TPTP Problem File: SLH0669^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FO_Theory_Rewriting/0065_LV_to_GTT/prob_00271_010725__18459346_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1956 ( 452 unt; 663 typ;   0 def)
%            Number of atoms       : 3721 ( 999 equ;   0 cnn)
%            Maximal formula atoms :   16 (   2 avg)
%            Number of connectives : 16010 ( 534   ~;  57   |; 245   &;13271   @)
%                                         (   0 <=>;1903  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   8 avg)
%            Number of types       :  133 ( 132 usr)
%            Number of type conns  : 1417 (1417   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  534 ( 531 usr;  29 con; 0-4 aty)
%            Number of variables   : 4695 ( 326   ^;4262   !; 107   ?;4695   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:01:22.468
%------------------------------------------------------------------------------
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thf(sy_c_Tree__Automata_Ogta__der_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata_Ogta__lang_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata_Oprod__ta_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_001tf__f_001tf__v,type,
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thf(sy_c_Tree__Automata_Oprod__ta_001tf__v_001tf__f_001t__Bot____Terms__Obot____term_Itf__f_J,type,
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thf(sy_c_Tree__Automata_Oprod__ta_001tf__v_001tf__f_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
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thf(sy_c_Tree__Automata_Oprod__ta_001tf__v_001tf__f_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
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thf(sy_c_Tree__Automata_Oprod__ta_001tf__v_001tf__f_001tf__v,type,
    tree_prod_ta_v_f_v: tree_ta_v_f > tree_ta_v_f > tree_t3265465668771388046_v_v_f ).

thf(sy_c_Tree__Automata_Ota_OTA_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota_OTA_001tf__v_001tf__f,type,
    tree_TA_v_f: fset_T5471790544740713358le_v_f > fset_P6509793526439625293od_v_v > tree_ta_v_f ).

thf(sy_c_Tree__Automata_Ota_Oeps_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota_Oeps_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota_Orules_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota_Orules_001tf__v_001tf__f,type,
    tree_rules_v_f: tree_ta_v_f > fset_T5471790544740713358le_v_f ).

thf(sy_c_Tree__Automata_Ota__contains__aux_001tf__f_001tf__v,type,
    tree_t3101685780840579086ux_f_v: set_Pr5245412377734449720_f_nat > fset_v > tree_ta_v_f > fset_v > $o ).

thf(sy_c_Tree__Automata_Ota__der_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
    tree_t4678214921166696585rm_f_f: tree_t2044664379607036063rm_f_f > term_f7969170442531588692term_f > fset_Bot_bot_term_f ).

thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    tree_t3429357085031708437_f_nat: tree_t5579808332917730531_f_nat > term_P1124376183965932676term_f > fset_P2882900458636368611term_f ).

thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__v_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_Mtf__v_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mtf__v_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_Itf__v_Mt__Bot____Terms__Obot____term_Itf__f_J_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001t__Product____Type__Oprod_Itf__v_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_H_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__der_H_001tf__v_001tf__f,type,
    tree_ta_der_v_f2: tree_ta_v_f > term_f_v > fset_term_f_v ).

thf(sy_c_Tree__Automata_Ota__productive_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__reachable_001tf__v_001tf__f,type,
    tree_t1547661003257875222le_v_f: tree_ta_v_f > fset_v ).

thf(sy_c_Tree__Automata_Ota__restrict_001tf__v_001tf__f,type,
    tree_ta_restrict_v_f: tree_ta_v_f > fset_v > tree_ta_v_f ).

thf(sy_c_Tree__Automata_Ota__rule_OTA__rule_001tf__f_001tf__v,type,
    tree_ta_TA_rule_f_v: f > list_v > v > tree_ta_rule_v_f ).

thf(sy_c_Tree__Automata_Ota__rule_Or__rhs_001tf__v_001tf__f,type,
    tree_ta_r_rhs_v_f: tree_ta_rule_v_f > v ).

thf(sy_c_Tree__Automata_Ota__sig_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__sig_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_001tf__f,type,
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thf(sy_c_Tree__Automata_Ota__sig_001tf__v_001tf__f,type,
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thf(sy_c_Tree__Automata__Complement_Opartially__completely__defined__on_001tf__v_001tf__f,type,
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thf(sy_c_Utils_Ofunas__trs_001tf__f_001tf__v_001tf__v,type,
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thf(sy_c_Utils_Olinear__term_001tf__f_001tf__v,type,
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thf(sy_c_Wfrec_Osame__fst_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
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thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__f_Mtf__v_J_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
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thf(sy_c_member_001t__Ground____Terms__Ogterm_Itf__f_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
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thf(sy_c_member_001tf__v,type,
    member_v: v > set_v > $o ).

thf(sy_v_R,type,
    r: fset_P8018961893305114765rm_f_v ).

thf(sy_v__092_060F_062,type,
    f2: fset_P6228066233360383026_f_nat ).

thf(sy_v_l____,type,
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thf(sy_v_p____,type,
    p: bot_bot_term_f ).

thf(sy_v_q____,type,
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thf(sy_v_r____,type,
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thf(sy_v_s____,type,
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thf(sy_v_t____,type,
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% Relevant facts (1277)
thf(fact_0_tm_I1_J,axiom,
    ( q
    = ( bot_te7903553565439629424rm_f_v @ l ) ) ).

% tm(1)
thf(fact_1_tm_I3_J,axiom,
    fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ l @ r2 ) @ r ).

% tm(3)
thf(fact_2_linear__term_Osimps_I1_J,axiom,
    ! [Uu: v] : ( linear_term_f_v @ ( var_v_f @ Uu ) ) ).

% linear_term.simps(1)
thf(fact_3_sm_I1_J,axiom,
    member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ l ) @ ( bot_te7903553565439629424rm_f_v @ ( ground1072145386804415820rm_f_v @ s ) ) ) @ bot_bless_eq_f ).

% sm(1)
thf(fact_4_tm_I2_J,axiom,
    ( p
    = ( bot_te7903553565439629424rm_f_v @ r2 ) ) ).

% tm(2)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_Ar_O_A_092_060lbrakk_062q_A_061_Al_092_060_094sup_062_092_060bottom_062_059_Ap_A_061_Ar_092_060_094sup_062_092_060bottom_062_059_A_Il_M_Ar_J_A_124_092_060in_062_124_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [L: term_f_v] :
        ( ( q
          = ( bot_te7903553565439629424rm_f_v @ L ) )
       => ! [R: term_f_v] :
            ( ( p
              = ( bot_te7903553565439629424rm_f_v @ R ) )
           => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ r ) ) ) ).

% \<open>\<And>thesis. (\<And>l r. \<lbrakk>q = l\<^sup>\<bottom>; p = r\<^sup>\<bottom>; (l, r) |\<in>| R\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_bot__term__of__gterm__conv,axiom,
    ! [S: ground_gterm_f] :
      ( ( bot_te7903553565439629424rm_f_v @ ( ground1072145386804415820rm_f_v @ S ) )
      = ( bot_te7903553565439629424rm_f_v @ ( ground1072145386804415820rm_f_v @ S ) ) ) ).

% bot_term_of_gterm_conv
thf(fact_7_sm_I2_J,axiom,
    member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ r2 ) @ ( bot_te7903553565439629424rm_f_v @ ( ground1072145386804415820rm_f_v @ t ) ) ) @ bot_bless_eq_f ).

% sm(2)
thf(fact_8_assms_I1_J,axiom,
    lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ r ) ).

% assms(1)
thf(fact_9_term_Oinject_I1_J,axiom,
    ! [X1: v,Y1: v] :
      ( ( ( var_v_f @ X1 )
        = ( var_v_f @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% term.inject(1)
thf(fact_10_prod_Oinject,axiom,
    ! [X1: term_f_v,X2: term_f_v,Y1: term_f_v,Y2: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ X1 @ X2 )
        = ( produc2510790566102465375rm_f_v @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: bot_bot_term_f,X2: bot_bot_term_f,Y1: bot_bot_term_f,Y2: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ X1 @ X2 )
        = ( produc2865564700782054965term_f @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: ground_gterm_f,X2: ground_gterm_f,Y1: ground_gterm_f,Y2: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ X1 @ X2 )
        = ( produc3560254623552331287term_f @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A2: bot_bot_term_f,B2: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ A @ B )
        = ( produc2865564700782054965term_f @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A2: ground_gterm_f,B2: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ A @ B )
        = ( produc3560254623552331287term_f @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_bless__eq__refl,axiom,
    ! [S: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ S ) @ bot_bless_eq_f ) ).

% bless_eq_refl
thf(fact_17_bless__eq__trans,axiom,
    ! [S: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_bless_eq_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ U ) @ bot_bless_eq_f ) ) ) ).

% bless_eq_trans
thf(fact_18_bless__eq__anti__sym,axiom,
    ! [S: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ S ) @ bot_bless_eq_f )
       => ( S = T ) ) ) ).

% bless_eq_anti_sym
thf(fact_19_st_I1_J,axiom,
    fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ q @ p ) @ ( lV_to_Rel_f_f_v @ r ) ).

% st(1)
thf(fact_20_eps__statesI_I2_J,axiom,
    ! [P: produc8027700073345061287rm_f_v,Q: produc8027700073345061287rm_f_v,Delta: fset_P7452185835880607565rm_f_v] :
      ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ P @ Q ) @ Delta )
     => ( fmembe774991991480784110rm_f_v @ Q @ ( tree_e6712122993333108694rm_f_v @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_21_eps__statesI_I2_J,axiom,
    ! [P: produc250624644672406269term_f,Q: produc250624644672406269term_f,Delta: fset_P616664398915176141term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ P @ Q ) @ Delta )
     => ( fmembe6950792022577883716term_f @ Q @ ( tree_e7667339219947091244term_f @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_22_eps__statesI_I2_J,axiom,
    ! [P: term_f_v,Q: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q ) @ Delta )
     => ( fmember_term_f_v @ Q @ ( tree_e8707155279755596916rm_f_v @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_23_eps__statesI_I2_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ Delta )
     => ( fmembe1418995490368054919term_f @ Q @ ( tree_e551949757491771807term_f @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_24_eps__statesI_I2_J,axiom,
    ! [P: ground_gterm_f,Q: ground_gterm_f,Delta: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ P @ Q ) @ Delta )
     => ( fmembe8264398993653236347term_f @ Q @ ( tree_e2075845989350546787term_f @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_25_eps__statesI_I1_J,axiom,
    ! [P: produc8027700073345061287rm_f_v,Q: produc8027700073345061287rm_f_v,Delta: fset_P7452185835880607565rm_f_v] :
      ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ P @ Q ) @ Delta )
     => ( fmembe774991991480784110rm_f_v @ P @ ( tree_e6712122993333108694rm_f_v @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_26_eps__statesI_I1_J,axiom,
    ! [P: produc250624644672406269term_f,Q: produc250624644672406269term_f,Delta: fset_P616664398915176141term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ P @ Q ) @ Delta )
     => ( fmembe6950792022577883716term_f @ P @ ( tree_e7667339219947091244term_f @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_27_eps__statesI_I1_J,axiom,
    ! [P: term_f_v,Q: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q ) @ Delta )
     => ( fmember_term_f_v @ P @ ( tree_e8707155279755596916rm_f_v @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_28_eps__statesI_I1_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ Delta )
     => ( fmembe1418995490368054919term_f @ P @ ( tree_e551949757491771807term_f @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_29_eps__statesI_I1_J,axiom,
    ! [P: ground_gterm_f,Q: ground_gterm_f,Delta: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ P @ Q ) @ Delta )
     => ( fmembe8264398993653236347term_f @ P @ ( tree_e2075845989350546787term_f @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_30_fId__onI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ A ) @ ( fSet_f4703993580257979951rm_f_v @ A3 ) ) ) ).

% fId_onI
thf(fact_31_fId__onI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ A ) @ ( fSet_f5727372830607458181term_f @ A3 ) ) ) ).

% fId_onI
thf(fact_32_fId__onI,axiom,
    ! [A: term_f_v,A3: fset_term_f_v] :
      ( ( fmember_term_f_v @ A @ A3 )
     => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( fSet_fId_on_term_f_v @ A3 ) ) ) ).

% fId_onI
thf(fact_33_fId__onI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ A ) @ ( fSet_f5231733434221412358term_f @ A3 ) ) ) ).

% fId_onI
thf(fact_34_fId__onI,axiom,
    ! [A: ground_gterm_f,A3: fset_Ground_gterm_f] :
      ( ( fmembe8264398993653236347term_f @ A @ A3 )
     => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ A ) @ ( fSet_f228041949194217916term_f @ A3 ) ) ) ).

% fId_onI
thf(fact_35_fId__on__eqI,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( A = B )
     => ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
       => ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( fSet_f4703993580257979951rm_f_v @ A3 ) ) ) ) ).

% fId_on_eqI
thf(fact_36_fId__on__eqI,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( A = B )
     => ( ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( fSet_f5727372830607458181term_f @ A3 ) ) ) ) ).

% fId_on_eqI
thf(fact_37_fId__on__eqI,axiom,
    ! [A: term_f_v,B: term_f_v,A3: fset_term_f_v] :
      ( ( A = B )
     => ( ( fmember_term_f_v @ A @ A3 )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_fId_on_term_f_v @ A3 ) ) ) ) ).

% fId_on_eqI
thf(fact_38_fId__on__eqI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( A = B )
     => ( ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f5231733434221412358term_f @ A3 ) ) ) ) ).

% fId_on_eqI
thf(fact_39_fId__on__eqI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: fset_Ground_gterm_f] :
      ( ( A = B )
     => ( ( fmembe8264398993653236347term_f @ A @ A3 )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f228041949194217916term_f @ A3 ) ) ) ) ).

% fId_on_eqI
thf(fact_40_fId__on__iff,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y ) @ ( fSet_f4703993580257979951rm_f_v @ A3 ) )
      = ( ( X = Y )
        & ( fmembe774991991480784110rm_f_v @ X @ A3 ) ) ) ).

% fId_on_iff
thf(fact_41_fId__on__iff,axiom,
    ! [X: produc250624644672406269term_f,Y: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ X @ Y ) @ ( fSet_f5727372830607458181term_f @ A3 ) )
      = ( ( X = Y )
        & ( fmembe6950792022577883716term_f @ X @ A3 ) ) ) ).

% fId_on_iff
thf(fact_42_fId__on__iff,axiom,
    ! [X: term_f_v,Y: term_f_v,A3: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_fId_on_term_f_v @ A3 ) )
      = ( ( X = Y )
        & ( fmember_term_f_v @ X @ A3 ) ) ) ).

% fId_on_iff
thf(fact_43_fId__on__iff,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f5231733434221412358term_f @ A3 ) )
      = ( ( X = Y )
        & ( fmembe1418995490368054919term_f @ X @ A3 ) ) ) ).

% fId_on_iff
thf(fact_44_fId__on__iff,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,A3: fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f228041949194217916term_f @ A3 ) )
      = ( ( X = Y )
        & ( fmembe8264398993653236347term_f @ X @ A3 ) ) ) ).

% fId_on_iff
thf(fact_45_fId__onE,axiom,
    ! [C: produc3406425829894461415rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe195806516701975982rm_f_v @ C @ ( fSet_f4703993580257979951rm_f_v @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
           => ( C
             != ( produc3874942763110801239rm_f_v @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_46_fId__onE,axiom,
    ! [C: produc6853483567480260455term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ C @ ( fSet_f5727372830607458181term_f @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
           => ( C
             != ( produc7121717381804241367term_f @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_47_fId__onE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( fSet_fId_on_term_f_v @ A3 ) )
     => ~ ! [X3: term_f_v] :
            ( ( fmember_term_f_v @ X3 @ A3 )
           => ( C
             != ( produc2510790566102465375rm_f_v @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_48_fId__onE,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( fSet_f5231733434221412358term_f @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
           => ( C
             != ( produc2865564700782054965term_f @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_49_fId__onE,axiom,
    ! [C: produc7245736746747425831term_f,A3: fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ C @ ( fSet_f228041949194217916term_f @ A3 ) )
     => ~ ! [X3: ground_gterm_f] :
            ( ( fmembe8264398993653236347term_f @ X3 @ A3 )
           => ( C
             != ( produc3560254623552331287term_f @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_50_eps__statesE,axiom,
    ! [P: produc8027700073345061287rm_f_v,Delta: fset_P7452185835880607565rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ P @ ( tree_e6712122993333108694rm_f_v @ Delta ) )
     => ~ ! [Q2: produc8027700073345061287rm_f_v] :
            ~ ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ P @ Q2 ) @ Delta )
              | ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ Q2 @ P ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_51_eps__statesE,axiom,
    ! [P: produc250624644672406269term_f,Delta: fset_P616664398915176141term_f] :
      ( ( fmembe6950792022577883716term_f @ P @ ( tree_e7667339219947091244term_f @ Delta ) )
     => ~ ! [Q2: produc250624644672406269term_f] :
            ~ ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ P @ Q2 ) @ Delta )
              | ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ Q2 @ P ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_52_eps__statesE,axiom,
    ! [P: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmember_term_f_v @ P @ ( tree_e8707155279755596916rm_f_v @ Delta ) )
     => ~ ! [Q2: term_f_v] :
            ~ ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q2 ) @ Delta )
              | ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Q2 @ P ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_53_eps__statesE,axiom,
    ! [P: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ P @ ( tree_e551949757491771807term_f @ Delta ) )
     => ~ ! [Q2: bot_bot_term_f] :
            ~ ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q2 ) @ Delta )
              | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ P ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_54_eps__statesE,axiom,
    ! [P: ground_gterm_f,Delta: fset_P5324236452418035597term_f] :
      ( ( fmembe8264398993653236347term_f @ P @ ( tree_e2075845989350546787term_f @ Delta ) )
     => ~ ! [Q2: ground_gterm_f] :
            ~ ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ P @ Q2 ) @ Delta )
              | ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Q2 @ P ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_55_Pair__inject,axiom,
    ! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_56_Pair__inject,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A2: bot_bot_term_f,B2: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ A @ B )
        = ( produc2865564700782054965term_f @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_57_Pair__inject,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A2: ground_gterm_f,B2: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ A @ B )
        = ( produc3560254623552331287term_f @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_58_prod__cases,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,P: produc8027700073345061287rm_f_v] :
      ( ! [A4: term_f_v,B3: term_f_v] : ( P2 @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_59_prod__cases,axiom,
    ! [P2: produc250624644672406269term_f > $o,P: produc250624644672406269term_f] :
      ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f] : ( P2 @ ( produc2865564700782054965term_f @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_60_prod__cases,axiom,
    ! [P2: produc7245736746747425831term_f > $o,P: produc7245736746747425831term_f] :
      ( ! [A4: ground_gterm_f,B3: ground_gterm_f] : ( P2 @ ( produc3560254623552331287term_f @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_61_surj__pair,axiom,
    ! [P: produc8027700073345061287rm_f_v] :
    ? [X3: term_f_v,Y3: term_f_v] :
      ( P
      = ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_62_surj__pair,axiom,
    ! [P: produc250624644672406269term_f] :
    ? [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
      ( P
      = ( produc2865564700782054965term_f @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_63_surj__pair,axiom,
    ! [P: produc7245736746747425831term_f] :
    ? [X3: ground_gterm_f,Y3: ground_gterm_f] :
      ( P
      = ( produc3560254623552331287term_f @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_64_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8027700073345061287rm_f_v] :
      ~ ! [A4: term_f_v,B3: term_f_v] :
          ( Y
         != ( produc2510790566102465375rm_f_v @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_65_old_Oprod_Oexhaust,axiom,
    ! [Y: produc250624644672406269term_f] :
      ~ ! [A4: bot_bot_term_f,B3: bot_bot_term_f] :
          ( Y
         != ( produc2865564700782054965term_f @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_66_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7245736746747425831term_f] :
      ~ ! [A4: ground_gterm_f,B3: ground_gterm_f] :
          ( Y
         != ( produc3560254623552331287term_f @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_67__092_060open_062_Is_M_At_J_A_092_060in_062_Apair__at__lang_A_Iroot__pair__automaton_A_092_060F_062_AR_J_A_IRel_092_060_094sub_062f_AR_J_092_060close_062,axiom,
    member848276444142703440term_f @ ( produc3560254623552331287term_f @ s @ t ) @ ( pair_p5686396426817080910rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) @ ( lV_to_Rel_f_f_v @ r ) ) ).

% \<open>(s, t) \<in> pair_at_lang (root_pair_automaton \<F> R) (Rel\<^sub>f R)\<close>
thf(fact_68_fmember__iff__member__fset,axiom,
    ( fmembe2803290242953082350term_f
    = ( ^ [X4: produc7245736746747425831term_f,A5: fset_P5324236452418035597term_f] : ( member848276444142703440term_f @ X4 @ ( fset_P5939342050162774678term_f @ A5 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_69_fmember__iff__member__fset,axiom,
    ( fmembe6950792022577883716term_f
    = ( ^ [X4: produc250624644672406269term_f,A5: fset_P2882900458636368611term_f] : ( member4687689947140789414term_f @ X4 @ ( fset_P8925168969356593900term_f @ A5 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_70_fmember__iff__member__fset,axiom,
    ( fmembe1418995490368054919term_f
    = ( ^ [X4: bot_bot_term_f,A5: fset_Bot_bot_term_f] : ( member2723211850833494437term_f @ X4 @ ( fset_Bot_bot_term_f2 @ A5 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_71_fmember__iff__member__fset,axiom,
    ( fmembe774991991480784110rm_f_v
    = ( ^ [X4: produc8027700073345061287rm_f_v,A5: fset_P8018961893305114765rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ ( fset_P4617584883882644886rm_f_v @ A5 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_72_notin__fset,axiom,
    ! [X: produc7245736746747425831term_f,S2: fset_P5324236452418035597term_f] :
      ( ( ~ ( fmembe2803290242953082350term_f @ X @ S2 ) )
      = ( ~ ( member848276444142703440term_f @ X @ ( fset_P5939342050162774678term_f @ S2 ) ) ) ) ).

% notin_fset
thf(fact_73_notin__fset,axiom,
    ! [X: produc250624644672406269term_f,S2: fset_P2882900458636368611term_f] :
      ( ( ~ ( fmembe6950792022577883716term_f @ X @ S2 ) )
      = ( ~ ( member4687689947140789414term_f @ X @ ( fset_P8925168969356593900term_f @ S2 ) ) ) ) ).

% notin_fset
thf(fact_74_notin__fset,axiom,
    ! [X: bot_bot_term_f,S2: fset_Bot_bot_term_f] :
      ( ( ~ ( fmembe1418995490368054919term_f @ X @ S2 ) )
      = ( ~ ( member2723211850833494437term_f @ X @ ( fset_Bot_bot_term_f2 @ S2 ) ) ) ) ).

% notin_fset
thf(fact_75_notin__fset,axiom,
    ! [X: produc8027700073345061287rm_f_v,S2: fset_P8018961893305114765rm_f_v] :
      ( ( ~ ( fmembe774991991480784110rm_f_v @ X @ S2 ) )
      = ( ~ ( member4316649861536941904rm_f_v @ X @ ( fset_P4617584883882644886rm_f_v @ S2 ) ) ) ) ).

% notin_fset
thf(fact_76_fmember_Orep__eq,axiom,
    ( fmembe2803290242953082350term_f
    = ( ^ [X4: produc7245736746747425831term_f,Xa: fset_P5324236452418035597term_f] : ( member848276444142703440term_f @ X4 @ ( fset_P5939342050162774678term_f @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_77_fmember_Orep__eq,axiom,
    ( fmembe6950792022577883716term_f
    = ( ^ [X4: produc250624644672406269term_f,Xa: fset_P2882900458636368611term_f] : ( member4687689947140789414term_f @ X4 @ ( fset_P8925168969356593900term_f @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_78_fmember_Orep__eq,axiom,
    ( fmembe1418995490368054919term_f
    = ( ^ [X4: bot_bot_term_f,Xa: fset_Bot_bot_term_f] : ( member2723211850833494437term_f @ X4 @ ( fset_Bot_bot_term_f2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_79_fmember_Orep__eq,axiom,
    ( fmembe774991991480784110rm_f_v
    = ( ^ [X4: produc8027700073345061287rm_f_v,Xa: fset_P8018961893305114765rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ ( fset_P4617584883882644886rm_f_v @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_80_fImageI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f7835841554439036636term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_81_fImageI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f,A3: fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( fmembe8264398993653236347term_f @ A @ A3 )
       => ( fmembe8264398993653236347term_f @ B @ ( fSet_f1232815550461614398term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_82_fImageI,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v,A3: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( ( fmember_term_f_v @ A @ A3 )
       => ( fmember_term_f_v @ B @ ( fSet_f3701177912297302022rm_f_v @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_83_fImageI,axiom,
    ! [A: produc250624644672406269term_f,B: bot_bot_term_f,R2: fset_P947404469294174954term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ A @ B ) @ R2 )
     => ( ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f6497789455392978765term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_84_fImageI,axiom,
    ! [A: bot_bot_term_f,B: produc250624644672406269term_f,R2: fset_P4484195161054225940term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ A @ B ) @ R2 )
     => ( ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f6435055801683725359term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_85_fImageI,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: bot_bot_term_f,R2: fset_P3141206054603096980term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ A @ B ) @ R2 )
     => ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f3925250297721860343term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_86_fImageI,axiom,
    ! [A: bot_bot_term_f,B: produc8027700073345061287rm_f_v,R2: fset_P4422861445534322366rm_f_v,A3: fset_Bot_bot_term_f] :
      ( ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ A @ B ) @ R2 )
     => ( ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f7620545100792095449rm_f_v @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_87_fImageI,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,R2: fset_P616664398915176141term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ R2 )
     => ( ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f4528015000127795454term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_88_fImageI,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc250624644672406269term_f,R2: fset_P989200333075663523term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ A @ B ) @ R2 )
     => ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f777551917965208532term_f @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_89_fImageI,axiom,
    ! [A: produc250624644672406269term_f,B: produc8027700073345061287rm_f_v,R2: fset_P7732689718384382839rm_f_v,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ A @ B ) @ R2 )
     => ( ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f569563426813526952rm_f_v @ R2 @ A3 ) ) ) ) ).

% fImageI
thf(fact_90_rename__subst__domain__range__Var__rhs,axiom,
    ! [Rho: v > term_f_v] :
      ( ( rename1918500867710583060_v_f_v @ Rho @ var_v_f )
      = var_v_f ) ).

% rename_subst_domain_range_Var_rhs
thf(fact_91_mem__Collect__eq,axiom,
    ! [A: produc250624644672406269term_f,P2: produc250624644672406269term_f > $o] :
      ( ( member4687689947140789414term_f @ A @ ( collec753421845813909992term_f @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
    ! [A: produc7245736746747425831term_f,P2: produc7245736746747425831term_f > $o] :
      ( ( member848276444142703440term_f @ A @ ( collec3209569126566747026term_f @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A3: set_Pr6841673932529763677term_f] :
      ( ( collec753421845813909992term_f
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A3: set_Pr989862937836626183term_f] :
      ( ( collec3209569126566747026term_f
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_95_rename__subst__domain__range__Var__lhs,axiom,
    ! [Sigma: v > term_f_v] :
      ( ( rename1918500867710583060_v_f_v @ var_v_f @ Sigma )
      = Sigma ) ).

% rename_subst_domain_range_Var_lhs
thf(fact_96_restrict__subst__domain__Var,axiom,
    ! [V: set_v] :
      ( ( restri8836950881698410593in_v_f @ V @ var_v_f )
      = var_v_f ) ).

% restrict_subst_domain_Var
thf(fact_97_fconverse__iff,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f82489520865301080rm_f_v @ R2 ) )
      = ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R2 ) ) ).

% fconverse_iff
thf(fact_98_fconverse__iff,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f5567478040060018990term_f @ R2 ) )
      = ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B @ A ) @ R2 ) ) ).

% fconverse_iff
thf(fact_99_fconverse__iff,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f7363521725891221136term_f @ R2 ) )
      = ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B @ A ) @ R2 ) ) ).

% fconverse_iff
thf(fact_100_fconverseI,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( fSet_f82489520865301080rm_f_v @ R2 ) ) ) ).

% fconverseI
thf(fact_101_fconverseI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B @ A ) @ ( fSet_f5567478040060018990term_f @ R2 ) ) ) ).

% fconverseI
thf(fact_102_fconverseI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B @ A ) @ ( fSet_f7363521725891221136term_f @ R2 ) ) ) ).

% fconverseI
thf(fact_103_restrict__subst__domain__def,axiom,
    ( restri8836950881698410593in_v_f
    = ( ^ [V2: set_v,Sigma2: v > term_f_v,X4: v] : ( if_term_f_v @ ( member_v @ X4 @ V2 ) @ ( Sigma2 @ X4 ) @ ( var_v_f @ X4 ) ) ) ) ).

% restrict_subst_domain_def
thf(fact_104_fset__eqI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ! [X3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
          = ( fmembe774991991480784110rm_f_v @ X3 @ B4 ) )
     => ( A3 = B4 ) ) ).

% fset_eqI
thf(fact_105_fset__eqI,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
          = ( fmembe6950792022577883716term_f @ X3 @ B4 ) )
     => ( A3 = B4 ) ) ).

% fset_eqI
thf(fact_106_fset__eqI,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
          = ( fmembe1418995490368054919term_f @ X3 @ B4 ) )
     => ( A3 = B4 ) ) ).

% fset_eqI
thf(fact_107_fequalityCE,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( A3 = B4 )
     => ( ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
         => ~ ( fmembe774991991480784110rm_f_v @ C @ B4 ) )
       => ~ ( ~ ( fmembe774991991480784110rm_f_v @ C @ A3 )
           => ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_108_fequalityCE,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f,C: produc250624644672406269term_f] :
      ( ( A3 = B4 )
     => ( ( ( fmembe6950792022577883716term_f @ C @ A3 )
         => ~ ( fmembe6950792022577883716term_f @ C @ B4 ) )
       => ~ ( ~ ( fmembe6950792022577883716term_f @ C @ A3 )
           => ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_109_fequalityCE,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f,C: bot_bot_term_f] :
      ( ( A3 = B4 )
     => ( ( ( fmembe1418995490368054919term_f @ C @ A3 )
         => ~ ( fmembe1418995490368054919term_f @ C @ B4 ) )
       => ~ ( ~ ( fmembe1418995490368054919term_f @ C @ A3 )
           => ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_110_eq__fmem__trans,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( A = B )
     => ( ( fmembe774991991480784110rm_f_v @ B @ A3 )
       => ( fmembe774991991480784110rm_f_v @ A @ A3 ) ) ) ).

% eq_fmem_trans
thf(fact_111_eq__fmem__trans,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( A = B )
     => ( ( fmembe6950792022577883716term_f @ B @ A3 )
       => ( fmembe6950792022577883716term_f @ A @ A3 ) ) ) ).

% eq_fmem_trans
thf(fact_112_eq__fmem__trans,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( A = B )
     => ( ( fmembe1418995490368054919term_f @ B @ A3 )
       => ( fmembe1418995490368054919term_f @ A @ A3 ) ) ) ).

