigma__Un__distrib2,axiom,
    ! [I: set_term_f_v,A3: term_f_v > set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ I
        @ ^ [I2: term_f_v] : ( sup_sup_set_term_f_v @ ( A3 @ I2 ) @ ( B4 @ I2 ) ) )
      = ( sup_su1994854688125285083rm_f_v @ ( produc6667182599969081488rm_f_v @ I @ A3 ) @ ( produc6667182599969081488rm_f_v @ I @ B4 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_859_Times__Un__distrib1,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,C4: set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ ( sup_sup_set_term_f_v @ A3 @ B4 )
        @ ^ [Uu: term_f_v] : C4 )
      = ( sup_su1994854688125285083rm_f_v
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : C4 )
        @ ( produc6667182599969081488rm_f_v @ B4
          @ ^ [Uu: term_f_v] : C4 ) ) ) ).

% Times_Un_distrib1
thf(fact_860_Times__Int__Times,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,D3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in3424906307742236341rm_f_v
        @ ( produc7856489909025154568rm_f_v @ A3
          @ ^ [Uu: produc8027700073345061287rm_f_v] : B4 )
        @ ( produc7856489909025154568rm_f_v @ C4
          @ ^ [Uu: produc8027700073345061287rm_f_v] : D3 ) )
      = ( produc7856489909025154568rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A3 @ C4 )
        @ ^ [Uu: produc8027700073345061287rm_f_v] : ( inf_in6135112614888068085rm_f_v @ B4 @ D3 ) ) ) ).

% Times_Int_Times
thf(fact_861_Times__Int__Times,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,C4: set_term_f_v,D3: set_term_f_v] :
      ( ( inf_in6135112614888068085rm_f_v
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : B4 )
        @ ( produc6667182599969081488rm_f_v @ C4
          @ ^ [Uu: term_f_v] : D3 ) )
      = ( produc6667182599969081488rm_f_v @ ( inf_inf_set_term_f_v @ A3 @ C4 )
        @ ^ [Uu: term_f_v] : ( inf_inf_set_term_f_v @ B4 @ D3 ) ) ) ).

% Times_Int_Times
thf(fact_862_Sigma__Int__distrib2,axiom,
    ! [I: set_term_f_v,A3: term_f_v > set_term_f_v,B4: term_f_v > set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ I
        @ ^ [I2: term_f_v] : ( inf_inf_set_term_f_v @ ( A3 @ I2 ) @ ( B4 @ I2 ) ) )
      = ( inf_in6135112614888068085rm_f_v @ ( produc6667182599969081488rm_f_v @ I @ A3 ) @ ( produc6667182599969081488rm_f_v @ I @ B4 ) ) ) ).

% Sigma_Int_distrib2
thf(fact_863_NFP__redp__def,axiom,
    ( rewrit4742200027381058679dp_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v,S6: term_f_v,T3: term_f_v] :
          ( ( member_term_f_v @ T3 @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S6 @ T3 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) ) ) ) ) ).

% NFP_redp_def
thf(fact_864_Restr__subset,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le947722214152454265rm_f_v @ A3 @ B4 )
     => ( ( inf_in6135112614888068085rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ R3
            @ ( produc6667182599969081488rm_f_v @ B4
              @ ^ [Uu: term_f_v] : B4 ) )
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : A3 ) )
        = ( inf_in6135112614888068085rm_f_v @ R3
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu: term_f_v] : A3 ) ) ) ) ).

% Restr_subset
thf(fact_865_Restr__subset,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( inf_in7383377402786884787_f_nat
          @ ( inf_in7383377402786884787_f_nat @ R3
            @ ( produc7667213020494768334_f_nat @ B4
              @ ^ [Uu: product_prod_f_nat] : B4 ) )
          @ ( produc7667213020494768334_f_nat @ A3
            @ ^ [Uu: product_prod_f_nat] : A3 ) )
        = ( inf_in7383377402786884787_f_nat @ R3
          @ ( produc7667213020494768334_f_nat @ A3
            @ ^ [Uu: product_prod_f_nat] : A3 ) ) ) ) ).

% Restr_subset
thf(fact_866_NE__redp__def,axiom,
    ( rewrit7960950296172055926dp_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v,S7: set_Pr8827868859434726151rm_f_v,S6: term_f_v,T3: term_f_v] :
          ( ( member_term_f_v @ T3 @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) )
         => ( ( member_term_f_v @ T3 @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S6 @ T3 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ S7 ) ) ) ) ) ) ) ) ).

% NE_redp_def
thf(fact_867_gsrstep__conversion__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra5640035136655339381rm_f_v
        @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) )
          @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
            @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) )
      = ( transi3028809021613410336rm_f_v
        @ ( inf_in6135112614888068085rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
          @ ( produc6667182599969081488rm_f_v @ ( collect_term_f_v @ term_ground_f_v )
            @ ^ [Uu: term_f_v] : ( collect_term_f_v @ term_ground_f_v ) ) ) ) ) ).

% gsrstep_conversion_dist
thf(fact_868_converse__Times,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v] :
      ( ( conver3099152742603637891rm_f_v
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : B4 ) )
      = ( produc6667182599969081488rm_f_v @ B4
        @ ^ [Uu: term_f_v] : A3 ) ) ).

% converse_Times
thf(fact_869_all__ctxt__closed__relcomp,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S4: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ R )
         => ( ( S5 != T4 )
           => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S5 ) @ F )
              & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T4 ) @ F ) ) ) )
     => ( ! [S5: term_f_v,T4: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ S4 )
           => ( ( S5 != T4 )
             => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S5 ) @ F )
                & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T4 ) @ F ) ) ) )
       => ( ( terms_512237927760305819ed_f_v @ F @ R )
         => ( ( terms_512237927760305819ed_f_v @ F @ S4 )
           => ( terms_512237927760305819ed_f_v @ F @ ( relcom2934258169684288428rm_f_v @ R @ S4 ) ) ) ) ) ) ).

% all_ctxt_closed_relcomp
thf(fact_870_lv__root__step__idep__subst,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,Sigma: v > term_f_v,Tau: v > term_f_v] :
      ( ( rewrit2771924964435581537_f_v_f @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
       => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X2 ) ) @ F )
         => ( ! [X2: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Tau @ X2 ) ) @ F )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Tau ) ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) ) ) ) ) ) ).

% lv_root_step_idep_subst
thf(fact_871_conversion__rtrancl,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( abstra5640035136655339381rm_f_v @ A3 ) )
      = ( abstra5640035136655339381rm_f_v @ A3 ) ) ).

% conversion_rtrancl
thf(fact_872_rrstep__basicI,axiom,
    ! [L: term_f_v,R3: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ R )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ ( rrstep_f_v @ R ) ) ) ).

% rrstep_basicI
thf(fact_873_conversion__O__conversion,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( abstra5640035136655339381rm_f_v @ A3 ) @ ( abstra5640035136655339381rm_f_v @ A3 ) )
      = ( abstra5640035136655339381rm_f_v @ A3 ) ) ).

% conversion_O_conversion
thf(fact_874_conversion__converse,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( abstra5640035136655339381rm_f_v @ A3 ) )
      = ( abstra5640035136655339381rm_f_v @ A3 ) ) ).

% conversion_converse
thf(fact_875_conversionI_H,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A3 ) ) ) ).

% conversionI'
thf(fact_876_all__ctxt__closed__sig__rsteps,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] : ( terms_512237927760305819ed_f_v @ F @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ).

% all_ctxt_closed_sig_rsteps
thf(fact_877_conversionI,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ ( conver3099152742603637891rm_f_v @ A3 ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A3 ) ) ) ).

% conversionI
thf(fact_878_conversion__refl,axiom,
    ! [A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( abstra5640035136655339381rm_f_v @ A3 ) ) ).

% conversion_refl
thf(fact_879_conversion__inv,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( abstra5640035136655339381rm_f_v @ R2 ) )
      = ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ X3 ) @ ( abstra5640035136655339381rm_f_v @ R2 ) ) ) ).

% conversion_inv
thf(fact_880_all__ctxt__closed__srstep__conversions,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] : ( terms_512237927760305819ed_f_v @ F @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ).

% all_ctxt_closed_srstep_conversions
thf(fact_881_rrstep__subst,axiom,
    ! [S: term_f_v,T: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rrstep_f_v @ R ) )
     => ~ ! [L4: term_f_v,R9: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L4 @ R9 ) @ R )
           => ! [Sigma2: v > term_f_v] :
                ( ( S
                  = ( subst_7901567856578221233_f_v_v @ L4 @ Sigma2 ) )
               => ( T
                 != ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) ) ) ) ) ).

% rrstep_subst
thf(fact_882_rrstep_Ointros,axiom,
    ! [L: term_f_v,R3: term_f_v,R: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R3 ) @ R )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ L @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ R3 @ Sigma ) ) @ ( rrstep_f_v @ R ) ) ) ).

% rrstep.intros
thf(fact_883_rrstep_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rrstep_f_v @ R ) )
      = ( ? [L5: term_f_v,R6: term_f_v,Sigma3: v > term_f_v] :
            ( ( A1
              = ( subst_7901567856578221233_f_v_v @ L5 @ Sigma3 ) )
            & ( A22
              = ( subst_7901567856578221233_f_v_v @ R6 @ Sigma3 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L5 @ R6 ) @ R ) ) ) ) ).

% rrstep.simps
thf(fact_884_rrstep_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rrstep_f_v @ R ) )
     => ~ ! [L4: term_f_v,R9: term_f_v,Sigma2: v > term_f_v] :
            ( ( A1
              = ( subst_7901567856578221233_f_v_v @ L4 @ Sigma2 ) )
           => ( ( A22
                = ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L4 @ R9 ) @ R ) ) ) ) ).

% rrstep.cases
thf(fact_885_srrstep__converse__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
      = ( sig_step_f_v @ F @ ( rrstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).

% srrstep_converse_dist
thf(fact_886_rrstep__rstep__mono,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( rrstep_f_v @ R ) @ ( rstep_f_v @ R ) ) ).

% rrstep_rstep_mono
thf(fact_887_conversion__def,axiom,
    ( abstra5640035136655339381rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v] : ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A5 @ ( conver3099152742603637891rm_f_v @ A5 ) ) ) ) ) ).

% conversion_def
thf(fact_888_srrstep__to__srestep,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ).

% srrstep_to_srestep
thf(fact_889_srrstep__monp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le8976984241387448984_f_nat @ F @ G )
     => ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) @ ( sig_step_f_v @ G @ ( rrstep_f_v @ R ) ) ) ) ).

% srrstep_monp
thf(fact_890_subset__CollectI,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,Q: produc8027700073345061287rm_f_v > $o,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( ord_le4559761987009501863rm_f_v @ B4 @ A3 )
     => ( ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ B4 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le4559761987009501863rm_f_v
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ X @ A3 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_891_subset__CollectI,axiom,
    ! [B4: set_term_f_v,A3: set_term_f_v,Q: term_f_v > $o,P2: term_f_v > $o] :
      ( ( ord_le947722214152454265rm_f_v @ B4 @ A3 )
     => ( ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ B4 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le947722214152454265rm_f_v
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( member_term_f_v @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( member_term_f_v @ X @ A3 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_892_subset__CollectI,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,Q: product_prod_f_nat > $o,P2: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ B4 @ A3 )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ B4 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A3 )
                & ( P2 @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_893_subset__Collect__iff,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( ord_le4559761987009501863rm_f_v @ B4 @ A3 )
     => ( ( ord_le4559761987009501863rm_f_v @ B4
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ X @ A3 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: produc8027700073345061287rm_f_v] :
              ( ( member4316649861536941904rm_f_v @ X @ B4 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_894_subset__Collect__iff,axiom,
    ! [B4: set_term_f_v,A3: set_term_f_v,P2: term_f_v > $o] :
      ( ( ord_le947722214152454265rm_f_v @ B4 @ A3 )
     => ( ( ord_le947722214152454265rm_f_v @ B4
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( member_term_f_v @ X @ A3 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: term_f_v] :
              ( ( member_term_f_v @ X @ B4 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_895_subset__Collect__iff,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A3: set_Pr5245412377734449720_f_nat,P2: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ B4 @ A3 )
     => ( ( ord_le8976984241387448984_f_nat @ B4
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A3 )
                & ( P2 @ X ) ) ) )
        = ( ! [X: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X @ B4 )
             => ( P2 @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_896_all__ctxt__closed__sig__reflE,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,T: term_f_v] :
      ( ( terms_512237927760305819ed_f_v @ F @ R )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ T ) @ R ) ) ) ).

% all_ctxt_closed_sig_reflE
thf(fact_897_conversionE,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ ( conver3099152742603637891rm_f_v @ A3 ) ) ) ) ) ).

% conversionE
thf(fact_898_srrstepD,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rrstep_f_v @ R ) )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
        & ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).

% srrstepD
thf(fact_899_llrg__rrsteps__groundness,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( rewrit8970090298225341052_v_f_v @ R )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
       => ( term_ground_f_v @ T ) ) ) ).

% llrg_rrsteps_groundness
thf(fact_900_symcl__srstep__conversion,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% symcl_srstep_conversion
thf(fact_901_sig__step__conversion__dist,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
      = ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ) ) ).

% sig_step_conversion_dist
thf(fact_902_srsteps__with__root__step__def,axiom,
    ( srstep3130564645414158319ep_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( sig_step_f_v @ F2 @ ( rrstep_f_v @ R4 ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) ) ) ) ) ).

% srsteps_with_root_step_def
thf(fact_903_symcl__srsteps__conversion,axiom,
    ! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% symcl_srsteps_conversion
thf(fact_904_reduction__join__relations__to__root__step,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,P2: term_f_v > term_f_v > $o,S: term_f_v,T: term_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ R ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) ) ) )
         => ( P2 @ S5 @ T4 ) )
     => ( ( terms_512237927760305819ed_f_v @ F
          @ ( collec7769483783292507282rm_f_v
            @ ^ [Uu: produc8027700073345061287rm_f_v] :
              ? [S6: term_f_v,T3: term_f_v] :
                ( ( Uu
                  = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
                & ( P2 @ S6 @ T3 ) ) ) )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) )
             => ( P2 @ S @ T ) ) ) ) ) ) ).

% reduction_join_relations_to_root_step
thf(fact_905_reduction__relations__to__root__step,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > term_f_v > $o,S: term_f_v,T: term_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
         => ( P2 @ S5 @ T4 ) )
     => ( ( terms_512237927760305819ed_f_v @ F
          @ ( collec7769483783292507282rm_f_v
            @ ^ [Uu: produc8027700073345061287rm_f_v] :
              ? [S6: term_f_v,T3: term_f_v] :
                ( ( Uu
                  = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
                & ( P2 @ S6 @ T3 ) ) ) )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
         => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) )
             => ( P2 @ S @ T ) ) ) ) ) ) ).

% reduction_relations_to_root_step
thf(fact_906_UNC__E,axiom,
    ! [A3: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( abstra6927431686333010263rm_f_v @ A3 )
     => ( ( A != B )
       => ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A3 ) )
         => ( ( member4316649861536941904rm_f_v @ B @ ( abstra349717802288577587rm_f_v @ A3 ) )
           => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra4572590880773903381rm_f_v @ A3 ) ) ) ) ) ) ).