% eq_fmem_trans
thf(fact_113_eqfset__imp__iff,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( A3 = B4 )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
        = ( fmembe774991991480784110rm_f_v @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_114_eqfset__imp__iff,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f,X: produc250624644672406269term_f] :
      ( ( A3 = B4 )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
        = ( fmembe6950792022577883716term_f @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_115_eqfset__imp__iff,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ( A3 = B4 )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
        = ( fmembe1418995490368054919term_f @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_116_if__split__fmem1,axiom,
    ! [Q3: $o,X: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,B: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( if_Pro8994587301352985965rm_f_v @ Q3 @ X @ Y ) @ B )
      = ( ( Q3
         => ( fmembe774991991480784110rm_f_v @ X @ B ) )
        & ( ~ Q3
         => ( fmembe774991991480784110rm_f_v @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_117_if__split__fmem1,axiom,
    ! [Q3: $o,X: produc250624644672406269term_f,Y: produc250624644672406269term_f,B: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( if_Pro1782623486134218435term_f @ Q3 @ X @ Y ) @ B )
      = ( ( Q3
         => ( fmembe6950792022577883716term_f @ X @ B ) )
        & ( ~ Q3
         => ( fmembe6950792022577883716term_f @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_118_if__split__fmem1,axiom,
    ! [Q3: $o,X: bot_bot_term_f,Y: bot_bot_term_f,B: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ ( if_Bot_bot_term_f @ Q3 @ X @ Y ) @ B )
      = ( ( Q3
         => ( fmembe1418995490368054919term_f @ X @ B ) )
        & ( ~ Q3
         => ( fmembe1418995490368054919term_f @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_119_if__split__fmem2,axiom,
    ! [A: produc8027700073345061287rm_f_v,Q3: $o,X: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ ( if_fse5575177004205410643rm_f_v @ Q3 @ X @ Y ) )
      = ( ( Q3
         => ( fmembe774991991480784110rm_f_v @ A @ X ) )
        & ( ~ Q3
         => ( fmembe774991991480784110rm_f_v @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_120_if__split__fmem2,axiom,
    ! [A: produc250624644672406269term_f,Q3: $o,X: fset_P2882900458636368611term_f,Y: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ ( if_fse2792004000006630313term_f @ Q3 @ X @ Y ) )
      = ( ( Q3
         => ( fmembe6950792022577883716term_f @ A @ X ) )
        & ( ~ Q3
         => ( fmembe6950792022577883716term_f @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_121_if__split__fmem2,axiom,
    ! [A: bot_bot_term_f,Q3: $o,X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ ( if_fse3524882727628502648term_f @ Q3 @ X @ Y ) )
      = ( ( Q3
         => ( fmembe1418995490368054919term_f @ A @ X ) )
        & ( ~ Q3
         => ( fmembe1418995490368054919term_f @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_122_eqfelem__imp__iff,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( X = Y )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
        = ( fmembe774991991480784110rm_f_v @ Y @ A3 ) ) ) ).

% eqfelem_imp_iff
thf(fact_123_eqfelem__imp__iff,axiom,
    ! [X: produc250624644672406269term_f,Y: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( X = Y )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
        = ( fmembe6950792022577883716term_f @ Y @ A3 ) ) ) ).

% eqfelem_imp_iff
thf(fact_124_eqfelem__imp__iff,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( X = Y )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
        = ( fmembe1418995490368054919term_f @ Y @ A3 ) ) ) ).

% eqfelem_imp_iff
thf(fact_125_fconverseD,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f82489520865301080rm_f_v @ R2 ) )
     => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R2 ) ) ).

% fconverseD
thf(fact_126_fconverseD,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f5567478040060018990term_f @ R2 ) )
     => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B @ A ) @ R2 ) ) ).

% fconverseD
thf(fact_127_fconverseD,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f7363521725891221136term_f @ R2 ) )
     => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B @ A ) @ R2 ) ) ).

% fconverseD
thf(fact_128_fconverseE,axiom,
    ! [Yx: produc8027700073345061287rm_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ Yx @ ( fSet_f82489520865301080rm_f_v @ R2 ) )
     => ~ ! [X3: term_f_v,Y3: term_f_v] :
            ( ( Yx
              = ( produc2510790566102465375rm_f_v @ Y3 @ X3 ) )
           => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 ) ) ) ).

% fconverseE
thf(fact_129_fconverseE,axiom,
    ! [Yx: produc250624644672406269term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ Yx @ ( fSet_f5567478040060018990term_f @ R2 ) )
     => ~ ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
            ( ( Yx
              = ( produc2865564700782054965term_f @ Y3 @ X3 ) )
           => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 ) ) ) ).

% fconverseE
thf(fact_130_fconverseE,axiom,
    ! [Yx: produc7245736746747425831term_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ Yx @ ( fSet_f7363521725891221136term_f @ R2 ) )
     => ~ ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
            ( ( Yx
              = ( produc3560254623552331287term_f @ Y3 @ X3 ) )
           => ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 ) ) ) ).

% fconverseE
thf(fact_131_fset__cong,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( ( fset_P4617584883882644886rm_f_v @ X )
        = ( fset_P4617584883882644886rm_f_v @ Y ) )
      = ( X = Y ) ) ).

% fset_cong
thf(fact_132_FSet__Utils_OImageE,axiom,
    ! [B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fSet_f7835841554439036636term_f @ R2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_133_FSet__Utils_OImageE,axiom,
    ! [B: ground_gterm_f,R2: fset_P5324236452418035597term_f,A3: fset_Ground_gterm_f] :
      ( ( fmembe8264398993653236347term_f @ B @ ( fSet_f1232815550461614398term_f @ R2 @ A3 ) )
     => ~ ! [X3: ground_gterm_f] :
            ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe8264398993653236347term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_134_FSet__Utils_OImageE,axiom,
    ! [B: term_f_v,R2: fset_P8018961893305114765rm_f_v,A3: fset_term_f_v] :
      ( ( fmember_term_f_v @ B @ ( fSet_f3701177912297302022rm_f_v @ R2 @ A3 ) )
     => ~ ! [X3: term_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ B ) @ R2 )
           => ~ ( fmember_term_f_v @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_135_FSet__Utils_OImageE,axiom,
    ! [B: produc250624644672406269term_f,R2: fset_P4484195161054225940term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fSet_f6435055801683725359term_f @ R2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_136_FSet__Utils_OImageE,axiom,
    ! [B: bot_bot_term_f,R2: fset_P947404469294174954term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fSet_f6497789455392978765term_f @ R2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_137_FSet__Utils_OImageE,axiom,
    ! [B: produc8027700073345061287rm_f_v,R2: fset_P4422861445534322366rm_f_v,A3: fset_Bot_bot_term_f] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f7620545100792095449rm_f_v @ R2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ X3 @ B ) @ R2 )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_138_FSet__Utils_OImageE,axiom,
    ! [B: bot_bot_term_f,R2: fset_P3141206054603096980term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fSet_f3925250297721860343term_f @ R2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_139_FSet__Utils_OImageE,axiom,
    ! [B: produc250624644672406269term_f,R2: fset_P616664398915176141term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fSet_f4528015000127795454term_f @ R2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_140_FSet__Utils_OImageE,axiom,
    ! [B: produc8027700073345061287rm_f_v,R2: fset_P7732689718384382839rm_f_v,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f569563426813526952rm_f_v @ R2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ X3 @ B ) @ R2 )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_141_FSet__Utils_OImageE,axiom,
    ! [B: produc250624644672406269term_f,R2: fset_P989200333075663523term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fSet_f777551917965208532term_f @ R2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ X3 @ B ) @ R2 )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_142_frev__ImageI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f7835841554439036636term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_143_frev__ImageI,axiom,
    ! [A: ground_gterm_f,A3: fset_Ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe8264398993653236347term_f @ A @ A3 )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
       => ( fmembe8264398993653236347term_f @ B @ ( fSet_f1232815550461614398term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_144_frev__ImageI,axiom,
    ! [A: term_f_v,A3: fset_term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmember_term_f_v @ A @ A3 )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
       => ( fmember_term_f_v @ B @ ( fSet_f3701177912297302022rm_f_v @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_145_frev__ImageI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: bot_bot_term_f,R2: fset_P947404469294174954term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ A @ B ) @ R2 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f6497789455392978765term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_146_frev__ImageI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc250624644672406269term_f,R2: fset_P4484195161054225940term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ A @ B ) @ R2 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f6435055801683725359term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_147_frev__ImageI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: bot_bot_term_f,R2: fset_P3141206054603096980term_f] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ A @ B ) @ R2 )
       => ( fmembe1418995490368054919term_f @ B @ ( fSet_f3925250297721860343term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_148_frev__ImageI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc8027700073345061287rm_f_v,R2: fset_P4422861445534322366rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ A @ B ) @ R2 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f7620545100792095449rm_f_v @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_149_frev__ImageI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc250624644672406269term_f,R2: fset_P616664398915176141term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ R2 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f4528015000127795454term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_150_frev__ImageI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: produc250624644672406269term_f,R2: fset_P989200333075663523term_f] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ A @ B ) @ R2 )
       => ( fmembe6950792022577883716term_f @ B @ ( fSet_f777551917965208532term_f @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_151_frev__ImageI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc8027700073345061287rm_f_v,R2: fset_P7732689718384382839rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ A @ B ) @ R2 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fSet_f569563426813526952rm_f_v @ R2 @ A3 ) ) ) ) ).

% frev_ImageI
thf(fact_152_assms_I2_J,axiom,
    ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ r ) @ f2 ).

% assms(2)
thf(fact_153_st_I2_J,axiom,
    fmembe1418995490368054919term_f @ q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ s ) ).

% st(2)
thf(fact_154_st_I3_J,axiom,
    fmembe1418995490368054919term_f @ p @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ t ) ).

% st(3)
thf(fact_155__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_Aq_O_A_092_060lbrakk_062_Iq_M_Ap_J_A_124_092_060in_062_124_ARel_092_060_094sub_062f_AR_059_Aq_A_124_092_060in_062_124_Agta__der_A_Ifst_A_Iroot__pair__automaton_A_092_060F_062_AR_J_J_As_059_Ap_A_124_092_060in_062_124_Agta__der_A_Isnd_A_Iroot__pair__automaton_A_092_060F_062_AR_J_J_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P3: bot_bot_term_f,Q2: bot_bot_term_f] :
        ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ P3 ) @ ( lV_to_Rel_f_f_v @ r ) )
       => ( ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ s ) )
         => ~ ( fmembe1418995490368054919term_f @ P3 @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ t ) ) ) ) ).

% \<open>\<And>thesis. (\<And>p q. \<lbrakk>(q, p) |\<in>| Rel\<^sub>f R; q |\<in>| gta_der (fst (root_pair_automaton \<F> R)) s; p |\<in>| gta_der (snd (root_pair_automaton \<F> R)) t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_156__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_Ap_O_A_092_060lbrakk_062_Iq_M_Ap_J_A_124_092_060in_062_124_ARel_092_060_094sub_062f_AR_059_Aq_A_124_092_060in_062_124_Agta__der_A_Ifst_A_Iroot__pair__automaton_A_092_060F_062_AR_J_J_As_059_Ap_A_124_092_060in_062_124_Agta__der_A_Isnd_A_Iroot__pair__automaton_A_092_060F_062_AR_J_J_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q2: bot_bot_term_f,P3: bot_bot_term_f] :
        ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ P3 ) @ ( lV_to_Rel_f_f_v @ r ) )
       => ( ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ s ) )
         => ~ ( fmembe1418995490368054919term_f @ P3 @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ f2 @ r ) ) @ t ) ) ) ) ).

% \<open>\<And>thesis. (\<And>q p. \<lbrakk>(q, p) |\<in>| Rel\<^sub>f R; q |\<in>| gta_der (fst (root_pair_automaton \<F> R)) s; p |\<in>| gta_der (snd (root_pair_automaton \<F> R)) t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_157_agtt__grrstep__def,axiom,
    ( lV_to_5725947919369604231ep_f_v
    = ( ^ [R3: fset_P8018961893305114765rm_f_v,F: fset_P6228066233360383026_f_nat] : ( pair_p1447136629559935937term_f @ ( lV_to_8065481040984897572_f_v_v @ F @ R3 ) @ ( lV_to_Rel_f_f_v @ R3 ) ) ) ) ).

% agtt_grrstep_def
thf(fact_158_frelcompI,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v,C: term_f_v,S: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ S )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( fSet_f1619609727995933463rm_f_v @ R2 @ S ) ) ) ) ).

% frelcompI
thf(fact_159_frelcompI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,C: bot_bot_term_f,S: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B @ C ) @ S )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( fSet_f8009933528932036120term_f @ R2 @ S ) ) ) ) ).

% frelcompI
thf(fact_160_frelcompI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f,C: ground_gterm_f,S: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B @ C ) @ S )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( fSet_f2546636934287035336term_f @ R2 @ S ) ) ) ) ).

% frelcompI
thf(fact_161_fId__on_Orep__eq,axiom,
    ! [X: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P8224949580060463702rm_f_v @ ( fSet_f4703993580257979951rm_f_v @ X ) )
      = ( id_on_7786124912184395972rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) ) ) ).

% fId_on.rep_eq
thf(fact_162_fId__on_Orep__eq,axiom,
    ! [X: fset_term_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fSet_fId_on_term_f_v @ X ) )
      = ( id_on_term_f_v @ ( fset_term_f_v2 @ X ) ) ) ).

% fId_on.rep_eq
thf(fact_163_bless__eq__to__instance,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f )
     => ( ( linear_term_f_v @ S )
       => ? [Sigma3: v > term_f_v] :
            ( ( subst_7901567856578221233_f_v_v @ S @ Sigma3 )
            = T ) ) ) ).

% bless_eq_to_instance
thf(fact_164_fsubset__antisym,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( ord_le1552505484586773650_f_nat @ B4 @ A3 )
       => ( A3 = B4 ) ) ) ).

% fsubset_antisym
thf(fact_165_fsubsetI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ! [X3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
         => ( fmembe774991991480784110rm_f_v @ X3 @ B4 ) )
     => ( ord_le4587745213494032429rm_f_v @ A3 @ B4 ) ) ).

% fsubsetI
thf(fact_166_fsubsetI,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
         => ( fmembe6950792022577883716term_f @ X3 @ B4 ) )
     => ( ord_le7092741666294821507term_f @ A3 @ B4 ) ) ).

% fsubsetI
thf(fact_167_fsubsetI,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
         => ( fmembe1418995490368054919term_f @ X3 @ B4 ) )
     => ( ord_le7572167314378612510term_f @ A3 @ B4 ) ) ).

% fsubsetI
thf(fact_168_fsubsetI,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ! [X3: product_prod_f_nat] :
          ( ( fmembe6795308708001954811_f_nat @ X3 @ A3 )
         => ( fmembe6795308708001954811_f_nat @ X3 @ B4 ) )
     => ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ).

% fsubsetI
thf(fact_169_subst__apply__term__empty,axiom,
    ! [T: term_f_v] :
      ( ( subst_7901567856578221233_f_v_v @ T @ var_v_f )
      = T ) ).

% subst_apply_term_empty
thf(fact_170_prod_Ocollapse,axiom,
    ! [Prod: produc250624644672406269term_f] :
      ( ( produc2865564700782054965term_f @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_171_prod_Ocollapse,axiom,
    ! [Prod: produc7245736746747425831term_f] :
      ( ( produc3560254623552331287term_f @ ( produc1239122367099457539term_f @ Prod ) @ ( produc1464490519189811525term_f @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_172_prod_Ocollapse,axiom,
    ! [Prod: produc6920895926601326823rm_f_f] :
      ( ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ Prod ) @ ( produc1110312829679706757rm_f_f @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_173_prod_Ocollapse,axiom,
    ! [Prod: produc8027700073345061287rm_f_v] :
      ( ( produc2510790566102465375rm_f_v @ ( produc1199688945740118859rm_f_v @ Prod ) @ ( produc5682565293565518221rm_f_v @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_174_fequalityE,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( A3 = B4 )
     => ~ ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
         => ~ ( ord_le1552505484586773650_f_nat @ B4 @ A3 ) ) ) ).

% fequalityE
thf(fact_175_fequalityD1,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( A3 = B4 )
     => ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ).

% fequalityD1
thf(fact_176_fequalityD2,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( A3 = B4 )
     => ( ord_le1552505484586773650_f_nat @ B4 @ A3 ) ) ).

% fequalityD2
thf(fact_177_fsubset__refl,axiom,
    ! [A3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A3 @ A3 ) ).

% fsubset_refl
thf(fact_178_fsubset__trans,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( ord_le1552505484586773650_f_nat @ B4 @ C2 )
       => ( ord_le1552505484586773650_f_nat @ A3 @ C2 ) ) ) ).

% fsubset_trans
thf(fact_179_fset__eq__fsubset,axiom,
    ( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y4 = Z ) )
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B5: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ A5 @ B5 )
          & ( ord_le1552505484586773650_f_nat @ B5 @ A5 ) ) ) ) ).

% fset_eq_fsubset
thf(fact_180_surjective__pairing,axiom,
    ! [T: produc250624644672406269term_f] :
      ( T
      = ( produc2865564700782054965term_f @ ( produc5789359262401207841term_f @ T ) @ ( produc656023363694804067term_f @ T ) ) ) ).

% surjective_pairing
thf(fact_181_surjective__pairing,axiom,
    ! [T: produc7245736746747425831term_f] :
      ( T
      = ( produc3560254623552331287term_f @ ( produc1239122367099457539term_f @ T ) @ ( produc1464490519189811525term_f @ T ) ) ) ).

% surjective_pairing
thf(fact_182_surjective__pairing,axiom,
    ! [T: produc6920895926601326823rm_f_f] :
      ( T
      = ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ T ) @ ( produc1110312829679706757rm_f_f @ T ) ) ) ).

% surjective_pairing
thf(fact_183_surjective__pairing,axiom,
    ! [T: produc8027700073345061287rm_f_v] :
      ( T
      = ( produc2510790566102465375rm_f_v @ ( produc1199688945740118859rm_f_v @ T ) @ ( produc5682565293565518221rm_f_v @ T ) ) ) ).

% surjective_pairing
thf(fact_184_prod_Oexhaust__sel,axiom,
    ! [Prod: produc250624644672406269term_f] :
      ( Prod
      = ( produc2865564700782054965term_f @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_185_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7245736746747425831term_f] :
      ( Prod
      = ( produc3560254623552331287term_f @ ( produc1239122367099457539term_f @ Prod ) @ ( produc1464490519189811525term_f @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_186_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6920895926601326823rm_f_f] :
      ( Prod
      = ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ Prod ) @ ( produc1110312829679706757rm_f_f @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_187_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8027700073345061287rm_f_v] :
      ( Prod
      = ( produc2510790566102465375rm_f_v @ ( produc1199688945740118859rm_f_v @ Prod ) @ ( produc5682565293565518221rm_f_v @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_188_prod_Oexpand,axiom,
    ! [Prod: produc6920895926601326823rm_f_f,Prod2: produc6920895926601326823rm_f_f] :
      ( ( ( ( produc1890379913328437059rm_f_f @ Prod )
          = ( produc1890379913328437059rm_f_f @ Prod2 ) )
        & ( ( produc1110312829679706757rm_f_f @ Prod )
          = ( produc1110312829679706757rm_f_f @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_189_prod_Oexpand,axiom,
    ! [Prod: produc8027700073345061287rm_f_v,Prod2: produc8027700073345061287rm_f_v] :
      ( ( ( ( produc1199688945740118859rm_f_v @ Prod )
          = ( produc1199688945740118859rm_f_v @ Prod2 ) )
        & ( ( produc5682565293565518221rm_f_v @ Prod )
          = ( produc5682565293565518221rm_f_v @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_190_prod__eqI,axiom,
    ! [P: produc6920895926601326823rm_f_f,Q: produc6920895926601326823rm_f_f] :
      ( ( ( produc1890379913328437059rm_f_f @ P )
        = ( produc1890379913328437059rm_f_f @ Q ) )
     => ( ( ( produc1110312829679706757rm_f_f @ P )
          = ( produc1110312829679706757rm_f_f @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_191_prod__eqI,axiom,
    ! [P: produc8027700073345061287rm_f_v,Q: produc8027700073345061287rm_f_v] :
      ( ( ( produc1199688945740118859rm_f_v @ P )
        = ( produc1199688945740118859rm_f_v @ Q ) )
     => ( ( ( produc5682565293565518221rm_f_v @ P )
          = ( produc5682565293565518221rm_f_v @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_192_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc6920895926601326823rm_f_f,Z: produc6920895926601326823rm_f_f] : ( Y4 = Z ) )
    = ( ^ [S3: produc6920895926601326823rm_f_f,T2: produc6920895926601326823rm_f_f] :
          ( ( ( produc1890379913328437059rm_f_f @ S3 )
            = ( produc1890379913328437059rm_f_f @ T2 ) )
          & ( ( produc1110312829679706757rm_f_f @ S3 )
            = ( produc1110312829679706757rm_f_f @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_193_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] : ( Y4 = Z ) )
    = ( ^ [S3: produc8027700073345061287rm_f_v,T2: produc8027700073345061287rm_f_v] :
          ( ( ( produc1199688945740118859rm_f_v @ S3 )
            = ( produc1199688945740118859rm_f_v @ T2 ) )
          & ( ( produc5682565293565518221rm_f_v @ S3 )
            = ( produc5682565293565518221rm_f_v @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_194_fImage__mono,axiom,
    ! [R4: fset_P7991872372518838603_f_nat,R2: fset_P7991872372518838603_f_nat,A6: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le6425377310842345707_f_nat @ R4 @ R2 )
     => ( ( ord_le1552505484586773650_f_nat @ A6 @ A3 )
       => ( ord_le1552505484586773650_f_nat @ ( fSet_f7437826027993719364_f_nat @ R4 @ A6 ) @ ( fSet_f7437826027993719364_f_nat @ R2 @ A3 ) ) ) ) ).

% fImage_mono
thf(fact_195_fImage__mono,axiom,
    ! [R4: fset_P6228066233360383026_f_nat,R2: fset_P6228066233360383026_f_nat,A6: fset_f,A3: fset_f] :
      ( ( ord_le1552505484586773650_f_nat @ R4 @ R2 )
     => ( ( ord_less_eq_fset_f @ A6 @ A3 )
       => ( ord_less_eq_fset_nat @ ( fSet_fImage_f_nat @ R4 @ A6 ) @ ( fSet_fImage_f_nat @ R2 @ A3 ) ) ) ) ).

% fImage_mono
thf(fact_196_eps__states__mono,axiom,
    ! [Delta: fset_P7991872372518838603_f_nat,Gamma: fset_P7991872372518838603_f_nat] :
      ( ( ord_le6425377310842345707_f_nat @ Delta @ Gamma )
     => ( ord_le1552505484586773650_f_nat @ ( tree_e2646172767733093139_f_nat @ Delta ) @ ( tree_e2646172767733093139_f_nat @ Gamma ) ) ) ).

% eps_states_mono
thf(fact_197_subst__apply__eq__Var,axiom,
    ! [S: term_f_v,Sigma: v > term_f_v,X: v] :
      ( ( ( subst_7901567856578221233_f_v_v @ S @ Sigma )
        = ( var_v_f @ X ) )
     => ~ ! [Y3: v] :
            ( ( S
              = ( var_v_f @ Y3 ) )
           => ( ( Sigma @ Y3 )
             != ( var_v_f @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_198_snd__conv,axiom,
    ! [X1: bot_bot_term_f,X2: bot_bot_term_f] :
      ( ( produc656023363694804067term_f @ ( produc2865564700782054965term_f @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_199_snd__conv,axiom,
    ! [X1: ground_gterm_f,X2: ground_gterm_f] :
      ( ( produc1464490519189811525term_f @ ( produc3560254623552331287term_f @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_200_snd__conv,axiom,
    ! [X1: tree_t2044664379607036063rm_f_f,X2: tree_t2044664379607036063rm_f_f] :
      ( ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_201_snd__conv,axiom,
    ! [X1: term_f_v,X2: term_f_v] :
      ( ( produc5682565293565518221rm_f_v @ ( produc2510790566102465375rm_f_v @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_202_snd__eqD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A: bot_bot_term_f] :
      ( ( ( produc656023363694804067term_f @ ( produc2865564700782054965term_f @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_203_snd__eqD,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,A: ground_gterm_f] :
      ( ( ( produc1464490519189811525term_f @ ( produc3560254623552331287term_f @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_204_snd__eqD,axiom,
    ! [X: tree_t2044664379607036063rm_f_f,Y: tree_t2044664379607036063rm_f_f,A: tree_t2044664379607036063rm_f_f] :
      ( ( ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_205_snd__eqD,axiom,
    ! [X: term_f_v,Y: term_f_v,A: term_f_v] :
      ( ( ( produc5682565293565518221rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_206_fst__conv,axiom,
    ! [X1: bot_bot_term_f,X2: bot_bot_term_f] :
      ( ( produc5789359262401207841term_f @ ( produc2865564700782054965term_f @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_207_fst__conv,axiom,
    ! [X1: ground_gterm_f,X2: ground_gterm_f] :
      ( ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_208_fst__conv,axiom,
    ! [X1: tree_t2044664379607036063rm_f_f,X2: tree_t2044664379607036063rm_f_f] :
      ( ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_209_fst__conv,axiom,
    ! [X1: term_f_v,X2: term_f_v] :
      ( ( produc1199688945740118859rm_f_v @ ( produc2510790566102465375rm_f_v @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_210_fst__eqD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A: bot_bot_term_f] :
      ( ( ( produc5789359262401207841term_f @ ( produc2865564700782054965term_f @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_211_fst__eqD,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,A: ground_gterm_f] :
      ( ( ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_212_fst__eqD,axiom,
    ! [X: tree_t2044664379607036063rm_f_f,Y: tree_t2044664379607036063rm_f_f,A: tree_t2044664379607036063rm_f_f] :
      ( ( ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_213_fst__eqD,axiom,
    ! [X: term_f_v,Y: term_f_v,A: term_f_v] :
      ( ( ( produc1199688945740118859rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_214_fsubsetD,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v,C: produc8027700073345061287rm_f_v] :
      ( ( ord_le4587745213494032429rm_f_v @ A3 @ B4 )
     => ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
       => ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_215_fsubsetD,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f,C: produc250624644672406269term_f] :
      ( ( ord_le7092741666294821507term_f @ A3 @ B4 )
     => ( ( fmembe6950792022577883716term_f @ C @ A3 )
       => ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_216_fsubsetD,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f,C: bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ A3 @ B4 )
     => ( ( fmembe1418995490368054919term_f @ C @ A3 )
       => ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_217_fsubsetD,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C: product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( fmembe6795308708001954811_f_nat @ C @ A3 )
       => ( fmembe6795308708001954811_f_nat @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_218_fin__mono,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( ord_le4587745213494032429rm_f_v @ A3 @ B4 )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
       => ( fmembe774991991480784110rm_f_v @ X @ B4 ) ) ) ).

% fin_mono
thf(fact_219_fin__mono,axiom,
    ! [A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f,X: produc250624644672406269term_f] :
      ( ( ord_le7092741666294821507term_f @ A3 @ B4 )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
       => ( fmembe6950792022577883716term_f @ X @ B4 ) ) ) ).

% fin_mono
thf(fact_220_fin__mono,axiom,
    ! [A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ A3 @ B4 )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
       => ( fmembe1418995490368054919term_f @ X @ B4 ) ) ) ).

% fin_mono
thf(fact_221_fin__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,X: product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( fmembe6795308708001954811_f_nat @ X @ A3 )
       => ( fmembe6795308708001954811_f_nat @ X @ B4 ) ) ) ).

% fin_mono
thf(fact_222_instance__to__bless__eq,axiom,
    ! [S: term_f_v,Sigma: v > term_f_v,T: term_f_v] :
      ( ( ( subst_7901567856578221233_f_v_v @ S @ Sigma )
        = T )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f ) ) ).

% instance_to_bless_eq
thf(fact_223_frelcompE,axiom,
    ! [Xz: produc8027700073345061287rm_f_v,R2: fset_P8018961893305114765rm_f_v,S: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ Xz @ ( fSet_f1619609727995933463rm_f_v @ R2 @ S ) )
     => ~ ! [X3: term_f_v,Y3: term_f_v,Z2: term_f_v] :
            ( ( Xz
              = ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) )
           => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
             => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_224_frelcompE,axiom,
    ! [Xz: produc250624644672406269term_f,R2: fset_P2882900458636368611term_f,S: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ Xz @ ( fSet_f8009933528932036120term_f @ R2 @ S ) )
     => ~ ! [X3: bot_bot_term_f,Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
            ( ( Xz
              = ( produc2865564700782054965term_f @ X3 @ Z2 ) )
           => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
             => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_225_frelcompE,axiom,
    ! [Xz: produc7245736746747425831term_f,R2: fset_P5324236452418035597term_f,S: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ Xz @ ( fSet_f2546636934287035336term_f @ R2 @ S ) )
     => ~ ! [X3: ground_gterm_f,Y3: ground_gterm_f,Z2: ground_gterm_f] :
            ( ( Xz
              = ( produc3560254623552331287term_f @ X3 @ Z2 ) )
           => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
             => ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_226_pair__at__langI,axiom,
    ! [Q: produc8027700073345061287rm_f_v,G: produc3419615073385449853_f_v_f,S: ground_gterm_f,P: produc8027700073345061287rm_f_v,T: ground_gterm_f,Q3: fset_P7452185835880607565rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
     => ( ( fmembe774991991480784110rm_f_v @ P @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) )
       => ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ Q @ P ) @ Q3 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p3958297607064521345_f_v_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_langI
thf(fact_227_pair__at__langI,axiom,
    ! [Q: produc250624644672406269term_f,G: produc4465884753748837329rm_f_f,S: ground_gterm_f,P: produc250624644672406269term_f,T: ground_gterm_f,Q3: fset_P616664398915176141term_f] :
      ( ( fmembe6950792022577883716term_f @ Q @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
     => ( ( fmembe6950792022577883716term_f @ P @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) )
       => ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ Q @ P ) @ Q3 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p4181219179613995691rm_f_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_langI
thf(fact_228_pair__at__langI,axiom,
    ! [Q: term_f_v,G: produc169004890939464999_f_v_f,S: ground_gterm_f,P: term_f_v,T: ground_gterm_f,Q3: fset_P8018961893305114765rm_f_v] :
      ( ( fmember_term_f_v @ Q @ ( tree_g4770736341938457127_f_v_f @ ( produc2716261656266771715_f_v_f @ G ) @ S ) )
     => ( ( fmember_term_f_v @ P @ ( tree_g4770736341938457127_f_v_f @ ( produc5187309876156436549_f_v_f @ G ) @ T ) )
       => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Q @ P ) @ Q3 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5494371807164803257_f_v_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_langI
thf(fact_229_pair__at__langI,axiom,
    ! [Q: ground_gterm_f,G: produc4554128051175246307rm_f_f,S: ground_gterm_f,P: ground_gterm_f,T: ground_gterm_f,Q3: fset_P5324236452418035597term_f] :
      ( ( fmembe8264398993653236347term_f @ Q @ ( tree_g2823239125338765382rm_f_f @ ( produc486686396505431687rm_f_f @ G ) @ S ) )
     => ( ( fmembe8264398993653236347term_f @ P @ ( tree_g2823239125338765382rm_f_f @ ( produc6723095485791902921rm_f_f @ G ) @ T ) )
       => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Q @ P ) @ Q3 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5324507160602824756rm_f_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_langI
thf(fact_230_pair__at__langI,axiom,
    ! [Q: bot_bot_term_f,G: produc6920895926601326823rm_f_f,S: ground_gterm_f,P: bot_bot_term_f,T: ground_gterm_f,Q3: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ P @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) )
       => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ P ) @ Q3 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5686396426817080910rm_f_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_langI
thf(fact_231_pair__at__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc3419615073385449853_f_v_f,Q3: fset_P7452185835880607565rm_f_v] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p3958297607064521345_f_v_f @ G @ Q3 ) )
     => ~ ! [Q2: produc8027700073345061287rm_f_v,P3: produc8027700073345061287rm_f_v] :
            ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ Q2 @ P3 ) @ Q3 )
           => ( ( fmembe774991991480784110rm_f_v @ Q2 @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
             => ~ ( fmembe774991991480784110rm_f_v @ P3 @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) ) ) ) ) ).