% UNC_E
thf(fact_907_UNC__E,axiom,
    ! [A3: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat] :
      ( ( abstra4133103602043500242_f_nat @ A3 )
     => ( ( A != B )
       => ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ A3 ) )
         => ( ( member2678038196313615385_f_nat @ B @ ( abstra8862684023087959798_f_nat @ A3 ) )
           => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra8410917409225108_f_nat @ A3 ) ) ) ) ) ) ).

% UNC_E
thf(fact_908_UNC__E,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
      ( ( abstra9159237726496465843rm_f_v @ A3 )
     => ( ( A != B )
       => ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A3 ) )
         => ( ( member_term_f_v @ B @ ( abstract_NF_term_f_v @ A3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A3 ) ) ) ) ) ) ).

% UNC_E
thf(fact_909_commute__redp__mctxt__cl,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( terms_512237927760305819ed_f_v @ F
      @ ( collec7769483783292507282rm_f_v
        @ ^ [Uu: produc8027700073345061287rm_f_v] :
          ? [S6: term_f_v,T3: term_f_v] :
            ( ( Uu
              = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
            & ( rewrit4152091719787942777dp_f_v @ F @ R @ S4 @ S6 @ T3 ) ) ) ) ).

% commute_redp_mctxt_cl
thf(fact_910_prop__mctxt__cl__NE__redp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( terms_512237927760305819ed_f_v @ F
      @ ( collec7769483783292507282rm_f_v
        @ ^ [Uu: produc8027700073345061287rm_f_v] :
          ? [S6: term_f_v,T3: term_f_v] :
            ( ( Uu
              = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
            & ( rewrit7960950296172055926dp_f_v @ F @ R @ S4 @ S6 @ T3 ) ) ) ) ).

% prop_mctxt_cl_NE_redp
thf(fact_911_prop__mctxt__cl__NFP__redp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( terms_512237927760305819ed_f_v @ F
      @ ( collec7769483783292507282rm_f_v
        @ ^ [Uu: produc8027700073345061287rm_f_v] :
          ? [S6: term_f_v,T3: term_f_v] :
            ( ( Uu
              = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
            & ( rewrit4742200027381058679dp_f_v @ F @ R @ S6 @ T3 ) ) ) ) ).

% prop_mctxt_cl_NFP_redp
thf(fact_912_prop__mctxt__cl__reflcl__on,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,P2: term_f_v > term_f_v > $o,S: term_f_v] :
      ( ( terms_512237927760305819ed_f_v @ F
        @ ( collec7769483783292507282rm_f_v
          @ ^ [Uu: produc8027700073345061287rm_f_v] :
            ? [S6: term_f_v,T3: term_f_v] :
              ( ( Uu
                = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
              & ( P2 @ S6 @ T3 ) ) ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
       => ( P2 @ S @ S ) ) ) ).

% prop_mctxt_cl_reflcl_on
thf(fact_913_prop__mctxt__cl__refl__on,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,P2: term_f_v > term_f_v > $o,T: term_f_v] :
      ( ( terms_512237927760305819ed_f_v @ F
        @ ( collec7769483783292507282rm_f_v
          @ ^ [Uu: produc8027700073345061287rm_f_v] :
            ? [S6: term_f_v,T3: term_f_v] :
              ( ( Uu
                = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
              & ( P2 @ S6 @ T3 ) ) ) )
     => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
       => ( P2 @ T @ T ) ) ) ).

% prop_mctxt_cl_refl_on
thf(fact_914_UNC__def,axiom,
    ( abstra6927431686333010263rm_f_v
    = ( ^ [A5: set_Pr7897041799657265351rm_f_v] :
        ! [A4: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] :
          ( ( ( member4316649861536941904rm_f_v @ A4 @ ( abstra349717802288577587rm_f_v @ A5 ) )
            & ( member4316649861536941904rm_f_v @ B3 @ ( abstra349717802288577587rm_f_v @ A5 ) )
            & ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A4 @ B3 ) @ ( abstra4572590880773903381rm_f_v @ A5 ) ) )
         => ( A4 = B3 ) ) ) ) ).

% UNC_def
thf(fact_915_UNC__def,axiom,
    ( abstra4133103602043500242_f_nat
    = ( ^ [A5: set_Pr516226060960033221_f_nat] :
        ! [A4: product_prod_f_nat,B3: product_prod_f_nat] :
          ( ( ( member2678038196313615385_f_nat @ A4 @ ( abstra8862684023087959798_f_nat @ A5 ) )
            & ( member2678038196313615385_f_nat @ B3 @ ( abstra8862684023087959798_f_nat @ A5 ) )
            & ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A4 @ B3 ) @ ( abstra8410917409225108_f_nat @ A5 ) ) )
         => ( A4 = B3 ) ) ) ) ).

% UNC_def
thf(fact_916_UNC__def,axiom,
    ( abstra9159237726496465843rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v] :
        ! [A4: term_f_v,B3: term_f_v] :
          ( ( ( member_term_f_v @ A4 @ ( abstract_NF_term_f_v @ A5 ) )
            & ( member_term_f_v @ B3 @ ( abstract_NF_term_f_v @ A5 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( abstra5640035136655339381rm_f_v @ A5 ) ) )
         => ( A4 = B3 ) ) ) ) ).

% UNC_def
thf(fact_917_UNC__I,axiom,
    ! [A3: set_Pr7897041799657265351rm_f_v] :
      ( ! [A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ A2 @ ( abstra349717802288577587rm_f_v @ A3 ) )
         => ( ( member4316649861536941904rm_f_v @ B2 @ ( abstra349717802288577587rm_f_v @ A3 ) )
           => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ B2 ) @ ( abstra4572590880773903381rm_f_v @ A3 ) )
             => ( A2 = B2 ) ) ) )
     => ( abstra6927431686333010263rm_f_v @ A3 ) ) ).

% UNC_I
thf(fact_918_UNC__I,axiom,
    ! [A3: set_Pr516226060960033221_f_nat] :
      ( ! [A2: product_prod_f_nat,B2: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ A2 @ ( abstra8862684023087959798_f_nat @ A3 ) )
         => ( ( member2678038196313615385_f_nat @ B2 @ ( abstra8862684023087959798_f_nat @ A3 ) )
           => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ B2 ) @ ( abstra8410917409225108_f_nat @ A3 ) )
             => ( A2 = B2 ) ) ) )
     => ( abstra4133103602043500242_f_nat @ A3 ) ) ).

% UNC_I
thf(fact_919_UNC__I,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ! [A2: term_f_v,B2: term_f_v] :
          ( ( member_term_f_v @ A2 @ ( abstract_NF_term_f_v @ A3 ) )
         => ( ( member_term_f_v @ B2 @ ( abstract_NF_term_f_v @ A3 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( abstra5640035136655339381rm_f_v @ A3 ) )
             => ( A2 = B2 ) ) ) )
     => ( abstra9159237726496465843rm_f_v @ A3 ) ) ).

% UNC_I
thf(fact_920_UNC__to__rrstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( abstra9159237726496465843rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
       => ( rewrit5493321224924644152dp_f_v @ F @ R @ S @ T ) ) ) ).

% UNC_to_rrstep
thf(fact_921_UNC__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
         => ( rewrit5493321224924644152dp_f_v @ F @ R @ S5 @ T4 ) )
     => ( abstra9159237726496465843rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ).

% UNC_rrstep_intro
thf(fact_922_prop__mctxt__cl__UN__redp,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( terms_512237927760305819ed_f_v @ F
      @ ( collec7769483783292507282rm_f_v
        @ ^ [Uu: produc8027700073345061287rm_f_v] :
          ? [S6: term_f_v,T3: term_f_v] :
            ( ( Uu
              = ( produc2510790566102465375rm_f_v @ S6 @ T3 ) )
            & ( rewrit5493321224924644152dp_f_v @ F @ R @ S6 @ T3 ) ) ) ) ).

% prop_mctxt_cl_UN_redp
thf(fact_923_UN__redp__def,axiom,
    ( rewrit5493321224924644152dp_f_v
    = ( ^ [F2: set_Pr5245412377734449720_f_nat,R4: set_Pr8827868859434726151rm_f_v,S6: term_f_v,T3: term_f_v] :
          ( ( ( member_term_f_v @ S6 @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) )
            & ( member_term_f_v @ T3 @ ( abstract_NF_term_f_v @ ( sig_step_f_v @ F2 @ ( rstep_f_v @ R4 ) ) ) ) )
         => ( S6 = T3 ) ) ) ) ).

% UN_redp_def
thf(fact_924_sig__step__rsteps__eq__dist,axiom,
    ! [R: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R ) @ F )
     => ( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R ) ) ) @ id_term_f_v )
        = ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ).

% sig_step_rsteps_eq_dist
thf(fact_925_NFP__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) )
         => ( rewrit4742200027381058679dp_f_v @ F @ R @ S5 @ T4 ) )
     => ( rewrit1590804194565402411rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v ) ) ).

% NFP_rrstep_intro
thf(fact_926_meetI,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra5602527437878040038rm_f_v @ A3 ) ) ) ) ).

% meetI
thf(fact_927_UNIV__I,axiom,
    ! [X3: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X3 @ top_to5755558563273943127rm_f_v ) ).

% UNIV_I
thf(fact_928_UNIV__I,axiom,
    ! [X3: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X3 @ top_to3664669310755125480_f_nat ) ).

% UNIV_I
thf(fact_929_boolean__algebra_Odisj__one__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ top_to5755558563273943127rm_f_v @ X3 )
      = top_to5755558563273943127rm_f_v ) ).

% boolean_algebra.disj_one_left
thf(fact_930_boolean__algebra_Odisj__one__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ top_to5755558563273943127rm_f_v )
      = top_to5755558563273943127rm_f_v ) ).

% boolean_algebra.disj_one_right
thf(fact_931_sup__top__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ top_to5755558563273943127rm_f_v @ X3 )
      = top_to5755558563273943127rm_f_v ) ).

% sup_top_left
thf(fact_932_sup__top__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ top_to5755558563273943127rm_f_v )
      = top_to5755558563273943127rm_f_v ) ).

% sup_top_right
thf(fact_933_inf__top__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ top_to5755558563273943127rm_f_v @ X3 )
      = X3 ) ).

% inf_top_left
thf(fact_934_inf__top__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ top_to5755558563273943127rm_f_v )
      = X3 ) ).

% inf_top_right
thf(fact_935_inf__eq__top__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 )
        = top_to5755558563273943127rm_f_v )
      = ( ( X3 = top_to5755558563273943127rm_f_v )
        & ( Y2 = top_to5755558563273943127rm_f_v ) ) ) ).

% inf_eq_top_iff
thf(fact_936_top__eq__inf__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( top_to5755558563273943127rm_f_v
        = ( inf_in6135112614888068085rm_f_v @ X3 @ Y2 ) )
      = ( ( X3 = top_to5755558563273943127rm_f_v )
        & ( Y2 = top_to5755558563273943127rm_f_v ) ) ) ).

% top_eq_inf_iff
thf(fact_937_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ A @ B )
        = top_to5755558563273943127rm_f_v )
      = ( ( A = top_to5755558563273943127rm_f_v )
        & ( B = top_to5755558563273943127rm_f_v ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_938_inf__top_Oleft__neutral,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ top_to5755558563273943127rm_f_v @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_939_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( top_to5755558563273943127rm_f_v
        = ( inf_in6135112614888068085rm_f_v @ A @ B ) )
      = ( ( A = top_to5755558563273943127rm_f_v )
        & ( B = top_to5755558563273943127rm_f_v ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_940_inf__top_Oright__neutral,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A @ top_to5755558563273943127rm_f_v )
      = A ) ).

% inf_top.right_neutral
thf(fact_941_Int__UNIV,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = top_to5755558563273943127rm_f_v )
      = ( ( A3 = top_to5755558563273943127rm_f_v )
        & ( B4 = top_to5755558563273943127rm_f_v ) ) ) ).

% Int_UNIV
thf(fact_942_IdI,axiom,
    ! [A: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ id_term_f_v ) ).

% IdI
thf(fact_943_pair__in__Id__conv,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ id_term_f_v )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_944_R__O__Id,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ R2 @ id_term_f_v )
      = R2 ) ).

% R_O_Id
thf(fact_945_Id__O__R,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ id_term_f_v @ R2 )
      = R2 ) ).

% Id_O_R
thf(fact_946_converse__Id,axiom,
    ( ( conver3099152742603637891rm_f_v @ id_term_f_v )
    = id_term_f_v ) ).

% converse_Id
thf(fact_947_UNIV__Times__UNIV,axiom,
    ( ( produc6667182599969081488rm_f_v @ top_top_set_term_f_v
      @ ^ [Uu: term_f_v] : top_top_set_term_f_v )
    = top_to5755558563273943127rm_f_v ) ).

% UNIV_Times_UNIV
thf(fact_948_rtrancl__reflcl,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ id_term_f_v ) )
      = ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% rtrancl_reflcl
thf(fact_949_rtrancl__reflcl__absorb,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R2 ) @ id_term_f_v )
      = ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% rtrancl_reflcl_absorb
thf(fact_950_trancl__reflcl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3208867851621559554rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) )
      = ( transi3028809021613410336rm_f_v @ R3 ) ) ).

% trancl_reflcl
thf(fact_951_IdD,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ id_term_f_v )
     => ( A = B ) ) ).

% IdD
thf(fact_952_subset__UNIV,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A3 @ top_to3664669310755125480_f_nat ) ).

% subset_UNIV
thf(fact_953_insert__UNIV,axiom,
    ! [X3: product_prod_f_nat] :
      ( ( insert5008453031520946354_f_nat @ X3 @ top_to3664669310755125480_f_nat )
      = top_to3664669310755125480_f_nat ) ).

% insert_UNIV
thf(fact_954_Un__UNIV__left,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ top_to5755558563273943127rm_f_v @ B4 )
      = top_to5755558563273943127rm_f_v ) ).

% Un_UNIV_left
thf(fact_955_Un__UNIV__right,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ top_to5755558563273943127rm_f_v )
      = top_to5755558563273943127rm_f_v ) ).

% Un_UNIV_right
thf(fact_956_UNIV__def,axiom,
    ( top_top_set_term_f_v
    = ( collect_term_f_v
      @ ^ [X: term_f_v] : $true ) ) ).

% UNIV_def
thf(fact_957_UNIV__eq__I,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X2 @ A3 )
     => ( top_to5755558563273943127rm_f_v = A3 ) ) ).

% UNIV_eq_I
thf(fact_958_UNIV__eq__I,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X2 @ A3 )
     => ( top_to3664669310755125480_f_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_959_UNIV__witness,axiom,
    ? [X2: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X2 @ top_to5755558563273943127rm_f_v ) ).

% UNIV_witness
thf(fact_960_UNIV__witness,axiom,
    ? [X2: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X2 @ top_to3664669310755125480_f_nat ) ).