% pair_at_langE
thf(fact_232_pair__at__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc4465884753748837329rm_f_f,Q3: fset_P616664398915176141term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p4181219179613995691rm_f_f @ G @ Q3 ) )
     => ~ ! [Q2: produc250624644672406269term_f,P3: produc250624644672406269term_f] :
            ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ Q2 @ P3 ) @ Q3 )
           => ( ( fmembe6950792022577883716term_f @ Q2 @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
             => ~ ( fmembe6950792022577883716term_f @ P3 @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) ) ) ) ) ).

% pair_at_langE
thf(fact_233_pair__at__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc169004890939464999_f_v_f,Q3: fset_P8018961893305114765rm_f_v] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5494371807164803257_f_v_f @ G @ Q3 ) )
     => ~ ! [Q2: term_f_v,P3: term_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Q2 @ P3 ) @ Q3 )
           => ( ( fmember_term_f_v @ Q2 @ ( tree_g4770736341938457127_f_v_f @ ( produc2716261656266771715_f_v_f @ G ) @ S ) )
             => ~ ( fmember_term_f_v @ P3 @ ( tree_g4770736341938457127_f_v_f @ ( produc5187309876156436549_f_v_f @ G ) @ T ) ) ) ) ) ).

% pair_at_langE
thf(fact_234_pair__at__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc4554128051175246307rm_f_f,Q3: fset_P5324236452418035597term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5324507160602824756rm_f_f @ G @ Q3 ) )
     => ~ ! [Q2: ground_gterm_f,P3: ground_gterm_f] :
            ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Q2 @ P3 ) @ Q3 )
           => ( ( fmembe8264398993653236347term_f @ Q2 @ ( tree_g2823239125338765382rm_f_f @ ( produc486686396505431687rm_f_f @ G ) @ S ) )
             => ~ ( fmembe8264398993653236347term_f @ P3 @ ( tree_g2823239125338765382rm_f_f @ ( produc6723095485791902921rm_f_f @ G ) @ T ) ) ) ) ) ).

% pair_at_langE
thf(fact_235_pair__at__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc6920895926601326823rm_f_f,Q3: fset_P2882900458636368611term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5686396426817080910rm_f_f @ G @ Q3 ) )
     => ~ ! [Q2: bot_bot_term_f,P3: bot_bot_term_f] :
            ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ P3 ) @ Q3 )
           => ( ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
             => ~ ( fmembe1418995490368054919term_f @ P3 @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) ) ) ) ) ).

% pair_at_langE
thf(fact_236_Id__onI,axiom,
    ! [A: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ A @ A3 )
     => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ A ) @ ( id_on_4928293012650018330term_f @ A3 ) ) ) ).

% Id_onI
thf(fact_237_Id__onI,axiom,
    ! [A: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ A @ A3 )
     => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ A ) @ ( id_on_5951252762333000388term_f @ A3 ) ) ) ).

% Id_onI
thf(fact_238_Id__onI,axiom,
    ! [A: term_f_v,A3: set_term_f_v] :
      ( ( member_term_f_v @ A @ A3 )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( id_on_term_f_v @ A3 ) ) ) ).

% Id_onI
thf(fact_239_Id__onI,axiom,
    ! [A: bot_bot_term_f,A3: set_Bot_bot_term_f] :
      ( ( member2723211850833494437term_f @ A @ A3 )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ A ) @ ( id_on_Bot_bot_term_f @ A3 ) ) ) ).

% Id_onI
thf(fact_240_Id__onI,axiom,
    ! [A: ground_gterm_f,A3: set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ A @ A3 )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ A ) @ ( id_on_Ground_gterm_f @ A3 ) ) ) ).

% Id_onI
thf(fact_241_exI__realizer,axiom,
    ! [P2: bot_bot_term_f > bot_bot_term_f > $o,Y: bot_bot_term_f,X: bot_bot_term_f] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc656023363694804067term_f @ ( produc2865564700782054965term_f @ X @ Y ) ) @ ( produc5789359262401207841term_f @ ( produc2865564700782054965term_f @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_242_exI__realizer,axiom,
    ! [P2: ground_gterm_f > ground_gterm_f > $o,Y: ground_gterm_f,X: ground_gterm_f] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc1464490519189811525term_f @ ( produc3560254623552331287term_f @ X @ Y ) ) @ ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_243_exI__realizer,axiom,
    ! [P2: tree_t2044664379607036063rm_f_f > tree_t2044664379607036063rm_f_f > $o,Y: tree_t2044664379607036063rm_f_f,X: tree_t2044664379607036063rm_f_f] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ X @ Y ) ) @ ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_244_exI__realizer,axiom,
    ! [P2: term_f_v > term_f_v > $o,Y: term_f_v,X: term_f_v] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc5682565293565518221rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) ) @ ( produc1199688945740118859rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_245_conjI__realizer,axiom,
    ! [P2: bot_bot_term_f > $o,P: bot_bot_term_f,Q3: bot_bot_term_f > $o,Q: bot_bot_term_f] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc5789359262401207841term_f @ ( produc2865564700782054965term_f @ P @ Q ) ) )
          & ( Q3 @ ( produc656023363694804067term_f @ ( produc2865564700782054965term_f @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_246_conjI__realizer,axiom,
    ! [P2: ground_gterm_f > $o,P: ground_gterm_f,Q3: ground_gterm_f > $o,Q: ground_gterm_f] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ P @ Q ) ) )
          & ( Q3 @ ( produc1464490519189811525term_f @ ( produc3560254623552331287term_f @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_247_conjI__realizer,axiom,
    ! [P2: tree_t2044664379607036063rm_f_f > $o,P: tree_t2044664379607036063rm_f_f,Q3: tree_t2044664379607036063rm_f_f > $o,Q: tree_t2044664379607036063rm_f_f] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ P @ Q ) ) )
          & ( Q3 @ ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_248_conjI__realizer,axiom,
    ! [P2: term_f_v > $o,P: term_f_v,Q3: term_f_v > $o,Q: term_f_v] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1199688945740118859rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q ) ) )
          & ( Q3 @ ( produc5682565293565518221rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_249_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: bot_bot_term_f > bot_bot_term_f > $o,X: bot_bot_term_f,Y: bot_bot_term_f,A: produc250624644672406269term_f] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2865564700782054965term_f @ X @ Y ) )
       => ( P2 @ ( produc5789359262401207841term_f @ A ) @ ( produc656023363694804067term_f @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_250_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ground_gterm_f > ground_gterm_f > $o,X: ground_gterm_f,Y: ground_gterm_f,A: produc7245736746747425831term_f] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc3560254623552331287term_f @ X @ Y ) )
       => ( P2 @ ( produc1239122367099457539term_f @ A ) @ ( produc1464490519189811525term_f @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_251_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: tree_t2044664379607036063rm_f_f > tree_t2044664379607036063rm_f_f > $o,X: tree_t2044664379607036063rm_f_f,Y: tree_t2044664379607036063rm_f_f,A: produc6920895926601326823rm_f_f] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc1864056441719732567rm_f_f @ X @ Y ) )
       => ( P2 @ ( produc1890379913328437059rm_f_f @ A ) @ ( produc1110312829679706757rm_f_f @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: term_f_v > term_f_v > $o,X: term_f_v,Y: term_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2510790566102465375rm_f_v @ X @ Y ) )
       => ( P2 @ ( produc1199688945740118859rm_f_v @ A ) @ ( produc5682565293565518221rm_f_v @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_253_dual__order_Orefl,axiom,
    ! [A: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A @ A ) ).

% dual_order.refl
thf(fact_254_order__refl,axiom,
    ! [X: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X @ X ) ).

% order_refl
thf(fact_255_Id__on__iff,axiom,
    ! [X: produc250624644672406269term_f,Y: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ X @ Y ) @ ( id_on_4928293012650018330term_f @ A3 ) )
      = ( ( X = Y )
        & ( member4687689947140789414term_f @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_256_Id__on__iff,axiom,
    ! [X: produc7245736746747425831term_f,Y: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ X @ Y ) @ ( id_on_5951252762333000388term_f @ A3 ) )
      = ( ( X = Y )
        & ( member848276444142703440term_f @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_257_Id__on__iff,axiom,
    ! [X: term_f_v,Y: term_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( id_on_term_f_v @ A3 ) )
      = ( ( X = Y )
        & ( member_term_f_v @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_258_Id__on__iff,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A3: set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( id_on_Bot_bot_term_f @ A3 ) )
      = ( ( X = Y )
        & ( member2723211850833494437term_f @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_259_Id__on__iff,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,A3: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( id_on_Ground_gterm_f @ A3 ) )
      = ( ( X = Y )
        & ( member5261315044688711901term_f @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_260_Id__on__eqI,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( A = B )
     => ( ( member4687689947140789414term_f @ A @ A3 )
       => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( id_on_4928293012650018330term_f @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_261_Id__on__eqI,axiom,
    ! [A: produc7245736746747425831term_f,B: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f] :
      ( ( A = B )
     => ( ( member848276444142703440term_f @ A @ A3 )
       => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ B ) @ ( id_on_5951252762333000388term_f @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_262_Id__on__eqI,axiom,
    ! [A: term_f_v,B: term_f_v,A3: set_term_f_v] :
      ( ( A = B )
     => ( ( member_term_f_v @ A @ A3 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( id_on_term_f_v @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_263_Id__on__eqI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Bot_bot_term_f] :
      ( ( A = B )
     => ( ( member2723211850833494437term_f @ A @ A3 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( id_on_Bot_bot_term_f @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_264_Id__on__eqI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Ground_gterm_f] :
      ( ( A = B )
     => ( ( member5261315044688711901term_f @ A @ A3 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( id_on_Ground_gterm_f @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_265_subrelI,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ S ) )
     => ( ord_le4559761987009501863rm_f_v @ R2 @ S ) ) ).

% subrelI
thf(fact_266_subrelI,axiom,
    ! [R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ S ) )
     => ( ord_le5176047027162639613term_f @ R2 @ S ) ) ).

% subrelI
thf(fact_267_subrelI,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ S ) )
     => ( ord_le263819222746101927term_f @ R2 @ S ) ) ).

% subrelI
thf(fact_268_less__eq__fset_Orep__eq,axiom,
    ( ord_le4587745213494032429rm_f_v
    = ( ^ [X4: fset_P8018961893305114765rm_f_v,Xa: fset_P8018961893305114765rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X4 ) @ ( fset_P4617584883882644886rm_f_v @ Xa ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_269_less__eq__fset_Orep__eq,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [X4: fset_P6228066233360383026_f_nat,Xa: fset_P6228066233360383026_f_nat] : ( ord_le8976984241387448984_f_nat @ ( fset_P3576968334923099475_f_nat @ X4 ) @ ( fset_P3576968334923099475_f_nat @ Xa ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_270_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y4 = Z ) )
    = ( ^ [X4: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ X4 @ Y5 )
          & ( ord_le1552505484586773650_f_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_271_ord__eq__le__trans,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( A = B )
     => ( ( ord_le1552505484586773650_f_nat @ B @ C )
       => ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_272_ord__le__eq__trans,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( B = C )
       => ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_273_order__antisym,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_274_order_Otrans,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( ord_le1552505484586773650_f_nat @ B @ C )
       => ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_275_order__trans,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ Y @ Z3 )
       => ( ord_le1552505484586773650_f_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_276_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y4 = Z ) )
    = ( ^ [A7: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ B6 @ A7 )
          & ( ord_le1552505484586773650_f_nat @ A7 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_277_dual__order_Oantisym,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( ( ord_le1552505484586773650_f_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_278_dual__order_Otrans,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( ( ord_le1552505484586773650_f_nat @ C @ B )
       => ( ord_le1552505484586773650_f_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_279_antisym,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( ord_le1552505484586773650_f_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_280_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y4 = Z ) )
    = ( ^ [A7: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ A7 @ B6 )
          & ( ord_le1552505484586773650_f_nat @ B6 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_281_order__subst1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ ( F2 @ B ) )
     => ( ( ord_le1552505484586773650_f_nat @ B @ C )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X3 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_282_order__subst2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( ord_le1552505484586773650_f_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X3 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_283_order__eq__refl,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( X = Y )
     => ( ord_le1552505484586773650_f_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_284_ord__eq__le__subst,axiom,
    ! [A: fset_P6228066233360383026_f_nat,F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_le1552505484586773650_f_nat @ B @ C )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X3 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_285_ord__le__eq__subst,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X3 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_286_order__antisym__conv,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X )
     => ( ( ord_le1552505484586773650_f_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_287_exE__realizer_H,axiom,
    ! [P2: tree_t2044664379607036063rm_f_f > tree_t2044664379607036063rm_f_f > $o,P: produc6920895926601326823rm_f_f] :
      ( ( P2 @ ( produc1110312829679706757rm_f_f @ P ) @ ( produc1890379913328437059rm_f_f @ P ) )
     => ~ ! [X3: tree_t2044664379607036063rm_f_f,Y3: tree_t2044664379607036063rm_f_f] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_288_exE__realizer_H,axiom,
    ! [P2: term_f_v > term_f_v > $o,P: produc8027700073345061287rm_f_v] :
      ( ( P2 @ ( produc5682565293565518221rm_f_v @ P ) @ ( produc1199688945740118859rm_f_v @ P ) )
     => ~ ! [X3: term_f_v,Y3: term_f_v] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_289_Id__onE,axiom,
    ! [C: produc6853483567480260455term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ C @ ( id_on_4928293012650018330term_f @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X3 @ A3 )
           => ( C
             != ( produc7121717381804241367term_f @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_290_Id__onE,axiom,
    ! [C: produc8979644124618241255term_f,A3: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ C @ ( id_on_5951252762333000388term_f @ A3 ) )
     => ~ ! [X3: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X3 @ A3 )
           => ( C
             != ( produc7348823797596265303term_f @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_291_Id__onE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ C @ ( id_on_term_f_v @ A3 ) )
     => ~ ! [X3: term_f_v] :
            ( ( member_term_f_v @ X3 @ A3 )
           => ( C
             != ( produc2510790566102465375rm_f_v @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_292_Id__onE,axiom,
    ! [C: produc250624644672406269term_f,A3: set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ C @ ( id_on_Bot_bot_term_f @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ X3 @ A3 )
           => ( C
             != ( produc2865564700782054965term_f @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_293_Id__onE,axiom,
    ! [C: produc7245736746747425831term_f,A3: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ C @ ( id_on_Ground_gterm_f @ A3 ) )
     => ~ ! [X3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X3 @ A3 )
           => ( C
             != ( produc3560254623552331287term_f @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_294_pair__comp__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G_1: produc6920895926601326823rm_f_f,Q_1: fset_P2882900458636368611term_f,U: ground_gterm_f,G_2: produc6920895926601326823rm_f_f,Q_2: fset_P2882900458636368611term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( pair_p5686396426817080910rm_f_f @ G_1 @ Q_1 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( pair_p5686396426817080910rm_f_f @ G_2 @ Q_2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ U ) @ ( pair_p5686396426817080910rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ ( produc1110312829679706757rm_f_f @ G_2 ) ) @ ( pair_D621854674755835117term_f @ G_1 @ Q_1 @ G_2 @ Q_2 ) ) ) ) ) ).

% pair_comp_sound1
thf(fact_295_pair__comp__sound2,axiom,
    ! [S: ground_gterm_f,U: ground_gterm_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f,Q_1: fset_P2882900458636368611term_f,Q_2: fset_P2882900458636368611term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ U ) @ ( pair_p5686396426817080910rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ ( produc1110312829679706757rm_f_f @ G_2 ) ) @ ( pair_D621854674755835117term_f @ G_1 @ Q_1 @ G_2 @ Q_2 ) ) )
     => ? [T3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T3 ) @ ( pair_p5686396426817080910rm_f_f @ G_1 @ Q_1 ) )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T3 @ U ) @ ( pair_p5686396426817080910rm_f_f @ G_2 @ Q_2 ) ) ) ) ).

% pair_comp_sound2
thf(fact_296_sndI,axiom,
    ! [X: produc250624644672406269term_f,Y: bot_bot_term_f,Z3: bot_bot_term_f] :
      ( ( X
        = ( produc2865564700782054965term_f @ Y @ Z3 ) )
     => ( ( produc656023363694804067term_f @ X )
        = Z3 ) ) ).

% sndI
thf(fact_297_sndI,axiom,
    ! [X: produc7245736746747425831term_f,Y: ground_gterm_f,Z3: ground_gterm_f] :
      ( ( X
        = ( produc3560254623552331287term_f @ Y @ Z3 ) )
     => ( ( produc1464490519189811525term_f @ X )
        = Z3 ) ) ).

% sndI
thf(fact_298_sndI,axiom,
    ! [X: produc6920895926601326823rm_f_f,Y: tree_t2044664379607036063rm_f_f,Z3: tree_t2044664379607036063rm_f_f] :
      ( ( X
        = ( produc1864056441719732567rm_f_f @ Y @ Z3 ) )
     => ( ( produc1110312829679706757rm_f_f @ X )
        = Z3 ) ) ).

% sndI
thf(fact_299_sndI,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: term_f_v,Z3: term_f_v] :
      ( ( X
        = ( produc2510790566102465375rm_f_v @ Y @ Z3 ) )
     => ( ( produc5682565293565518221rm_f_v @ X )
        = Z3 ) ) ).

% sndI
thf(fact_300_eq__snd__iff,axiom,
    ! [B: bot_bot_term_f,P: produc250624644672406269term_f] :
      ( ( B
        = ( produc656023363694804067term_f @ P ) )
      = ( ? [A7: bot_bot_term_f] :
            ( P
            = ( produc2865564700782054965term_f @ A7 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_301_eq__snd__iff,axiom,
    ! [B: ground_gterm_f,P: produc7245736746747425831term_f] :
      ( ( B
        = ( produc1464490519189811525term_f @ P ) )
      = ( ? [A7: ground_gterm_f] :
            ( P
            = ( produc3560254623552331287term_f @ A7 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_302_eq__snd__iff,axiom,
    ! [B: tree_t2044664379607036063rm_f_f,P: produc6920895926601326823rm_f_f] :
      ( ( B
        = ( produc1110312829679706757rm_f_f @ P ) )
      = ( ? [A7: tree_t2044664379607036063rm_f_f] :
            ( P
            = ( produc1864056441719732567rm_f_f @ A7 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_303_eq__snd__iff,axiom,
    ! [B: term_f_v,P: produc8027700073345061287rm_f_v] :
      ( ( B
        = ( produc5682565293565518221rm_f_v @ P ) )
      = ( ? [A7: term_f_v] :
            ( P
            = ( produc2510790566102465375rm_f_v @ A7 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_304_fstI,axiom,
    ! [X: produc250624644672406269term_f,Y: bot_bot_term_f,Z3: bot_bot_term_f] :
      ( ( X
        = ( produc2865564700782054965term_f @ Y @ Z3 ) )
     => ( ( produc5789359262401207841term_f @ X )
        = Y ) ) ).

% fstI
thf(fact_305_fstI,axiom,
    ! [X: produc7245736746747425831term_f,Y: ground_gterm_f,Z3: ground_gterm_f] :
      ( ( X
        = ( produc3560254623552331287term_f @ Y @ Z3 ) )
     => ( ( produc1239122367099457539term_f @ X )
        = Y ) ) ).

% fstI
thf(fact_306_fstI,axiom,
    ! [X: produc6920895926601326823rm_f_f,Y: tree_t2044664379607036063rm_f_f,Z3: tree_t2044664379607036063rm_f_f] :
      ( ( X
        = ( produc1864056441719732567rm_f_f @ Y @ Z3 ) )
     => ( ( produc1890379913328437059rm_f_f @ X )
        = Y ) ) ).

% fstI
thf(fact_307_fstI,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: term_f_v,Z3: term_f_v] :
      ( ( X
        = ( produc2510790566102465375rm_f_v @ Y @ Z3 ) )
     => ( ( produc1199688945740118859rm_f_v @ X )
        = Y ) ) ).

% fstI
thf(fact_308_eq__fst__iff,axiom,
    ! [A: bot_bot_term_f,P: produc250624644672406269term_f] :
      ( ( A
        = ( produc5789359262401207841term_f @ P ) )
      = ( ? [B6: bot_bot_term_f] :
            ( P
            = ( produc2865564700782054965term_f @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_309_eq__fst__iff,axiom,
    ! [A: ground_gterm_f,P: produc7245736746747425831term_f] :
      ( ( A
        = ( produc1239122367099457539term_f @ P ) )
      = ( ? [B6: ground_gterm_f] :
            ( P
            = ( produc3560254623552331287term_f @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_310_eq__fst__iff,axiom,
    ! [A: tree_t2044664379607036063rm_f_f,P: produc6920895926601326823rm_f_f] :
      ( ( A
        = ( produc1890379913328437059rm_f_f @ P ) )
      = ( ? [B6: tree_t2044664379607036063rm_f_f] :
            ( P
            = ( produc1864056441719732567rm_f_f @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_311_eq__fst__iff,axiom,
    ! [A: term_f_v,P: produc8027700073345061287rm_f_v] :
      ( ( A
        = ( produc1199688945740118859rm_f_v @ P ) )
      = ( ? [B6: term_f_v] :
            ( P
            = ( produc2510790566102465375rm_f_v @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_312_ffunas__trs_Orep__eq,axiom,
    ! [X: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P3576968334923099475_f_nat @ ( lV_to_ffunas_trs_f_v @ X ) )
      = ( funas_trs_f_v_v @ ( fset_P4617584883882644886rm_f_v @ X ) ) ) ).

% ffunas_trs.rep_eq
thf(fact_313_pair__trancl__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc3419615073385449853_f_v_f,Q3: fset_P7452185835880607565rm_f_v] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p3958297607064521345_f_v_f @ G @ Q3 ) ) )
     => ? [Q2: produc8027700073345061287rm_f_v,P3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ P3 @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
          & ( fmembe774991991480784110rm_f_v @ Q2 @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) )
          & ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ P3 @ Q2 ) @ ( pair_D1533242965815047241_f_v_f @ Q3 @ ( produc1990533172842203297_f_v_f @ G ) @ ( produc2866064201297546979_f_v_f @ G ) ) ) ) ) ).

% pair_trancl_sound1
thf(fact_314_pair__trancl__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc4465884753748837329rm_f_f,Q3: fset_P616664398915176141term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p4181219179613995691rm_f_f @ G @ Q3 ) ) )
     => ? [Q2: produc250624644672406269term_f,P3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ P3 @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
          & ( fmembe6950792022577883716term_f @ Q2 @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) )
          & ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ P3 @ Q2 ) @ ( pair_D1844654262279195763rm_f_f @ Q3 @ ( produc8105410132398821365rm_f_f @ G ) @ ( produc8657598210785773111rm_f_f @ G ) ) ) ) ) ).

% pair_trancl_sound1
thf(fact_315_pair__trancl__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc169004890939464999_f_v_f,Q3: fset_P8018961893305114765rm_f_v] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5494371807164803257_f_v_f @ G @ Q3 ) ) )
     => ? [Q2: term_f_v,P3: term_f_v] :
          ( ( fmember_term_f_v @ P3 @ ( tree_g4770736341938457127_f_v_f @ ( produc2716261656266771715_f_v_f @ G ) @ S ) )
          & ( fmember_term_f_v @ Q2 @ ( tree_g4770736341938457127_f_v_f @ ( produc5187309876156436549_f_v_f @ G ) @ T ) )
          & ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P3 @ Q2 ) @ ( pair_D1570434042789516529_f_v_f @ Q3 @ ( produc2716261656266771715_f_v_f @ G ) @ ( produc5187309876156436549_f_v_f @ G ) ) ) ) ) ).

% pair_trancl_sound1
thf(fact_316_pair__trancl__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc4554128051175246307rm_f_f,Q3: fset_P5324236452418035597term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5324507160602824756rm_f_f @ G @ Q3 ) ) )
     => ? [Q2: ground_gterm_f,P3: ground_gterm_f] :
          ( ( fmembe8264398993653236347term_f @ P3 @ ( tree_g2823239125338765382rm_f_f @ ( produc486686396505431687rm_f_f @ G ) @ S ) )
          & ( fmembe8264398993653236347term_f @ Q2 @ ( tree_g2823239125338765382rm_f_f @ ( produc6723095485791902921rm_f_f @ G ) @ T ) )
          & ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ P3 @ Q2 ) @ ( pair_D8441016924214782460rm_f_f @ Q3 @ ( produc486686396505431687rm_f_f @ G ) @ ( produc6723095485791902921rm_f_f @ G ) ) ) ) ) ).

% pair_trancl_sound1
thf(fact_317_pair__trancl__sound1,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc6920895926601326823rm_f_f,Q3: fset_P2882900458636368611term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5686396426817080910rm_f_f @ G @ Q3 ) ) )
     => ? [Q2: bot_bot_term_f,P3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ P3 @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
          & ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) )
          & ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P3 @ Q2 ) @ ( pair_D5922714117594991238rm_f_f @ Q3 @ ( produc1890379913328437059rm_f_f @ G ) @ ( produc1110312829679706757rm_f_f @ G ) ) ) ) ) ).

% pair_trancl_sound1
thf(fact_318_pair__trancl__sound2,axiom,
    ! [P: produc8027700073345061287rm_f_v,Q: produc8027700073345061287rm_f_v,Q3: fset_P7452185835880607565rm_f_v,G: produc3419615073385449853_f_v_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe195806516701975982rm_f_v @ ( produc3874942763110801239rm_f_v @ P @ Q ) @ ( pair_D1533242965815047241_f_v_f @ Q3 @ ( produc1990533172842203297_f_v_f @ G ) @ ( produc2866064201297546979_f_v_f @ G ) ) )
     => ( ( fmembe774991991480784110rm_f_v @ P @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
       => ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p3958297607064521345_f_v_f @ G @ Q3 ) ) ) ) ) ) ).

% pair_trancl_sound2
thf(fact_319_pair__trancl__sound2,axiom,
    ! [P: produc250624644672406269term_f,Q: produc250624644672406269term_f,Q3: fset_P616664398915176141term_f,G: produc4465884753748837329rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ P @ Q ) @ ( pair_D1844654262279195763rm_f_f @ Q3 @ ( produc8105410132398821365rm_f_f @ G ) @ ( produc8657598210785773111rm_f_f @ G ) ) )
     => ( ( fmembe6950792022577883716term_f @ P @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
       => ( ( fmembe6950792022577883716term_f @ Q @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p4181219179613995691rm_f_f @ G @ Q3 ) ) ) ) ) ) ).

% pair_trancl_sound2
thf(fact_320_pair__trancl__sound2,axiom,
    ! [P: term_f_v,Q: term_f_v,Q3: fset_P8018961893305114765rm_f_v,G: produc169004890939464999_f_v_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P @ Q ) @ ( pair_D1570434042789516529_f_v_f @ Q3 @ ( produc2716261656266771715_f_v_f @ G ) @ ( produc5187309876156436549_f_v_f @ G ) ) )
     => ( ( fmember_term_f_v @ P @ ( tree_g4770736341938457127_f_v_f @ ( produc2716261656266771715_f_v_f @ G ) @ S ) )
       => ( ( fmember_term_f_v @ Q @ ( tree_g4770736341938457127_f_v_f @ ( produc5187309876156436549_f_v_f @ G ) @ T ) )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5494371807164803257_f_v_f @ G @ Q3 ) ) ) ) ) ) ).

% pair_trancl_sound2
thf(fact_321_pair__trancl__sound2,axiom,
    ! [P: ground_gterm_f,Q: ground_gterm_f,Q3: fset_P5324236452418035597term_f,G: produc4554128051175246307rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ P @ Q ) @ ( pair_D8441016924214782460rm_f_f @ Q3 @ ( produc486686396505431687rm_f_f @ G ) @ ( produc6723095485791902921rm_f_f @ G ) ) )
     => ( ( fmembe8264398993653236347term_f @ P @ ( tree_g2823239125338765382rm_f_f @ ( produc486686396505431687rm_f_f @ G ) @ S ) )
       => ( ( fmembe8264398993653236347term_f @ Q @ ( tree_g2823239125338765382rm_f_f @ ( produc6723095485791902921rm_f_f @ G ) @ T ) )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5324507160602824756rm_f_f @ G @ Q3 ) ) ) ) ) ) ).

% pair_trancl_sound2
thf(fact_322_pair__trancl__sound2,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,Q3: fset_P2882900458636368611term_f,G: produc6920895926601326823rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ ( pair_D5922714117594991238rm_f_f @ Q3 @ ( produc1890379913328437059rm_f_f @ G ) @ ( produc1110312829679706757rm_f_f @ G ) ) )
     => ( ( fmembe1418995490368054919term_f @ P @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
       => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ ( pair_p5686396426817080910rm_f_f @ G @ Q3 ) ) ) ) ) ) ).

% pair_trancl_sound2
thf(fact_323_pair__trancl__sound,axiom,
    ! [G: produc6920895926601326823rm_f_f,Q3: fset_P2882900458636368611term_f] :
      ( ( transi8854406268384802389term_f @ ( pair_p5686396426817080910rm_f_f @ G @ Q3 ) )
      = ( pair_p5686396426817080910rm_f_f @ G @ ( pair_D5922714117594991238rm_f_f @ Q3 @ ( produc1890379913328437059rm_f_f @ G ) @ ( produc1110312829679706757rm_f_f @ G ) ) ) ) ).

% pair_trancl_sound
thf(fact_324_subsetI,axiom,
    ! [A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ X3 @ A3 )
         => ( member4687689947140789414term_f @ X3 @ B4 ) )
     => ( ord_le5176047027162639613term_f @ A3 @ B4 ) ) ).

% subsetI
thf(fact_325_subsetI,axiom,
    ! [A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f] :
      ( ! [X3: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X3 @ A3 )
         => ( member848276444142703440term_f @ X3 @ B4 ) )
     => ( ord_le263819222746101927term_f @ A3 @ B4 ) ) ).

% subsetI
thf(fact_326_trancl__mono,axiom,
    ! [P: produc250624644672406269term_f,R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ P @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ( ord_le5176047027162639613term_f @ R2 @ S )
       => ( member4687689947140789414term_f @ P @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_mono
thf(fact_327_trancl__mono,axiom,
    ! [P: produc7245736746747425831term_f,R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ P @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ( ord_le263819222746101927term_f @ R2 @ S )
       => ( member848276444142703440term_f @ P @ ( transi8854406268384802389term_f @ S ) ) ) ) ).