% UNIV_witness
thf(fact_961_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ top_to3664669310755125480_f_nat @ A )
     => ( A = top_to3664669310755125480_f_nat ) ) ).

% top.extremum_uniqueI
thf(fact_962_top_Oextremum__unique,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ top_to3664669310755125480_f_nat @ A )
      = ( A = top_to3664669310755125480_f_nat ) ) ).

% top.extremum_unique
thf(fact_963_top__greatest,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ top_to3664669310755125480_f_nat ) ).

% top_greatest
thf(fact_964_Int__UNIV__left,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ top_to5755558563273943127rm_f_v @ B4 )
      = B4 ) ).

% Int_UNIV_left
thf(fact_965_Int__UNIV__right,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ top_to5755558563273943127rm_f_v )
      = A3 ) ).

% Int_UNIV_right
thf(fact_966_IdE,axiom,
    ! [P: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P @ id_term_f_v )
     => ~ ! [X2: term_f_v] :
            ( P
           != ( produc2510790566102465375rm_f_v @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_967_boolean__algebra_Oconj__one__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ top_to5755558563273943127rm_f_v )
      = X3 ) ).

% boolean_algebra.conj_one_right
thf(fact_968_NFP__stepD,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( rewrit585724738297129951rm_f_v @ R3 @ top_to5755558563273943127rm_f_v )
     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ C @ ( abstra349717802288577587rm_f_v @ R3 ) )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ) ) ).

% NFP_stepD
thf(fact_969_NFP__stepD,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( rewrit8036794773242160714_f_nat @ R3 @ top_to3664669310755125480_f_nat )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi8903367562120364223_f_nat @ R3 ) )
         => ( ( member2678038196313615385_f_nat @ C @ ( abstra8862684023087959798_f_nat @ R3 ) )
           => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( transi8903367562120364223_f_nat @ R3 ) ) ) ) ) ) ).

% NFP_stepD
thf(fact_970_NFP__stepD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( rewrit1590804194565402411rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( ( member_term_f_v @ C @ ( abstract_NF_term_f_v @ R3 ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ) ) ).

% NFP_stepD
thf(fact_971_Id__def,axiom,
    ( id_term_f_v
    = ( collec7769483783292507282rm_f_v
      @ ^ [P3: produc8027700073345061287rm_f_v] :
        ? [X: term_f_v] :
          ( P3
          = ( produc2510790566102465375rm_f_v @ X @ X ) ) ) ) ).

% Id_def
thf(fact_972_const__replace__closed__symcl,axiom,
    ! [C: f,U4: set_Pr8827868859434726151rm_f_v] :
      ( ( replac3904333054701138457ed_f_v @ C @ U4 )
     => ( replac3904333054701138457ed_f_v @ C @ ( sup_su1994854688125285083rm_f_v @ U4 @ id_term_f_v ) ) ) ).

% const_replace_closed_symcl
thf(fact_973_rtrancl__unfold,axiom,
    ( transi3028809021613410336rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ id_term_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R6 ) @ R6 ) ) ) ) ).

% rtrancl_unfold
thf(fact_974_rtrancl__trancl__reflcl,axiom,
    ( transi3028809021613410336rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ ( transi3208867851621559554rm_f_v @ R6 ) @ id_term_f_v ) ) ) ).

% rtrancl_trancl_reflcl
thf(fact_975_trancl__o__refl__is__trancl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) )
      = ( transi3208867851621559554rm_f_v @ R3 ) ) ).

% trancl_o_refl_is_trancl
thf(fact_976_reflcl__set__eq,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su8963822174138643348_f_v_o
        @ ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R3 )
        @ ^ [Y4: term_f_v,Z3: term_f_v] : ( Y4 = Z3 ) )
      = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) ) ) ) ).

% reflcl_set_eq
thf(fact_977_NFP__lift__to__conversion,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,S: produc8027700073345061287rm_f_v,T: produc8027700073345061287rm_f_v] :
      ( ( rewrit585724738297129951rm_f_v @ R3 @ top_to5755558563273943127rm_f_v )
     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ S @ T ) @ ( transi1062743709496199978rm_f_v @ ( sup_su6770080740296300187rm_f_v @ R3 @ ( conver3687159706446585467rm_f_v @ R3 ) ) ) )
       => ( ( member4316649861536941904rm_f_v @ T @ ( abstra349717802288577587rm_f_v @ R3 ) )
         => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ S @ T ) @ ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ) ).

% NFP_lift_to_conversion
thf(fact_978_NFP__lift__to__conversion,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,S: product_prod_f_nat,T: product_prod_f_nat] :
      ( ( rewrit8036794773242160714_f_nat @ R3 @ top_to3664669310755125480_f_nat )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ S @ T ) @ ( transi8903367562120364223_f_nat @ ( sup_su8886893976626623897_f_nat @ R3 @ ( conver7859330309926320577_f_nat @ R3 ) ) ) )
       => ( ( member2678038196313615385_f_nat @ T @ ( abstra8862684023087959798_f_nat @ R3 ) )
         => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ S @ T ) @ ( transi8903367562120364223_f_nat @ R3 ) ) ) ) ) ).

% NFP_lift_to_conversion
thf(fact_979_NFP__lift__to__conversion,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit1590804194565402411rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ ( conver3099152742603637891rm_f_v @ R3 ) ) ) )
       => ( ( member_term_f_v @ T @ ( abstract_NF_term_f_v @ R3 ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ) ).

% NFP_lift_to_conversion
thf(fact_980_rtrancl__Int__subset,axiom,
    ! [S: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ id_term_f_v @ S )
     => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( inf_in6135112614888068085rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ S ) @ R3 ) @ S )
       => ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ S ) ) ) ).

% rtrancl_Int_subset
thf(fact_981_meetD,axiom,
    ! [B: term_f_v,C: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra5602527437878040038rm_f_v @ A3 ) )
     => ? [A2: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% meetD
thf(fact_982_meetE,axiom,
    ! [B: term_f_v,C: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra5602527437878040038rm_f_v @ A3 ) )
     => ~ ! [A2: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% meetE
thf(fact_983_NFP__to__rrstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit1590804194565402411rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
       => ( rewrit4742200027381058679dp_f_v @ F @ R @ S @ T ) ) ) ).

% NFP_to_rrstep
thf(fact_984_meet__def,axiom,
    ( abstra5602527437878040038rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ A5 ) ) @ ( transi3028809021613410336rm_f_v @ A5 ) ) ) ) ).

% meet_def
thf(fact_985_SCR__on__def,axiom,
    ( rewrit252611673853975797rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v,C2: term_f_v] :
              ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ R6 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ C2 ) @ R6 ) )
             => ? [D2: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ D2 ) @ ( sup_su1994854688125285083rm_f_v @ R6 @ id_term_f_v ) )
                  & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C2 @ D2 ) @ ( transi3028809021613410336rm_f_v @ R6 ) ) ) ) ) ) ) ).

% SCR_on_def
thf(fact_986_SCR__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v,U3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ U3 ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
           => ? [V3: term_f_v] :
                ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ V3 ) @ ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ id_term_f_v ) )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U3 @ V3 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) )
     => ( ! [S5: term_f_v,T4: term_f_v,U3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ U3 ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
             => ? [V3: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ V3 ) @ ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ id_term_f_v ) )
                  & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U3 @ V3 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) )
       => ( rewrit252611673853975797rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v ) ) ) ).

% SCR_rrstep_intro
thf(fact_987_SCE__to__rrstep_I1_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,U2: term_f_v] :
      ( ( rewrit252611673853975797rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U2 ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
         => ? [V4: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ V4 ) @ ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ id_term_f_v ) )
              & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U2 @ V4 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% SCE_to_rrstep(1)
thf(fact_988_converse__UNIV,axiom,
    ( ( conver3099152742603637891rm_f_v @ top_to5755558563273943127rm_f_v )
    = top_to5755558563273943127rm_f_v ) ).

% converse_UNIV
thf(fact_989_top__empty__eq,axiom,
    ( top_to8437117464082123078_f_v_o
    = ( ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ top_to5755558563273943127rm_f_v ) ) ) ).

% top_empty_eq
thf(fact_990_top__empty__eq,axiom,
    ( top_to6006982329135709597_nat_o
    = ( ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ top_to3664669310755125480_f_nat ) ) ) ).

% top_empty_eq
thf(fact_991_top__set__def,axiom,
    ( top_top_set_term_f_v
    = ( collect_term_f_v @ top_top_term_f_v_o ) ) ).

% top_set_def
thf(fact_992_top__empty__eq2,axiom,
    ( top_to7863253239933653264_f_v_o
    = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ top_to5755558563273943127rm_f_v ) ) ) ).

% top_empty_eq2
thf(fact_993_top__empty__eq2,axiom,
    ( top_top_f_nat_o
    = ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ top_to3664669310755125480_f_nat ) ) ) ).

% top_empty_eq2
thf(fact_994_SCE__to__rrstep_I2_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,U2: term_f_v] :
      ( ( rewrit252611673853975797rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U2 ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
         => ? [V4: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ V4 ) @ ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ id_term_f_v ) )
              & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U2 @ V4 ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ) ) ).

% SCE_to_rrstep(2)
thf(fact_995_strongly__confluentI,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: term_f_v,Y: term_f_v,Z: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y ) @ R3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Z ) @ R3 )
           => ? [U5: term_f_v] :
                ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ U5 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ U5 ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) ) ) ) )
     => ( abstra950190854684145090rm_f_v @ R3 @ top_top_set_term_f_v ) ) ).

% strongly_confluentI
thf(fact_996_UNF__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [T4: term_f_v,U3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ U3 ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) )
         => ( rewrit5493321224924644152dp_f_v @ F @ R @ T4 @ U3 ) )
     => ( abstra8129321985590711349rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v ) ) ).

% UNF_rrstep_intro
thf(fact_997_UNF__to__rrstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( abstra8129321985590711349rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) )
       => ( rewrit5493321224924644152dp_f_v @ F @ R @ S @ T ) ) ) ).

% UNF_to_rrstep
thf(fact_998_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v,Z2: produc8027700073345061287rm_f_v] :
      ( ( abstra8507947352017634312rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ R3 )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ R3 )
           => ? [U3: produc8027700073345061287rm_f_v] :
                ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y2 @ U3 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
                & ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z2 @ U3 ) @ ( sup_su6770080740296300187rm_f_v @ R3 @ id_Pro1935643801653256451rm_f_v ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_999_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat,Y2: product_prod_f_nat,Z2: product_prod_f_nat] :
      ( ( abstra2269792502866851169_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ X3 @ A3 )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ R3 )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ R3 )
           => ? [U3: product_prod_f_nat] :
                ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y2 @ U3 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
                & ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Z2 @ U3 ) @ ( sup_su8886893976626623897_f_nat @ R3 @ id_Pro2160904946306395686_f_nat ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_1000_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,X3: term_f_v,Y2: term_f_v,Z2: term_f_v] :
      ( ( abstra950190854684145090rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ X3 @ A3 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ R3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R3 )
           => ? [U3: term_f_v] :
                ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ U3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ U3 ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_1001_strongly__confluent__on__def,axiom,
    ( abstra950190854684145090rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [Y3: term_f_v,Z4: term_f_v] :
              ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R6 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z4 ) @ R6 ) )
             => ? [U: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ U ) @ ( transi3028809021613410336rm_f_v @ R6 ) )
                  & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z4 @ U ) @ ( sup_su1994854688125285083rm_f_v @ R6 @ id_term_f_v ) ) ) ) ) ) ) ).

% strongly_confluent_on_def
thf(fact_1002_CE__to__rrstep_I2_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit5169762006700492917rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ S4 @ ( conver3099152742603637891rm_f_v @ S4 ) ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) ) ).

% CE_to_rrstep(2)
thf(fact_1003_CE__to__rrstep_I1_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit5169762006700492917rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) ) ) ).

% CE_to_rrstep(1)
thf(fact_1004_CE__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R @ ( conver3099152742603637891rm_f_v @ R ) ) ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) ) )
     => ( ! [S5: term_f_v,T4: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ S4 @ ( conver3099152742603637891rm_f_v @ S4 ) ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) )
       => ( rewrit5169762006700492917rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v ) ) ) ).

% CE_rrstep_intro
thf(fact_1005_CE__on__def,axiom,
    ( rewrit5169762006700492917rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( abstra5640035136655339381rm_f_v @ R6 ) )
              = ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( abstra5640035136655339381rm_f_v @ S6 ) ) ) ) ) ) ).

% CE_on_def
thf(fact_1006_NE__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
         => ( rewrit7960950296172055926dp_f_v @ F @ R @ S4 @ S5 @ T4 ) )
     => ( ! [S5: term_f_v,T4: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) )
           => ( rewrit7960950296172055926dp_f_v @ F @ S4 @ R @ S5 @ T4 ) )
       => ( ( ( abstract_NF_term_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
            = ( abstract_NF_term_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) ) )
         => ( rewrit2656876241799709610rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v ) ) ) ) ).

% NE_rrstep_intro
thf(fact_1007_NE__to__rrstep_I1_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit2656876241799709610rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) )
       => ( rewrit7960950296172055926dp_f_v @ F @ R @ S4 @ S @ T ) ) ) ).

% NE_to_rrstep(1)
thf(fact_1008_NE__to__rrstep_I2_J,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( rewrit2656876241799709610rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ S4 ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( srstep3130564645414158319ep_f_v @ F @ S4 ) )
       => ( rewrit7960950296172055926dp_f_v @ F @ S4 @ R @ S @ T ) ) ) ).

% NE_to_rrstep(2)
thf(fact_1009_some__NF__UNF,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v] :
      ( ( abstra8974104706389925717rm_f_v @ R3 @ top_to5755558563273943127rm_f_v )
     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ Y2 @ ( abstra349717802288577587rm_f_v @ R3 ) )
         => ( Y2
            = ( abstra6650587300220611071rm_f_v @ R3 @ X3 ) ) ) ) ) ).

% some_NF_UNF
thf(fact_1010_some__NF__UNF,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat,Y2: product_prod_f_nat] :
      ( ( abstra4786933130458578516_f_nat @ R3 @ top_to3664669310755125480_f_nat )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
       => ( ( member2678038196313615385_f_nat @ Y2 @ ( abstra8862684023087959798_f_nat @ R3 ) )
         => ( Y2
            = ( abstra2867534111362060330_f_nat @ R3 @ X3 ) ) ) ) ) ).

% some_NF_UNF
thf(fact_1011_some__NF__UNF,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ( abstra8129321985590711349rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( ( member_term_f_v @ Y2 @ ( abstract_NF_term_f_v @ R3 ) )
         => ( Y2
            = ( abstra7098836743531038987rm_f_v @ R3 @ X3 ) ) ) ) ) ).

% some_NF_UNF
thf(fact_1012_CR__to__rrstep,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ ( conver3099152742603637891rm_f_v @ R ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) ) )
       => ( rewrit4152091719787942777dp_f_v @ F @ R @ R @ S @ T ) ) ) ).

% CR_to_rrstep
thf(fact_1013_CR__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) @ ( srstep3130564645414158319ep_f_v @ F @ R ) ) )
         => ( rewrit4152091719787942777dp_f_v @ F @ R @ R @ S5 @ T4 ) )
     => ( abstra3735013631728912813rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v ) ) ).

% CR_rrstep_intro
thf(fact_1014_diamond__imp__CR_H,axiom,
    ! [S: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra3437181289681318589rm_f_v @ S )
     => ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
       => ( ( ord_le4559761987009501863rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v ) ) ) ) ).

% diamond_imp_CR'
thf(fact_1015_image2__def,axiom,
    ( bNF_Gr1346902064820140821rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,F4: produc8027700073345061287rm_f_v > term_f_v,G3: produc8027700073345061287rm_f_v > term_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [Uu: produc8027700073345061287rm_f_v] :
            ? [A4: produc8027700073345061287rm_f_v] :
              ( ( Uu
                = ( produc2510790566102465375rm_f_v @ ( F4 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member4316649861536941904rm_f_v @ A4 @ A5 ) ) ) ) ) ).

% image2_def
thf(fact_1016_image2__def,axiom,
    ( bNF_Gr6519777854805764516rm_f_v
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,F4: product_prod_f_nat > term_f_v,G3: product_prod_f_nat > term_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [Uu: produc8027700073345061287rm_f_v] :
            ? [A4: product_prod_f_nat] :
              ( ( Uu
                = ( produc2510790566102465375rm_f_v @ ( F4 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member2678038196313615385_f_nat @ A4 @ A5 ) ) ) ) ) ).

% image2_def
thf(fact_1017_image2__def,axiom,
    ( bNF_Gr8759763996313099078_f_nat
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v,F4: produc8027700073345061287rm_f_v > f,G3: produc8027700073345061287rm_f_v > nat] :
          ( collec1417984143148964183_f_nat
          @ ^ [Uu: product_prod_f_nat] :
            ? [A4: produc8027700073345061287rm_f_v] :
              ( ( Uu
                = ( product_Pair_f_nat @ ( F4 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member4316649861536941904rm_f_v @ A4 @ A5 ) ) ) ) ) ).

% image2_def
thf(fact_1018_image2__def,axiom,
    ( bNF_Gr5097439811561032567_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,F4: product_prod_f_nat > f,G3: product_prod_f_nat > nat] :
          ( collec1417984143148964183_f_nat
          @ ^ [Uu: product_prod_f_nat] :
            ? [A4: product_prod_f_nat] :
              ( ( Uu
                = ( product_Pair_f_nat @ ( F4 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member2678038196313615385_f_nat @ A4 @ A5 ) ) ) ) ) ).

% image2_def
thf(fact_1019_semi__confluence__imp__CR,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
     => ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v ) ) ).

% semi_confluence_imp_CR
thf(fact_1020_joinI,axiom,
    ! [A: term_f_v,C: term_f_v,A3: set_Pr8827868859434726151rm_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ) ).

% joinI
thf(fact_1021_joinI__left,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ).

% joinI_left
thf(fact_1022_joinI__right,axiom,
    ! [B: term_f_v,A: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ).

% joinI_right
thf(fact_1023_CR__onI,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ! [A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ A2 @ A3 )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ B2 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
           => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ C3 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
             => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B2 @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R3 ) ) ) ) )
     => ( abstra5806628924625096669rm_f_v @ R3 @ A3 ) ) ).

% CR_onI
thf(fact_1024_CR__onI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ! [A2: product_prod_f_nat,B2: product_prod_f_nat,C3: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ A2 @ A3 )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ B2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
           => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ C3 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
             => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B2 @ C3 ) @ ( abstra7058733697175803464_f_nat @ R3 ) ) ) ) )
     => ( abstra4917929278449670092_f_nat @ R3 @ A3 ) ) ).

% CR_onI
thf(fact_1025_CR__onI,axiom,
    ! [A3: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [A2: term_f_v,B2: term_f_v,C3: term_f_v] :
          ( ( member_term_f_v @ A2 @ A3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ C3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
             => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) )
     => ( abstra3735013631728912813rm_f_v @ R3 @ A3 ) ) ).

% CR_onI
thf(fact_1026_join__sym,axiom,
    ! [S: term_f_v,T: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ).

% join_sym
thf(fact_1027_joinD,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
     => ? [C3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% joinD
thf(fact_1028_joinE,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
     => ~ ! [C3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% joinE
thf(fact_1029_join__rtrancl__join,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ) ).

% join_rtrancl_join
thf(fact_1030_rtrancl__join__join,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ) ).

% rtrancl_join_join
thf(fact_1031_join__NF__imp__eq,axiom,
    ! [X3: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( abstra3336617290665100897rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ X3 @ ( abstra349717802288577587rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ Y2 @ ( abstra349717802288577587rm_f_v @ R3 ) )
         => ( X3 = Y2 ) ) ) ) ).

% join_NF_imp_eq
thf(fact_1032_join__NF__imp__eq,axiom,
    ! [X3: product_prod_f_nat,Y2: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( abstra7058733697175803464_f_nat @ R3 ) )
     => ( ( member2678038196313615385_f_nat @ X3 @ ( abstra8862684023087959798_f_nat @ R3 ) )
       => ( ( member2678038196313615385_f_nat @ Y2 @ ( abstra8862684023087959798_f_nat @ R3 ) )
         => ( X3 = Y2 ) ) ) ) ).

% join_NF_imp_eq
thf(fact_1033_join__NF__imp__eq,axiom,
    ! [X3: term_f_v,Y2: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
     => ( ( member_term_f_v @ X3 @ ( abstract_NF_term_f_v @ R3 ) )
       => ( ( member_term_f_v @ Y2 @ ( abstract_NF_term_f_v @ R3 ) )
         => ( X3 = Y2 ) ) ) ) ).

% join_NF_imp_eq
thf(fact_1034_image2__eqI,axiom,
    ! [B: term_f_v,F3: produc8027700073345061287rm_f_v > term_f_v,X3: produc8027700073345061287rm_f_v,C: term_f_v,G2: produc8027700073345061287rm_f_v > term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( B
        = ( F3 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( bNF_Gr1346902064820140821rm_f_v @ A3 @ F3 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_1035_image2__eqI,axiom,
    ! [B: term_f_v,F3: product_prod_f_nat > term_f_v,X3: product_prod_f_nat,C: term_f_v,G2: product_prod_f_nat > term_f_v,A3: set_Pr5245412377734449720_f_nat] :
      ( ( B
        = ( F3 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member2678038196313615385_f_nat @ X3 @ A3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( bNF_Gr6519777854805764516rm_f_v @ A3 @ F3 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_1036_image2__eqI,axiom,
    ! [B: f,F3: produc8027700073345061287rm_f_v > f,X3: produc8027700073345061287rm_f_v,C: nat,G2: produc8027700073345061287rm_f_v > nat,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( B
        = ( F3 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member4316649861536941904rm_f_v @ X3 @ A3 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ C ) @ ( bNF_Gr8759763996313099078_f_nat @ A3 @ F3 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_1037_image2__eqI,axiom,
    ! [B: f,F3: product_prod_f_nat > f,X3: product_prod_f_nat,C: nat,G2: product_prod_f_nat > nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( B
        = ( F3 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member2678038196313615385_f_nat @ X3 @ A3 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ C ) @ ( bNF_Gr5097439811561032567_f_nat @ A3 @ F3 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_1038_NF__join__imp__reach,axiom,
    ! [Y2: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Y2 @ ( abstra349717802288577587rm_f_v @ A3 ) )
     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( abstra3336617290665100897rm_f_v @ A3 ) )
       => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( transi1062743709496199978rm_f_v @ A3 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_1039_NF__join__imp__reach,axiom,
    ! [Y2: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat] :
      ( ( member2678038196313615385_f_nat @ Y2 @ ( abstra8862684023087959798_f_nat @ A3 ) )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( abstra7058733697175803464_f_nat @ A3 ) )
       => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( transi8903367562120364223_f_nat @ A3 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_1040_NF__join__imp__reach,axiom,
    ! [Y2: term_f_v,A3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( member_term_f_v @ Y2 @ ( abstract_NF_term_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( abstra8605546704478008873rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_1041_join__def,axiom,
    ( abstra8605546704478008873rm_f_v
    = ( ^ [A5: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A5 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ A5 ) ) ) ) ) ).

% join_def
thf(fact_1042_CR__onD,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra5806628924625096669rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( abstra3336617290665100897rm_f_v @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_1043_CR__onD,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra4917929278449670092_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi8903367562120364223_f_nat @ R3 ) )
           => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( abstra7058733697175803464_f_nat @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_1044_CR__onD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_1045_CR__onE,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra5806628924625096669rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( abstra3336617290665100897rm_f_v @ R3 ) )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
           => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_1046_CR__onE,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra4917929278449670092_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( abstra7058733697175803464_f_nat @ R3 ) )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
           => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi8903367562120364223_f_nat @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_1047_CR__onE,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_1048_CR__on__def,axiom,
    ( abstra3735013631728912813rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v,C2: term_f_v] :
              ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( transi3028809021613410336rm_f_v @ R6 ) )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ C2 ) @ ( transi3028809021613410336rm_f_v @ R6 ) ) )
             => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C2 ) @ ( abstra8605546704478008873rm_f_v @ R6 ) ) ) ) ) ) ).

% CR_on_def
thf(fact_1049_CR__divergence__imp__join,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v,Z2: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ).

% CR_divergence_imp_join
thf(fact_1050_partially__localize__CR,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v )
      = ( ! [X: term_f_v,Y3: term_f_v,Z4: term_f_v] :
            ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R3 )
              & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z4 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z4 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ).

% partially_localize_CR
thf(fact_1051_CR__join__right__I,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v,Z2: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ).

% CR_join_right_I
thf(fact_1052_CR__join__left__I,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v,Z2: term_f_v] :
      ( ( abstra3735013631728912813rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y2 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ).

% CR_join_left_I
thf(fact_1053_diamond__imp__semi__confluence,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra3437181289681318589rm_f_v @ R3 )
     => ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ).

% diamond_imp_semi_confluence
thf(fact_1054_WCR__rrstep__intro,axiom,
    ! [F: set_Pr5245412377734449720_f_nat,R: set_Pr8827868859434726151rm_f_v] :
      ( ! [S5: term_f_v,T4: term_f_v,U3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ T4 ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R ) ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S5 @ U3 ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ U3 ) @ ( abstra8605546704478008873rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) ) ) ) )
     => ( abstra312943320106832654rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R ) ) @ top_top_set_term_f_v ) ) ).

% WCR_rrstep_intro
thf(fact_1055_WCR__alt__def,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra312943320106832654rm_f_v @ A3 @ top_top_set_term_f_v )
      = ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ A3 ) @ A3 ) @ ( abstra8605546704478008873rm_f_v @ A3 ) ) ) ).

% WCR_alt_def
thf(fact_1056_CR__on__singletonI,axiom,
    ! [A: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ! [B2: product_prod_f_nat,C3: product_prod_f_nat] :
          ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
           => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B2 @ C3 ) @ ( abstra7058733697175803464_f_nat @ R3 ) ) ) )
     => ( abstra4917929278449670092_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ).

% CR_on_singletonI
thf(fact_1057_CR__on__singletonI,axiom,
    ! [A: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [B2: term_f_v,C3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) )
     => ( abstra3735013631728912813rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ).

% CR_on_singletonI
thf(fact_1058_empty__Collect__eq,axiom,
    ! [P2: term_f_v > $o] :
      ( ( bot_bot_set_term_f_v
        = ( collect_term_f_v @ P2 ) )
      = ( ! [X: term_f_v] :
            ~ ( P2 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1059_Collect__empty__eq,axiom,
    ! [P2: term_f_v > $o] :
      ( ( ( collect_term_f_v @ P2 )
        = bot_bot_set_term_f_v )
      = ( ! [X: term_f_v] :
            ~ ( P2 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1060_all__not__in__conv,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ! [X: produc8027700073345061287rm_f_v] :
            ~ ( member4316649861536941904rm_f_v @ X @ A3 ) )
      = ( A3 = bot_bo4638812505780724595rm_f_v ) ) ).

% all_not_in_conv
thf(fact_1061_all__not__in__conv,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( ! [X: product_prod_f_nat] :
            ~ ( member2678038196313615385_f_nat @ X @ A3 ) )
      = ( A3 = bot_bo136713897782694092_f_nat ) ) ).

% all_not_in_conv
thf(fact_1062_empty__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v] :
      ~ ( member4316649861536941904rm_f_v @ C @ bot_bo4638812505780724595rm_f_v ) ).

% empty_iff
thf(fact_1063_empty__iff,axiom,
    ! [C: product_prod_f_nat] :
      ~ ( member2678038196313615385_f_nat @ C @ bot_bo136713897782694092_f_nat ) ).

% empty_iff
thf(fact_1064_relcomp__empty1,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ bot_bo4638812505780724595rm_f_v @ R2 )
      = bot_bo4638812505780724595rm_f_v ) ).

% relcomp_empty1
thf(fact_1065_relcomp__empty2,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( relcom2934258169684288428rm_f_v @ R2 @ bot_bo4638812505780724595rm_f_v )
      = bot_bo4638812505780724595rm_f_v ) ).

% relcomp_empty2
thf(fact_1066_converse__empty,axiom,
    ( ( conver3099152742603637891rm_f_v @ bot_bo4638812505780724595rm_f_v )
    = bot_bo4638812505780724595rm_f_v ) ).

% converse_empty
thf(fact_1067_trancl__empty,axiom,
    ( ( transi3208867851621559554rm_f_v @ bot_bo4638812505780724595rm_f_v )
    = bot_bo4638812505780724595rm_f_v ) ).

% trancl_empty
thf(fact_1068_empty__subsetI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A3 ) ).

% empty_subsetI
thf(fact_1069_subset__empty,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ bot_bo136713897782694092_f_nat )
      = ( A3 = bot_bo136713897782694092_f_nat ) ) ).

% subset_empty
thf(fact_1070_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A @ bot_bo4638812505780724595rm_f_v )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1071_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( bot_bo4638812505780724595rm_f_v
        = ( sup_su1994854688125285083rm_f_v @ A @ B ) )
      = ( ( A = bot_bo4638812505780724595rm_f_v )
        & ( B = bot_bo4638812505780724595rm_f_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1072_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ bot_bo4638812505780724595rm_f_v @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1073_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v] :
      ( ( ( sup_su1994854688125285083rm_f_v @ A @ B )
        = bot_bo4638812505780724595rm_f_v )
      = ( ( A = bot_bo4638812505780724595rm_f_v )
        & ( B = bot_bo4638812505780724595rm_f_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1074_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ( X3 = bot_bo4638812505780724595rm_f_v )
        & ( Y2 = bot_bo4638812505780724595rm_f_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_1075_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( bot_bo4638812505780724595rm_f_v
        = ( sup_su1994854688125285083rm_f_v @ X3 @ Y2 ) )
      = ( ( X3 = bot_bo4638812505780724595rm_f_v )
        & ( Y2 = bot_bo4638812505780724595rm_f_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_1076_sup__bot__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v )
      = X3 ) ).