% trancl_mono
thf(fact_328_trancl_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ R2 )
       => ~ ! [B3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_329_trancl_Ocases,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ R2 )
       => ~ ! [B3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ B3 ) @ ( transi2774326432670905901term_f @ R2 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B3 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_330_trancl_Ocases,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ R2 )
       => ~ ! [B3: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ B3 ) @ ( transi8854406268384802389term_f @ R2 ) )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B3 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_331_trancl_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
      = ( ? [A7: term_f_v,B6: term_f_v] :
            ( ( A1 = A7 )
            & ( A22 = B6 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ R2 ) )
        | ? [A7: term_f_v,B6: term_f_v,C3: term_f_v] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B6 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_332_trancl_Osimps,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi2774326432670905901term_f @ R2 ) )
      = ( ? [A7: bot_bot_term_f,B6: bot_bot_term_f] :
            ( ( A1 = A7 )
            & ( A22 = B6 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A7 @ B6 ) @ R2 ) )
        | ? [A7: bot_bot_term_f,B6: bot_bot_term_f,C3: bot_bot_term_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A7 @ B6 ) @ ( transi2774326432670905901term_f @ R2 ) )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B6 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_333_trancl_Osimps,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( transi8854406268384802389term_f @ R2 ) )
      = ( ? [A7: ground_gterm_f,B6: ground_gterm_f] :
            ( ( A1 = A7 )
            & ( A22 = B6 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A7 @ B6 ) @ R2 ) )
        | ? [A7: ground_gterm_f,B6: ground_gterm_f,C3: ground_gterm_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A7 @ B6 ) @ ( transi8854406268384802389term_f @ R2 ) )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B6 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_334_trancl_Or__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_335_trancl_Or__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_336_trancl_Or__into__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_337_trancl__map,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,F2: term_f_v > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_338_trancl__map,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,F2: term_f_v > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_339_trancl__map,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,F2: term_f_v > ground_gterm_f,S: set_Pr989862937836626183term_f,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi8854406268384802389term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_340_trancl__map,axiom,
    ! [R2: set_Pr6841673932529763677term_f,F2: bot_bot_term_f > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_341_trancl__map,axiom,
    ! [R2: set_Pr6841673932529763677term_f,F2: bot_bot_term_f > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_342_trancl__map,axiom,
    ! [R2: set_Pr6841673932529763677term_f,F2: bot_bot_term_f > ground_gterm_f,S: set_Pr989862937836626183term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi8854406268384802389term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_343_trancl__map,axiom,
    ! [R2: set_Pr989862937836626183term_f,F2: ground_gterm_f > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_344_trancl__map,axiom,
    ! [R2: set_Pr989862937836626183term_f,F2: ground_gterm_f > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_345_trancl__map,axiom,
    ! [R2: set_Pr989862937836626183term_f,F2: ground_gterm_f > ground_gterm_f,S: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi8854406268384802389term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_346_trancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R2: 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 @ R2 ) )
     => ( ! [A4: term_f_v,B3: term_f_v] :
            ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: term_f_v,B3: 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 @ A4 @ B3 ) ) @ ( transi7658437189833989320rm_f_v @ R2 ) )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_347_trancl__induct2,axiom,
    ! [Ax: bot_bot_term_f,Ay: bot_bot_term_f,Bx: bot_bot_term_f,By: bot_bot_term_f,R2: set_Pr4896805198020829255term_f,P2: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ Bx @ By ) ) @ ( transi2747239000429906462term_f @ R2 ) )
     => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f] :
            ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ A4 @ B3 ) ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f,Aa: bot_bot_term_f,Ba: bot_bot_term_f] :
              ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ A4 @ B3 ) ) @ ( transi2747239000429906462term_f @ R2 ) )
             => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ ( produc2865564700782054965term_f @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_348_trancl__induct2,axiom,
    ! [Ax: ground_gterm_f,Ay: ground_gterm_f,Bx: ground_gterm_f,By: ground_gterm_f,R2: set_Pr1766941801908210887term_f,P2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ Bx @ By ) ) @ ( transi1490960240026123464term_f @ R2 ) )
     => ( ! [A4: ground_gterm_f,B3: ground_gterm_f] :
            ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ A4 @ B3 ) ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: ground_gterm_f,B3: ground_gterm_f,Aa: ground_gterm_f,Ba: ground_gterm_f] :
              ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ A4 @ B3 ) ) @ ( transi1490960240026123464term_f @ R2 ) )
             => ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ ( produc3560254623552331287term_f @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_349_in__mono,axiom,
    ! [A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f,X: produc250624644672406269term_f] :
      ( ( ord_le5176047027162639613term_f @ A3 @ B4 )
     => ( ( member4687689947140789414term_f @ X @ A3 )
       => ( member4687689947140789414term_f @ X @ B4 ) ) ) ).

% in_mono
thf(fact_350_in__mono,axiom,
    ! [A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f,X: produc7245736746747425831term_f] :
      ( ( ord_le263819222746101927term_f @ A3 @ B4 )
     => ( ( member848276444142703440term_f @ X @ A3 )
       => ( member848276444142703440term_f @ X @ B4 ) ) ) ).

% in_mono
thf(fact_351_subsetD,axiom,
    ! [A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f,C: produc250624644672406269term_f] :
      ( ( ord_le5176047027162639613term_f @ A3 @ B4 )
     => ( ( member4687689947140789414term_f @ C @ A3 )
       => ( member4687689947140789414term_f @ C @ B4 ) ) ) ).

% subsetD
thf(fact_352_subsetD,axiom,
    ! [A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f,C: produc7245736746747425831term_f] :
      ( ( ord_le263819222746101927term_f @ A3 @ B4 )
     => ( ( member848276444142703440term_f @ C @ A3 )
       => ( member848276444142703440term_f @ C @ B4 ) ) ) ).

% subsetD
thf(fact_353_subset__eq,axiom,
    ( ord_le5176047027162639613term_f
    = ( ^ [A5: set_Pr6841673932529763677term_f,B5: set_Pr6841673932529763677term_f] :
        ! [X4: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ X4 @ A5 )
         => ( member4687689947140789414term_f @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_354_subset__eq,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A5: set_Pr989862937836626183term_f,B5: set_Pr989862937836626183term_f] :
        ! [X4: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X4 @ A5 )
         => ( member848276444142703440term_f @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_355_subset__iff,axiom,
    ( ord_le5176047027162639613term_f
    = ( ^ [A5: set_Pr6841673932529763677term_f,B5: set_Pr6841673932529763677term_f] :
        ! [T2: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ T2 @ A5 )
         => ( member4687689947140789414term_f @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_356_subset__iff,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A5: set_Pr989862937836626183term_f,B5: set_Pr989862937836626183term_f] :
        ! [T2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ T2 @ A5 )
         => ( member848276444142703440term_f @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_357_converse__trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ! [Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: term_f_v,Z2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R2 )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_358_converse__trancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,P2: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Z2 @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_359_converse__trancl__induct,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,P2: ground_gterm_f > $o] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ! [Y3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Z2 @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_360_trancl__trans__induct,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: set_Pr8827868859434726151rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ! [X3: term_f_v,Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: term_f_v,Y3: term_f_v,Z2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_361_trancl__trans__induct,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,P2: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ ( transi2774326432670905901term_f @ R2 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ ( transi2774326432670905901term_f @ R2 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_362_trancl__trans__induct,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: set_Pr989862937836626183term_f,P2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: ground_gterm_f,Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ ( transi8854406268384802389term_f @ R2 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ ( transi8854406268384802389term_f @ R2 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_363_trancl__into__trancl2,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 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_364_trancl__into__trancl2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_365_trancl__into__trancl2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_366_Transitive__Closure_Otrancl__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 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R2 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_367_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R2 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_368_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ R2 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_369_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_370_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] :
          ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ X3 ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_371_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f] :
          ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ X3 ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_372_converse__tranclE,axiom,
    ! [X: term_f_v,Z3: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_373_converse__tranclE,axiom,
    ! [X: bot_bot_term_f,Z3: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ R2 )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_374_converse__tranclE,axiom,
    ! [X: ground_gterm_f,Z3: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y3 ) @ R2 )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_375_r__r__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R5: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R5 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R5 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R5 ) ) ) ) ).

% r_r_into_trancl
thf(fact_376_r__r__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R5: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R5 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R5 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R5 ) ) ) ) ).

% r_r_into_trancl
thf(fact_377_r__r__into__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R5: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R5 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ R5 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R5 ) ) ) ) ).

% r_r_into_trancl
thf(fact_378_trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ! [Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: term_f_v,Z2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_379_trancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,P2: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ ( transi2774326432670905901term_f @ R2 ) )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_380_trancl__induct,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,P2: ground_gterm_f > $o] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ! [Y3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ Y3 ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ Y3 ) @ ( transi8854406268384802389term_f @ R2 ) )
             => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_381_trancl__trans,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: set_Pr8827868859434726151rm_f_v,Z3: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_382_trancl__trans,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,Z3: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_383_trancl__trans,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: set_Pr989862937836626183term_f,Z3: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_384_tranclE,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
       => ~ ! [C4: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C4 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C4 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_385_tranclE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
       => ~ ! [C4: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C4 ) @ ( transi2774326432670905901term_f @ R2 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ C4 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_386_tranclE,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
       => ~ ! [C4: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C4 ) @ ( transi8854406268384802389term_f @ R2 ) )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ C4 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_387_pair__comp__sound,axiom,
    ! [G_1: produc6920895926601326823rm_f_f,Q_1: fset_P2882900458636368611term_f,G_2: produc6920895926601326823rm_f_f,Q_2: fset_P2882900458636368611term_f] :
      ( ( relcom4354988593214781939term_f @ ( pair_p5686396426817080910rm_f_f @ G_1 @ Q_1 ) @ ( pair_p5686396426817080910rm_f_f @ G_2 @ Q_2 ) )
      = ( pair_p5686396426817080910rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ ( produc1110312829679706757rm_f_f @ G_2 ) ) @ ( pair_D621854674755835117term_f @ G_1 @ Q_1 @ G_2 @ Q_2 ) ) ) ).

% pair_comp_sound
thf(fact_388_fId__on__frelcomp__id2,axiom,
    ! [R5: fset_P4253400765599247181rm_f_f,S2: fset_T2564841199962184965rm_f_f] :
      ( ( ord_le5296964664799249573rm_f_f @ ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f @ R5 ) @ S2 )
     => ( ( fSet_f2413528405311631859rm_f_f @ R5 @ ( fSet_f3475564561967329575rm_f_f @ S2 ) )
        = R5 ) ) ).

% fId_on_frelcomp_id2
thf(fact_389_fId__on__frelcomp__id2,axiom,
    ! [R5: fset_P8018961893305114765rm_f_v,S2: fset_term_f_v] :
      ( ( ord_le6659761365940488051rm_f_v @ ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v @ R5 ) @ S2 )
     => ( ( fSet_f1619609727995933463rm_f_v @ R5 @ ( fSet_fId_on_term_f_v @ S2 ) )
        = R5 ) ) ).

% fId_on_frelcomp_id2
thf(fact_390_fId__on__frelcomp__id,axiom,
    ! [R5: fset_P4253400765599247181rm_f_f,S2: fset_T2564841199962184965rm_f_f] :
      ( ( ord_le5296964664799249573rm_f_f @ ( fimage7671923670337682677rm_f_f @ produc1890379913328437059rm_f_f @ R5 ) @ S2 )
     => ( ( fSet_f2413528405311631859rm_f_f @ ( fSet_f3475564561967329575rm_f_f @ S2 ) @ R5 )
        = R5 ) ) ).

% fId_on_frelcomp_id
thf(fact_391_fId__on__frelcomp__id,axiom,
    ! [R5: fset_P8018961893305114765rm_f_v,S2: fset_term_f_v] :
      ( ( ord_le6659761365940488051rm_f_v @ ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v @ R5 ) @ S2 )
     => ( ( fSet_f1619609727995933463rm_f_v @ ( fSet_fId_on_term_f_v @ S2 ) @ R5 )
        = R5 ) ) ).

% fId_on_frelcomp_id
thf(fact_392_agtt__langI,axiom,
    ! [Q: produc8027700073345061287rm_f_v,G: produc3419615073385449853_f_v_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
     => ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l3936741131039298827_f_v_f @ G ) ) ) ) ).

% agtt_langI
thf(fact_393_agtt__langI,axiom,
    ! [Q: produc250624644672406269term_f,G: produc4465884753748837329rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe6950792022577883716term_f @ Q @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
     => ( ( fmembe6950792022577883716term_f @ Q @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l4583084979836839221rm_f_f @ G ) ) ) ) ).

% agtt_langI
thf(fact_394_agtt__langI,axiom,
    ! [Q: bot_bot_term_f,G: produc6920895926601326823rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l7696934014624939844rm_f_f @ G ) ) ) ) ).

% agtt_langI
thf(fact_395_agtt__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc3419615073385449853_f_v_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l3936741131039298827_f_v_f @ G ) )
     => ~ ! [Q2: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ Q2 @ ( tree_g8599241186450568851_f_v_f @ ( produc1990533172842203297_f_v_f @ G ) @ S ) )
           => ~ ( fmembe774991991480784110rm_f_v @ Q2 @ ( tree_g8599241186450568851_f_v_f @ ( produc2866064201297546979_f_v_f @ G ) @ T ) ) ) ) ).

% agtt_langE
thf(fact_396_agtt__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc4465884753748837329rm_f_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l4583084979836839221rm_f_f @ G ) )
     => ~ ! [Q2: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ Q2 @ ( tree_g3700709089741334205rm_f_f @ ( produc8105410132398821365rm_f_f @ G ) @ S ) )
           => ~ ( fmembe6950792022577883716term_f @ Q2 @ ( tree_g3700709089741334205rm_f_f @ ( produc8657598210785773111rm_f_f @ G ) @ T ) ) ) ) ).

% agtt_langE
thf(fact_397_agtt__langE,axiom,
    ! [S: ground_gterm_f,T: ground_gterm_f,G: produc6920895926601326823rm_f_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( agtt_l7696934014624939844rm_f_f @ G ) )
     => ~ ! [Q2: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ S ) )
           => ~ ( fmembe1418995490368054919term_f @ Q2 @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ T ) ) ) ) ).

% agtt_langE
thf(fact_398_prod_Oswap__def,axiom,
    ( produc7831899462348520405term_f
    = ( ^ [P4: produc250624644672406269term_f] : ( produc2865564700782054965term_f @ ( produc656023363694804067term_f @ P4 ) @ ( produc5789359262401207841term_f @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_399_prod_Oswap__def,axiom,
    ( produc8604619745253505975term_f
    = ( ^ [P4: produc7245736746747425831term_f] : ( produc3560254623552331287term_f @ ( produc1464490519189811525term_f @ P4 ) @ ( produc1239122367099457539term_f @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_400_prod_Oswap__def,axiom,
    ( produc2883289087695599351rm_f_f
    = ( ^ [P4: produc6920895926601326823rm_f_f] : ( produc1864056441719732567rm_f_f @ ( produc1110312829679706757rm_f_f @ P4 ) @ ( produc1890379913328437059rm_f_f @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_401_prod_Oswap__def,axiom,
    ( produc7255811591395797247rm_f_v
    = ( ^ [P4: produc8027700073345061287rm_f_v] : ( produc2510790566102465375rm_f_v @ ( produc5682565293565518221rm_f_v @ P4 ) @ ( produc1199688945740118859rm_f_v @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_402_fimage__eqI,axiom,
    ! [B: fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
       => ( fmembe7677060439241084203_f_nat @ B @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_403_fimage__eqI,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v,X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage7135986912551575357rm_f_v @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_404_fimage__eqI,axiom,
    ! [B: produc250624644672406269term_f,F2: produc8027700073345061287rm_f_v > produc250624644672406269term_f,X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage392373404974708883term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_405_fimage__eqI,axiom,
    ! [B: bot_bot_term_f,F2: produc8027700073345061287rm_f_v > bot_bot_term_f,X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage3302344361971725688term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_406_fimage__eqI,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: produc250624644672406269term_f > produc8027700073345061287rm_f_v,X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage184384913823027303rm_f_v @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_407_fimage__eqI,axiom,
    ! [B: produc250624644672406269term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f,X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage494449270717284797term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_408_fimage__eqI,axiom,
    ! [B: bot_bot_term_f,F2: produc250624644672406269term_f > bot_bot_term_f,X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe6950792022577883716term_f @ X @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage6436382725501958094term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_409_fimage__eqI,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: bot_bot_term_f > produc8027700073345061287rm_f_v,X: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage6997639165041960794rm_f_v @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_410_fimage__eqI,axiom,
    ! [B: produc250624644672406269term_f,F2: bot_bot_term_f > produc250624644672406269term_f,X: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage6373649071792704688term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_411_fimage__eqI,axiom,
    ! [B: bot_bot_term_f,F2: bot_bot_term_f > bot_bot_term_f,X: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( fmembe1418995490368054919term_f @ X @ A3 )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage8971441074658983195term_f @ F2 @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_412_swap__simp,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( produc7255811591395797247rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) )
      = ( produc2510790566102465375rm_f_v @ Y @ X ) ) ).

% swap_simp
thf(fact_413_swap__simp,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( produc7831899462348520405term_f @ ( produc2865564700782054965term_f @ X @ Y ) )
      = ( produc2865564700782054965term_f @ Y @ X ) ) ).

% swap_simp
thf(fact_414_swap__simp,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f] :
      ( ( produc8604619745253505975term_f @ ( produc3560254623552331287term_f @ X @ Y ) )
      = ( produc3560254623552331287term_f @ Y @ X ) ) ).

% swap_simp
thf(fact_415_fst__swap,axiom,
    ! [X: produc6920895926601326823rm_f_f] :
      ( ( produc1890379913328437059rm_f_f @ ( produc2883289087695599351rm_f_f @ X ) )
      = ( produc1110312829679706757rm_f_f @ X ) ) ).

% fst_swap
thf(fact_416_fst__swap,axiom,
    ! [X: produc8027700073345061287rm_f_v] :
      ( ( produc1199688945740118859rm_f_v @ ( produc7255811591395797247rm_f_v @ X ) )
      = ( produc5682565293565518221rm_f_v @ X ) ) ).

% fst_swap
thf(fact_417_snd__swap,axiom,
    ! [X: produc6920895926601326823rm_f_f] :
      ( ( produc1110312829679706757rm_f_f @ ( produc2883289087695599351rm_f_f @ X ) )
      = ( produc1890379913328437059rm_f_f @ X ) ) ).

% snd_swap
thf(fact_418_snd__swap,axiom,
    ! [X: produc8027700073345061287rm_f_v] :
      ( ( produc5682565293565518221rm_f_v @ ( produc7255811591395797247rm_f_v @ X ) )
      = ( produc1199688945740118859rm_f_v @ X ) ) ).

% snd_swap
thf(fact_419_rev__fimage__eqI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe7677060439241084203_f_nat @ B @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_420_rev__fimage__eqI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: produc8027700073345061287rm_f_v,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage7135986912551575357rm_f_v @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_421_rev__fimage__eqI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: produc250624644672406269term_f,F2: produc8027700073345061287rm_f_v > produc250624644672406269term_f] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage392373404974708883term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_422_rev__fimage__eqI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: bot_bot_term_f,F2: produc8027700073345061287rm_f_v > bot_bot_term_f] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage3302344361971725688term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_423_rev__fimage__eqI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc8027700073345061287rm_f_v,F2: produc250624644672406269term_f > produc8027700073345061287rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage184384913823027303rm_f_v @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_424_rev__fimage__eqI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc250624644672406269term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage494449270717284797term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_425_rev__fimage__eqI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: bot_bot_term_f,F2: produc250624644672406269term_f > bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage6436382725501958094term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_426_rev__fimage__eqI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc8027700073345061287rm_f_v,F2: bot_bot_term_f > produc8027700073345061287rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe774991991480784110rm_f_v @ B @ ( fimage6997639165041960794rm_f_v @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_427_rev__fimage__eqI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc250624644672406269term_f,F2: bot_bot_term_f > produc250624644672406269term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe6950792022577883716term_f @ B @ ( fimage6373649071792704688term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_428_rev__fimage__eqI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: bot_bot_term_f,F2: bot_bot_term_f > bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( ( B
          = ( F2 @ X ) )
       => ( fmembe1418995490368054919term_f @ B @ ( fimage8971441074658983195term_f @ F2 @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_429_fimage__cong,axiom,
    ! [M: fset_P8018961893305114765rm_f_v,N: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,G2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( M = N )
     => ( ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X3 @ N )
           => ( ( F2 @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( fimage84417724776064284_f_nat @ F2 @ M )
          = ( fimage84417724776064284_f_nat @ G2 @ N ) ) ) ) ).

% fimage_cong
thf(fact_430_fimageI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( fmembe7677060439241084203_f_nat @ ( F2 @ X ) @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_431_fimageI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( fmembe774991991480784110rm_f_v @ ( F2 @ X ) @ ( fimage7135986912551575357rm_f_v @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_432_fimageI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > produc250624644672406269term_f] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( fmembe6950792022577883716term_f @ ( F2 @ X ) @ ( fimage392373404974708883term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_433_fimageI,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > bot_bot_term_f] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( fmembe1418995490368054919term_f @ ( F2 @ X ) @ ( fimage3302344361971725688term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_434_fimageI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > produc8027700073345061287rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( fmembe774991991480784110rm_f_v @ ( F2 @ X ) @ ( fimage184384913823027303rm_f_v @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_435_fimageI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( fmembe6950792022577883716term_f @ ( F2 @ X ) @ ( fimage494449270717284797term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_436_fimageI,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( fmembe1418995490368054919term_f @ ( F2 @ X ) @ ( fimage6436382725501958094term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_437_fimageI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > produc8027700073345061287rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( fmembe774991991480784110rm_f_v @ ( F2 @ X ) @ ( fimage6997639165041960794rm_f_v @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_438_fimageI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > produc250624644672406269term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( fmembe6950792022577883716term_f @ ( F2 @ X ) @ ( fimage6373649071792704688term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_439_fimageI,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( fmembe1418995490368054919term_f @ ( F2 @ X ) @ ( fimage8971441074658983195term_f @ F2 @ A3 ) ) ) ).

% fimageI
thf(fact_440_fimageE,axiom,
    ! [B: fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe7677060439241084203_f_nat @ B @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_441_fimageE,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( fimage7135986912551575357rm_f_v @ F2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_442_fimageE,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: produc250624644672406269term_f > produc8027700073345061287rm_f_v,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( fimage184384913823027303rm_f_v @ F2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_443_fimageE,axiom,
    ! [B: produc8027700073345061287rm_f_v,F2: bot_bot_term_f > produc8027700073345061287rm_f_v,A3: fset_Bot_bot_term_f] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( fimage6997639165041960794rm_f_v @ F2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_444_fimageE,axiom,
    ! [B: produc250624644672406269term_f,F2: produc8027700073345061287rm_f_v > produc250624644672406269term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fimage392373404974708883term_f @ F2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_445_fimageE,axiom,
    ! [B: produc250624644672406269term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fimage494449270717284797term_f @ F2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_446_fimageE,axiom,
    ! [B: produc250624644672406269term_f,F2: bot_bot_term_f > produc250624644672406269term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ B @ ( fimage6373649071792704688term_f @ F2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_447_fimageE,axiom,
    ! [B: bot_bot_term_f,F2: produc8027700073345061287rm_f_v > bot_bot_term_f,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fimage3302344361971725688term_f @ F2 @ A3 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe774991991480784110rm_f_v @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_448_fimageE,axiom,
    ! [B: bot_bot_term_f,F2: produc250624644672406269term_f > bot_bot_term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fimage6436382725501958094term_f @ F2 @ A3 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe6950792022577883716term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_449_fimageE,axiom,
    ! [B: bot_bot_term_f,F2: bot_bot_term_f > bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fimage8971441074658983195term_f @ F2 @ A3 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe1418995490368054919term_f @ X3 @ A3 ) ) ) ).

% fimageE
thf(fact_450_fset_Omap__ident__strong,axiom,
    ! [T: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f] :
      ( ! [Z2: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ Z2 @ ( fset_P8925168969356593900term_f @ T ) )
         => ( ( F2 @ Z2 )
            = Z2 ) )
     => ( ( fimage494449270717284797term_f @ F2 @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_451_fset_Omap__ident__strong,axiom,
    ! [T: fset_P5324236452418035597term_f,F2: produc7245736746747425831term_f > produc7245736746747425831term_f] :
      ( ! [Z2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ Z2 @ ( fset_P5939342050162774678term_f @ T ) )
         => ( ( F2 @ Z2 )
            = Z2 ) )
     => ( ( fimage448765646422269245term_f @ F2 @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_452_fset_Omap__ident__strong,axiom,
    ! [T: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v] :
      ( ! [Z2: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ Z2 @ ( fset_P4617584883882644886rm_f_v @ T ) )
         => ( ( F2 @ Z2 )
            = Z2 ) )
     => ( ( fimage7135986912551575357rm_f_v @ F2 @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_453_fset_Oinj__map__strong,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,Fa: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ! [Z2: produc8027700073345061287rm_f_v,Za: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ Z2 @ ( fset_P4617584883882644886rm_f_v @ X ) )
         => ( ( member4316649861536941904rm_f_v @ Za @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) )
           => ( ( ( F2 @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( fimage84417724776064284_f_nat @ F2 @ X )
          = ( fimage84417724776064284_f_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% fset.inj_map_strong
thf(fact_454_fset_Omap__cong0,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,G2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ! [Z2: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ Z2 @ ( fset_P4617584883882644886rm_f_v @ X ) )
         => ( ( F2 @ Z2 )
            = ( G2 @ Z2 ) ) )
     => ( ( fimage84417724776064284_f_nat @ F2 @ X )
        = ( fimage84417724776064284_f_nat @ G2 @ X ) ) ) ).

% fset.map_cong0
thf(fact_455_fset_Omap__cong,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Ya: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,G2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( X = Ya )
     => ( ! [Z2: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ Z2 @ ( fset_P4617584883882644886rm_f_v @ Ya ) )
           => ( ( F2 @ Z2 )
              = ( G2 @ Z2 ) ) )
       => ( ( fimage84417724776064284_f_nat @ F2 @ X )
          = ( fimage84417724776064284_f_nat @ G2 @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_456_fimage__mono,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( ord_le4587745213494032429rm_f_v @ A3 @ B4 )
     => ( ord_le8062249421014450114_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) @ ( fimage84417724776064284_f_nat @ F2 @ B4 ) ) ) ).

% fimage_mono
thf(fact_457_fimage__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,F2: product_prod_f_nat > product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ord_le1552505484586773650_f_nat @ ( fimage736527463273833859_f_nat @ F2 @ A3 ) @ ( fimage736527463273833859_f_nat @ F2 @ B4 ) ) ) ).

% fimage_mono
thf(fact_458_subset__fimage__iff,axiom,
    ! [B4: fset_f1705081619141295970_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ord_le8062249421014450114_f_nat @ B4 @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) )
      = ( ? [AA: fset_P8018961893305114765rm_f_v] :
            ( ( ord_le4587745213494032429rm_f_v @ AA @ A3 )
            & ( B4
              = ( fimage84417724776064284_f_nat @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_459_subset__fimage__iff,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,F2: product_prod_f_nat > product_prod_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B4 @ ( fimage736527463273833859_f_nat @ F2 @ A3 ) )
      = ( ? [AA: fset_P6228066233360383026_f_nat] :
            ( ( ord_le1552505484586773650_f_nat @ AA @ A3 )
            & ( B4
              = ( fimage736527463273833859_f_nat @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_460_relcompEpair,axiom,
    ! [A: term_f_v,C: term_f_v,R2: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( relcom2934258169684288428rm_f_v @ R2 @ S ) )
     => ~ ! [B3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ R2 )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_461_relcompEpair,axiom,
    ! [A: bot_bot_term_f,C: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( relcom2207653138174173357term_f @ R2 @ S ) )
     => ~ ! [B3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B3 ) @ R2 )
           => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_462_relcompEpair,axiom,
    ! [A: ground_gterm_f,C: ground_gterm_f,R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( relcom4354988593214781939term_f @ R2 @ S ) )
     => ~ ! [B3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B3 ) @ R2 )
           => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_463_relcompE,axiom,
    ! [Xz: produc8027700073345061287rm_f_v,R2: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Xz @ ( relcom2934258169684288428rm_f_v @ R2 @ S ) )
     => ~ ! [X3: term_f_v,Y3: term_f_v,Z2: term_f_v] :
            ( ( Xz
              = ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_464_relcompE,axiom,
    ! [Xz: produc250624644672406269term_f,R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ Xz @ ( relcom2207653138174173357term_f @ R2 @ S ) )
     => ~ ! [X3: bot_bot_term_f,Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
            ( ( Xz
              = ( produc2865564700782054965term_f @ X3 @ Z2 ) )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_465_relcompE,axiom,
    ! [Xz: produc7245736746747425831term_f,R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ Xz @ ( relcom4354988593214781939term_f @ R2 @ S ) )
     => ~ ! [X3: ground_gterm_f,Y3: ground_gterm_f,Z2: ground_gterm_f] :
            ( ( Xz
              = ( produc3560254623552331287term_f @ X3 @ Z2 ) )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_466_relcomp_OrelcompI,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,C: term_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ S )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( relcom2934258169684288428rm_f_v @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_467_relcomp_OrelcompI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ S )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( relcom2207653138174173357term_f @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_468_relcomp_OrelcompI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ S )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( relcom4354988593214781939term_f @ R2 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_469_relcomp_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ S ) )
      = ( ? [A7: term_f_v,B6: term_f_v,C3: term_f_v] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ R2 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B6 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_470_relcomp_Osimps,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( relcom2207653138174173357term_f @ R2 @ S ) )
      = ( ? [A7: bot_bot_term_f,B6: bot_bot_term_f,C3: bot_bot_term_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A7 @ B6 ) @ R2 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B6 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_471_relcomp_Osimps,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( relcom4354988593214781939term_f @ R2 @ S ) )
      = ( ? [A7: ground_gterm_f,B6: ground_gterm_f,C3: ground_gterm_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A7 @ B6 ) @ R2 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B6 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_472_relcomp_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R2 @ S ) )
     => ~ ! [B3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ R2 )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_473_relcomp_Ocases,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( relcom2207653138174173357term_f @ R2 @ S ) )
     => ~ ! [B3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ B3 ) @ R2 )
           => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_474_relcomp_Ocases,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( relcom4354988593214781939term_f @ R2 @ S ) )
     => ~ ! [B3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ B3 ) @ R2 )
           => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_475_relcomp3__I,axiom,
    ! [T: term_f_v,U: term_f_v,A3: set_Pr8827868859434726151rm_f_v,S: term_f_v,B4: set_Pr8827868859434726151rm_f_v,V3: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ A3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ B4 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ V3 ) @ B4 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ V3 ) @ ( relcom2934258169684288428rm_f_v @ B4 @ ( relcom2934258169684288428rm_f_v @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_476_relcomp3__I,axiom,
    ! [T: bot_bot_term_f,U: bot_bot_term_f,A3: set_Pr6841673932529763677term_f,S: bot_bot_term_f,B4: set_Pr6841673932529763677term_f,V3: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ A3 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ B4 )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ U @ V3 ) @ B4 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ V3 ) @ ( relcom2207653138174173357term_f @ B4 @ ( relcom2207653138174173357term_f @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_477_relcomp3__I,axiom,
    ! [T: ground_gterm_f,U: ground_gterm_f,A3: set_Pr989862937836626183term_f,S: ground_gterm_f,B4: set_Pr989862937836626183term_f,V3: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ A3 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ B4 )
       => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ U @ V3 ) @ B4 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ V3 ) @ ( relcom4354988593214781939term_f @ B4 @ ( relcom4354988593214781939term_f @ A3 @ B4 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_478_fimage__fsubsetI,axiom,
    ! [A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
         => ( fmembe1418995490368054919term_f @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le7572167314378612510term_f @ ( fimage8971441074658983195term_f @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_479_fimage__fsubsetI,axiom,
    ! [A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > product_prod_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
         => ( fmembe6795308708001954811_f_nat @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le1552505484586773650_f_nat @ ( fimage3913130732273524879_f_nat @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_480_fimage__fsubsetI,axiom,
    ! [A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
         => ( fmembe1418995490368054919term_f @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le7572167314378612510term_f @ ( fimage6436382725501958094term_f @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_481_fimage__fsubsetI,axiom,
    ! [A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > produc250624644672406269term_f,B4: fset_P2882900458636368611term_f] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
         => ( fmembe6950792022577883716term_f @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le7092741666294821507term_f @ ( fimage6373649071792704688term_f @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_482_fimage__fsubsetI,axiom,
    ! [A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > product_prod_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
         => ( fmembe6795308708001954811_f_nat @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le1552505484586773650_f_nat @ ( fimage1700273135149255618_f_nat @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_483_fimage__fsubsetI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ! [X3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
         => ( fmembe1418995490368054919term_f @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le7572167314378612510term_f @ ( fimage3302344361971725688term_f @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_484_fimage__fsubsetI,axiom,
    ! [A3: fset_Bot_bot_term_f,F2: bot_bot_term_f > produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ! [X3: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
         => ( fmembe774991991480784110rm_f_v @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le4587745213494032429rm_f_v @ ( fimage6997639165041960794rm_f_v @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_485_fimage__fsubsetI,axiom,
    ! [A3: fset_P2882900458636368611term_f,F2: produc250624644672406269term_f > produc250624644672406269term_f,B4: fset_P2882900458636368611term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
         => ( fmembe6950792022577883716term_f @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le7092741666294821507term_f @ ( fimage494449270717284797term_f @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_486_fimage__fsubsetI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > product_prod_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ! [X3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
         => ( fmembe6795308708001954811_f_nat @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le1552505484586773650_f_nat @ ( fimage5122521059875848300_f_nat @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_487_fimage__fsubsetI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,B4: fset_f1705081619141295970_f_nat] :
      ( ! [X3: produc8027700073345061287rm_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
         => ( fmembe7677060439241084203_f_nat @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le8062249421014450114_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_488_frelcomp_Orep__eq,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fSet_f1619609727995933463rm_f_v @ X @ Xa2 ) )
      = ( relcom2934258169684288428rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) ) ) ).

% frelcomp.rep_eq
thf(fact_489_pair__at__agtt__cost,axiom,
    ( pair_p5686396426817080910rm_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f,Q4: fset_P2882900458636368611term_f] : ( agtt_l8776386655999288343rm_f_f @ ( pair_p1447136629559935937term_f @ G3 @ Q4 ) ) ) ) ).

% pair_at_agtt_cost
thf(fact_490_pair__at__agtt,axiom,
    ( agtt_l7696934014624939844rm_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f] : ( pair_p5686396426817080910rm_f_f @ G3 @ ( fSet_f5231733434221412358term_f @ ( gtt_in8205028897947813884_f_f_f @ G3 ) ) ) ) ) ).

% pair_at_agtt
thf(fact_491_gtt__eps__free__def,axiom,
    ( gtt_ep64433474226639880rm_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f] : ( produc1864056441719732567rm_f_f @ ( tree_e1506877361127215193rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) @ ( tree_e1506877361127215193rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) ) ) ) ) ).

% gtt_eps_free_def
thf(fact_492_eps__states__def,axiom,
    ( tree_e1645133391170978382rm_f_f
    = ( ^ [Delta_epsilon: fset_P4253400765599247181rm_f_f] : ( sup_su7125214341635615961rm_f_f @ ( fimage7671923670337682677rm_f_f @ produc1890379913328437059rm_f_f @ Delta_epsilon ) @ ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f @ Delta_epsilon ) ) ) ) ).

% eps_states_def
thf(fact_493_eps__states__def,axiom,
    ( tree_e8707155279755596916rm_f_v
    = ( ^ [Delta_epsilon: fset_P8018961893305114765rm_f_v] : ( sup_su89973778935984191rm_f_v @ ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v @ Delta_epsilon ) @ ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v @ Delta_epsilon ) ) ) ) ).

% eps_states_def
thf(fact_494_eps__states__def,axiom,
    ( tree_e2646172767733093139_f_nat
    = ( ^ [Delta_epsilon: fset_P7991872372518838603_f_nat] : ( sup_su8375471814421675870_f_nat @ ( fimage2899573712749927210_f_nat @ produc4150312202723698825_f_nat @ Delta_epsilon ) @ ( fimage2899573712749927210_f_nat @ produc4732230679739841739_f_nat @ Delta_epsilon ) ) ) ) ).

% eps_states_def
thf(fact_495_same__fstI,axiom,
    ! [P2: term_f_v > $o,X: term_f_v,Y6: term_f_v,Y: term_f_v,R5: term_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( P2 @ X )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ Y ) @ ( R5 @ X ) )
       => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y6 ) @ ( produc2510790566102465375rm_f_v @ X @ Y ) ) @ ( same_f5557695934196986804rm_f_v @ P2 @ R5 ) ) ) ) ).

% same_fstI
thf(fact_496_same__fstI,axiom,
    ! [P2: bot_bot_term_f > $o,X: bot_bot_term_f,Y6: bot_bot_term_f,Y: bot_bot_term_f,R5: bot_bot_term_f > set_Pr6841673932529763677term_f] :
      ( ( P2 @ X )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y6 @ Y ) @ ( R5 @ X ) )
       => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ X @ Y6 ) @ ( produc2865564700782054965term_f @ X @ Y ) ) @ ( same_f1003344001368109706term_f @ P2 @ R5 ) ) ) ) ).

% same_fstI
thf(fact_497_same__fstI,axiom,
    ! [P2: ground_gterm_f > $o,X: ground_gterm_f,Y6: ground_gterm_f,Y: ground_gterm_f,R5: ground_gterm_f > set_Pr989862937836626183term_f] :
      ( ( P2 @ X )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y6 @ Y ) @ ( R5 @ X ) )
       => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ X @ Y6 ) @ ( produc3560254623552331287term_f @ X @ Y ) ) @ ( same_f6493960152301490668term_f @ P2 @ R5 ) ) ) ) ).

% same_fstI
thf(fact_498_funion__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
        | ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ).

% funion_iff
thf(fact_499_funion__iff,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( sup_su1809081536757948599term_f @ A3 @ B4 ) )
      = ( ( fmembe6950792022577883716term_f @ C @ A3 )
        | ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ).

% funion_iff
thf(fact_500_funion__iff,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( sup_su6586017679312891882term_f @ A3 @ B4 ) )
      = ( ( fmembe1418995490368054919term_f @ C @ A3 )
        | ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ).

% funion_iff
thf(fact_501_funion__iff,axiom,
    ! [C: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe6795308708001954811_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
      = ( ( fmembe6795308708001954811_f_nat @ C @ A3 )
        | ( fmembe6795308708001954811_f_nat @ C @ B4 ) ) ) ).

% funion_iff
thf(fact_502_funionCI,axiom,
    ! [C: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ~ ( fmembe774991991480784110rm_f_v @ C @ B4 )
       => ( fmembe774991991480784110rm_f_v @ C @ A3 ) )
     => ( fmembe774991991480784110rm_f_v @ C @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) ) ) ).

% funionCI
thf(fact_503_funionCI,axiom,
    ! [C: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f,A3: fset_P2882900458636368611term_f] :
      ( ( ~ ( fmembe6950792022577883716term_f @ C @ B4 )
       => ( fmembe6950792022577883716term_f @ C @ A3 ) )
     => ( fmembe6950792022577883716term_f @ C @ ( sup_su1809081536757948599term_f @ A3 @ B4 ) ) ) ).

% funionCI
thf(fact_504_funionCI,axiom,
    ! [C: bot_bot_term_f,B4: fset_Bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( ~ ( fmembe1418995490368054919term_f @ C @ B4 )
       => ( fmembe1418995490368054919term_f @ C @ A3 ) )
     => ( fmembe1418995490368054919term_f @ C @ ( sup_su6586017679312891882term_f @ A3 @ B4 ) ) ) ).

% funionCI
thf(fact_505_funionCI,axiom,
    ! [C: product_prod_f_nat,B4: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( ~ ( fmembe6795308708001954811_f_nat @ C @ B4 )
       => ( fmembe6795308708001954811_f_nat @ C @ A3 ) )
     => ( fmembe6795308708001954811_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ) ).

% funionCI
thf(fact_506_eps__states__union,axiom,
    ! [Delta: fset_P7991872372518838603_f_nat,Gamma: fset_P7991872372518838603_f_nat] :
      ( ( tree_e2646172767733093139_f_nat @ ( sup_su8668618379824644895_f_nat @ Delta @ Gamma ) )
      = ( sup_su8375471814421675870_f_nat @ ( tree_e2646172767733093139_f_nat @ Delta ) @ ( tree_e2646172767733093139_f_nat @ Gamma ) ) ) ).

% eps_states_union
thf(fact_507_funion__assoc,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) @ C2 )
      = ( sup_su8375471814421675870_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) ) ) ).

% funion_assoc
thf(fact_508_funion__absorb,axiom,
    ! [A3: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ A3 )
      = A3 ) ).

% funion_absorb
thf(fact_509_funion__commute,axiom,
    ( sup_su8375471814421675870_f_nat
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B5: fset_P6228066233360383026_f_nat] : ( sup_su8375471814421675870_f_nat @ B5 @ A5 ) ) ) ).

% funion_commute
thf(fact_510_funion__left__absorb,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
      = ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ).

% funion_left_absorb
thf(fact_511_funion__left__commute,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) )
      = ( sup_su8375471814421675870_f_nat @ B4 @ ( sup_su8375471814421675870_f_nat @ A3 @ C2 ) ) ) ).

% funion_left_commute
thf(fact_512_fUn__fImage,axiom,
    ! [R5: fset_P6228066233360383026_f_nat,S2: fset_P6228066233360383026_f_nat,A3: fset_f] :
      ( ( fSet_fImage_f_nat @ ( sup_su8375471814421675870_f_nat @ R5 @ S2 ) @ A3 )
      = ( sup_sup_fset_nat @ ( fSet_fImage_f_nat @ R5 @ A3 ) @ ( fSet_fImage_f_nat @ S2 @ A3 ) ) ) ).

% fUn_fImage
thf(fact_513_funionI2,axiom,
    ! [C: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ B4 )
     => ( fmembe774991991480784110rm_f_v @ C @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) ) ) ).

% funionI2
thf(fact_514_funionI2,axiom,
    ! [C: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ B4 )
     => ( fmembe6950792022577883716term_f @ C @ ( sup_su1809081536757948599term_f @ A3 @ B4 ) ) ) ).

% funionI2
thf(fact_515_funionI2,axiom,
    ! [C: bot_bot_term_f,B4: fset_Bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ B4 )
     => ( fmembe1418995490368054919term_f @ C @ ( sup_su6586017679312891882term_f @ A3 @ B4 ) ) ) ).

% funionI2
thf(fact_516_funionI2,axiom,
    ! [C: product_prod_f_nat,B4: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( fmembe6795308708001954811_f_nat @ C @ B4 )
     => ( fmembe6795308708001954811_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ) ).

% funionI2
thf(fact_517_funionI1,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
     => ( fmembe774991991480784110rm_f_v @ C @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) ) ) ).

% funionI1
thf(fact_518_funionI1,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ A3 )
     => ( fmembe6950792022577883716term_f @ C @ ( sup_su1809081536757948599term_f @ A3 @ B4 ) ) ) ).

% funionI1
thf(fact_519_funionI1,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ A3 )
     => ( fmembe1418995490368054919term_f @ C @ ( sup_su6586017679312891882term_f @ A3 @ B4 ) ) ) ).

% funionI1
thf(fact_520_funionI1,axiom,
    ! [C: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe6795308708001954811_f_nat @ C @ A3 )
     => ( fmembe6795308708001954811_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ) ).

% funionI1
thf(fact_521_funionE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) )
     => ( ~ ( fmembe774991991480784110rm_f_v @ C @ A3 )
       => ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ).

% funionE
thf(fact_522_funionE,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( sup_su1809081536757948599term_f @ A3 @ B4 ) )
     => ( ~ ( fmembe6950792022577883716term_f @ C @ A3 )
       => ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ).

% funionE
thf(fact_523_funionE,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( sup_su6586017679312891882term_f @ A3 @ B4 ) )
     => ( ~ ( fmembe1418995490368054919term_f @ C @ A3 )
       => ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ).

% funionE
thf(fact_524_funionE,axiom,
    ! [C: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe6795308708001954811_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
     => ( ~ ( fmembe6795308708001954811_f_nat @ C @ A3 )
       => ( fmembe6795308708001954811_f_nat @ C @ B4 ) ) ) ).

% funionE
thf(fact_525_fimage__funion,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fimage84417724776064284_f_nat @ F2 @ ( sup_su641650120996271713rm_f_v @ A3 @ B4 ) )
      = ( sup_su1580556572264692878_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) @ ( fimage84417724776064284_f_nat @ F2 @ B4 ) ) ) ).

% fimage_funion
thf(fact_526_fimage__funion,axiom,
    ! [F2: product_prod_f_nat > product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fimage736527463273833859_f_nat @ F2 @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
      = ( sup_su8375471814421675870_f_nat @ ( fimage736527463273833859_f_nat @ F2 @ A3 ) @ ( fimage736527463273833859_f_nat @ F2 @ B4 ) ) ) ).

% fimage_funion
thf(fact_527_funion__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,D: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ C2 )
     => ( ( ord_le1552505484586773650_f_nat @ B4 @ D )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) @ ( sup_su8375471814421675870_f_nat @ C2 @ D ) ) ) ) ).

% funion_mono
thf(fact_528_funion__least,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ C2 )
     => ( ( ord_le1552505484586773650_f_nat @ B4 @ C2 )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) @ C2 ) ) ) ).

% funion_least
thf(fact_529_funion__upper1,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ).

% funion_upper1
thf(fact_530_funion__upper2,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ B4 @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ).

% funion_upper2
thf(fact_531_funion__absorb1,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( sup_su8375471814421675870_f_nat @ A3 @ B4 )
        = B4 ) ) ).

% funion_absorb1
thf(fact_532_funion__absorb2,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B4 @ A3 )
     => ( ( sup_su8375471814421675870_f_nat @ A3 @ B4 )
        = A3 ) ) ).

% funion_absorb2
thf(fact_533_fsubset__funion__eq,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B5: fset_P6228066233360383026_f_nat] :
          ( ( sup_su8375471814421675870_f_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% fsubset_funion_eq
thf(fact_534_fImage__fUn,axiom,
    ! [R5: fset_P7991872372518838603_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fSet_f7437826027993719364_f_nat @ R5 @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
      = ( sup_su8375471814421675870_f_nat @ ( fSet_f7437826027993719364_f_nat @ R5 @ A3 ) @ ( fSet_f7437826027993719364_f_nat @ R5 @ B4 ) ) ) ).

% fImage_fUn
thf(fact_535_le__sup__iff,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ X @ Y ) @ Z3 )
      = ( ( ord_le1552505484586773650_f_nat @ X @ Z3 )
        & ( ord_le1552505484586773650_f_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_536_sup_Obounded__iff,axiom,
    ! [B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ B @ C ) @ A )
      = ( ( ord_le1552505484586773650_f_nat @ B @ A )
        & ( ord_le1552505484586773650_f_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_537_gtt__comp__complete__semi_H_I1_J,axiom,
    ! [Q: bot_bot_term_f,G_2: produc6920895926601326823rm_f_f,S: ground_gterm_f,U: ground_gterm_f,G_1: produc6920895926601326823rm_f_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G_2 ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) @ U ) )
       => ( ( gtt_ac2236489977636718847rm_f_f @ G_1 @ T @ U )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ S ) ) ) ) ) ).

% gtt_comp_complete_semi'(1)
thf(fact_538_gtt__comp__complete__semi_H_I2_J,axiom,
    ! [Q: bot_bot_term_f,G_2: produc6920895926601326823rm_f_f,S: ground_gterm_f,U: ground_gterm_f,G_1: produc6920895926601326823rm_f_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G_2 ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) @ U ) )
       => ( ( gtt_ac2236489977636718847rm_f_f @ G_1 @ T @ U )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ T ) ) ) ) ) ).

% gtt_comp_complete_semi'(2)
thf(fact_539_gtt__comp__complete__semi_I1_J,axiom,
    ! [Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,S: ground_gterm_f,U: ground_gterm_f,G_2: produc6920895926601326823rm_f_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G_1 ) @ U ) )
       => ( ( gtt_ac2236489977636718847rm_f_f @ G_2 @ U @ T )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ S ) ) ) ) ) ).

% gtt_comp_complete_semi(1)
thf(fact_540_gtt__comp__complete__semi_I2_J,axiom,
    ! [Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,S: ground_gterm_f,U: ground_gterm_f,G_2: produc6920895926601326823rm_f_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ S ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G_1 ) @ U ) )
       => ( ( gtt_ac2236489977636718847rm_f_f @ G_2 @ U @ T )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ T ) ) ) ) ) ).

% gtt_comp_complete_semi(2)
thf(fact_541_sup_OcoboundedI2,axiom,
    ! [C: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C @ B )
     => ( ord_le1552505484586773650_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_542_union__fset,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( sup_su641650120996271713rm_f_v @ X @ Xa2 ) )
      = ( sup_su1994854688125285083rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) ) ) ).

% union_fset
thf(fact_543_union__fset,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Xa2: fset_P6228066233360383026_f_nat] :
      ( ( fset_P3576968334923099475_f_nat @ ( sup_su8375471814421675870_f_nat @ X @ Xa2 ) )
      = ( sup_su770888850772321124_f_nat @ ( fset_P3576968334923099475_f_nat @ X ) @ ( fset_P3576968334923099475_f_nat @ Xa2 ) ) ) ).

% union_fset
thf(fact_544_inf__sup__ord_I4_J,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ Y @ ( sup_su8375471814421675870_f_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_545_inf__sup__ord_I3_J,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X @ ( sup_su8375471814421675870_f_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_546_le__supE,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ A @ B ) @ X )
     => ~ ( ( ord_le1552505484586773650_f_nat @ A @ X )
         => ~ ( ord_le1552505484586773650_f_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_547_le__supI,axiom,
    ! [A: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ X )
     => ( ( ord_le1552505484586773650_f_nat @ B @ X )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_548_sup__ge1,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X @ ( sup_su8375471814421675870_f_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_549_sup__ge2,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ Y @ ( sup_su8375471814421675870_f_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_550_le__supI1,axiom,
    ! [X: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ A )
     => ( ord_le1552505484586773650_f_nat @ X @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_551_le__supI2,axiom,
    ! [X: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ B )
     => ( ord_le1552505484586773650_f_nat @ X @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_552_sup_Omono,axiom,
    ! [C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,D2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C @ A )
     => ( ( ord_le1552505484586773650_f_nat @ D2 @ B )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ C @ D2 ) @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_553_sup__mono,axiom,
    ! [A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,D2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ C )
     => ( ( ord_le1552505484586773650_f_nat @ B @ D2 )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ A @ B ) @ ( sup_su8375471814421675870_f_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_554_sup__least,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X )
     => ( ( ord_le1552505484586773650_f_nat @ Z3 @ X )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_555_le__iff__sup,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [X4: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
          ( ( sup_su8375471814421675870_f_nat @ X4 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_556_sup_OorderE,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( A
        = ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_557_sup_OorderI,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( A
        = ( sup_su8375471814421675870_f_nat @ A @ B ) )
     => ( ord_le1552505484586773650_f_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_558_sup__unique,axiom,
    ! [F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ Y3 @ X3 )
             => ( ( ord_le1552505484586773650_f_nat @ Z2 @ X3 )
               => ( ord_le1552505484586773650_f_nat @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_su8375471814421675870_f_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_559_sup_Oabsorb1,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( ( sup_su8375471814421675870_f_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_560_sup_Oabsorb2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( sup_su8375471814421675870_f_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_561_sup__absorb1,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X )
     => ( ( sup_su8375471814421675870_f_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_562_sup__absorb2,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ Y )
     => ( ( sup_su8375471814421675870_f_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_563_sup_OboundedE,axiom,
    ! [B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ B @ C ) @ A )
     => ~ ( ( ord_le1552505484586773650_f_nat @ B @ A )
         => ~ ( ord_le1552505484586773650_f_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_564_sup_OboundedI,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( ( ord_le1552505484586773650_f_nat @ C @ A )
       => ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_565_sup_Oorder__iff,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [B6: fset_P6228066233360383026_f_nat,A7: fset_P6228066233360383026_f_nat] :
          ( A7
          = ( sup_su8375471814421675870_f_nat @ A7 @ B6 ) ) ) ) ).

% sup.order_iff
thf(fact_566_sup_Ocobounded1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_567_sup_Ocobounded2,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ B @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_568_sup_Oabsorb__iff1,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [B6: fset_P6228066233360383026_f_nat,A7: fset_P6228066233360383026_f_nat] :
          ( ( sup_su8375471814421675870_f_nat @ A7 @ B6 )
          = A7 ) ) ) ).

% sup.absorb_iff1
thf(fact_569_sup_Oabsorb__iff2,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A7: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( sup_su8375471814421675870_f_nat @ A7 @ B6 )
          = B6 ) ) ) ).

% sup.absorb_iff2
thf(fact_570_sup_OcoboundedI1,axiom,
    ! [C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C @ A )
     => ( ord_le1552505484586773650_f_nat @ C @ ( sup_su8375471814421675870_f_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_571_gtt__accept__exI,axiom,
    ! [G: produc6920895926601326823rm_f_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( gtt_ac2236489977636718847rm_f_f @ G @ S @ T )
     => ? [U2: term_f7969170442531588692term_f] :
          ( ( fmembe7587999265257414299term_f @ U2 @ ( tree_t2575918229523990688rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) @ ( ground5944018991622673045term_f @ S ) ) )
          & ( fmembe7587999265257414299term_f @ U2 @ ( tree_t2575918229523990688rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) @ ( ground5944018991622673045term_f @ T ) ) ) ) ) ).

% gtt_accept_exI
thf(fact_572_gtt__accept__exI,axiom,
    ! [G: produc3767436553304384037ta_v_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( gtt_accept_v_f @ G @ S @ T )
     => ? [U2: term_f_v] :
          ( ( fmember_term_f_v @ U2 @ ( tree_ta_der_v_f2 @ ( produc9004591791807015497ta_v_f @ G ) @ ( ground1072145386804415820rm_f_v @ S ) ) )
          & ( fmember_term_f_v @ U2 @ ( tree_ta_der_v_f2 @ ( produc592183213271074443ta_v_f @ G ) @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% gtt_accept_exI
thf(fact_573_gtt__nr__states__def,axiom,
    ( gtt_nr2273603058196838529_f_f_f
    = ( ^ [G4: produc6920895926601326823rm_f_f] : ( sup_su6586017679312891882term_f @ ( ta_nr_3843931555992929701rm_f_f @ ( produc1890379913328437059rm_f_f @ G4 ) ) @ ( ta_nr_3843931555992929701rm_f_f @ ( produc1110312829679706757rm_f_f @ G4 ) ) ) ) ) ).

% gtt_nr_states_def
thf(fact_574_comp__rtrancl__trancl,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R5 @ S2 ) @ S2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R5 @ S2 ) ) @ S2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ S2 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_575_comp__rtrancl__trancl,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,S: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( ord_le5176047027162639613term_f @ ( relcom2207653138174173357term_f @ R5 @ S2 ) @ S2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ ( sup_su1358690925027996977term_f @ R5 @ S2 ) ) @ S2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ ( transi2774326432670905901term_f @ S2 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_576_comp__rtrancl__trancl,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ R5 @ S2 ) @ S2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ ( sup_su8241624573636005595term_f @ R5 @ S2 ) ) @ S2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( transi8854406268384802389term_f @ S2 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_577_gtt__syms__GTT__comp,axiom,
    ! [A3: produc6920895926601326823rm_f_f,B4: produc6920895926601326823rm_f_f] :
      ( ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ A3 @ B4 ) ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ A3 @ B4 ) ) ) )
      = ( sup_su8375471814421675870_f_nat @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ A3 ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ A3 ) ) ) @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ B4 ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ B4 ) ) ) ) ) ).

% gtt_syms_GTT_comp
thf(fact_578_converse__rtrancl__into__rtrancl,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 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_579_converse__rtrancl__into__rtrancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( transi6299148271647642955term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_580_converse__rtrancl__into__rtrancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( transi7074228675819176119term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_581_converse__rtrancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( P2 @ B )
       => ( ! [Y3: term_f_v,Z2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R2 )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_582_converse__rtrancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,P2: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( P2 @ B )
       => ( ! [Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Z2 @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_583_converse__rtrancl__induct,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,P2: ground_gterm_f > $o] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( P2 @ B )
       => ( ! [Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Z2 @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_584_converse__rtranclE,axiom,
    ! [X: term_f_v,Z3: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( X != Z3 )
       => ~ ! [Y3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_585_converse__rtranclE,axiom,
    ! [X: bot_bot_term_f,Z3: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( X != Z3 )
       => ~ ! [Y3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ R2 )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_586_converse__rtranclE,axiom,
    ! [X: ground_gterm_f,Z3: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( X != Z3 )
       => ~ ! [Y3: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y3 ) @ R2 )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z3 ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_587_rtrancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( P2 @ A )
       => ( ! [Y3: term_f_v,Z2: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_588_rtrancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,P2: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( P2 @ A )
       => ( ! [Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ ( transi6299148271647642955term_f @ R2 ) )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_589_rtrancl__induct,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,P2: ground_gterm_f > $o] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( P2 @ A )
       => ( ! [Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ Y3 ) @ ( transi7074228675819176119term_f @ R2 ) )
             => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_590_rtrancl__trans,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: set_Pr8827868859434726151rm_f_v,Z3: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_591_rtrancl__trans,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,Z3: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ ( transi6299148271647642955term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_592_rtrancl__trans,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: set_Pr989862937836626183term_f,Z3: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ ( transi7074228675819176119term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ) ).

% rtrancl_trans
thf(fact_593_rtranclE,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 )
       => ~ ! [Y3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_594_rtranclE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ ( transi6299148271647642955term_f @ R2 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_595_rtranclE,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( A != B )
       => ~ ! [Y3: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ Y3 ) @ ( transi7074228675819176119term_f @ R2 ) )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ B ) @ R2 ) ) ) ) ).

% rtranclE
thf(fact_596_rtrancl_Ortrancl__into__rtrancl,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 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R2 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_597_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R2 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_598_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ R2 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_599_rtrancl_Ortrancl__refl,axiom,
    ! [A: term_f_v,R2: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_600_rtrancl_Ortrancl__refl,axiom,
    ! [A: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ A ) @ ( transi6299148271647642955term_f @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_601_rtrancl_Ortrancl__refl,axiom,
    ! [A: ground_gterm_f,R2: set_Pr989862937836626183term_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ A ) @ ( transi7074228675819176119term_f @ R2 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_602_rtrancl_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
      = ( ? [A7: term_f_v] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: term_f_v,B6: term_f_v,C3: term_f_v] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B6 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_603_rtrancl_Osimps,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi6299148271647642955term_f @ R2 ) )
      = ( ? [A7: bot_bot_term_f] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: bot_bot_term_f,B6: bot_bot_term_f,C3: bot_bot_term_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A7 @ B6 ) @ ( transi6299148271647642955term_f @ R2 ) )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B6 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_604_rtrancl_Osimps,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( transi7074228675819176119term_f @ R2 ) )
      = ( ? [A7: ground_gterm_f] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: ground_gterm_f,B6: ground_gterm_f,C3: ground_gterm_f] :
            ( ( A1 = A7 )
            & ( A22 = C3 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A7 @ B6 ) @ ( transi7074228675819176119term_f @ R2 ) )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B6 @ C3 ) @ R2 ) ) ) ) ).

% rtrancl.simps
thf(fact_605_rtrancl_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_606_rtrancl_Ocases,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ B3 ) @ ( transi6299148271647642955term_f @ R2 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B3 @ A22 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_607_rtrancl_Ocases,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: ground_gterm_f] :
              ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ B3 ) @ ( transi7074228675819176119term_f @ R2 ) )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B3 @ A22 ) @ R2 ) ) ) ) ).