% sup_bot_right
thf(fact_1077_sup__bot__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ bot_bo4638812505780724595rm_f_v @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1078_inf__bot__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v )
      = bot_bo4638812505780724595rm_f_v ) ).

% inf_bot_right
thf(fact_1079_inf__bot__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ bot_bo4638812505780724595rm_f_v @ X3 )
      = bot_bo4638812505780724595rm_f_v ) ).

% inf_bot_left
thf(fact_1080_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v )
      = bot_bo4638812505780724595rm_f_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_1081_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ bot_bo4638812505780724595rm_f_v @ X3 )
      = bot_bo4638812505780724595rm_f_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_1082_singletonI,axiom,
    ! [A: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ) ).

% singletonI
thf(fact_1083_singletonI,axiom,
    ! [A: product_prod_f_nat] : ( member2678038196313615385_f_nat @ A @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ).

% singletonI
thf(fact_1084_Un__empty,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( sup_su1994854688125285083rm_f_v @ A3 @ B4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ( A3 = bot_bo4638812505780724595rm_f_v )
        & ( B4 = bot_bo4638812505780724595rm_f_v ) ) ) ).

% Un_empty
thf(fact_1085_rtrancl__empty,axiom,
    ( ( transi3028809021613410336rm_f_v @ bot_bo4638812505780724595rm_f_v )
    = id_term_f_v ) ).

% rtrancl_empty
thf(fact_1086_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_term_f_v
            @ ^ [S6: term_f_v] : P2 )
          = top_top_set_term_f_v ) )
      & ( ~ P2
       => ( ( collect_term_f_v
            @ ^ [S6: term_f_v] : P2 )
          = bot_bot_set_term_f_v ) ) ) ).

% Collect_const
thf(fact_1087_singleton__conv,axiom,
    ! [A: product_prod_f_nat] :
      ( ( collec1417984143148964183_f_nat
        @ ^ [X: product_prod_f_nat] : ( X = A ) )
      = ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ).

% singleton_conv
thf(fact_1088_singleton__conv,axiom,
    ! [A: term_f_v] :
      ( ( collect_term_f_v
        @ ^ [X: term_f_v] : ( X = A ) )
      = ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ).

% singleton_conv
thf(fact_1089_singleton__conv2,axiom,
    ! [A: product_prod_f_nat] :
      ( ( collec1417984143148964183_f_nat
        @ ( ^ [Y4: product_prod_f_nat,Z3: product_prod_f_nat] : ( Y4 = Z3 )
          @ A ) )
      = ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ).

% singleton_conv2
thf(fact_1090_singleton__conv2,axiom,
    ! [A: term_f_v] :
      ( ( collect_term_f_v
        @ ( ^ [Y4: term_f_v,Z3: term_f_v] : ( Y4 = Z3 )
          @ A ) )
      = ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ).

% singleton_conv2
thf(fact_1091_Times__empty,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v] :
      ( ( ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : B4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ( A3 = bot_bot_set_term_f_v )
        | ( B4 = bot_bot_set_term_f_v ) ) ) ).

% Times_empty
thf(fact_1092_Sigma__empty2,axiom,
    ! [A3: set_term_f_v] :
      ( ( produc6667182599969081488rm_f_v @ A3
        @ ^ [Uu: term_f_v] : bot_bot_set_term_f_v )
      = bot_bo4638812505780724595rm_f_v ) ).

% Sigma_empty2
thf(fact_1093_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_f_nat,A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat )
        = ( insert5008453031520946354_f_nat @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_le8976984241387448984_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1094_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
      ( ( ( insert5008453031520946354_f_nat @ A @ A3 )
        = ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) )
      = ( ( A = B )
        & ( ord_le8976984241387448984_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1095_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ A3 ) @ B4 )
        = bot_bo136713897782694092_f_nat )
      = ( ~ ( member2678038196313615385_f_nat @ A @ B4 )
        & ( ( inf_in4124010953442585290_f_nat @ A3 @ B4 )
          = bot_bo136713897782694092_f_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1096_insert__disjoint_I1_J,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ A3 ) @ B4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ~ ( member4316649861536941904rm_f_v @ A @ B4 )
        & ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
          = bot_bo4638812505780724595rm_f_v ) ) ) ).

% insert_disjoint(1)
thf(fact_1097_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( bot_bo136713897782694092_f_nat
        = ( inf_in4124010953442585290_f_nat @ ( insert5008453031520946354_f_nat @ A @ A3 ) @ B4 ) )
      = ( ~ ( member2678038196313615385_f_nat @ A @ B4 )
        & ( bot_bo136713897782694092_f_nat
          = ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1098_insert__disjoint_I2_J,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( bot_bo4638812505780724595rm_f_v
        = ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ A3 ) @ B4 ) )
      = ( ~ ( member4316649861536941904rm_f_v @ A @ B4 )
        & ( bot_bo4638812505780724595rm_f_v
          = ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1099_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( ( inf_in4124010953442585290_f_nat @ B4 @ ( insert5008453031520946354_f_nat @ A @ A3 ) )
        = bot_bo136713897782694092_f_nat )
      = ( ~ ( member2678038196313615385_f_nat @ A @ B4 )
        & ( ( inf_in4124010953442585290_f_nat @ B4 @ A3 )
          = bot_bo136713897782694092_f_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1100_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ B4 @ ( insert5456475782666790775rm_f_v @ A @ A3 ) )
        = bot_bo4638812505780724595rm_f_v )
      = ( ~ ( member4316649861536941904rm_f_v @ A @ B4 )
        & ( ( inf_in6135112614888068085rm_f_v @ B4 @ A3 )
          = bot_bo4638812505780724595rm_f_v ) ) ) ).

% disjoint_insert(1)
thf(fact_1101_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( bot_bo136713897782694092_f_nat
        = ( inf_in4124010953442585290_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ B @ B4 ) ) )
      = ( ~ ( member2678038196313615385_f_nat @ B @ A3 )
        & ( bot_bo136713897782694092_f_nat
          = ( inf_in4124010953442585290_f_nat @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1102_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( bot_bo4638812505780724595rm_f_v
        = ( inf_in6135112614888068085rm_f_v @ A3 @ ( insert5456475782666790775rm_f_v @ B @ B4 ) ) )
      = ( ~ ( member4316649861536941904rm_f_v @ B @ A3 )
        & ( bot_bo4638812505780724595rm_f_v
          = ( inf_in6135112614888068085rm_f_v @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1103_WCR__onI,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ! [A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ A2 @ A3 )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ B2 ) @ R3 )
           => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ C3 ) @ R3 )
             => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B2 @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R3 ) ) ) ) )
     => ( abstra8545463541035707836rm_f_v @ R3 @ A3 ) ) ).

% WCR_onI
thf(fact_1104_WCR__onI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ! [A2: product_prod_f_nat,B2: product_prod_f_nat,C3: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ A2 @ A3 )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ B2 ) @ R3 )
           => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ C3 ) @ R3 )
             => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B2 @ C3 ) @ ( abstra7058733697175803464_f_nat @ R3 ) ) ) ) )
     => ( abstra3602624719130115245_f_nat @ R3 @ A3 ) ) ).

% WCR_onI
thf(fact_1105_WCR__onI,axiom,
    ! [A3: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [A2: term_f_v,B2: term_f_v,C3: term_f_v] :
          ( ( member_term_f_v @ A2 @ A3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ R3 )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ C3 ) @ R3 )
             => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B2 @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) )
     => ( abstra312943320106832654rm_f_v @ R3 @ A3 ) ) ).

% WCR_onI
thf(fact_1106_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ! [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ A3 )
           => ! [Y3: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ Y3 @ B4 )
               => ( X != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1107_Int__empty__right,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ A3 @ bot_bo4638812505780724595rm_f_v )
      = bot_bo4638812505780724595rm_f_v ) ).

% Int_empty_right
thf(fact_1108_Int__empty__left,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in6135112614888068085rm_f_v @ bot_bo4638812505780724595rm_f_v @ B4 )
      = bot_bo4638812505780724595rm_f_v ) ).

% Int_empty_left
thf(fact_1109_disjoint__iff,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ( inf_in4124010953442585290_f_nat @ A3 @ B4 )
        = bot_bo136713897782694092_f_nat )
      = ( ! [X: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X @ A3 )
           => ~ ( member2678038196313615385_f_nat @ X @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1110_disjoint__iff,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ! [X: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X @ A3 )
           => ~ ( member4316649861536941904rm_f_v @ X @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1111_Int__emptyI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ X2 @ A3 )
         => ~ ( member2678038196313615385_f_nat @ X2 @ B4 ) )
     => ( ( inf_in4124010953442585290_f_nat @ A3 @ B4 )
        = bot_bo136713897782694092_f_nat ) ) ).

% Int_emptyI
thf(fact_1112_Int__emptyI,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
         => ~ ( member4316649861536941904rm_f_v @ X2 @ B4 ) )
     => ( ( inf_in6135112614888068085rm_f_v @ A3 @ B4 )
        = bot_bo4638812505780724595rm_f_v ) ) ).

% Int_emptyI
thf(fact_1113_Collect__conv__if,axiom,
    ! [P2: product_prod_f_nat > $o,A: product_prod_f_nat] :
      ( ( ( P2 @ A )
       => ( ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( X = A )
                & ( P2 @ X ) ) )
          = ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( X = A )
                & ( P2 @ X ) ) )
          = bot_bo136713897782694092_f_nat ) ) ) ).

% Collect_conv_if
thf(fact_1114_Collect__conv__if,axiom,
    ! [P2: term_f_v > $o,A: term_f_v] :
      ( ( ( P2 @ A )
       => ( ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( X = A )
                & ( P2 @ X ) ) )
          = ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( X = A )
                & ( P2 @ X ) ) )
          = bot_bot_set_term_f_v ) ) ) ).

% Collect_conv_if
thf(fact_1115_Collect__conv__if2,axiom,
    ! [P2: product_prod_f_nat > $o,A: product_prod_f_nat] :
      ( ( ( P2 @ A )
       => ( ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( A = X )
                & ( P2 @ X ) ) )
          = ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( A = X )
                & ( P2 @ X ) ) )
          = bot_bo136713897782694092_f_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1116_Collect__conv__if2,axiom,
    ! [P2: term_f_v > $o,A: term_f_v] :
      ( ( ( P2 @ A )
       => ( ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( A = X )
                & ( P2 @ X ) ) )
          = ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_term_f_v
            @ ^ [X: term_f_v] :
                ( ( A = X )
                & ( P2 @ X ) ) )
          = bot_bot_set_term_f_v ) ) ) ).

% Collect_conv_if2
thf(fact_1117_bot_Oextremum,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A ) ).

% bot.extremum
thf(fact_1118_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1119_bot_Oextremum__unique,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat )
      = ( A = bot_bo136713897782694092_f_nat ) ) ).

% bot.extremum_unique
thf(fact_1120_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1121_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat )
     => ( A = bot_bo136713897782694092_f_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1122_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1123_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_1124_ex__in__conv,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ? [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A3 ) )
      = ( A3 != bot_bo4638812505780724595rm_f_v ) ) ).

% ex_in_conv
thf(fact_1125_ex__in__conv,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( ? [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A3 ) )
      = ( A3 != bot_bo136713897782694092_f_nat ) ) ).

% ex_in_conv
thf(fact_1126_equals0I,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ! [Y: produc8027700073345061287rm_f_v] :
          ~ ( member4316649861536941904rm_f_v @ Y @ A3 )
     => ( A3 = bot_bo4638812505780724595rm_f_v ) ) ).

% equals0I
thf(fact_1127_equals0I,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ! [Y: product_prod_f_nat] :
          ~ ( member2678038196313615385_f_nat @ Y @ A3 )
     => ( A3 = bot_bo136713897782694092_f_nat ) ) ).

% equals0I
thf(fact_1128_equals0D,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( A3 = bot_bo4638812505780724595rm_f_v )
     => ~ ( member4316649861536941904rm_f_v @ A @ A3 ) ) ).

% equals0D
thf(fact_1129_equals0D,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
      ( ( A3 = bot_bo136713897782694092_f_nat )
     => ~ ( member2678038196313615385_f_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_1130_emptyE,axiom,
    ! [A: produc8027700073345061287rm_f_v] :
      ~ ( member4316649861536941904rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ).

% emptyE
thf(fact_1131_emptyE,axiom,
    ! [A: product_prod_f_nat] :
      ~ ( member2678038196313615385_f_nat @ A @ bot_bo136713897782694092_f_nat ) ).

% emptyE
thf(fact_1132_empty__def,axiom,
    ( bot_bot_set_term_f_v
    = ( collect_term_f_v
      @ ^ [X: term_f_v] : $false ) ) ).

% empty_def
thf(fact_1133_Un__empty__right,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ A3 @ bot_bo4638812505780724595rm_f_v )
      = A3 ) ).

% Un_empty_right
thf(fact_1134_Un__empty__left,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su1994854688125285083rm_f_v @ bot_bo4638812505780724595rm_f_v @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1135_singletonD,axiom,
    ! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ B @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1136_singletonD,axiom,
    ! [B: product_prod_f_nat,A: product_prod_f_nat] :
      ( ( member2678038196313615385_f_nat @ B @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1137_singleton__iff,axiom,
    ! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ B @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1138_singleton__iff,axiom,
    ! [B: product_prod_f_nat,A: product_prod_f_nat] :
      ( ( member2678038196313615385_f_nat @ B @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1139_doubleton__eq__iff,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat,D: product_prod_f_nat] :
      ( ( ( insert5008453031520946354_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) )
        = ( insert5008453031520946354_f_nat @ C @ ( insert5008453031520946354_f_nat @ D @ bot_bo136713897782694092_f_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1140_insert__not__empty,axiom,
    ! [A: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat] :
      ( ( insert5008453031520946354_f_nat @ A @ A3 )
     != bot_bo136713897782694092_f_nat ) ).

% insert_not_empty
thf(fact_1141_singleton__inject,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat] :
      ( ( ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat )
        = ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_1142_Sigma__empty__iff,axiom,
    ! [I: set_term_f_v,X4: term_f_v > set_term_f_v] :
      ( ( ( produc6667182599969081488rm_f_v @ I @ X4 )
        = bot_bo4638812505780724595rm_f_v )
      = ( ! [X: term_f_v] :
            ( ( member_term_f_v @ X @ I )
           => ( ( X4 @ X )
              = bot_bot_set_term_f_v ) ) ) ) ).

% Sigma_empty_iff
thf(fact_1143_times__eq__iff,axiom,
    ! [A3: set_term_f_v,B4: set_term_f_v,C4: set_term_f_v,D3: set_term_f_v] :
      ( ( ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : B4 )
        = ( produc6667182599969081488rm_f_v @ C4
          @ ^ [Uu: term_f_v] : D3 ) )
      = ( ( ( A3 = C4 )
          & ( B4 = D3 ) )
        | ( ( ( A3 = bot_bot_set_term_f_v )
            | ( B4 = bot_bot_set_term_f_v ) )
          & ( ( C4 = bot_bot_set_term_f_v )
            | ( D3 = bot_bot_set_term_f_v ) ) ) ) ) ).