% rtrancl.cases
thf(fact_608_rtrancl__Un__separator__converseE,axiom,
    ! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q3 ) ) )
     => ( ! [X3: term_f_v,Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X3 ) @ Q3 )
             => ( Y3 = X3 ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_609_rtrancl__Un__separator__converseE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,P2: set_Pr6841673932529763677term_f,Q3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ ( sup_su1358690925027996977term_f @ P2 @ Q3 ) ) )
     => ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ B ) @ ( transi6299148271647642955term_f @ P2 ) )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ X3 ) @ Q3 )
             => ( Y3 = X3 ) ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_610_rtrancl__Un__separator__converseE,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,P2: set_Pr989862937836626183term_f,Q3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ ( sup_su8241624573636005595term_f @ P2 @ Q3 ) ) )
     => ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ B ) @ ( transi7074228675819176119term_f @ P2 ) )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ X3 ) @ Q3 )
             => ( Y3 = X3 ) ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_611_rtrancl__Un__separatorE,axiom,
    ! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q3 ) ) )
     => ( ! [X3: term_f_v,Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ X3 ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ Q3 )
             => ( X3 = Y3 ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_612_rtrancl__Un__separatorE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,P2: set_Pr6841673932529763677term_f,Q3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ ( sup_su1358690925027996977term_f @ P2 @ Q3 ) ) )
     => ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ X3 ) @ ( transi6299148271647642955term_f @ P2 ) )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ Q3 )
             => ( X3 = Y3 ) ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_613_rtrancl__Un__separatorE,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,P2: set_Pr989862937836626183term_f,Q3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ ( sup_su8241624573636005595term_f @ P2 @ Q3 ) ) )
     => ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ X3 ) @ ( transi7074228675819176119term_f @ P2 ) )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ Q3 )
             => ( X3 = Y3 ) ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_614_steps__map,axiom,
    ! [P2: term_f_v > $o,Q3: set_Pr8827868859434726151rm_f_v > $o,F2: term_f_v > term_f_v,G2: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v,T: term_f_v,R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,U: term_f_v] :
      ( ! [T3: term_f_v,U2: term_f_v,R6: set_Pr8827868859434726151rm_f_v] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) )
              & ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R5 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R5 ) ) ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) @ ( relcom2934258169684288428rm_f_v @ ( G2 @ S2 ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_615_steps__map,axiom,
    ! [P2: term_f_v > $o,Q3: set_Pr8827868859434726151rm_f_v > $o,F2: term_f_v > bot_bot_term_f,G2: set_Pr8827868859434726151rm_f_v > set_Pr6841673932529763677term_f,T: term_f_v,R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,U: term_f_v] :
      ( ! [T3: term_f_v,U2: term_f_v,R6: set_Pr8827868859434726151rm_f_v] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R5 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R5 ) ) ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) @ ( relcom2207653138174173357term_f @ ( G2 @ S2 ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_616_steps__map,axiom,
    ! [P2: term_f_v > $o,Q3: set_Pr8827868859434726151rm_f_v > $o,F2: term_f_v > ground_gterm_f,G2: set_Pr8827868859434726151rm_f_v > set_Pr989862937836626183term_f,T: term_f_v,R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,U: term_f_v] :
      ( ! [T3: term_f_v,U2: term_f_v,R6: set_Pr8827868859434726151rm_f_v] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R5 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R5 ) ) ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) @ ( relcom4354988593214781939term_f @ ( G2 @ S2 ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_617_steps__map,axiom,
    ! [P2: bot_bot_term_f > $o,Q3: set_Pr6841673932529763677term_f > $o,F2: bot_bot_term_f > term_f_v,G2: set_Pr6841673932529763677term_f > set_Pr8827868859434726151rm_f_v,T: bot_bot_term_f,R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,U: bot_bot_term_f] :
      ( ! [T3: bot_bot_term_f,U2: bot_bot_term_f,R6: set_Pr6841673932529763677term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( transi6299148271647642955term_f @ R5 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) )
              & ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ R5 ) @ ( relcom2207653138174173357term_f @ S2 @ ( transi6299148271647642955term_f @ R5 ) ) ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) @ ( relcom2934258169684288428rm_f_v @ ( G2 @ S2 ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_618_steps__map,axiom,
    ! [P2: bot_bot_term_f > $o,Q3: set_Pr6841673932529763677term_f > $o,F2: bot_bot_term_f > bot_bot_term_f,G2: set_Pr6841673932529763677term_f > set_Pr6841673932529763677term_f,T: bot_bot_term_f,R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,U: bot_bot_term_f] :
      ( ! [T3: bot_bot_term_f,U2: bot_bot_term_f,R6: set_Pr6841673932529763677term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( transi6299148271647642955term_f @ R5 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ R5 ) @ ( relcom2207653138174173357term_f @ S2 @ ( transi6299148271647642955term_f @ R5 ) ) ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) @ ( relcom2207653138174173357term_f @ ( G2 @ S2 ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_619_steps__map,axiom,
    ! [P2: bot_bot_term_f > $o,Q3: set_Pr6841673932529763677term_f > $o,F2: bot_bot_term_f > ground_gterm_f,G2: set_Pr6841673932529763677term_f > set_Pr989862937836626183term_f,T: bot_bot_term_f,R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,U: bot_bot_term_f] :
      ( ! [T3: bot_bot_term_f,U2: bot_bot_term_f,R6: set_Pr6841673932529763677term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( transi6299148271647642955term_f @ R5 ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ R5 ) @ ( relcom2207653138174173357term_f @ S2 @ ( transi6299148271647642955term_f @ R5 ) ) ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) @ ( relcom4354988593214781939term_f @ ( G2 @ S2 ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_620_steps__map,axiom,
    ! [P2: ground_gterm_f > $o,Q3: set_Pr989862937836626183term_f > $o,F2: ground_gterm_f > term_f_v,G2: set_Pr989862937836626183term_f > set_Pr8827868859434726151rm_f_v,T: ground_gterm_f,R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,U: ground_gterm_f] :
      ( ! [T3: ground_gterm_f,U2: ground_gterm_f,R6: set_Pr989862937836626183term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( transi7074228675819176119term_f @ R5 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) )
              & ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ R5 ) @ ( relcom4354988593214781939term_f @ S2 @ ( transi7074228675819176119term_f @ R5 ) ) ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) @ ( relcom2934258169684288428rm_f_v @ ( G2 @ S2 ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_621_steps__map,axiom,
    ! [P2: ground_gterm_f > $o,Q3: set_Pr989862937836626183term_f > $o,F2: ground_gterm_f > bot_bot_term_f,G2: set_Pr989862937836626183term_f > set_Pr6841673932529763677term_f,T: ground_gterm_f,R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,U: ground_gterm_f] :
      ( ! [T3: ground_gterm_f,U2: ground_gterm_f,R6: set_Pr989862937836626183term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( transi7074228675819176119term_f @ R5 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ R5 ) @ ( relcom4354988593214781939term_f @ S2 @ ( transi7074228675819176119term_f @ R5 ) ) ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) @ ( relcom2207653138174173357term_f @ ( G2 @ S2 ) @ ( transi6299148271647642955term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_622_steps__map,axiom,
    ! [P2: ground_gterm_f > $o,Q3: set_Pr989862937836626183term_f > $o,F2: ground_gterm_f > ground_gterm_f,G2: set_Pr989862937836626183term_f > set_Pr989862937836626183term_f,T: ground_gterm_f,R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,U: ground_gterm_f] :
      ( ! [T3: ground_gterm_f,U2: ground_gterm_f,R6: set_Pr989862937836626183term_f] :
          ( ( P2 @ T3 )
         => ( ( Q3 @ R6 )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T3 @ U2 ) @ R6 )
             => ( ( P2 @ U2 )
                & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q3 @ R5 )
         => ( ( Q3 @ S2 )
           => ( ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( transi7074228675819176119term_f @ R5 ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) )
              & ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ R5 ) @ ( relcom4354988593214781939term_f @ S2 @ ( transi7074228675819176119term_f @ R5 ) ) ) )
               => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) @ ( relcom4354988593214781939term_f @ ( G2 @ S2 ) @ ( transi7074228675819176119term_f @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_623_right__comp__S,axiom,
    ! [X: term_f_v,Y: term_f_v,S2: set_Pr8827868859434726151rm_f_v,NS: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( 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 @ X @ Y ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_624_right__comp__S,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,S2: set_Pr6841673932529763677term_f,NS: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( relcom2207653138174173357term_f @ S2 @ ( sup_su1358690925027996977term_f @ ( relcom2207653138174173357term_f @ S2 @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ S2 ) @ ( transi6299148271647642955term_f @ NS ) ) ) @ ( transi6299148271647642955term_f @ NS ) ) ) )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( relcom2207653138174173357term_f @ S2 @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ S2 ) @ ( transi6299148271647642955term_f @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_625_right__comp__S,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,S2: set_Pr989862937836626183term_f,NS: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( relcom4354988593214781939term_f @ S2 @ ( sup_su8241624573636005595term_f @ ( relcom4354988593214781939term_f @ S2 @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ S2 ) @ ( transi7074228675819176119term_f @ NS ) ) ) @ ( transi7074228675819176119term_f @ NS ) ) ) )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( relcom4354988593214781939term_f @ S2 @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ S2 ) @ ( transi7074228675819176119term_f @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_626_first__step__O,axiom,
    ! [C2: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ( C2
        = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ C2 ) @ B4 ) )
       => ? [Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ B4 ) ) ) ) ).

% first_step_O
thf(fact_627_first__step__O,axiom,
    ! [C2: set_Pr6841673932529763677term_f,A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( C2
        = ( sup_su1358690925027996977term_f @ A3 @ B4 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ C2 ) @ B4 ) )
       => ? [Y3: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ A3 ) @ B4 ) ) ) ) ).

% first_step_O
thf(fact_628_first__step__O,axiom,
    ! [C2: set_Pr989862937836626183term_f,A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ( C2
        = ( sup_su8241624573636005595term_f @ A3 @ B4 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ C2 ) @ B4 ) )
       => ? [Y3: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y3 ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ A3 ) @ B4 ) ) ) ) ).

% first_step_O
thf(fact_629_first__step,axiom,
    ! [C2: set_Pr8827868859434726151rm_f_v,A3: set_Pr8827868859434726151rm_f_v,B4: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v,Z3: term_f_v] :
      ( ( C2
        = ( sup_su1994854688125285083rm_f_v @ A3 @ B4 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ C2 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ B4 )
         => ? [Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A3 ) @ B4 ) ) ) ) ) ).

% first_step
thf(fact_630_first__step,axiom,
    ! [C2: set_Pr6841673932529763677term_f,A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f,Z3: bot_bot_term_f] :
      ( ( C2
        = ( sup_su1358690925027996977term_f @ A3 @ B4 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ C2 ) )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ B4 )
         => ? [Y3: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ A3 ) @ B4 ) ) ) ) ) ).

% first_step
thf(fact_631_first__step,axiom,
    ! [C2: set_Pr989862937836626183term_f,A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f,Z3: ground_gterm_f] :
      ( ( C2
        = ( sup_su8241624573636005595term_f @ A3 @ B4 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ C2 ) )
       => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ B4 )
         => ? [Y3: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y3 ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ A3 ) @ B4 ) ) ) ) ) ).

% first_step
thf(fact_632_firstStep,axiom,
    ! [L2: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,R5: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ( L2
        = ( sup_su1994854688125285083rm_f_v @ S2 @ R5 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ L2 ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
          | ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R5 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ L2 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_633_firstStep,axiom,
    ! [L2: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,R5: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( L2
        = ( sup_su1358690925027996977term_f @ S2 @ R5 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ L2 ) )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ R5 ) )
          | ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ R5 ) @ ( relcom2207653138174173357term_f @ S2 @ ( transi6299148271647642955term_f @ L2 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_634_firstStep,axiom,
    ! [L2: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,R5: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ( L2
        = ( sup_su8241624573636005595term_f @ S2 @ R5 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ L2 ) )
       => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ R5 ) )
          | ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ R5 ) @ ( relcom4354988593214781939term_f @ S2 @ ( transi7074228675819176119term_f @ L2 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_635_trancl__rtrancl__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 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_636_trancl__rtrancl__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( transi6299148271647642955term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_637_trancl__rtrancl__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( transi7074228675819176119term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_638_rtrancl__trancl__trancl,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: set_Pr8827868859434726151rm_f_v,Z3: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_639_rtrancl__trancl__trancl,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,Z3: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_640_rtrancl__trancl__trancl,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: set_Pr989862937836626183term_f,Z3: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_641_rtrancl__into__trancl2,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 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_642_rtrancl__into__trancl2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( transi6299148271647642955term_f @ R2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_643_rtrancl__into__trancl2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( transi7074228675819176119term_f @ R2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_644_rtrancl__into__trancl1,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 ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R2 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R2 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_645_rtrancl__into__trancl1,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R2 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R2 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_646_rtrancl__into__trancl1,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ R2 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi8854406268384802389term_f @ R2 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_647_rtrancl__eq__or__trancl,axiom,
    ! [X: term_f_v,Y: term_f_v,R5: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R5 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_648_rtrancl__eq__or__trancl,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R5: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ R5 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R5 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_649_rtrancl__eq__or__trancl,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R5: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ R5 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R5 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_650_trancl__into__rtrancl,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R2 ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ).

% trancl_into_rtrancl
thf(fact_651_trancl__into__rtrancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R2 ) )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ).

% trancl_into_rtrancl
thf(fact_652_trancl__into__rtrancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R2 ) )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ).

% trancl_into_rtrancl
thf(fact_653_tranclD2,axiom,
    ! [X: term_f_v,Y: term_f_v,R5: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R5 ) )
     => ? [Z2: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ Y ) @ R5 ) ) ) ).

% tranclD2
thf(fact_654_tranclD2,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R5: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R5 ) )
     => ? [Z2: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z2 ) @ ( transi6299148271647642955term_f @ R5 ) )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Z2 @ Y ) @ R5 ) ) ) ).

% tranclD2
thf(fact_655_tranclD2,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R5: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R5 ) )
     => ? [Z2: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z2 ) @ ( transi7074228675819176119term_f @ R5 ) )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Z2 @ Y ) @ R5 ) ) ) ).

% tranclD2
thf(fact_656_rtranclD,axiom,
    ! [A: term_f_v,B: term_f_v,R5: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R5 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R5 ) ) ) ) ) ).

% rtranclD
thf(fact_657_rtranclD,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R5: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R5 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R5 ) ) ) ) ) ).

% rtranclD
thf(fact_658_rtranclD,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R5: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R5 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi8854406268384802389term_f @ R5 ) ) ) ) ) ).

% rtranclD
thf(fact_659_tranclD,axiom,
    ! [X: term_f_v,Y: term_f_v,R5: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R5 ) )
     => ? [Z2: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z2 ) @ R5 )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ Y ) @ ( transi3028809021613410336rm_f_v @ R5 ) ) ) ) ).

% tranclD
thf(fact_660_tranclD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R5: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R5 ) )
     => ? [Z2: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z2 ) @ R5 )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Z2 @ Y ) @ ( transi6299148271647642955term_f @ R5 ) ) ) ) ).

% tranclD
thf(fact_661_tranclD,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R5: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi8854406268384802389term_f @ R5 ) )
     => ? [Z2: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Z2 ) @ R5 )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Z2 @ Y ) @ ( transi7074228675819176119term_f @ R5 ) ) ) ) ).

% tranclD
thf(fact_662_converse__rtrancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R2: 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 @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: term_f_v,B3: term_f_v,Aa: term_f_v,Ba: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R2 )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_663_converse__rtrancl__induct2,axiom,
    ! [Ax: bot_bot_term_f,Ay: bot_bot_term_f,Bx: bot_bot_term_f,By: bot_bot_term_f,R2: set_Pr4896805198020829255term_f,P2: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ Bx @ By ) ) @ ( transi4676799229066866816term_f @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f,Aa: bot_bot_term_f,Ba: bot_bot_term_f] :
              ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ ( produc2865564700782054965term_f @ Aa @ Ba ) ) @ R2 )
             => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Aa @ Ba ) @ ( produc2865564700782054965term_f @ Bx @ By ) ) @ ( transi4676799229066866816term_f @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_664_converse__rtrancl__induct2,axiom,
    ! [Ax: ground_gterm_f,Ay: ground_gterm_f,Bx: ground_gterm_f,By: ground_gterm_f,R2: set_Pr1766941801908210887term_f,P2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ Bx @ By ) ) @ ( transi834741307384780330term_f @ R2 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: ground_gterm_f,B3: ground_gterm_f,Aa: ground_gterm_f,Ba: ground_gterm_f] :
              ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ ( produc3560254623552331287term_f @ Aa @ Ba ) ) @ R2 )
             => ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Aa @ Ba ) @ ( produc3560254623552331287term_f @ Bx @ By ) ) @ ( transi834741307384780330term_f @ R2 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_665_converse__rtranclE2,axiom,
    ! [Xa2: term_f_v,Xb: term_f_v,Za2: term_f_v,Zb: term_f_v,R2: set_Pr7897041799657265351rm_f_v] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa2 @ Xb ) @ ( produc2510790566102465375rm_f_v @ Za2 @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R2 ) )
     => ( ( ( produc2510790566102465375rm_f_v @ Xa2 @ Xb )
         != ( produc2510790566102465375rm_f_v @ Za2 @ Zb ) )
       => ~ ! [A4: term_f_v,B3: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa2 @ Xb ) @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) ) @ R2 )
             => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Za2 @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_666_converse__rtranclE2,axiom,
    ! [Xa2: bot_bot_term_f,Xb: bot_bot_term_f,Za2: bot_bot_term_f,Zb: bot_bot_term_f,R2: set_Pr4896805198020829255term_f] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Xa2 @ Xb ) @ ( produc2865564700782054965term_f @ Za2 @ Zb ) ) @ ( transi4676799229066866816term_f @ R2 ) )
     => ( ( ( produc2865564700782054965term_f @ Xa2 @ Xb )
         != ( produc2865564700782054965term_f @ Za2 @ Zb ) )
       => ~ ! [A4: bot_bot_term_f,B3: bot_bot_term_f] :
              ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Xa2 @ Xb ) @ ( produc2865564700782054965term_f @ A4 @ B3 ) ) @ R2 )
             => ~ ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ ( produc2865564700782054965term_f @ Za2 @ Zb ) ) @ ( transi4676799229066866816term_f @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_667_converse__rtranclE2,axiom,
    ! [Xa2: ground_gterm_f,Xb: ground_gterm_f,Za2: ground_gterm_f,Zb: ground_gterm_f,R2: set_Pr1766941801908210887term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Xa2 @ Xb ) @ ( produc3560254623552331287term_f @ Za2 @ Zb ) ) @ ( transi834741307384780330term_f @ R2 ) )
     => ( ( ( produc3560254623552331287term_f @ Xa2 @ Xb )
         != ( produc3560254623552331287term_f @ Za2 @ Zb ) )
       => ~ ! [A4: ground_gterm_f,B3: ground_gterm_f] :
              ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Xa2 @ Xb ) @ ( produc3560254623552331287term_f @ A4 @ B3 ) ) @ R2 )
             => ~ ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ ( produc3560254623552331287term_f @ Za2 @ Zb ) ) @ ( transi834741307384780330term_f @ R2 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_668_rtrancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R2: 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 @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: term_f_v,B3: 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 @ A4 @ B3 ) ) @ ( transi1062743709496199978rm_f_v @ R2 ) )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_669_rtrancl__induct2,axiom,
    ! [Ax: bot_bot_term_f,Ay: bot_bot_term_f,Bx: bot_bot_term_f,By: bot_bot_term_f,R2: set_Pr4896805198020829255term_f,P2: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ Bx @ By ) ) @ ( transi4676799229066866816term_f @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f,Aa: bot_bot_term_f,Ba: bot_bot_term_f] :
              ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ A4 @ B3 ) ) @ ( transi4676799229066866816term_f @ R2 ) )
             => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ ( produc2865564700782054965term_f @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_670_rtrancl__induct2,axiom,
    ! [Ax: ground_gterm_f,Ay: ground_gterm_f,Bx: ground_gterm_f,By: ground_gterm_f,R2: set_Pr1766941801908210887term_f,P2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ Bx @ By ) ) @ ( transi834741307384780330term_f @ R2 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: ground_gterm_f,B3: ground_gterm_f,Aa: ground_gterm_f,Ba: ground_gterm_f] :
              ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ Ax @ Ay ) @ ( produc3560254623552331287term_f @ A4 @ B3 ) ) @ ( transi834741307384780330term_f @ R2 ) )
             => ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ ( produc3560254623552331287term_f @ Aa @ Ba ) ) @ R2 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_671_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_672_non__strict__into__strict,axiom,
    ! [NS: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,S: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( ord_le5176047027162639613term_f @ ( relcom2207653138174173357term_f @ NS @ S2 ) @ S2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ NS ) @ S2 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ S2 ) ) ) ).

% non_strict_into_strict
thf(fact_673_non__strict__into__strict,axiom,
    ! [NS: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ NS @ S2 ) @ S2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ NS ) @ S2 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S @ T ) @ S2 ) ) ) ).

% non_strict_into_strict
thf(fact_674_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( 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_675_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( ord_le5176047027162639613term_f @ ( relcom2207653138174173357term_f @ NS @ S2 ) @ S2 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ ( sup_su1358690925027996977term_f @ NS @ S2 ) ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( sup_su1358690925027996977term_f @ ( relcom2207653138174173357term_f @ S2 @ ( relcom2207653138174173357term_f @ ( transi6299148271647642955term_f @ S2 ) @ ( transi6299148271647642955term_f @ NS ) ) ) @ ( transi6299148271647642955term_f @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_676_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ NS @ S2 ) @ S2 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ ( sup_su8241624573636005595term_f @ NS @ S2 ) ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( sup_su8241624573636005595term_f @ ( relcom4354988593214781939term_f @ S2 @ ( relcom4354988593214781939term_f @ ( transi7074228675819176119term_f @ S2 ) @ ( transi7074228675819176119term_f @ NS ) ) ) @ ( transi7074228675819176119term_f @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_677_trim__gtt__prod__syms,axiom,
    ! [G5: produc6920895926601326823rm_f_f] : ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ ( trim_g8862173965670262857rm_f_f @ G5 ) ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ ( trim_g8862173965670262857rm_f_f @ G5 ) ) ) ) @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ G5 ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ G5 ) ) ) ) ).

% trim_gtt_prod_syms
thf(fact_678_gtt__only__reach__syms,axiom,
    ! [G: produc6920895926601326823rm_f_f] : ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gtt_on1102194929509959736rm_f_f @ G ) ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gtt_on1102194929509959736rm_f_f @ G ) ) ) ) @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) ) ) ).

% gtt_only_reach_syms
thf(fact_679_gtt__only__prod__syms,axiom,
    ! [G: produc6920895926601326823rm_f_f] : ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gtt_on676859095552228228rm_f_f @ G ) ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gtt_on676859095552228228rm_f_f @ G ) ) ) ) @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) ) ) ).

% gtt_only_prod_syms
thf(fact_680_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_681_meetI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ A3 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi6299148271647642955term_f @ A3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( abstra5633755342312003345term_f @ A3 ) ) ) ) ).

% meetI
thf(fact_682_meetI,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Pr989862937836626183term_f,C: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ A3 ) )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( transi7074228675819176119term_f @ A3 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( abstra1295463143952312945term_f @ A3 ) ) ) ) ).

% meetI
thf(fact_683_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 ) )
     => ? [A4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% meetD
thf(fact_684_meetD,axiom,
    ! [B: bot_bot_term_f,C: bot_bot_term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( abstra5633755342312003345term_f @ A3 ) )
     => ? [A4: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A4 @ B ) @ ( transi6299148271647642955term_f @ A3 ) )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A4 @ C ) @ ( transi6299148271647642955term_f @ A3 ) ) ) ) ).

% meetD
thf(fact_685_meetD,axiom,
    ! [B: ground_gterm_f,C: ground_gterm_f,A3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( abstra1295463143952312945term_f @ A3 ) )
     => ? [A4: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A4 @ B ) @ ( transi7074228675819176119term_f @ A3 ) )
          & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A4 @ C ) @ ( transi7074228675819176119term_f @ A3 ) ) ) ) ).

% meetD
thf(fact_686_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 ) )
     => ~ ! [A4: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B ) @ ( transi3028809021613410336rm_f_v @ A3 ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ C ) @ ( transi3028809021613410336rm_f_v @ A3 ) ) ) ) ).

% meetE
thf(fact_687_meetE,axiom,
    ! [B: bot_bot_term_f,C: bot_bot_term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( abstra5633755342312003345term_f @ A3 ) )
     => ~ ! [A4: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A4 @ B ) @ ( transi6299148271647642955term_f @ A3 ) )
           => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A4 @ C ) @ ( transi6299148271647642955term_f @ A3 ) ) ) ) ).

% meetE
thf(fact_688_meetE,axiom,
    ! [B: ground_gterm_f,C: ground_gterm_f,A3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B @ C ) @ ( abstra1295463143952312945term_f @ A3 ) )
     => ~ ! [A4: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A4 @ B ) @ ( transi7074228675819176119term_f @ A3 ) )
           => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A4 @ C ) @ ( transi7074228675819176119term_f @ A3 ) ) ) ) ).

% meetE
thf(fact_689_gtt__interface__def,axiom,
    ( gtt_in8205028897947813884_f_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f] : ( inf_in2113687546666124112term_f @ ( tree_Q1727860138906775108rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) @ ( tree_Q1727860138906775108rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) ) ) ) ) ).

% gtt_interface_def
thf(fact_690_le__inf__iff,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ ( inf_in6532419608991314884_f_nat @ Y @ Z3 ) )
      = ( ( ord_le1552505484586773650_f_nat @ X @ Y )
        & ( ord_le1552505484586773650_f_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_691_inf_Obounded__iff,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ ( inf_in6532419608991314884_f_nat @ B @ C ) )
      = ( ( ord_le1552505484586773650_f_nat @ A @ B )
        & ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_692_finter__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
        & ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ).

% finter_iff
thf(fact_693_finter__iff,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) )
      = ( ( fmembe6950792022577883716term_f @ C @ A3 )
        & ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ).

% finter_iff
thf(fact_694_finter__iff,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) )
      = ( ( fmembe1418995490368054919term_f @ C @ A3 )
        & ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ).

% finter_iff
thf(fact_695_finterI,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
     => ( ( fmembe774991991480784110rm_f_v @ C @ B4 )
       => ( fmembe774991991480784110rm_f_v @ C @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) ) ) ).

% finterI
thf(fact_696_finterI,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ A3 )
     => ( ( fmembe6950792022577883716term_f @ C @ B4 )
       => ( fmembe6950792022577883716term_f @ C @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) ) ) ) ).

% finterI
thf(fact_697_finterI,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ A3 )
     => ( ( fmembe1418995490368054919term_f @ C @ B4 )
       => ( fmembe1418995490368054919term_f @ C @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) ) ) ) ).

% finterI
thf(fact_698_inf__sup__ord_I2_J,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_699_inf__sup__ord_I1_J,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_700_inf__le1,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_701_inf__le2,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_702_le__infE,axiom,
    ! [X: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ ( inf_in6532419608991314884_f_nat @ A @ B ) )
     => ~ ( ( ord_le1552505484586773650_f_nat @ X @ A )
         => ~ ( ord_le1552505484586773650_f_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_703_le__infI,axiom,
    ! [X: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ A )
     => ( ( ord_le1552505484586773650_f_nat @ X @ B )
       => ( ord_le1552505484586773650_f_nat @ X @ ( inf_in6532419608991314884_f_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_704_inf__mono,axiom,
    ! [A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,D2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ C )
     => ( ( ord_le1552505484586773650_f_nat @ B @ D2 )
       => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ ( inf_in6532419608991314884_f_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_705_le__infI1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ X )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_706_le__infI2,axiom,
    ! [B: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ X )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_707_inf_OorderE,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( A
        = ( inf_in6532419608991314884_f_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_708_inf_OorderI,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( A
        = ( inf_in6532419608991314884_f_nat @ A @ B ) )
     => ( ord_le1552505484586773650_f_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_709_inf__unique,axiom,
    ! [F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X3 @ Y3 )
             => ( ( ord_le1552505484586773650_f_nat @ X3 @ Z2 )
               => ( ord_le1552505484586773650_f_nat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_in6532419608991314884_f_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_710_le__iff__inf,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [X4: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ X4 @ Y5 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_711_inf_Oabsorb1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( inf_in6532419608991314884_f_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_712_inf_Oabsorb2,axiom,
    ! [B: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ A )
     => ( ( inf_in6532419608991314884_f_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_713_inf__absorb1,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ Y )
     => ( ( inf_in6532419608991314884_f_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_714_inf__absorb2,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X )
     => ( ( inf_in6532419608991314884_f_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_715_inf_OboundedE,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ ( inf_in6532419608991314884_f_nat @ B @ C ) )
     => ~ ( ( ord_le1552505484586773650_f_nat @ A @ B )
         => ~ ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_716_inf_OboundedI,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B )
     => ( ( ord_le1552505484586773650_f_nat @ A @ C )
       => ( ord_le1552505484586773650_f_nat @ A @ ( inf_in6532419608991314884_f_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_717_inf__greatest,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ X @ Z3 )
       => ( ord_le1552505484586773650_f_nat @ X @ ( inf_in6532419608991314884_f_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_718_inf_Oorder__iff,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A7: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( A7
          = ( inf_in6532419608991314884_f_nat @ A7 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_719_inf_Ocobounded1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_720_inf_Ocobounded2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_721_inf_Oabsorb__iff1,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A7: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ A7 @ B6 )
          = A7 ) ) ) ).

% inf.absorb_iff1
thf(fact_722_inf_Oabsorb__iff2,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [B6: fset_P6228066233360383026_f_nat,A7: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ A7 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_723_inf_OcoboundedI1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ C )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_724_inf_OcoboundedI2,axiom,
    ! [B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B @ C )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_725_finterD2,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) )
     => ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ).

% finterD2
thf(fact_726_finterD2,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) )
     => ( fmembe6950792022577883716term_f @ C @ B4 ) ) ).

% finterD2
thf(fact_727_finterD2,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) )
     => ( fmembe1418995490368054919term_f @ C @ B4 ) ) ).

% finterD2
thf(fact_728_finterD1,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) )
     => ( fmembe774991991480784110rm_f_v @ C @ A3 ) ) ).

% finterD1
thf(fact_729_finterD1,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) )
     => ( fmembe6950792022577883716term_f @ C @ A3 ) ) ).

% finterD1
thf(fact_730_finterD1,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) )
     => ( fmembe1418995490368054919term_f @ C @ A3 ) ) ).

% finterD1
thf(fact_731_finterE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) )
     => ~ ( ( fmembe774991991480784110rm_f_v @ C @ A3 )
         => ~ ( fmembe774991991480784110rm_f_v @ C @ B4 ) ) ) ).

% finterE
thf(fact_732_finterE,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe6950792022577883716term_f @ C @ A3 )
         => ~ ( fmembe6950792022577883716term_f @ C @ B4 ) ) ) ).

% finterE
thf(fact_733_finterE,axiom,
    ! [C: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ C @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe1418995490368054919term_f @ C @ A3 )
         => ~ ( fmembe1418995490368054919term_f @ C @ B4 ) ) ) ).

% finterE
thf(fact_734_finter__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,D: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ C2 )
     => ( ( ord_le1552505484586773650_f_nat @ B4 @ D )
       => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ ( inf_in6532419608991314884_f_nat @ C2 @ D ) ) ) ) ).

% finter_mono
thf(fact_735_finter__lower1,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ A3 ) ).

% finter_lower1
thf(fact_736_finter__lower2,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ B4 ) ).

% finter_lower2
thf(fact_737_finter__absorb1,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B4 @ A3 )
     => ( ( inf_in6532419608991314884_f_nat @ A3 @ B4 )
        = B4 ) ) ).

% finter_absorb1
thf(fact_738_finter__absorb2,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ( inf_in6532419608991314884_f_nat @ A3 @ B4 )
        = A3 ) ) ).

% finter_absorb2
thf(fact_739_finter__greatest,axiom,
    ! [C2: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C2 @ A3 )
     => ( ( ord_le1552505484586773650_f_nat @ C2 @ B4 )
       => ( ord_le1552505484586773650_f_nat @ C2 @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) ) ) ) ).

% finter_greatest
thf(fact_740_funion__finter__crazy,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ ( inf_in6532419608991314884_f_nat @ B4 @ C2 ) ) @ ( inf_in6532419608991314884_f_nat @ C2 @ A3 ) )
      = ( inf_in6532419608991314884_f_nat @ ( inf_in6532419608991314884_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) ) @ ( sup_su8375471814421675870_f_nat @ C2 @ A3 ) ) ) ).

% funion_finter_crazy
thf(fact_741_finter__funion__distrib,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( inf_in6532419608991314884_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) )
      = ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ ( inf_in6532419608991314884_f_nat @ A3 @ C2 ) ) ) ).

% finter_funion_distrib
thf(fact_742_funion__finter__distrib,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ ( inf_in6532419608991314884_f_nat @ B4 @ C2 ) )
      = ( inf_in6532419608991314884_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) @ ( sup_su8375471814421675870_f_nat @ A3 @ C2 ) ) ) ).

% funion_finter_distrib
thf(fact_743_finter__funion__distrib2,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( inf_in6532419608991314884_f_nat @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) @ A3 )
      = ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ B4 @ A3 ) @ ( inf_in6532419608991314884_f_nat @ C2 @ A3 ) ) ) ).

% finter_funion_distrib2
thf(fact_744_funion__finter__distrib2,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,A3: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ B4 @ C2 ) @ A3 )
      = ( inf_in6532419608991314884_f_nat @ ( sup_su8375471814421675870_f_nat @ B4 @ A3 ) @ ( sup_su8375471814421675870_f_nat @ C2 @ A3 ) ) ) ).

% funion_finter_distrib2
thf(fact_745_distrib__inf__le,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ X @ Y ) @ ( inf_in6532419608991314884_f_nat @ X @ Z3 ) ) @ ( inf_in6532419608991314884_f_nat @ X @ ( sup_su8375471814421675870_f_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_746_distrib__sup__le,axiom,
    ! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( sup_su8375471814421675870_f_nat @ X @ ( inf_in6532419608991314884_f_nat @ Y @ Z3 ) ) @ ( inf_in6532419608991314884_f_nat @ ( sup_su8375471814421675870_f_nat @ X @ Y ) @ ( sup_su8375471814421675870_f_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_747_fimage__finter__fsubset,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] : ( ord_le8062249421014450114_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) @ ( inf_in4830817821989094900_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) @ ( fimage84417724776064284_f_nat @ F2 @ B4 ) ) ) ).

% fimage_finter_fsubset
thf(fact_748_funion__finter__assoc__eq,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ( sup_su8375471814421675870_f_nat @ ( inf_in6532419608991314884_f_nat @ A3 @ B4 ) @ C2 )
        = ( inf_in6532419608991314884_f_nat @ A3 @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) ) )
      = ( ord_le1552505484586773650_f_nat @ C2 @ A3 ) ) ).

% funion_finter_assoc_eq
thf(fact_749_gtt__states__def,axiom,
    ( gtt_st6132224256192451901_f_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f] : ( sup_su6586017679312891882term_f @ ( tree_Q1727860138906775108rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) @ ( tree_Q1727860138906775108rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) ) ) ) ) ).

% gtt_states_def
thf(fact_750_ta__der__to__ta__der_H,axiom,
    ! [Q: v,A8: tree_ta_v_f,S: term_f_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ S ) )
      = ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A8 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_751_ta__der_H__to__ta__der,axiom,
    ! [Q: v,A8: tree_ta_v_f,S: term_f_v] :
      ( ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A8 @ S ) )
     => ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_752_gtt__comp__sound__semi,axiom,
    ! [G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f,Q: bot_bot_term_f,S: ground_gterm_f,T: ground_gterm_f] :
      ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
        = bot_bo4668628460741225810term_f )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ S ) )
       => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ T ) )
         => ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
           => ? [U2: ground_gterm_f] :
                ( ( fmembe1418995490368054919term_f @ Q @ ( tree_g2054044545600476092rm_f_f @ ( produc1110312829679706757rm_f_f @ G_1 ) @ U2 ) )
                & ( gtt_ac2236489977636718847rm_f_f @ G_2 @ U2 @ T ) ) ) ) ) ) ).