% times_eq_iff
thf(fact_1144_subset__singletonD,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ( ( A3 = bot_bo136713897782694092_f_nat )
        | ( A3
          = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% subset_singletonD
thf(fact_1145_subset__singleton__iff,axiom,
    ! [X4: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X4 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
      = ( ( X4 = bot_bo136713897782694092_f_nat )
        | ( X4
          = ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_1146_singleton__Un__iff,axiom,
    ! [X3: product_prod_f_nat,A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
      ( ( ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat )
        = ( sup_su770888850772321124_f_nat @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bo136713897782694092_f_nat )
          & ( B4
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) )
        | ( ( A3
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
          & ( B4 = bot_bo136713897782694092_f_nat ) )
        | ( ( A3
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
          & ( B4
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1147_singleton__Un__iff,axiom,
    ! [X3: produc8027700073345061287rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v )
        = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bo4638812505780724595rm_f_v )
          & ( B4
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) )
        | ( ( A3
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
          & ( B4 = bot_bo4638812505780724595rm_f_v ) )
        | ( ( A3
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
          & ( B4
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1148_Un__singleton__iff,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( ( sup_su770888850772321124_f_nat @ A3 @ B4 )
        = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
      = ( ( ( A3 = bot_bo136713897782694092_f_nat )
          & ( B4
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) )
        | ( ( A3
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
          & ( B4 = bot_bo136713897782694092_f_nat ) )
        | ( ( A3
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
          & ( B4
            = ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1149_Un__singleton__iff,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( ( sup_su1994854688125285083rm_f_v @ A3 @ B4 )
        = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
      = ( ( ( A3 = bot_bo4638812505780724595rm_f_v )
          & ( B4
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) )
        | ( ( A3
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
          & ( B4 = bot_bo4638812505780724595rm_f_v ) )
        | ( ( A3
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
          & ( B4
            = ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1150_insert__is__Un,axiom,
    ( insert5008453031520946354_f_nat
    = ( ^ [A4: product_prod_f_nat] : ( sup_su770888850772321124_f_nat @ ( insert5008453031520946354_f_nat @ A4 @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1151_insert__is__Un,axiom,
    ( insert5456475782666790775rm_f_v
    = ( ^ [A4: produc8027700073345061287rm_f_v] : ( sup_su1994854688125285083rm_f_v @ ( insert5456475782666790775rm_f_v @ A4 @ bot_bo4638812505780724595rm_f_v ) ) ) ) ).

% insert_is_Un
thf(fact_1152_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr8827868859434726151rm_f_v,X3: set_Pr8827868859434726151rm_f_v,Y2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ A @ X3 )
        = bot_bo4638812505780724595rm_f_v )
     => ( ( ( sup_su1994854688125285083rm_f_v @ A @ X3 )
          = top_to5755558563273943127rm_f_v )
       => ( ( ( inf_in6135112614888068085rm_f_v @ A @ Y2 )
            = bot_bo4638812505780724595rm_f_v )
         => ( ( ( sup_su1994854688125285083rm_f_v @ A @ Y2 )
              = top_to5755558563273943127rm_f_v )
           => ( X3 = Y2 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1153_rtrancl__Un2__separatorE,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ( relcom2934258169684288428rm_f_v @ B4 @ A3 )
        = bot_bo4638812505780724595rm_f_v )
     => ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
        = ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ ( transi3028809021613410336rm_f_v @ B4 ) ) ) ) ) ).

% rtrancl_Un2_separatorE
thf(fact_1154_trancl__Un2__separatorE,axiom,
    ! [B4: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( ( relcom2934258169684288428rm_f_v @ B4 @ A3 )
        = bot_bo4638812505780724595rm_f_v )
     => ( ( transi3208867851621559554rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
        = ( sup_su1994854688125285083rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( transi3208867851621559554rm_f_v @ A3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3208867851621559554rm_f_v @ A3 ) @ ( transi3208867851621559554rm_f_v @ B4 ) ) ) @ ( transi3208867851621559554rm_f_v @ B4 ) ) ) ) ).

% trancl_Un2_separatorE
thf(fact_1155_times__subset__iff,axiom,
    ! [A3: set_term_f_v,C4: set_term_f_v,B4: set_term_f_v,D3: set_term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu: term_f_v] : C4 )
        @ ( produc6667182599969081488rm_f_v @ B4
          @ ^ [Uu: term_f_v] : D3 ) )
      = ( ( A3 = bot_bot_set_term_f_v )
        | ( C4 = bot_bot_set_term_f_v )
        | ( ( ord_le947722214152454265rm_f_v @ A3 @ B4 )
          & ( ord_le947722214152454265rm_f_v @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_1156_times__subset__iff,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,C4: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,D3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8946695678602162533_f_nat
        @ ( produc7667213020494768334_f_nat @ A3
          @ ^ [Uu: product_prod_f_nat] : C4 )
        @ ( produc7667213020494768334_f_nat @ B4
          @ ^ [Uu: product_prod_f_nat] : D3 ) )
      = ( ( A3 = bot_bo136713897782694092_f_nat )
        | ( C4 = bot_bo136713897782694092_f_nat )
        | ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
          & ( ord_le8976984241387448984_f_nat @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_1157_times__subset__iff,axiom,
    ! [A3: set_f,C4: set_nat,B4: set_f,D3: set_nat] :
      ( ( ord_le8976984241387448984_f_nat
        @ ( product_Sigma_f_nat @ A3
          @ ^ [Uu: f] : C4 )
        @ ( product_Sigma_f_nat @ B4
          @ ^ [Uu: f] : D3 ) )
      = ( ( A3 = bot_bot_set_f )
        | ( C4 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_f @ A3 @ B4 )
          & ( ord_less_eq_set_nat @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_1158_WCR__on__def,axiom,
    ( abstra312943320106832654rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v,C2: term_f_v] :
              ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ R6 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ C2 ) @ R6 ) )
             => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C2 ) @ ( abstra8605546704478008873rm_f_v @ R6 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_1159_WCR__onE,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra8545463541035707836rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( abstra3336617290665100897rm_f_v @ R3 ) )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R3 )
           => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_1160_WCR__onE,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra3602624719130115245_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( abstra7058733697175803464_f_nat @ R3 ) )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R3 )
           => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_1161_WCR__onE,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra312943320106832654rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra8605546704478008873rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_1162_WCR__onD,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra8545463541035707836rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R3 )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ R3 )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( abstra3336617290665100897rm_f_v @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_1163_WCR__onD,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra3602624719130115245_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R3 )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ R3 )
           => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( abstra7058733697175803464_f_nat @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_1164_WCR__onD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra312943320106832654rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ R3 )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra8605546704478008873rm_f_v @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_1165_irrefl__tranclI,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( ( inf_in6135112614888068085rm_f_v @ ( conver3099152742603637891rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
        = bot_bo4638812505780724595rm_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% irrefl_tranclI
thf(fact_1166_the__elem__eq,axiom,
    ! [X3: product_prod_f_nat] :
      ( ( the_el4348547134654534221_f_nat @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_1167_some__NF__WCR,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
     => ( ( abstra8545463541035707836rm_f_v @ R3
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] : ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X @ bot_bo4638812505780724595rm_f_v ) ) ) )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y2 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ Y2 @ ( abstra349717802288577587rm_f_v @ R3 ) )
           => ( Y2
              = ( abstra6650587300220611071rm_f_v @ R3 @ X3 ) ) ) ) ) ) ).

% some_NF_WCR
thf(fact_1168_some__NF__WCR,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat,Y2: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ( ( abstra3602624719130115245_f_nat @ R3
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
         => ( ( member2678038196313615385_f_nat @ Y2 @ ( abstra8862684023087959798_f_nat @ R3 ) )
           => ( Y2
              = ( abstra2867534111362060330_f_nat @ R3 @ X3 ) ) ) ) ) ) ).

% some_NF_WCR
thf(fact_1169_some__NF__WCR,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y2: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
     => ( ( abstra312943320106832654rm_f_v @ R3
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] : ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X @ bot_bot_set_term_f_v ) ) ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
         => ( ( member_term_f_v @ Y2 @ ( abstract_NF_term_f_v @ R3 ) )
           => ( Y2
              = ( abstra7098836743531038987rm_f_v @ R3 @ X3 ) ) ) ) ) ) ).

% some_NF_WCR
thf(fact_1170_bot__empty__eq2,axiom,
    ( bot_bo2204096811554796588_f_v_o
    = ( ^ [X: term_f_v,Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ bot_bo4638812505780724595rm_f_v ) ) ) ).

% bot_empty_eq2
thf(fact_1171_bot__empty__eq2,axiom,
    ( bot_bot_f_nat_o
    = ( ^ [X: f,Y3: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ bot_bo136713897782694092_f_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1172_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1173_bot__set__def,axiom,
    ( bot_bot_set_term_f_v
    = ( collect_term_f_v @ bot_bot_term_f_v_o ) ) ).

% bot_set_def
thf(fact_1174_bot__empty__eq,axiom,
    ( bot_bo7198560532584249642_f_v_o
    = ( ^ [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ bot_bo4638812505780724595rm_f_v ) ) ) ).

% bot_empty_eq
thf(fact_1175_bot__empty__eq,axiom,
    ( bot_bo5409878808861784761_nat_o
    = ( ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) ).

% bot_empty_eq
thf(fact_1176_SN__on__trancl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ A3 )
     => ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) @ A3 ) ) ).

% SN_on_trancl
thf(fact_1177_SN__on__trancl__imp__SN__on,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,T2: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) @ T2 )
     => ( abstra6117554785717145rm_f_v @ R2 @ T2 ) ) ).

% SN_on_trancl_imp_SN_on
thf(fact_1178_SN__on__trancl__SN__on__conv,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,T2: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) @ T2 )
      = ( abstra6117554785717145rm_f_v @ R2 @ T2 ) ) ).

% SN_on_trancl_SN_on_conv
thf(fact_1179_SN__on__Un2,axiom,
    ! [R2: set_Pr7897041799657265351rm_f_v,X4: set_Pr8827868859434726151rm_f_v,Y5: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R2 @ X4 )
     => ( ( abstra845488452567719153rm_f_v @ R2 @ Y5 )
       => ( abstra845488452567719153rm_f_v @ R2 @ ( sup_su1994854688125285083rm_f_v @ X4 @ Y5 ) ) ) ) ).

% SN_on_Un2
thf(fact_1180_SN__on__subset2,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A3 @ B4 )
     => ( ( abstra5607565530240152504_f_nat @ R3 @ B4 )
       => ( abstra5607565530240152504_f_nat @ R3 @ A3 ) ) ) ).

% SN_on_subset2
thf(fact_1181_SN__on__induct,axiom,
    ! [R2: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,S: produc8027700073345061287rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
      ( ( abstra845488452567719153rm_f_v @ R2 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ S @ A3 )
       => ( ! [T4: produc8027700073345061287rm_f_v] :
              ( ! [U5: produc8027700073345061287rm_f_v] :
                  ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ T4 @ U5 ) @ R2 )
                 => ( P2 @ U5 ) )
             => ( P2 @ T4 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1182_SN__on__induct,axiom,
    ! [R2: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,S: product_prod_f_nat,P2: product_prod_f_nat > $o] :
      ( ( abstra5607565530240152504_f_nat @ R2 @ A3 )
     => ( ( member2678038196313615385_f_nat @ S @ A3 )
       => ( ! [T4: product_prod_f_nat] :
              ( ! [U5: product_prod_f_nat] :
                  ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ T4 @ U5 ) @ R2 )
                 => ( P2 @ U5 ) )
             => ( P2 @ T4 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1183_SN__on__induct,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,S: term_f_v,P2: term_f_v > $o] :
      ( ( abstra6117554785717145rm_f_v @ R2 @ A3 )
     => ( ( member_term_f_v @ S @ A3 )
       => ( ! [T4: term_f_v] :
              ( ! [U5: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ U5 ) @ R2 )
                 => ( P2 @ U5 ) )
             => ( P2 @ T4 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1184_SN__on__irrefl,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ A3 )
     => ! [X5: term_f_v] :
          ( ( member_term_f_v @ X5 @ A3 )
         => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X5 @ X5 ) @ R3 ) ) ) ).

% SN_on_irrefl
thf(fact_1185_SN__on__weakening,axiom,
    ! [R1: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,R22: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R1 @ A3 )
     => ( abstra6117554785717145rm_f_v @ ( inf_in6135112614888068085rm_f_v @ R1 @ R22 ) @ A3 ) ) ).

% SN_on_weakening
thf(fact_1186_SN__imp__SN__trancl,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R2 @ top_top_set_term_f_v )
     => ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) @ top_top_set_term_f_v ) ) ).

% SN_imp_SN_trancl
thf(fact_1187_SN__trancl__imp__SN,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) @ top_top_set_term_f_v )
     => ( abstra6117554785717145rm_f_v @ R2 @ top_top_set_term_f_v ) ) ).

% SN_trancl_imp_SN
thf(fact_1188_SN__trancl__SN__conv,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ ( transi3208867851621559554rm_f_v @ R2 ) @ top_top_set_term_f_v )
      = ( abstra6117554785717145rm_f_v @ R2 @ top_top_set_term_f_v ) ) ).

% SN_trancl_SN_conv
thf(fact_1189_SN__O__comm,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ top_top_set_term_f_v )
      = ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ top_top_set_term_f_v ) ) ).

% SN_O_comm
thf(fact_1190_refl__not__SN,axiom,
    ! [X3: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ R2 )
     => ~ ( abstra6117554785717145rm_f_v @ R2 @ top_top_set_term_f_v ) ) ).

% refl_not_SN
thf(fact_1191_SN__imp__minimal,axiom,
    ! [A3: set_Pr7897041799657265351rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ A3 @ top_to5755558563273943127rm_f_v )
     => ! [Q2: set_Pr8827868859434726151rm_f_v] :
          ( ? [X5: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X5 @ Q2 )
         => ? [X2: produc8027700073345061287rm_f_v] :
              ( ( member4316649861536941904rm_f_v @ X2 @ Q2 )
              & ! [Y6: produc8027700073345061287rm_f_v] :
                  ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X2 @ Y6 ) @ A3 )
                 => ~ ( member4316649861536941904rm_f_v @ Y6 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1192_SN__imp__minimal,axiom,
    ! [A3: set_Pr516226060960033221_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ A3 @ top_to3664669310755125480_f_nat )
     => ! [Q2: set_Pr5245412377734449720_f_nat] :
          ( ? [X5: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X5 @ Q2 )
         => ? [X2: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X2 @ Q2 )
              & ! [Y6: product_prod_f_nat] :
                  ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X2 @ Y6 ) @ A3 )
                 => ~ ( member2678038196313615385_f_nat @ Y6 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1193_SN__imp__minimal,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ A3 @ top_top_set_term_f_v )
     => ! [Q2: set_term_f_v] :
          ( ? [X5: term_f_v] : ( member_term_f_v @ X5 @ Q2 )
         => ? [X2: term_f_v] :
              ( ( member_term_f_v @ X2 @ Q2 )
              & ! [Y6: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y6 ) @ A3 )
                 => ~ ( member_term_f_v @ Y6 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1194_SN__induct__rule,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ! [A2: term_f_v] :
            ( ! [B9: term_f_v] :
                ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B9 ) @ R3 )
               => ( P2 @ B9 ) )
           => ( P2 @ A2 ) )
       => ( P2 @ A ) ) ) ).

% SN_induct_rule
thf(fact_1195_SN__on__all__reducts__SN__on__conv,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
      = ( ! [B3: product_prod_f_nat] :
            ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B3 ) @ R3 )
           => ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ B3 @ bot_bo136713897782694092_f_nat ) ) ) ) ) ).

% SN_on_all_reducts_SN_on_conv
thf(fact_1196_SN__on__all__reducts__SN__on__conv,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
      = ( ! [B3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ R3 )
           => ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ B3 @ bot_bot_set_term_f_v ) ) ) ) ) ).

% SN_on_all_reducts_SN_on_conv
thf(fact_1197_lower__set__imp__not__SN__on,axiom,
    ! [S: produc8027700073345061287rm_f_v,X4: set_Pr8827868859434726151rm_f_v,R2: set_Pr7897041799657265351rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ S @ X4 )
     => ( ! [X2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ X4 )
           => ? [Xa2: produc8027700073345061287rm_f_v] :
                ( ( member4316649861536941904rm_f_v @ Xa2 @ X4 )
                & ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X2 @ Xa2 ) @ R2 ) ) )
       => ~ ( abstra845488452567719153rm_f_v @ R2 @ ( insert5456475782666790775rm_f_v @ S @ bot_bo4638812505780724595rm_f_v ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_1198_lower__set__imp__not__SN__on,axiom,
    ! [S: product_prod_f_nat,X4: set_Pr5245412377734449720_f_nat,R2: set_Pr516226060960033221_f_nat] :
      ( ( member2678038196313615385_f_nat @ S @ X4 )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ X4 )
           => ? [Xa2: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Xa2 @ X4 )
                & ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X2 @ Xa2 ) @ R2 ) ) )
       => ~ ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ S @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_1199_lower__set__imp__not__SN__on,axiom,
    ! [S: term_f_v,X4: set_term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member_term_f_v @ S @ X4 )
     => ( ! [X2: term_f_v] :
            ( ( member_term_f_v @ X2 @ X4 )
           => ? [Xa2: term_f_v] :
                ( ( member_term_f_v @ Xa2 @ X4 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Xa2 ) @ R2 ) ) )
       => ~ ( abstra6117554785717145rm_f_v @ R2 @ ( insert_term_f_v @ S @ bot_bot_set_term_f_v ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_1200_SN__on__induct__acc__style,axiom,
    ! [R2: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,P2: product_prod_f_nat > $o] :
      ( ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
     => ( ! [X2: product_prod_f_nat] :
            ( ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ X2 @ bot_bo136713897782694092_f_nat ) )
           => ( ! [Y6: product_prod_f_nat] :
                  ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X2 @ Y6 ) @ R2 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% SN_on_induct_acc_style
thf(fact_1201_SN__on__induct__acc__style,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,A: term_f_v,P2: term_f_v > $o] :
      ( ( abstra6117554785717145rm_f_v @ R2 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
     => ( ! [X2: term_f_v] :
            ( ( abstra6117554785717145rm_f_v @ R2 @ ( insert_term_f_v @ X2 @ bot_bot_set_term_f_v ) )
           => ( ! [Y6: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y6 ) @ R2 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% SN_on_induct_acc_style
thf(fact_1202_step__preserves__SN__on,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R3 )
     => ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
       => ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% step_preserves_SN_on
thf(fact_1203_step__preserves__SN__on,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
       => ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) ) ) ).

% step_preserves_SN_on
thf(fact_1204_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
     => ! [Q2: set_Pr8827868859434726151rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ X3 @ Q2 )
         => ? [X2: produc8027700073345061287rm_f_v] :
              ( ( member4316649861536941904rm_f_v @ X2 @ Q2 )
              & ! [Y6: produc8027700073345061287rm_f_v] :
                  ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X2 @ Y6 ) @ R3 )
                 => ~ ( member4316649861536941904rm_f_v @ Y6 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1205_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ! [Q2: set_Pr5245412377734449720_f_nat] :
          ( ( member2678038196313615385_f_nat @ X3 @ Q2 )
         => ? [X2: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X2 @ Q2 )
              & ! [Y6: product_prod_f_nat] :
                  ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X2 @ Y6 ) @ R3 )
                 => ~ ( member2678038196313615385_f_nat @ Y6 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1206_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
     => ! [Q2: set_term_f_v] :
          ( ( member_term_f_v @ X3 @ Q2 )
         => ? [X2: term_f_v] :
              ( ( member_term_f_v @ X2 @ Q2 )
              & ! [Y6: term_f_v] :
                  ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2 @ Y6 ) @ R3 )
                 => ~ ( member_term_f_v @ Y6 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1207_step__reflects__SN__on,axiom,
    ! [A: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ! [B2: product_prod_f_nat] :
          ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B2 ) @ R3 )
         => ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ B2 @ bot_bo136713897782694092_f_nat ) ) )
     => ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ).

% step_reflects_SN_on
thf(fact_1208_step__reflects__SN__on,axiom,
    ! [A: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [B2: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B2 ) @ R3 )
         => ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ B2 @ bot_bot_set_term_f_v ) ) )
     => ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ).

% step_reflects_SN_on
thf(fact_1209_SN__on__induct_H,axiom,
    ! [R2: set_Pr516226060960033221_f_nat,S: product_prod_f_nat,P2: product_prod_f_nat > $o] :
      ( ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ S @ bot_bo136713897782694092_f_nat ) )
     => ( ! [T4: product_prod_f_nat] :
            ( ! [U5: product_prod_f_nat] :
                ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ T4 @ U5 ) @ R2 )
               => ( P2 @ U5 ) )
           => ( P2 @ T4 ) )
       => ( P2 @ S ) ) ) ).

% SN_on_induct'
thf(fact_1210_SN__on__induct_H,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S: term_f_v,P2: term_f_v > $o] :
      ( ( abstra6117554785717145rm_f_v @ R2 @ ( insert_term_f_v @ S @ bot_bot_set_term_f_v ) )
     => ( ! [T4: term_f_v] :
            ( ! [U5: term_f_v] :
                ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T4 @ U5 ) @ R2 )
               => ( P2 @ U5 ) )
           => ( P2 @ T4 ) )
       => ( P2 @ S ) ) ) ).

% SN_on_induct'
thf(fact_1211_SN__I,axiom,
    ! [A3: set_Pr516226060960033221_f_nat] :
      ( ! [A2: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A2 @ bot_bo136713897782694092_f_nat ) )
     => ( abstra5607565530240152504_f_nat @ A3 @ top_to3664669310755125480_f_nat ) ) ).

% SN_I
thf(fact_1212_SN__def,axiom,
    ! [R3: set_Pr516226060960033221_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ top_to3664669310755125480_f_nat )
      = ( ! [X: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% SN_def
thf(fact_1213_SN__on__O__push,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,R2: set_Pr8827868859434726151rm_f_v,X4: set_term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R2 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
     => ( ( abstra6117554785717145rm_f_v @ R2 @ X4 )
       => ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ ( transi3028809021613410336rm_f_v @ S2 ) ) @ X4 ) ) ) ).

% SN_on_O_push
thf(fact_1214_NF__imp__SN__on,axiom,
    ! [A: produc8027700073345061287rm_f_v,R2: set_Pr7897041799657265351rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ R2 ) )
     => ( abstra845488452567719153rm_f_v @ R2 @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ) ) ).

% NF_imp_SN_on
thf(fact_1215_NF__imp__SN__on,axiom,
    ! [A: product_prod_f_nat,R2: set_Pr516226060960033221_f_nat] :
      ( ( member2678038196313615385_f_nat @ A @ ( abstra8862684023087959798_f_nat @ R2 ) )
     => ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ).

% NF_imp_SN_on
thf(fact_1216_SN__on__subset__SN__terms,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,X4: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R2 @ X4 )
     => ( ord_le947722214152454265rm_f_v @ X4
        @ ( collect_term_f_v
          @ ^ [X: term_f_v] : ( abstra6117554785717145rm_f_v @ R2 @ ( insert_term_f_v @ X @ bot_bot_set_term_f_v ) ) ) ) ) ).

% SN_on_subset_SN_terms
thf(fact_1217_SN__on__subset__SN__terms,axiom,
    ! [R2: set_Pr516226060960033221_f_nat,X4: set_Pr5245412377734449720_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R2 @ X4 )
     => ( ord_le8976984241387448984_f_nat @ X4
        @ ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ R2 @ ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) ) ) ).

% SN_on_subset_SN_terms
thf(fact_1218_quasi__commute__imp__SN,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ top_top_set_term_f_v )
     => ( ( abstra6117554785717145rm_f_v @ S @ top_top_set_term_f_v )
       => ( ( abstra8826850394716133222rm_f_v @ R3 @ S )
         => ( abstra6117554785717145rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S ) @ top_top_set_term_f_v ) ) ) ) ).

% quasi_commute_imp_SN
thf(fact_1219_steps__reflect__SN__on,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,B: product_prod_f_nat,A: product_prod_f_nat] :
      ( ~ ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
       => ~ ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% steps_reflect_SN_on
thf(fact_1220_steps__reflect__SN__on,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,B: term_f_v,A: term_f_v] :
      ( ~ ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
       => ~ ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ) ).

% steps_reflect_SN_on
thf(fact_1221_steps__preserve__SN__on,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ A3 ) )
     => ( ( abstra5607565530240152504_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
       => ( abstra5607565530240152504_f_nat @ A3 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% steps_preserve_SN_on
thf(fact_1222_steps__preserve__SN__on,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( abstra6117554785717145rm_f_v @ A3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
       => ( abstra6117554785717145rm_f_v @ A3 @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) ) ) ).

% steps_preserve_SN_on
thf(fact_1223_compatible__SN,axiom,
    ! [S2: set_Pr8827868859434726151rm_f_v,NS: set_Pr8827868859434726151rm_f_v] :
      ( ( abstra6117554785717145rm_f_v @ S2 @ top_top_set_term_f_v )
     => ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
       => ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ top_top_set_term_f_v ) ) ) ).

% compatible_SN
thf(fact_1224_Newman__local,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X4: set_Pr5245412377734449720_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ X4 )
     => ( ( abstra3602624719130115245_f_nat @ R3
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) )
       => ( abstra4917929278449670092_f_nat @ R3 @ X4 ) ) ) ).

% Newman_local
thf(fact_1225_Newman__local,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X4: set_term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ X4 )
     => ( ( abstra312943320106832654rm_f_v @ R3
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] : ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X @ bot_bot_set_term_f_v ) ) ) )
       => ( abstra3735013631728912813rm_f_v @ R3 @ X4 ) ) ) ).

% Newman_local
thf(fact_1226_SN__reaches__NF,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
     => ? [Y: produc8027700073345061287rm_f_v] :
          ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
          & ( member4316649861536941904rm_f_v @ Y @ ( abstra349717802288577587rm_f_v @ R3 ) ) ) ) ).

% SN_reaches_NF
thf(fact_1227_SN__reaches__NF,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ? [Y: product_prod_f_nat] :
          ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi8903367562120364223_f_nat @ R3 ) )
          & ( member2678038196313615385_f_nat @ Y @ ( abstra8862684023087959798_f_nat @ R3 ) ) ) ) ).

% SN_reaches_NF
thf(fact_1228_SN__reaches__NF,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
     => ? [Y: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
          & ( member_term_f_v @ Y @ ( abstract_NF_term_f_v @ R3 ) ) ) ) ).

% SN_reaches_NF
thf(fact_1229_compatible__SN_H,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ( abstra6117554785717145rm_f_v @ S2 @ top_top_set_term_f_v )
       => ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) ) ) @ top_top_set_term_f_v ) ) ) ).

% compatible_SN'
thf(fact_1230_qc__SN__relto__iff,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S ) @ ( relcom2934258169684288428rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S @ R3 ) ) ) )
     => ( ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( relcom2934258169684288428rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) @ top_top_set_term_f_v )
        = ( abstra6117554785717145rm_f_v @ S @ top_top_set_term_f_v ) ) ) ).

% qc_SN_relto_iff
thf(fact_1231_some__NF,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ ( abstra6650587300220611071rm_f_v @ R3 @ X3 ) ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
        & ( member4316649861536941904rm_f_v @ ( abstra6650587300220611071rm_f_v @ R3 @ X3 ) @ ( abstra349717802288577587rm_f_v @ R3 ) ) ) ) ).

% some_NF
thf(fact_1232_some__NF,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ ( abstra2867534111362060330_f_nat @ R3 @ X3 ) ) @ ( transi8903367562120364223_f_nat @ R3 ) )
        & ( member2678038196313615385_f_nat @ ( abstra2867534111362060330_f_nat @ R3 @ X3 ) @ ( abstra8862684023087959798_f_nat @ R3 ) ) ) ) ).

% some_NF
thf(fact_1233_some__NF,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ ( abstra7098836743531038987rm_f_v @ R3 @ X3 ) ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
        & ( member_term_f_v @ ( abstra7098836743531038987rm_f_v @ R3 @ X3 ) @ ( abstract_NF_term_f_v @ R3 ) ) ) ) ).

% some_NF
thf(fact_1234_SN__WCR__reaches__NF,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
     => ( ( abstra8545463541035707836rm_f_v @ R3
          @ ( collec7769483783292507282rm_f_v
            @ ^ [X: produc8027700073345061287rm_f_v] : ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ X @ bot_bo4638812505780724595rm_f_v ) ) ) )
       => ? [X2: produc8027700073345061287rm_f_v] :
            ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ X2 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
            & ( member4316649861536941904rm_f_v @ X2 @ ( abstra349717802288577587rm_f_v @ R3 ) )
            & ! [Y6: produc8027700073345061287rm_f_v] :
                ( ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y6 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
                  & ( member4316649861536941904rm_f_v @ Y6 @ ( abstra349717802288577587rm_f_v @ R3 ) ) )
               => ( Y6 = X2 ) ) ) ) ) ).

% SN_WCR_reaches_NF
thf(fact_1235_SN__WCR__reaches__NF,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
     => ( ( abstra3602624719130115245_f_nat @ R3
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] : ( abstra5607565530240152504_f_nat @ R3 @ ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) )
       => ? [X2: product_prod_f_nat] :
            ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ X2 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
            & ( member2678038196313615385_f_nat @ X2 @ ( abstra8862684023087959798_f_nat @ R3 ) )
            & ! [Y6: product_prod_f_nat] :
                ( ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y6 ) @ ( transi8903367562120364223_f_nat @ R3 ) )
                  & ( member2678038196313615385_f_nat @ Y6 @ ( abstra8862684023087959798_f_nat @ R3 ) ) )
               => ( Y6 = X2 ) ) ) ) ) ).