% gtt_comp_sound_semi
thf(fact_753_relChain__def,axiom,
    ( bNF_Ca8507058823937871547_f_nat
    = ( ^ [R7: set_Pr8827868859434726151rm_f_v,As: term_f_v > fset_P6228066233360383026_f_nat] :
        ! [I: term_f_v,J: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R7 )
         => ( ord_le1552505484586773650_f_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_754_relChain__def,axiom,
    ( bNF_Ca6591412532971531110_f_nat
    = ( ^ [R7: set_Pr6841673932529763677term_f,As: bot_bot_term_f > fset_P6228066233360383026_f_nat] :
        ! [I: bot_bot_term_f,J: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ I @ J ) @ R7 )
         => ( ord_le1552505484586773650_f_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_755_relChain__def,axiom,
    ( bNF_Ca5107865095533045826_f_nat
    = ( ^ [R7: set_Pr989862937836626183term_f,As: ground_gterm_f > fset_P6228066233360383026_f_nat] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R7 )
         => ( ord_le1552505484586773650_f_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_756_fempty__iff,axiom,
    ! [C: produc8027700073345061287rm_f_v] :
      ~ ( fmembe774991991480784110rm_f_v @ C @ bot_bo4614643899460950777rm_f_v ) ).

% fempty_iff
thf(fact_757_fempty__iff,axiom,
    ! [C: produc250624644672406269term_f] :
      ~ ( fmembe6950792022577883716term_f @ C @ bot_bo6180437703705224527term_f ) ).

% fempty_iff
thf(fact_758_fempty__iff,axiom,
    ! [C: bot_bot_term_f] :
      ~ ( fmembe1418995490368054919term_f @ C @ bot_bo4668628460741225810term_f ) ).

% fempty_iff
thf(fact_759_all__not__fin__conv,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v] :
      ( ( ! [X4: produc8027700073345061287rm_f_v] :
            ~ ( fmembe774991991480784110rm_f_v @ X4 @ A3 ) )
      = ( A3 = bot_bo4614643899460950777rm_f_v ) ) ).

% all_not_fin_conv
thf(fact_760_all__not__fin__conv,axiom,
    ! [A3: fset_P2882900458636368611term_f] :
      ( ( ! [X4: produc250624644672406269term_f] :
            ~ ( fmembe6950792022577883716term_f @ X4 @ A3 ) )
      = ( A3 = bot_bo6180437703705224527term_f ) ) ).

% all_not_fin_conv
thf(fact_761_all__not__fin__conv,axiom,
    ! [A3: fset_Bot_bot_term_f] :
      ( ( ! [X4: bot_bot_term_f] :
            ~ ( fmembe1418995490368054919term_f @ X4 @ A3 ) )
      = ( A3 = bot_bo4668628460741225810term_f ) ) ).

% all_not_fin_conv
thf(fact_762_fimage__is__fempty,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ( fimage84417724776064284_f_nat @ F2 @ A3 )
        = bot_bo2413912301557604854_f_nat )
      = ( A3 = bot_bo4614643899460950777rm_f_v ) ) ).

% fimage_is_fempty
thf(fact_763_fempty__is__fimage,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( bot_bo2413912301557604854_f_nat
        = ( fimage84417724776064284_f_nat @ F2 @ A3 ) )
      = ( A3 = bot_bo4614643899460950777rm_f_v ) ) ).

% fempty_is_fimage
thf(fact_764_fimage__fempty,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( fimage84417724776064284_f_nat @ F2 @ bot_bo4614643899460950777rm_f_v )
      = bot_bo2413912301557604854_f_nat ) ).

% fimage_fempty
thf(fact_765_fempty__fsubsetI,axiom,
    ! [X: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ bot_bo3974441229639268550_f_nat @ X ) ).

% fempty_fsubsetI
thf(fact_766_fsubset__fempty,axiom,
    ! [A3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ bot_bo3974441229639268550_f_nat )
      = ( A3 = bot_bo3974441229639268550_f_nat ) ) ).

% fsubset_fempty
thf(fact_767_funion__fempty,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ( sup_su8375471814421675870_f_nat @ A3 @ B4 )
        = bot_bo3974441229639268550_f_nat )
      = ( ( A3 = bot_bo3974441229639268550_f_nat )
        & ( B4 = bot_bo3974441229639268550_f_nat ) ) ) ).

% funion_fempty
thf(fact_768_inter__fset,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( inf_in6563521360087198331rm_f_v @ X @ Xa2 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) ) ) ).

% inter_fset
thf(fact_769_bot_Oextremum__uniqueI,axiom,
    ! [A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ bot_bo3974441229639268550_f_nat )
     => ( A = bot_bo3974441229639268550_f_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_770_bot_Oextremum__unique,axiom,
    ! [A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ bot_bo3974441229639268550_f_nat )
      = ( A = bot_bo3974441229639268550_f_nat ) ) ).

% bot.extremum_unique
thf(fact_771_bot_Oextremum,axiom,
    ! [A: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ bot_bo3974441229639268550_f_nat @ A ) ).

% bot.extremum
thf(fact_772_Int__Collect__mono,axiom,
    ! [A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f,P2: produc250624644672406269term_f > $o,Q3: produc250624644672406269term_f > $o] :
      ( ( ord_le5176047027162639613term_f @ A3 @ B4 )
     => ( ! [X3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q3 @ X3 ) ) )
       => ( ord_le5176047027162639613term_f @ ( inf_in5033374358511726155term_f @ A3 @ ( collec753421845813909992term_f @ P2 ) ) @ ( inf_in5033374358511726155term_f @ B4 @ ( collec753421845813909992term_f @ Q3 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_773_Int__Collect__mono,axiom,
    ! [A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f,P2: produc7245736746747425831term_f > $o,Q3: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ A3 @ B4 )
     => ( ! [X3: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q3 @ X3 ) ) )
       => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A3 @ ( collec3209569126566747026term_f @ P2 ) ) @ ( inf_in6246359444536921845term_f @ B4 @ ( collec3209569126566747026term_f @ Q3 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_774_femptyE,axiom,
    ! [A: produc8027700073345061287rm_f_v] :
      ~ ( fmembe774991991480784110rm_f_v @ A @ bot_bo4614643899460950777rm_f_v ) ).

% femptyE
thf(fact_775_femptyE,axiom,
    ! [A: produc250624644672406269term_f] :
      ~ ( fmembe6950792022577883716term_f @ A @ bot_bo6180437703705224527term_f ) ).

% femptyE
thf(fact_776_femptyE,axiom,
    ! [A: bot_bot_term_f] :
      ~ ( fmembe1418995490368054919term_f @ A @ bot_bo4668628460741225810term_f ) ).

% femptyE
thf(fact_777_ex__fin__conv,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v] :
      ( ( ? [X4: produc8027700073345061287rm_f_v] : ( fmembe774991991480784110rm_f_v @ X4 @ A3 ) )
      = ( A3 != bot_bo4614643899460950777rm_f_v ) ) ).

% ex_fin_conv
thf(fact_778_ex__fin__conv,axiom,
    ! [A3: fset_P2882900458636368611term_f] :
      ( ( ? [X4: produc250624644672406269term_f] : ( fmembe6950792022577883716term_f @ X4 @ A3 ) )
      = ( A3 != bot_bo6180437703705224527term_f ) ) ).

% ex_fin_conv
thf(fact_779_ex__fin__conv,axiom,
    ! [A3: fset_Bot_bot_term_f] :
      ( ( ? [X4: bot_bot_term_f] : ( fmembe1418995490368054919term_f @ X4 @ A3 ) )
      = ( A3 != bot_bo4668628460741225810term_f ) ) ).

% ex_fin_conv
thf(fact_780_equalsffemptyD,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( A3 = bot_bo4614643899460950777rm_f_v )
     => ~ ( fmembe774991991480784110rm_f_v @ A @ A3 ) ) ).

% equalsffemptyD
thf(fact_781_equalsffemptyD,axiom,
    ! [A3: fset_P2882900458636368611term_f,A: produc250624644672406269term_f] :
      ( ( A3 = bot_bo6180437703705224527term_f )
     => ~ ( fmembe6950792022577883716term_f @ A @ A3 ) ) ).

% equalsffemptyD
thf(fact_782_equalsffemptyD,axiom,
    ! [A3: fset_Bot_bot_term_f,A: bot_bot_term_f] :
      ( ( A3 = bot_bo4668628460741225810term_f )
     => ~ ( fmembe1418995490368054919term_f @ A @ A3 ) ) ).

% equalsffemptyD
thf(fact_783_equalsffemptyI,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v] :
      ( ! [Y3: produc8027700073345061287rm_f_v] :
          ~ ( fmembe774991991480784110rm_f_v @ Y3 @ A3 )
     => ( A3 = bot_bo4614643899460950777rm_f_v ) ) ).

% equalsffemptyI
thf(fact_784_equalsffemptyI,axiom,
    ! [A3: fset_P2882900458636368611term_f] :
      ( ! [Y3: produc250624644672406269term_f] :
          ~ ( fmembe6950792022577883716term_f @ Y3 @ A3 )
     => ( A3 = bot_bo6180437703705224527term_f ) ) ).

% equalsffemptyI
thf(fact_785_equalsffemptyI,axiom,
    ! [A3: fset_Bot_bot_term_f] :
      ( ! [Y3: bot_bot_term_f] :
          ~ ( fmembe1418995490368054919term_f @ Y3 @ A3 )
     => ( A3 = bot_bo4668628460741225810term_f ) ) ).

% equalsffemptyI
thf(fact_786_funion__fempty__right,axiom,
    ! [A3: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ bot_bo3974441229639268550_f_nat )
      = A3 ) ).

% funion_fempty_right
thf(fact_787_funion__fempty__left,axiom,
    ! [B4: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ bot_bo3974441229639268550_f_nat @ B4 )
      = B4 ) ).

% funion_fempty_left
thf(fact_788_GTT__comp__first,axiom,
    ! [Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f,T: term_f7969170442531588692term_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ T ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
       => ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
            = bot_bo4668628460741225810term_f )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) @ T ) ) ) ) ) ).

% GTT_comp_first
thf(fact_789_GTT__comp__second,axiom,
    ! [G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f,Q: bot_bot_term_f,T: term_f7969170442531588692term_f] :
      ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
        = bot_bo4668628460741225810term_f )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
       => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( produc1110312829679706757rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) @ T ) )
         => ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( produc1110312829679706757rm_f_f @ G_2 ) @ T ) ) ) ) ) ).

% GTT_comp_second
thf(fact_790_gta__der__def,axiom,
    ( tree_g2054044545600476092rm_f_f
    = ( ^ [A9: tree_t2044664379607036063rm_f_f,T2: ground_gterm_f] : ( tree_t4678214921166696585rm_f_f @ A9 @ ( ground5944018991622673045term_f @ T2 ) ) ) ) ).

% gta_der_def
thf(fact_791_gta__der__def,axiom,
    ( tree_gta_der_v_f
    = ( ^ [A9: tree_ta_v_f,T2: ground_gterm_f] : ( tree_ta_der_v_f @ A9 @ ( ground1072145386804415820rm_f_v @ T2 ) ) ) ) ).

% gta_der_def
thf(fact_792_ta__res__eps__free,axiom,
    ! [A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( tree_ta_der_v_f @ ( tree_eps_free_v_f @ A8 ) @ ( ground1072145386804415820rm_f_v @ T ) )
      = ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ).

% ta_res_eps_free
thf(fact_793__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc5356295345560192723_f_v_v,A8: tree_t3174935524175603918_f_v_f,T: ground_gterm_f,B7: tree_ta_v_f] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc5298642213761690799_f_v_v @ Q ) @ ( tree_t2840613050917066246_f_v_f @ A8 @ ( ground3045865834483327584rm_f_v @ T ) ) )
     => ( ( fmember_v @ ( produc5880560690777833713_f_v_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ( fmembe7467772765099240218_f_v_v @ Q @ ( tree_t6251634673181180122_v_v_f @ ( tree_p2447607495132258648_v_f_v @ A8 @ B7 ) @ ( ground5619038790573094540_f_v_v @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_794__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc1773071398423524093rm_f_v,A8: tree_t4992178625403595000rm_f_f,T: ground_gterm_f,B7: tree_ta_v_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc4378590996728212697rm_f_v @ Q ) @ ( tree_t3517544178730711600rm_f_f @ A8 @ ( ground7936190122226446262term_f @ T ) ) )
     => ( ( fmember_v @ ( produc8355533809759549723rm_f_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ( fmembe4170856815034497860rm_f_v @ Q @ ( tree_t5203022314908419888_f_v_f @ ( tree_p9154792766393723822_f_f_v @ A8 @ B7 ) @ ( ground1845306247924645558rm_f_v @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_795__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc8536843054531535994rm_f_v,A8: tree_t2044664379607036063rm_f_f,T: ground_gterm_f,B7: tree_ta_v_f] :
      ( ( fmembe1418995490368054919term_f @ ( produc2366976734799249088rm_f_v @ Q ) @ ( tree_t4678214921166696585rm_f_f @ A8 @ ( ground5944018991622673045term_f @ T ) ) )
     => ( ( fmember_v @ ( produc525386408182734846rm_f_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ( fmembe8694731190162490227rm_f_v @ Q @ ( tree_t8726655337028372637_f_v_f @ ( tree_p6542812980490587197_f_f_v @ A8 @ B7 ) @ ( ground8772242327322964353rm_f_v @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_796__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc3903039606312699131rm_f_v,A8: tree_ta_v_f,T: ground_gterm_f,B7: tree_t3174935524175603918_f_v_f] :
      ( ( fmember_v @ ( produc4238978668135062231rm_f_v @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc4820897145151205145rm_f_v @ Q ) @ ( tree_t2840613050917066246_f_v_f @ B7 @ ( ground3045865834483327584rm_f_v @ T ) ) )
       => ( fmembe6014517025851746626rm_f_v @ Q @ ( tree_t4971506711592017074_f_v_f @ ( tree_p6789627866815570776rm_f_v @ A8 @ B7 ) @ ( ground4165783051325600948rm_f_v @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_797__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc4996097768051197777term_f,A8: tree_ta_v_f,T: ground_gterm_f,B7: tree_t4992178625403595000rm_f_f] :
      ( ( fmember_v @ ( produc4977177953880607789term_f @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc8954120766911944815term_f @ Q ) @ ( tree_t3517544178730711600rm_f_f @ B7 @ ( ground7936190122226446262term_f @ T ) ) )
       => ( fmembe7393883184662171544term_f @ Q @ ( tree_t4831870343989751772rm_f_f @ ( tree_p7435737321913684142term_f @ A8 @ B7 ) @ ( ground5068332617552319242term_f @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_798__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: produc6011825036148574416term_f,A8: tree_ta_v_f,T: ground_gterm_f,B7: tree_t2044664379607036063rm_f_f] :
      ( ( fmember_v @ ( produc693125514166837598term_f @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ( fmembe1418995490368054919term_f @ ( produc8074907224405099164term_f @ Q ) @ ( tree_t4678214921166696585rm_f_f @ B7 @ ( ground5944018991622673045term_f @ T ) ) )
       => ( fmembe6169713171779528649term_f @ Q @ ( tree_t3127172571005150663rm_f_f @ ( tree_p8102344957843274653term_f @ A8 @ B7 ) @ ( ground6247224308940002775term_f @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_799__092_060A_062___092_060B_062__der__to__prod__ta,axiom,
    ! [Q: product_prod_v_v,A8: tree_ta_v_f,T: ground_gterm_f,B7: tree_ta_v_f] :
      ( ( fmember_v @ ( product_fst_v_v @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ( fmember_v @ ( product_snd_v_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ( fmembe4541753606430338222od_v_v @ Q @ ( tree_t9194619691912400710_v_v_f @ ( tree_prod_ta_v_f_v @ A8 @ B7 ) @ ( ground5056950390923078432od_v_v @ T ) ) ) ) ) ).

% \<A>_\<B>_der_to_prod_ta
thf(fact_800_prod__ta__der,axiom,
    ! [Q: produc5356295345560192723_f_v_v,A8: tree_t3174935524175603918_f_v_f,B7: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmembe7467772765099240218_f_v_v @ Q @ ( tree_t6251634673181180122_v_v_f @ ( tree_p2447607495132258648_v_f_v @ A8 @ B7 ) @ ( ground5619038790573094540_f_v_v @ T ) ) )
      = ( ( fmembe774991991480784110rm_f_v @ ( produc5298642213761690799_f_v_v @ Q ) @ ( tree_t2840613050917066246_f_v_f @ A8 @ ( ground3045865834483327584rm_f_v @ T ) ) )
        & ( fmember_v @ ( produc5880560690777833713_f_v_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_801_prod__ta__der,axiom,
    ! [Q: produc1773071398423524093rm_f_v,A8: tree_t4992178625403595000rm_f_f,B7: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmembe4170856815034497860rm_f_v @ Q @ ( tree_t5203022314908419888_f_v_f @ ( tree_p9154792766393723822_f_f_v @ A8 @ B7 ) @ ( ground1845306247924645558rm_f_v @ T ) ) )
      = ( ( fmembe6950792022577883716term_f @ ( produc4378590996728212697rm_f_v @ Q ) @ ( tree_t3517544178730711600rm_f_f @ A8 @ ( ground7936190122226446262term_f @ T ) ) )
        & ( fmember_v @ ( produc8355533809759549723rm_f_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_802_prod__ta__der,axiom,
    ! [Q: produc8536843054531535994rm_f_v,A8: tree_t2044664379607036063rm_f_f,B7: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmembe8694731190162490227rm_f_v @ Q @ ( tree_t8726655337028372637_f_v_f @ ( tree_p6542812980490587197_f_f_v @ A8 @ B7 ) @ ( ground8772242327322964353rm_f_v @ T ) ) )
      = ( ( fmembe1418995490368054919term_f @ ( produc2366976734799249088rm_f_v @ Q ) @ ( tree_t4678214921166696585rm_f_f @ A8 @ ( ground5944018991622673045term_f @ T ) ) )
        & ( fmember_v @ ( produc525386408182734846rm_f_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_803_prod__ta__der,axiom,
    ! [Q: produc3903039606312699131rm_f_v,A8: tree_ta_v_f,B7: tree_t3174935524175603918_f_v_f,T: ground_gterm_f] :
      ( ( fmembe6014517025851746626rm_f_v @ Q @ ( tree_t4971506711592017074_f_v_f @ ( tree_p6789627866815570776rm_f_v @ A8 @ B7 ) @ ( ground4165783051325600948rm_f_v @ T ) ) )
      = ( ( fmember_v @ ( produc4238978668135062231rm_f_v @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
        & ( fmembe774991991480784110rm_f_v @ ( produc4820897145151205145rm_f_v @ Q ) @ ( tree_t2840613050917066246_f_v_f @ B7 @ ( ground3045865834483327584rm_f_v @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_804_prod__ta__der,axiom,
    ! [Q: produc4996097768051197777term_f,A8: tree_ta_v_f,B7: tree_t4992178625403595000rm_f_f,T: ground_gterm_f] :
      ( ( fmembe7393883184662171544term_f @ Q @ ( tree_t4831870343989751772rm_f_f @ ( tree_p7435737321913684142term_f @ A8 @ B7 ) @ ( ground5068332617552319242term_f @ T ) ) )
      = ( ( fmember_v @ ( produc4977177953880607789term_f @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
        & ( fmembe6950792022577883716term_f @ ( produc8954120766911944815term_f @ Q ) @ ( tree_t3517544178730711600rm_f_f @ B7 @ ( ground7936190122226446262term_f @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_805_prod__ta__der,axiom,
    ! [Q: produc6011825036148574416term_f,A8: tree_ta_v_f,B7: tree_t2044664379607036063rm_f_f,T: ground_gterm_f] :
      ( ( fmembe6169713171779528649term_f @ Q @ ( tree_t3127172571005150663rm_f_f @ ( tree_p8102344957843274653term_f @ A8 @ B7 ) @ ( ground6247224308940002775term_f @ T ) ) )
      = ( ( fmember_v @ ( produc693125514166837598term_f @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
        & ( fmembe1418995490368054919term_f @ ( produc8074907224405099164term_f @ Q ) @ ( tree_t4678214921166696585rm_f_f @ B7 @ ( ground5944018991622673045term_f @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_806_prod__ta__der,axiom,
    ! [Q: product_prod_v_v,A8: tree_ta_v_f,B7: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmembe4541753606430338222od_v_v @ Q @ ( tree_t9194619691912400710_v_v_f @ ( tree_prod_ta_v_f_v @ A8 @ B7 ) @ ( ground5056950390923078432od_v_v @ T ) ) )
      = ( ( fmember_v @ ( product_fst_v_v @ Q ) @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
        & ( fmember_v @ ( product_snd_v_v @ Q ) @ ( tree_ta_der_v_f @ B7 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% prod_ta_der
thf(fact_807_ta__der__not__stateD,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: term_f_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ T ) )
     => ( ~ ( fmember_v @ Q @ ( tree_Q_v_f @ A8 ) )
       => ( T
          = ( var_v_f @ Q ) ) ) ) ).

% ta_der_not_stateD
thf(fact_808_gterm__ta__der__states,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( fmember_v @ Q @ ( tree_Q_v_f @ A8 ) ) ) ).

% gterm_ta_der_states
thf(fact_809_ta__der__gterm__sig,axiom,
    ! [Q: produc8027700073345061287rm_f_v,A8: tree_t3174935524175603918_f_v_f,T: ground_gterm_f] :
      ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_t2840613050917066246_f_v_f @ A8 @ ( ground3045865834483327584rm_f_v @ T ) ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ ( tree_t9133019675285782406_f_v_f @ A8 ) ) ) ).

% ta_der_gterm_sig
thf(fact_810_ta__der__gterm__sig,axiom,
    ! [Q: produc250624644672406269term_f,A8: tree_t4992178625403595000rm_f_f,T: ground_gterm_f] :
      ( ( fmembe6950792022577883716term_f @ Q @ ( tree_t3517544178730711600rm_f_f @ A8 @ ( ground7936190122226446262term_f @ T ) ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ ( tree_t1292370339873334192rm_f_f @ A8 ) ) ) ).

% ta_der_gterm_sig
thf(fact_811_ta__der__gterm__sig,axiom,
    ! [Q: bot_bot_term_f,A8: tree_t2044664379607036063rm_f_f,T: ground_gterm_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A8 @ ( ground5944018991622673045term_f @ T ) ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ ( tree_t4125170225404765961rm_f_f @ A8 ) ) ) ).

% ta_der_gterm_sig
thf(fact_812_ta__der__gterm__sig,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ ( tree_ta_sig_v_f @ A8 ) ) ) ).

% ta_der_gterm_sig
thf(fact_813_ta__der__term__sig,axiom,
    ! [Q: produc8027700073345061287rm_f_v,A8: tree_t3174935524175603918_f_v_f,T: term_f1991478400439048311rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_t2840613050917066246_f_v_f @ A8 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f7595012570863997060rm_f_v @ T ) @ ( tree_t9133019675285782406_f_v_f @ A8 ) ) ) ).

% ta_der_term_sig
thf(fact_814_ta__der__term__sig,axiom,
    ! [Q: produc250624644672406269term_f,A8: tree_t4992178625403595000rm_f_f,T: term_f3944004038363720397term_f] :
      ( ( fmembe6950792022577883716term_f @ Q @ ( tree_t3517544178730711600rm_f_f @ A8 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f8146329828441968090term_f @ T ) @ ( tree_t1292370339873334192rm_f_f @ A8 ) ) ) ).

% ta_der_term_sig
thf(fact_815_ta__der__term__sig,axiom,
    ! [Q: bot_bot_term_f,A8: tree_t2044664379607036063rm_f_f,T: term_f7969170442531588692term_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A8 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f6305927314145560433term_f @ T ) @ ( tree_t4125170225404765961rm_f_f @ A8 ) ) ) ).

% ta_der_term_sig
thf(fact_816_ta__der__term__sig,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: term_f_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_term_f_v @ T ) @ ( tree_ta_sig_v_f @ A8 ) ) ) ).

% ta_der_term_sig
thf(fact_817_GTT__comp__eps__fst__statesD_I2_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) )
     => ( fmembe1418995490368054919term_f @ Q @ ( sup_su6586017679312891882term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ).

% GTT_comp_eps_fst_statesD(2)
thf(fact_818_bot__fset_Orep__eq,axiom,
    ( ( fset_P4617584883882644886rm_f_v @ bot_bo4614643899460950777rm_f_v )
    = bot_bo4638812505780724595rm_f_v ) ).

% bot_fset.rep_eq
thf(fact_819_ffunas__term__simps_I1_J,axiom,
    ! [X: v] :
      ( ( fSet_ffunas_term_f_v @ ( var_v_f @ X ) )
      = bot_bo3974441229639268550_f_nat ) ).

% ffunas_term_simps(1)
thf(fact_820_restrict__subst__domain__empty,axiom,
    ! [Sigma: v > term_f_v] :
      ( ( restri8836950881698410593in_v_f @ bot_bot_set_v @ Sigma )
      = var_v_f ) ).

% restrict_subst_domain_empty
thf(fact_821_gta__langI,axiom,
    ! [Q: v,Q3: fset_v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmember_v @ Q @ Q3 )
     => ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ( member5261315044688711901term_f @ T @ ( tree_gta_lang_v_f @ Q3 @ A8 ) ) ) ) ).

% gta_langI
thf(fact_822_ffunas__term__of__gterm__conv,axiom,
    ! [T: ground_gterm_f] :
      ( ( fSet_ffunas_term_f_v @ ( ground1072145386804415820rm_f_v @ T ) )
      = ( fSet_ffunas_gterm_f @ T ) ) ).

% ffunas_term_of_gterm_conv
thf(fact_823_gta__langE,axiom,
    ! [T: ground_gterm_f,Q3: fset_v,A8: tree_ta_v_f] :
      ( ( member5261315044688711901term_f @ T @ ( tree_gta_lang_v_f @ Q3 @ A8 ) )
     => ~ ! [Q2: v] :
            ( ( fmember_v @ Q2 @ Q3 )
           => ~ ( fmember_v @ Q2 @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% gta_langE
thf(fact_824_GTT__comp__eps__fst__statesD_I1_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) )
     => ( fmembe1418995490368054919term_f @ P @ ( sup_su6586017679312891882term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ).

% GTT_comp_eps_fst_statesD(1)
thf(fact_825_is__gtt__eps__free__def,axiom,
    ( is_gtt4146148574938235922rm_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f] :
          ( ( ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) )
            = bot_bo6180437703705224527term_f )
          & ( ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) )
            = bot_bo6180437703705224527term_f ) ) ) ) ).

% is_gtt_eps_free_def
thf(fact_826_pattern__automaton__ground__instance__reach,axiom,
    ! [L3: term_f_v,R5: fset_term_f_v,Sigma: v > term_f_v,T: ground_gterm_f,F3: fset_P6228066233360383026_f_nat] :
      ( ( fmember_term_f_v @ L3 @ R5 )
     => ( ( ( subst_7901567856578221233_f_v_v @ L3 @ Sigma )
          = ( ground1072145386804415820rm_f_v @ T ) )
       => ( ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ F3 )
         => ( fmembe1418995490368054919term_f @ ( bot_te7903553565439629424rm_f_v @ L3 ) @ ( tree_t4678214921166696585rm_f_f @ ( lV_to_2851878803089403142on_f_v @ F3 @ R5 ) @ ( ground5944018991622673045term_f @ T ) ) ) ) ) ) ).

% pattern_automaton_ground_instance_reach
thf(fact_827__092_060Delta_062_092_060_094sub_062_092_060epsilon_062__steps__from___092_060G_062_092_060_094sub_0621___092_060G_062_092_060_094sub_0622,axiom,
    ! [Q: bot_bot_term_f,Q5: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( sup_su6586017679312891882term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) )
       => ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
            = bot_bo4668628460741225810term_f )
         => ( ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
             => ( ( fmembe1418995490368054919term_f @ Q5 @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
               => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) ) ) ) ) )
           => ( ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
               => ( ( fmembe1418995490368054919term_f @ Q5 @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
                 => ! [P3: bot_bot_term_f] :
                      ( ( ( Q = P3 )
                        | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ P3 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) ) ) ) )
                     => ! [P5: bot_bot_term_f] :
                          ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P3 @ P5 ) @ ( gTT_De7106193943236547076rm_f_f @ ( produc1110312829679706757rm_f_f @ G_1 ) @ ( produc1890379913328437059rm_f_f @ G_2 ) ) )
                         => ~ ( ( P5 = Q5 )
                              | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P5 @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) ) ) ) ) ) ) ) )
             => ~ ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
                 => ~ ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) ) ) )
                      & ( fmembe1418995490368054919term_f @ Q5 @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ) ) ) ) ) ).

% \<Delta>\<^sub>\<epsilon>_steps_from_\<G>\<^sub>1_\<G>\<^sub>2
thf(fact_828_fr__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
     => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) ) ) ).

% fr_into_trancl
thf(fact_829_fr__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
     => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7983512424977124800term_f @ R2 ) ) ) ).

% fr_into_trancl
thf(fact_830_fr__into__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
     => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f5255319743886086274term_f @ R2 ) ) ) ).

% fr_into_trancl
thf(fact_831_ta__der__Var,axiom,
    ! [Q: v,A8: tree_ta_v_f,X: v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( var_v_f @ X ) ) )
      = ( ( X = Q )
        | ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ X @ Q ) @ ( fSet_ftrancl_v @ ( tree_eps_v_f @ A8 ) ) ) ) ) ).

% ta_der_Var
thf(fact_832_ta__der_H__Var,axiom,
    ! [Q: v,A8: tree_ta_v_f,X: v] :
      ( ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A8 @ ( var_v_f @ X ) ) )
      = ( ( X = Q )
        | ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ X @ Q ) @ ( fSet_ftrancl_v @ ( tree_eps_v_f @ A8 ) ) ) ) ) ).

% ta_der'_Var
thf(fact_833_bot__empty__eq,axiom,
    ( bot_bo329238025660093012rm_f_o
    = ( ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ bot_bo5323853609601926601term_f ) ) ) ).

% bot_empty_eq
thf(fact_834_bot__empty__eq,axiom,
    ( bot_bo6175410582289378090rm_f_o
    = ( ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ bot_bo9223291547480392563term_f ) ) ) ).

% bot_empty_eq
thf(fact_835_ftranclD,axiom,
    ! [X: term_f_v,Y: term_f_v,R5: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R5 ) )
     => ? [Z2: term_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z2 ) @ R5 )
          & ( ( Z2 = Y )
            | ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R5 ) ) ) ) ) ).

% ftranclD
thf(fact_836_ftranclD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R5: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R5 ) )
     => ? [Z2: bot_bot_term_f] :
          ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z2 ) @ R5 )
          & ( ( Z2 = Y )
            | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Z2 @ Y ) @ ( fSet_f7983512424977124800term_f @ R5 ) ) ) ) ) ).

% ftranclD
thf(fact_837_ftranclD,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R5: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R5 ) )
     => ? [Z2: ground_gterm_f] :
          ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z2 ) @ R5 )
          & ( ( Z2 = Y )
            | ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Z2 @ Y ) @ ( fSet_f5255319743886086274term_f @ R5 ) ) ) ) ) ).

% ftranclD
thf(fact_838_ftranclE,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 )
       => ~ ! [C4: term_f_v] :
              ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C4 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
             => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ C4 @ B ) @ R2 ) ) ) ) ).

% ftranclE
thf(fact_839_ftranclE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 )
       => ~ ! [C4: bot_bot_term_f] :
              ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ C4 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
             => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ C4 @ B ) @ R2 ) ) ) ) ).