% SN_WCR_reaches_NF
thf(fact_1236_SN__WCR__reaches__NF,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
     => ( ( abstra312943320106832654rm_f_v @ R3
          @ ( collect_term_f_v
            @ ^ [X: term_f_v] : ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X @ bot_bot_set_term_f_v ) ) ) )
       => ? [X2: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X2 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
            & ( member_term_f_v @ X2 @ ( abstract_NF_term_f_v @ R3 ) )
            & ! [Y6: term_f_v] :
                ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y6 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
                  & ( member_term_f_v @ Y6 @ ( abstract_NF_term_f_v @ R3 ) ) )
               => ( Y6 = X2 ) ) ) ) ) ).

% SN_WCR_reaches_NF
thf(fact_1237_restrict__SN__def,axiom,
    ( abstra4165953444401048309_f_nat
    = ( ^ [R6: set_Pr516226060960033221_f_nat,S6: set_Pr516226060960033221_f_nat] :
          ( collec8116569655616119888_f_nat
          @ ^ [Uu: produc2676906022994773093_f_nat] :
            ? [A4: product_prod_f_nat,B3: product_prod_f_nat] :
              ( ( Uu
                = ( produc5861087217798509725_f_nat @ A4 @ B3 ) )
              & ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A4 @ B3 ) @ R6 )
              & ( abstra5607565530240152504_f_nat @ S6 @ ( insert5008453031520946354_f_nat @ A4 @ bot_bo136713897782694092_f_nat ) ) ) ) ) ) ).

% restrict_SN_def
thf(fact_1238_restrict__SN__def,axiom,
    ( abstra219248723679151446rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v] :
          ( collec7769483783292507282rm_f_v
          @ ^ [Uu: produc8027700073345061287rm_f_v] :
            ? [A4: term_f_v,B3: term_f_v] :
              ( ( Uu
                = ( produc2510790566102465375rm_f_v @ A4 @ B3 ) )
              & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ R6 )
              & ( abstra6117554785717145rm_f_v @ S6 @ ( insert_term_f_v @ A4 @ bot_bot_set_term_f_v ) ) ) ) ) ) ).

% restrict_SN_def
thf(fact_1239_restrict__SN__trancl__simp,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( transi3208867851621559554rm_f_v @ ( abstra219248723679151446rm_f_v @ A3 @ A3 ) )
      = ( abstra219248723679151446rm_f_v @ ( transi3208867851621559554rm_f_v @ A3 ) @ A3 ) ) ).

% restrict_SN_trancl_simp
thf(fact_1240_union__comp__emptyL,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( relcom2934258169684288428rm_f_v @ A3 @ C4 )
        = bot_bo4638812505780724595rm_f_v )
     => ( ( ( relcom2934258169684288428rm_f_v @ B4 @ C4 )
          = bot_bo4638812505780724595rm_f_v )
       => ( ( relcom2934258169684288428rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) @ C4 )
          = bot_bo4638812505780724595rm_f_v ) ) ) ).

% union_comp_emptyL
thf(fact_1241_union__comp__emptyR,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,C4: set_Pr8827868859434726151rm_f_v] :
      ( ( ( relcom2934258169684288428rm_f_v @ A3 @ B4 )
        = bot_bo4638812505780724595rm_f_v )
     => ( ( ( relcom2934258169684288428rm_f_v @ A3 @ C4 )
          = bot_bo4638812505780724595rm_f_v )
       => ( ( relcom2934258169684288428rm_f_v @ A3 @ ( sup_su1994854688125285083rm_f_v @ B4 @ C4 ) )
          = bot_bo4638812505780724595rm_f_v ) ) ) ).

% union_comp_emptyR
thf(fact_1242_the__NF__conv,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ A3 @ top_top_set_term_f_v )
     => ( ( abstra3735013631728912813rm_f_v @ A3 @ top_top_set_term_f_v )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A3 ) )
         => ( ( abstra6752248511232870546rm_f_v @ A3 @ A )
            = ( abstra6752248511232870546rm_f_v @ A3 @ B ) ) ) ) ) ).

% the_NF_conv
thf(fact_1243_the__NF__steps,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ A3 @ top_top_set_term_f_v )
     => ( ( abstra3735013631728912813rm_f_v @ A3 @ top_top_set_term_f_v )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
         => ( ( abstra6752248511232870546rm_f_v @ A3 @ A )
            = ( abstra6752248511232870546rm_f_v @ A3 @ B ) ) ) ) ) ).

% the_NF_steps
thf(fact_1244_the__NF__step,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ A3 @ top_top_set_term_f_v )
     => ( ( abstra3735013631728912813rm_f_v @ A3 @ top_top_set_term_f_v )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ A3 )
         => ( ( abstra6752248511232870546rm_f_v @ A3 @ A )
            = ( abstra6752248511232870546rm_f_v @ A3 @ B ) ) ) ) ) ).

% the_NF_step
thf(fact_1245_the__NF__NF,axiom,
    ! [A3: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( abstra845488452567719153rm_f_v @ A3 @ top_to5755558563273943127rm_f_v )
     => ( ( abstra5806628924625096669rm_f_v @ A3 @ top_to5755558563273943127rm_f_v )
       => ( member4316649861536941904rm_f_v @ ( abstra7419483983937885496rm_f_v @ A3 @ A ) @ ( abstra349717802288577587rm_f_v @ A3 ) ) ) ) ).

% the_NF_NF
thf(fact_1246_the__NF__NF,axiom,
    ! [A3: set_Pr516226060960033221_f_nat,A: product_prod_f_nat] :
      ( ( abstra5607565530240152504_f_nat @ A3 @ top_to3664669310755125480_f_nat )
     => ( ( abstra4917929278449670092_f_nat @ A3 @ top_to3664669310755125480_f_nat )
       => ( member2678038196313615385_f_nat @ ( abstra3853861087784312881_f_nat @ A3 @ A ) @ ( abstra8862684023087959798_f_nat @ A3 ) ) ) ) ).

% the_NF_NF
thf(fact_1247_is__singleton__the__elem,axiom,
    ( is_sin4592364260436481678_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat] :
          ( A5
          = ( insert5008453031520946354_f_nat @ ( the_el4348547134654534221_f_nat @ A5 ) @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1248_min__ext__compat,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ R2 )
     => ( ord_le4674940582310090771rm_f_v @ ( relcom6328679179271033038rm_f_v @ ( min_ext_term_f_v @ R2 ) @ ( sup_su3258276822682701895rm_f_v @ ( min_ext_term_f_v @ S2 ) @ ( insert2525147299836527331rm_f_v @ ( produc8754436188854700491rm_f_v @ bot_bot_set_term_f_v @ bot_bot_set_term_f_v ) @ bot_bo8555627264450986207rm_f_v ) ) ) @ ( min_ext_term_f_v @ R2 ) ) ) ).

% min_ext_compat
thf(fact_1249_is__singletonI,axiom,
    ! [X3: product_prod_f_nat] : ( is_sin4592364260436481678_f_nat @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) ).

% is_singletonI
thf(fact_1250_is__singletonI_H,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v] :
      ( ( A3 != bot_bo4638812505780724595rm_f_v )
     => ( ! [X2: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X2 @ A3 )
           => ( ( member4316649861536941904rm_f_v @ Y @ A3 )
             => ( X2 = Y ) ) )
       => ( is_sin8002852355414267547rm_f_v @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1251_is__singletonI_H,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( A3 != bot_bo136713897782694092_f_nat )
     => ( ! [X2: product_prod_f_nat,Y: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A3 )
           => ( ( member2678038196313615385_f_nat @ Y @ A3 )
             => ( X2 = Y ) ) )
       => ( is_sin4592364260436481678_f_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1252_is__singleton__def,axiom,
    ( is_sin4592364260436481678_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat] :
        ? [X: product_prod_f_nat] :
          ( A5
          = ( insert5008453031520946354_f_nat @ X @ bot_bo136713897782694092_f_nat ) ) ) ) ).

% is_singleton_def
thf(fact_1253_is__singletonE,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat] :
      ( ( is_sin4592364260436481678_f_nat @ A3 )
     => ~ ! [X2: product_prod_f_nat] :
            ( A3
           != ( insert5008453031520946354_f_nat @ X2 @ bot_bo136713897782694092_f_nat ) ) ) ).

% is_singletonE
thf(fact_1254_max__ext__compat,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R2 @ S2 ) @ R2 )
     => ( ord_le4674940582310090771rm_f_v @ ( relcom6328679179271033038rm_f_v @ ( max_ext_term_f_v @ R2 ) @ ( sup_su3258276822682701895rm_f_v @ ( max_ext_term_f_v @ S2 ) @ ( insert2525147299836527331rm_f_v @ ( produc8754436188854700491rm_f_v @ bot_bot_set_term_f_v @ bot_bot_set_term_f_v ) @ bot_bo8555627264450986207rm_f_v ) ) ) @ ( max_ext_term_f_v @ R2 ) ) ) ).

% max_ext_compat
thf(fact_1255_the__NF,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
      ( ( abstra6117554785717145rm_f_v @ A3 @ top_top_set_term_f_v )
     => ( ( abstra3735013631728912813rm_f_v @ A3 @ top_top_set_term_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ ( abstra6752248511232870546rm_f_v @ A3 @ A ) ) @ ( abstra7482117481303700155rm_f_v @ A3 ) ) ) ) ).

% the_NF
thf(fact_1256_normalizability__I_H,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( abstra7482117481303700155rm_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( abstra7482117481303700155rm_f_v @ A3 ) ) ) ) ).

% normalizability_I'
thf(fact_1257_UNF__onI,axiom,
    ! [A3: set_Pr8827868859434726151rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ! [A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ A2 @ A3 )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ B2 ) @ ( abstra9166317777578425423rm_f_v @ R3 ) )
           => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A2 @ C3 ) @ ( abstra9166317777578425423rm_f_v @ R3 ) )
             => ( B2 = C3 ) ) ) )
     => ( abstra8974104706389925717rm_f_v @ R3 @ A3 ) ) ).

% UNF_onI
thf(fact_1258_UNF__onI,axiom,
    ! [A3: set_Pr5245412377734449720_f_nat,R3: set_Pr516226060960033221_f_nat] :
      ( ! [A2: product_prod_f_nat,B2: product_prod_f_nat,C3: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ A2 @ A3 )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ B2 ) @ ( abstra786810888989762010_f_nat @ R3 ) )
           => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A2 @ C3 ) @ ( abstra786810888989762010_f_nat @ R3 ) )
             => ( B2 = C3 ) ) ) )
     => ( abstra4786933130458578516_f_nat @ R3 @ A3 ) ) ).

% UNF_onI
thf(fact_1259_UNF__onI,axiom,
    ! [A3: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ! [A2: term_f_v,B2: term_f_v,C3: term_f_v] :
          ( ( member_term_f_v @ A2 @ A3 )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) @ ( abstra7482117481303700155rm_f_v @ R3 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A2 @ C3 ) @ ( abstra7482117481303700155rm_f_v @ R3 ) )
             => ( B2 = C3 ) ) ) )
     => ( abstra8129321985590711349rm_f_v @ R3 @ A3 ) ) ).

% UNF_onI
thf(fact_1260_normalizability__I,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: set_Pr7897041799657265351rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A3 ) )
     => ( ( member4316649861536941904rm_f_v @ B @ ( abstra349717802288577587rm_f_v @ A3 ) )
       => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ A3 ) ) ) ) ).

% normalizability_I
thf(fact_1261_normalizability__I,axiom,
    ! [A: product_prod_f_nat,B: product_prod_f_nat,A3: set_Pr516226060960033221_f_nat] :
      ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ A3 ) )
     => ( ( member2678038196313615385_f_nat @ B @ ( abstra8862684023087959798_f_nat @ A3 ) )
       => ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra786810888989762010_f_nat @ A3 ) ) ) ) ).

% normalizability_I
thf(fact_1262_normalizability__I,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
     => ( ( member_term_f_v @ B @ ( abstract_NF_term_f_v @ A3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ A3 ) ) ) ) ).

% normalizability_I
thf(fact_1263_UNF__onD,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra8974104706389925717rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ R3 ) )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( abstra9166317777578425423rm_f_v @ R3 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_1264_UNF__onD,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra4786933130458578516_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra786810888989762010_f_nat @ R3 ) )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( abstra786810888989762010_f_nat @ R3 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_1265_UNF__onD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra8129321985590711349rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ R3 ) )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( abstra7482117481303700155rm_f_v @ R3 ) )
           => ( B = C ) ) ) ) ) ).

% UNF_onD
thf(fact_1266_UNF__onE,axiom,
    ! [R3: set_Pr7897041799657265351rm_f_v,A3: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( abstra8974104706389925717rm_f_v @ R3 @ A3 )
     => ( ( member4316649861536941904rm_f_v @ A @ A3 )
       => ( ( B != C )
         => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ R3 ) )
           => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( abstra9166317777578425423rm_f_v @ R3 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_1267_UNF__onE,axiom,
    ! [R3: set_Pr516226060960033221_f_nat,A3: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,C: product_prod_f_nat] :
      ( ( abstra4786933130458578516_f_nat @ R3 @ A3 )
     => ( ( member2678038196313615385_f_nat @ A @ A3 )
       => ( ( B != C )
         => ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra786810888989762010_f_nat @ R3 ) )
           => ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( abstra786810888989762010_f_nat @ R3 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_1268_UNF__onE,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v,A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( abstra8129321985590711349rm_f_v @ R3 @ A3 )
     => ( ( member_term_f_v @ A @ A3 )
       => ( ( B != C )
         => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ R3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( abstra7482117481303700155rm_f_v @ R3 ) ) ) ) ) ) ).

% UNF_onE
thf(fact_1269_UNF__on__def,axiom,
    ( abstra8129321985590711349rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v,C2: term_f_v] :
              ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R6 ) )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ C2 ) @ ( abstra7482117481303700155rm_f_v @ R6 ) ) )
             => ( B3 = C2 ) ) ) ) ) ).

% UNF_on_def
thf(fact_1270_NE__on__def,axiom,
    ( rewrit2656876241799709610rm_f_v
    = ( ^ [R6: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v,A5: set_term_f_v] :
        ! [X: term_f_v] :
          ( ( member_term_f_v @ X @ A5 )
         => ! [B3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R6 ) )
              = ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B3 ) @ ( abstra7482117481303700155rm_f_v @ S6 ) ) ) ) ) ) ).

% NE_on_def
thf(fact_1271_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1272_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1273_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1274_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'

% Conjectures (1)
thf(conj_0,conjecture,
    ( member4316649861536941904rm_f_v
    @ ( produc2510790566102465375rm_f_v
      @ ( subst_7901567856578221233_f_v_v @ s
        @ ^ [X: v] : ( fun_f_v @ c @ nil_term_f_v ) )
      @ ( subst_7901567856578221233_f_v_v @ t
        @ ^ [X: v] : ( fun_f_v @ d @ nil_term_f_v ) ) )
    @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ r ) ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ c @ zero_zero_nat ) @ ( insert5008453031520946354_f_nat @ ( product_Pair_f_nat @ d @ zero_zero_nat ) @ f2 ) ) @ ( rstep_f_v @ r ) ) ) ) ) ).

%------------------------------------------------------------------------------