% ftranclE
thf(fact_840_ftranclE,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 )
       => ~ ! [C4: ground_gterm_f] :
              ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ C4 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
             => ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ C4 @ B ) @ R2 ) ) ) ) ).

% ftranclE
thf(fact_841_ftranclD2,axiom,
    ! [X: term_f_v,Y: term_f_v,R5: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R5 ) )
     => ? [Z2: term_f_v] :
          ( ( ( X = Z2 )
            | ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z2 ) @ ( fSet_f1510795870837090965rm_f_v @ R5 ) ) )
          & ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ Y ) @ R5 ) ) ) ).

% ftranclD2
thf(fact_842_ftranclD2,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R5: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R5 ) )
     => ? [Z2: bot_bot_term_f] :
          ( ( ( X = Z2 )
            | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z2 ) @ ( fSet_f7983512424977124800term_f @ R5 ) ) )
          & ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Z2 @ Y ) @ R5 ) ) ) ).

% ftranclD2
thf(fact_843_ftranclD2,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R5: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R5 ) )
     => ? [Z2: ground_gterm_f] :
          ( ( ( X = Z2 )
            | ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z2 ) @ ( fSet_f5255319743886086274term_f @ R5 ) ) )
          & ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Z2 @ Y ) @ R5 ) ) ) ).

% ftranclD2
thf(fact_844_ftrancl__map,axiom,
    ! [R2: fset_P8018961893305114765rm_f_v,F2: term_f_v > term_f_v,S: fset_P8018961893305114765rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f1510795870837090965rm_f_v @ S ) ) ) ) ).

% ftrancl_map
thf(fact_845_ftrancl__map,axiom,
    ! [R2: fset_P8018961893305114765rm_f_v,F2: term_f_v > bot_bot_term_f,S: fset_P2882900458636368611term_f,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f7983512424977124800term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_846_ftrancl__map,axiom,
    ! [R2: fset_P8018961893305114765rm_f_v,F2: term_f_v > ground_gterm_f,S: fset_P5324236452418035597term_f,X: term_f_v,Y: term_f_v] :
      ( ! [X3: term_f_v,Y3: term_f_v] :
          ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R2 )
         => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f5255319743886086274term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_847_ftrancl__map,axiom,
    ! [R2: fset_P2882900458636368611term_f,F2: bot_bot_term_f > term_f_v,S: fset_P8018961893305114765rm_f_v,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f1510795870837090965rm_f_v @ S ) ) ) ) ).

% ftrancl_map
thf(fact_848_ftrancl__map,axiom,
    ! [R2: fset_P2882900458636368611term_f,F2: bot_bot_term_f > bot_bot_term_f,S: fset_P2882900458636368611term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f7983512424977124800term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_849_ftrancl__map,axiom,
    ! [R2: fset_P2882900458636368611term_f,F2: bot_bot_term_f > ground_gterm_f,S: fset_P5324236452418035597term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f5255319743886086274term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_850_ftrancl__map,axiom,
    ! [R2: fset_P5324236452418035597term_f,F2: ground_gterm_f > term_f_v,S: fset_P8018961893305114765rm_f_v,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f1510795870837090965rm_f_v @ S ) ) ) ) ).

% ftrancl_map
thf(fact_851_ftrancl__map,axiom,
    ! [R2: fset_P5324236452418035597term_f,F2: ground_gterm_f > bot_bot_term_f,S: fset_P2882900458636368611term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f7983512424977124800term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_852_ftrancl__map,axiom,
    ! [R2: fset_P5324236452418035597term_f,F2: ground_gterm_f > ground_gterm_f,S: fset_P5324236452418035597term_f,X: ground_gterm_f,Y: ground_gterm_f] :
      ( ! [X3: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X3 @ Y3 ) @ R2 )
         => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( fSet_f5255319743886086274term_f @ S ) ) ) ) ).

% ftrancl_map
thf(fact_853_ftrancl__trans,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: fset_P8018961893305114765rm_f_v,Z3: term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) ) ) ) ).

% ftrancl_trans
thf(fact_854_ftrancl__trans,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: fset_P2882900458636368611term_f,Z3: bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) ) ) ) ).

% ftrancl_trans
thf(fact_855_ftrancl__trans,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: fset_P5324236452418035597term_f,Z3: ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) ) ) ) ).

% ftrancl_trans
thf(fact_856_ftrancl__induct,axiom,
    ! [X1: term_f_v,X2: term_f_v,R2: fset_P8018961893305114765rm_f_v,P2: term_f_v > term_f_v > $o] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X1 @ X2 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ! [A4: term_f_v,B3: term_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: term_f_v,B3: term_f_v,C4: term_f_v] :
              ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A4 @ B3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
             => ( ( P2 @ A4 @ B3 )
               => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C4 ) @ R2 )
                 => ( P2 @ A4 @ C4 ) ) ) )
         => ( P2 @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_857_ftrancl__induct,axiom,
    ! [X1: bot_bot_term_f,X2: bot_bot_term_f,R2: fset_P2882900458636368611term_f,P2: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X1 @ X2 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f] :
            ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: bot_bot_term_f,B3: bot_bot_term_f,C4: bot_bot_term_f] :
              ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A4 @ B3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
             => ( ( P2 @ A4 @ B3 )
               => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B3 @ C4 ) @ R2 )
                 => ( P2 @ A4 @ C4 ) ) ) )
         => ( P2 @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_858_ftrancl__induct,axiom,
    ! [X1: ground_gterm_f,X2: ground_gterm_f,R2: fset_P5324236452418035597term_f,P2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X1 @ X2 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ! [A4: ground_gterm_f,B3: ground_gterm_f] :
            ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ R2 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: ground_gterm_f,B3: ground_gterm_f,C4: ground_gterm_f] :
              ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A4 @ B3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
             => ( ( P2 @ A4 @ B3 )
               => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B3 @ C4 ) @ R2 )
                 => ( P2 @ A4 @ C4 ) ) ) )
         => ( P2 @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_859_not__ftrancl__into,axiom,
    ! [X: term_f_v,Z3: term_f_v,R2: fset_P8018961893305114765rm_f_v,Y: term_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z3 ) @ R2 )
       => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) ) ) ) ).

% not_ftrancl_into
thf(fact_860_not__ftrancl__into,axiom,
    ! [X: bot_bot_term_f,Z3: bot_bot_term_f,R2: fset_P2882900458636368611term_f,Y: bot_bot_term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y @ Z3 ) @ R2 )
       => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f7983512424977124800term_f @ R2 ) ) ) ) ).

% not_ftrancl_into
thf(fact_861_not__ftrancl__into,axiom,
    ! [X: ground_gterm_f,Z3: ground_gterm_f,R2: fset_P5324236452418035597term_f,Y: ground_gterm_f] :
      ( ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y @ Z3 ) @ R2 )
       => ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( fSet_f5255319743886086274term_f @ R2 ) ) ) ) ).

% not_ftrancl_into
thf(fact_862_converse__ftranclE,axiom,
    ! [X: term_f_v,Z3: term_f_v,R2: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: term_f_v] :
              ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R2 )
             => ~ ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) ) ) ) ) ).

% converse_ftranclE
thf(fact_863_converse__ftranclE,axiom,
    ! [X: bot_bot_term_f,Z3: bot_bot_term_f,R2: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: bot_bot_term_f] :
              ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ R2 )
             => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ ( fSet_f7983512424977124800term_f @ R2 ) ) ) ) ) ).

% converse_ftranclE
thf(fact_864_converse__ftranclE,axiom,
    ! [X: ground_gterm_f,Z3: ground_gterm_f,R2: fset_P5324236452418035597term_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: ground_gterm_f] :
              ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ X @ Y3 ) @ R2 )
             => ~ ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y3 @ Z3 ) @ ( fSet_f5255319743886086274term_f @ R2 ) ) ) ) ) ).

% converse_ftranclE
thf(fact_865_ftrancl__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v,C: term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R2 )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_866_ftrancl__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,C: bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R2 )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( fSet_f7983512424977124800term_f @ R2 ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_867_ftrancl__into__trancl,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f,C: ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ B @ C ) @ R2 )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ C ) @ ( fSet_f5255319743886086274term_f @ R2 ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_868_converse__ftrancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R2: fset_P8018961893305114765rm_f_v,P2: term_f_v > $o] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ! [Y3: term_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: term_f_v,Z2: term_f_v] :
              ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R2 )
             => ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ B ) @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_869_converse__ftrancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,P2: bot_bot_term_f > $o] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: bot_bot_term_f,Z2: bot_bot_term_f] :
              ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Z2 @ B ) @ ( fSet_f7983512424977124800term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_870_converse__ftrancl__induct,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: fset_P5324236452418035597term_f,P2: ground_gterm_f > $o] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
     => ( ! [Y3: ground_gterm_f] :
            ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: ground_gterm_f,Z2: ground_gterm_f] :
              ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Y3 @ Z2 ) @ R2 )
             => ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ Z2 @ B ) @ ( fSet_f5255319743886086274term_f @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_871_ftrancl__mono,axiom,
    ! [P: produc8027700073345061287rm_f_v,R2: fset_P8018961893305114765rm_f_v,S: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ P @ ( fSet_f1510795870837090965rm_f_v @ R2 ) )
     => ( ( ord_le4587745213494032429rm_f_v @ R2 @ S )
       => ( fmembe774991991480784110rm_f_v @ P @ ( fSet_f1510795870837090965rm_f_v @ S ) ) ) ) ).

% ftrancl_mono
thf(fact_872_ftrancl__mono,axiom,
    ! [P: produc250624644672406269term_f,R2: fset_P2882900458636368611term_f,S: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ P @ ( fSet_f7983512424977124800term_f @ R2 ) )
     => ( ( ord_le7092741666294821507term_f @ R2 @ S )
       => ( fmembe6950792022577883716term_f @ P @ ( fSet_f7983512424977124800term_f @ S ) ) ) ) ).

% ftrancl_mono
thf(fact_873_ftrancl_Orep__eq,axiom,
    ! [X: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fSet_f1510795870837090965rm_f_v @ X ) )
      = ( transi3208867851621559554rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) ) ) ).

% ftrancl.rep_eq
thf(fact_874_in__ftrancl__UnI,axiom,
    ! [X: produc8027700073345061287rm_f_v,R5: fset_P8018961893305114765rm_f_v,S2: fset_P8018961893305114765rm_f_v] :
      ( ( ( fmembe774991991480784110rm_f_v @ X @ ( fSet_f1510795870837090965rm_f_v @ R5 ) )
        | ( fmembe774991991480784110rm_f_v @ X @ ( fSet_f1510795870837090965rm_f_v @ S2 ) ) )
     => ( fmembe774991991480784110rm_f_v @ X @ ( fSet_f1510795870837090965rm_f_v @ ( sup_su641650120996271713rm_f_v @ R5 @ S2 ) ) ) ) ).

% in_ftrancl_UnI
thf(fact_875_in__ftrancl__UnI,axiom,
    ! [X: produc250624644672406269term_f,R5: fset_P2882900458636368611term_f,S2: fset_P2882900458636368611term_f] :
      ( ( ( fmembe6950792022577883716term_f @ X @ ( fSet_f7983512424977124800term_f @ R5 ) )
        | ( fmembe6950792022577883716term_f @ X @ ( fSet_f7983512424977124800term_f @ S2 ) ) )
     => ( fmembe6950792022577883716term_f @ X @ ( fSet_f7983512424977124800term_f @ ( sup_su1809081536757948599term_f @ R5 @ S2 ) ) ) ) ).

% in_ftrancl_UnI
thf(fact_876_root__pair__automaton__def,axiom,
    ( lV_to_8065481040984897572_f_v_v
    = ( ^ [F: fset_P6228066233360383026_f_nat,R8: fset_P8018961893305114765rm_f_v] : ( produc1864056441719732567rm_f_f @ ( lV_to_2851878803089403142on_f_v @ F @ ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v @ R8 ) ) @ ( lV_to_2851878803089403142on_f_v @ F @ ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v @ R8 ) ) ) ) ) ).

% root_pair_automaton_def
thf(fact_877_ta__der_H__eps,axiom,
    ! [P: v,A8: tree_ta_v_f,S: term_f_v,Q: v] :
      ( ( fmember_term_f_v @ ( var_v_f @ P ) @ ( tree_ta_der_v_f2 @ A8 @ S ) )
     => ( ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ P @ Q ) @ ( fSet_ftrancl_v @ ( tree_eps_v_f @ A8 ) ) )
       => ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A8 @ S ) ) ) ) ).

% ta_der'_eps
thf(fact_878_GTT__comp__eps__ftrancl__fst__statesD_I2_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) ) )
     => ( fmembe1418995490368054919term_f @ Q @ ( sup_su6586017679312891882term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ).

% GTT_comp_eps_ftrancl_fst_statesD(2)
thf(fact_879_GTT__comp__eps__ftrancl__fst__statesD_I1_J,axiom,
    ! [P: bot_bot_term_f,Q: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ Q ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) ) )
     => ( fmembe1418995490368054919term_f @ P @ ( sup_su6586017679312891882term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ).

% GTT_comp_eps_ftrancl_fst_statesD(1)
thf(fact_880__092_060Delta_062_092_060_094sub_062_092_060epsilon_062__steps__from___092_060G_062_092_060_094sub_0622,axiom,
    ! [Q: bot_bot_term_f,Q5: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) ) )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
       => ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
            = bot_bo4668628460741225810term_f )
         => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q @ Q5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) ) ) )
            & ( fmembe1418995490368054919term_f @ Q5 @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) ) ) ) ) ) ).

% \<Delta>\<^sub>\<epsilon>_steps_from_\<G>\<^sub>2
thf(fact_881__092_060Delta_062_092_060_094sub_062_092_060epsilon_062__steps__from___092_060G_062_092_060_094sub_0621,axiom,
    ! [P: bot_bot_term_f,R2: bot_bot_term_f,G_1: produc6920895926601326823rm_f_f,G_2: produc6920895926601326823rm_f_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ R2 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ ( gTT_GT8578867084790779380rm_f_f @ G_1 @ G_2 ) ) ) ) )
     => ( ( fmembe1418995490368054919term_f @ P @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
       => ( ( ( inf_in2113687546666124112term_f @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
            = bot_bo4668628460741225810term_f )
         => ( ( ( fmembe1418995490368054919term_f @ R2 @ ( gtt_st6132224256192451901_f_f_f @ G_1 ) )
             => ~ ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ R2 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) ) ) ) )
           => ~ ( ( fmembe1418995490368054919term_f @ R2 @ ( gtt_st6132224256192451901_f_f_f @ G_2 ) )
               => ! [Q2: bot_bot_term_f,P5: bot_bot_term_f] :
                    ( ( ( P = P5 )
                      | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P @ P5 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_1 ) ) ) ) )
                   => ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P5 @ Q2 ) @ ( gTT_De7106193943236547076rm_f_f @ ( produc1110312829679706757rm_f_f @ G_1 ) @ ( produc1890379913328437059rm_f_f @ G_2 ) ) )
                     => ~ ( ( Q2 = R2 )
                          | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ R2 ) @ ( fSet_f7983512424977124800term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% \<Delta>\<^sub>\<epsilon>_steps_from_\<G>\<^sub>1
thf(fact_882_pattern__automaton__reach__smaller__term,axiom,
    ! [L3: term_f_v,R5: fset_term_f_v,S: term_f_v,T: ground_gterm_f,F3: fset_P6228066233360383026_f_nat] :
      ( ( fmember_term_f_v @ L3 @ R5 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L3 @ S ) @ subter5033437232688913882eq_f_v )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( bot_te7903553565439629424rm_f_v @ ( ground1072145386804415820rm_f_v @ T ) ) ) @ bot_bless_eq_f )
         => ( ( ord_le1552505484586773650_f_nat @ ( fSet_ffunas_gterm_f @ T ) @ F3 )
           => ( fmembe1418995490368054919term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( tree_t4678214921166696585rm_f_f @ ( lV_to_2851878803089403142on_f_v @ F3 @ R5 ) @ ( ground5944018991622673045term_f @ T ) ) ) ) ) ) ) ).

% pattern_automaton_reach_smaller_term
thf(fact_883_term__to__bot__term_Osimps_I1_J,axiom,
    ! [Uu: v] :
      ( ( bot_te7903553565439629424rm_f_v @ ( var_v_f @ Uu ) )
      = bot_bot_Bot_f ) ).

% term_to_bot_term.simps(1)
thf(fact_884_bless__eq_Obase__l,axiom,
    ! [T: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ bot_bot_Bot_f @ T ) @ bot_bless_eq_f ) ).

% bless_eq.base_l
thf(fact_885_supteq__subst,axiom,
    ! [S: term_f_v,T: term_f_v,Sigma: v > term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ subter5033437232688913882eq_f_v ) ) ).

% supteq_subst
thf(fact_886_fstates__fmemberI,axiom,
    ! [S: term_f_v,R5: fset_term_f_v,U: term_f_v] :
      ( ( fmember_term_f_v @ S @ R5 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subter5033437232688913882eq_f_v )
       => ( fmembe1418995490368054919term_f @ ( bot_te7903553565439629424rm_f_v @ U ) @ ( lV_to_fstates_f_v @ R5 ) ) ) ) ).

% fstates_fmemberI
thf(fact_887_supteq__var__imp__eq,axiom,
    ! [X: v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( var_v_f @ X ) @ T ) @ subter5033437232688913882eq_f_v )
      = ( T
        = ( var_v_f @ X ) ) ) ).

% supteq_var_imp_eq
thf(fact_888_fstates__fmemberE,axiom,
    ! [T: bot_bot_term_f,R5: fset_term_f_v] :
      ( ( fmembe1418995490368054919term_f @ T @ ( lV_to_fstates_f_v @ R5 ) )
     => ~ ! [S4: term_f_v,U2: term_f_v] :
            ~ ( ( fmember_term_f_v @ S4 @ R5 )
              & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S4 @ U2 ) @ subter5033437232688913882eq_f_v )
              & ( ( bot_te7903553565439629424rm_f_v @ U2 )
                = T ) ) ) ).

% fstates_fmemberE
thf(fact_889_supteq_Orefl,axiom,
    ! [T: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ T ) @ subter5033437232688913882eq_f_v ) ).

% supteq.refl
thf(fact_890_subterm_Oeq__refl,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( X = Y )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v ) ) ).

% subterm.eq_refl
thf(fact_891_subterm_Oorder__refl,axiom,
    ! [X: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ subter5033437232688913882eq_f_v ) ).

% subterm.order_refl
thf(fact_892_subterm_Oorder_Otrans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.order.trans
thf(fact_893_subterm_Oorder__trans,axiom,
    ! [Y: term_f_v,X: term_f_v,Z3: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z3 @ Y ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z3 @ X ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.order_trans
thf(fact_894_subterm_Oantisym__conv,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv
thf(fact_895_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y4: term_f_v,Z: term_f_v] : ( Y4 = Z ) )
    = ( ^ [A7: term_f_v,B6: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B6 @ A7 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_896_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: term_f_v,Z: term_f_v] : ( Y4 = Z ) )
    = ( ^ [X4: term_f_v,Y5: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_897_subterm_Oorder_Oantisym,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_898_subterm_Oorder__antisym,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v )
       => ( X = Y ) ) ) ).

% subterm.order_antisym
thf(fact_899_subterm_Odual__order_Orefl,axiom,
    ! [A: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ subter5033437232688913882eq_f_v ) ).

% subterm.dual_order.refl
thf(fact_900_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( A = B )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_901_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( B = C )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_902_subterm_Odual__order_Otrans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.dual_order.trans
thf(fact_903_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: term_f_v,Z: term_f_v] : ( Y4 = Z ) )
    = ( ^ [A7: term_f_v,B6: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A7 @ B6 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B6 @ A7 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_904_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_905_eq__supteq,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( S = T )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v ) ) ).

% eq_supteq
thf(fact_906_supteq__trans,axiom,
    ! [S: term_f_v,T: term_f_v,U: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subter5033437232688913882eq_f_v ) ) ) ).

% supteq_trans
thf(fact_907_supteq__antisym,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ subter5033437232688913882eq_f_v )
       => ( S = T ) ) ) ).

% supteq_antisym
thf(fact_908_fstates__def_H,axiom,
    ! [T: bot_bot_term_f,R5: fset_term_f_v] :
      ( ( fmembe1418995490368054919term_f @ T @ ( lV_to_fstates_f_v @ R5 ) )
      = ( ? [S3: term_f_v,U3: term_f_v] :
            ( ( fmember_term_f_v @ S3 @ R5 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S3 @ U3 ) @ subter5033437232688913882eq_f_v )
            & ( ( bot_te7903553565439629424rm_f_v @ U3 )
              = T ) ) ) ) ).

% fstates_def'
thf(fact_909_fst__pair__cl__res__aux,axiom,
    ! [A8: tree_ta_v_f,Q3: fset_P6509793526439625293od_v_v,Q: v,T: ground_gterm_f] :
      ( ( ( inf_inf_fset_v @ ( tree_Q_v_f @ A8 ) @ ( fimage5203632385572561001_v_v_v @ product_snd_v_v @ Q3 ) )
        = bot_bot_fset_v )
     => ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ ( tree_TA_v_f @ ( tree_rules_v_f @ A8 ) @ ( sup_su1419116364459815457od_v_v @ ( tree_eps_v_f @ A8 ) @ ( fSet_frelcomp_v_v_v @ ( fSet_fId_on_v @ ( tree_Q_v_f @ A8 ) ) @ Q3 ) ) ) @ ( ground1072145386804415820rm_f_v @ T ) ) )
       => ? [P3: v] :
            ( ( fmember_v @ P3 @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
            & ( ~ ( fmember_v @ Q @ ( tree_Q_v_f @ A8 ) )
             => ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ P3 @ Q ) @ Q3 ) )
            & ( ( fmember_v @ Q @ ( tree_Q_v_f @ A8 ) )
             => ( P3 = Q ) ) ) ) ) ).

% fst_pair_cl_res_aux
thf(fact_910_partially__completely__defined__on__def,axiom,
    ( tree_A7658842082072087876on_v_f
    = ( ^ [A9: tree_ta_v_f,F: fset_P6228066233360383026_f_nat] :
        ! [T2: ground_gterm_f] :
          ( ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ T2 ) @ ( fset_P3576968334923099475_f_nat @ F ) )
          = ( ? [Q6: v] : ( fmember_v @ Q6 @ ( tree_ta_der_v_f @ A9 @ ( ground1072145386804415820rm_f_v @ T2 ) ) ) ) ) ) ) ).

% partially_completely_defined_on_def
thf(fact_911_agtt__lang__mono,axiom,
    ! [G: produc6920895926601326823rm_f_f,G6: produc6920895926601326823rm_f_f] :
      ( ( ord_le2151090855036312617rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) ) @ ( tree_r5405720787295496115rm_f_f @ ( produc1890379913328437059rm_f_f @ G6 ) ) )
     => ( ( ord_le7092741666294821507term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G6 ) ) )
       => ( ( ord_le2151090855036312617rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) @ ( tree_r5405720787295496115rm_f_f @ ( produc1110312829679706757rm_f_f @ G6 ) ) )
         => ( ( ord_le7092741666294821507term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G6 ) ) )
           => ( ord_le263819222746101927term_f @ ( agtt_l7696934014624939844rm_f_f @ G ) @ ( agtt_l7696934014624939844rm_f_f @ G6 ) ) ) ) ) ) ).

% agtt_lang_mono
thf(fact_912_ta__der__rule__empty,axiom,
    ! [Q: v,Delta_epsilon2: fset_P6509793526439625293od_v_v,T: term_f_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ ( tree_TA_v_f @ bot_bo3218165541019598882le_v_f @ Delta_epsilon2 ) @ T ) )
     => ~ ! [P3: v] :
            ( ( T
              = ( var_v_f @ P3 ) )
           => ~ ( ( P3 = Q )
                | ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ P3 @ Q ) @ ( fSet_ftrancl_v @ Delta_epsilon2 ) ) ) ) ) ).

% ta_der_rule_empty
thf(fact_913_ta__der__gterm__sig__fset,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ T ) @ ( fset_P3576968334923099475_f_nat @ ( tree_ta_sig_v_f @ A8 ) ) ) ) ).

% ta_der_gterm_sig_fset
thf(fact_914_pair__at__to__agtt__def,axiom,
    ( pair_p7034665317573588219rm_f_f
    = ( ^ [G3: produc6920895926601326823rm_f_f,Q4: fset_P2882900458636368611term_f] : ( produc1864056441719732567rm_f_f @ ( tree_T6922284161482903311rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) @ ( sup_su1809081536757948599term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) @ ( fSet_f8009933528932036120term_f @ ( fSet_f5231733434221412358term_f @ ( tree_Q1727860138906775108rm_f_f @ ( produc1890379913328437059rm_f_f @ G3 ) ) ) @ Q4 ) ) ) @ ( tree_T6922284161482903311rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G3 ) ) ) ) ) ) ).

% pair_at_to_agtt_def
thf(fact_915_subterm__eq__pres__ground,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( term_ground_f_v @ S )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
       => ( term_ground_f_v @ T ) ) ) ).

% subterm_eq_pres_ground
thf(fact_916_ta__der__not__reach,axiom,
    ! [A8: tree_ta_v_f,Q: v,T: ground_gterm_f] :
      ( ! [R: tree_ta_rule_v_f] :
          ( ( fmembe7939848548214878551le_v_f @ R @ ( tree_rules_v_f @ A8 ) )
         => ( ( tree_ta_r_rhs_v_f @ R )
           != Q ) )
     => ( ! [E: product_prod_v_v] :
            ( ( fmembe4541753606430338222od_v_v @ E @ ( tree_eps_v_f @ A8 ) )
           => ( ( product_snd_v_v @ E )
             != Q ) )
       => ~ ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ).

% ta_der_not_reach
thf(fact_917_ground_Osimps_I1_J,axiom,
    ! [X: v] :
      ~ ( term_ground_f_v @ ( var_v_f @ X ) ) ).

% ground.simps(1)
thf(fact_918_ta__der__funs__ta,axiom,
    ! [Q: produc8027700073345061287rm_f_v,A3: tree_t4061174609071262093_f_nat,T: term_P6739010288670303790rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ Q @ ( tree_t3566615653518211007_f_nat @ A3 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f5104597619784224358rm_f_v @ T ) @ ( tree_f5659650137421131470_f_nat @ A3 ) ) ) ).

% ta_der_funs_ta
thf(fact_919_ta__der__funs__ta,axiom,
    ! [Q: produc250624644672406269term_f,A3: tree_t5579808332917730531_f_nat,T: term_P1124376183965932676term_f] :
      ( ( fmembe6950792022577883716term_f @ Q @ ( tree_t3429357085031708437_f_nat @ A3 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f5632489650104615868term_f @ T ) @ ( tree_f906818546516505124_f_nat @ A3 ) ) ) ).

% ta_der_funs_ta
thf(fact_920_ta__der__funs__ta,axiom,
    ! [Q: bot_bot_term_f,A3: tree_t2770727371288418428_f_nat,T: term_P4696149571059479517term_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t3163304490811198844_f_nat @ A3 @ T ) )
     => ( ord_le1552505484586773650_f_nat @ ( fSet_f8822967322155677455term_f @ T ) @ ( tree_f3968442923031183277_f_nat @ A3 ) ) ) ).

% ta_der_funs_ta
thf(fact_921_ground__term__to__gtermD,axiom,
    ! [T: term_f_v] :
      ( ( term_ground_f_v @ T )
     => ? [T4: ground_gterm_f] :
          ( T
          = ( ground1072145386804415820rm_f_v @ T4 ) ) ) ).

% ground_term_to_gtermD
thf(fact_922_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_f] : ( term_ground_f_v @ ( ground1072145386804415820rm_f_v @ S ) ) ).

% ground_term_of_gterm
thf(fact_923_ffuns__term__Var,axiom,
    ! [X: v] :
      ( ( fSet_ffuns_term_f_v @ ( var_v_f @ X ) )
      = bot_bot_fset_f ) ).

% ffuns_term_Var
thf(fact_924_ta__der__eps__ctxt,axiom,
    ! [P: v,A3: tree_ta_v_f,C2: subterm_and_ctxt_f_v,Q5: v,Q: v] :
      ( ( fmember_v @ P @ ( tree_ta_der_v_f @ A3 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ Q5 ) ) ) )
     => ( ( fmembe4541753606430338222od_v_v @ ( product_Pair_v_v @ Q @ Q5 ) @ ( fSet_ftrancl_v @ ( tree_eps_v_f @ A3 ) ) )
       => ( fmember_v @ P @ ( tree_ta_der_v_f @ A3 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ Q ) ) ) ) ) ) ).

% ta_der_eps_ctxt
thf(fact_925_ctxt__supteq,axiom,
    ! [S: term_f_v,C2: subterm_and_ctxt_f_v,T: term_f_v] :
      ( ( S
        = ( subter6886040689604212261rm_f_v @ C2 @ T ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v ) ) ).

% ctxt_supteq
thf(fact_926_supteq__ctxt__conv,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
      = ( ? [C5: subterm_and_ctxt_f_v] :
            ( S
            = ( subter6886040689604212261rm_f_v @ C5 @ T ) ) ) ) ).

% supteq_ctxt_conv
thf(fact_927_ctxt__imp__supteq,axiom,
    ! [C2: subterm_and_ctxt_f_v,T: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ T ) @ T ) @ subter5033437232688913882eq_f_v ) ).

% ctxt_imp_supteq
thf(fact_928_supteq__ctxtE,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ~ ! [C6: subterm_and_ctxt_f_v] :
            ( S
           != ( subter6886040689604212261rm_f_v @ C6 @ T ) ) ) ).

% supteq_ctxtE
thf(fact_929_ta__der__ctxt__decompose,axiom,
    ! [Q: v,A8: tree_ta_v_f,C2: subterm_and_ctxt_f_v,T: term_f_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) )
     => ? [P3: v] :
          ( ( fmember_v @ P3 @ ( tree_ta_der_v_f @ A8 @ T ) )
          & ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ P3 ) ) ) ) ) ) ).

% ta_der_ctxt_decompose
thf(fact_930_ta__der__ctxt,axiom,
    ! [P: v,A8: tree_ta_v_f,T: term_f_v,Q: v,C2: subterm_and_ctxt_f_v] :
      ( ( fmember_v @ P @ ( tree_ta_der_v_f @ A8 @ T ) )
     => ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ P ) ) ) )
       => ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) ) ) ) ).

% ta_der_ctxt
thf(fact_931_subset__emptyI,axiom,
    ! [A3: set_Pr6841673932529763677term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ~ ( member4687689947140789414term_f @ X3 @ A3 )
     => ( ord_le5176047027162639613term_f @ A3 @ bot_bo5323853609601926601term_f ) ) ).

% subset_emptyI
thf(fact_932_subset__emptyI,axiom,
    ! [A3: set_Pr989862937836626183term_f] :
      ( ! [X3: produc7245736746747425831term_f] :
          ~ ( member848276444142703440term_f @ X3 @ A3 )
     => ( ord_le263819222746101927term_f @ A3 @ bot_bo9223291547480392563term_f ) ) ).

% subset_emptyI
thf(fact_933_fvars__term__of__gterm,axiom,
    ! [T: ground_gterm_f] :
      ( ( fSet_fvars_term_f_v @ ( ground1072145386804415820rm_f_v @ T ) )
      = bot_bot_fset_v ) ).

% fvars_term_of_gterm
thf(fact_934_ssubst__Pair__rhs,axiom,
    ! [R2: term_f_v,S: term_f_v,R5: set_Pr8827868859434726151rm_f_v,S5: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ R2 @ S ) @ R5 )
     => ( ( S5 = S )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ R2 @ S5 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_935_ssubst__Pair__rhs,axiom,
    ! [R2: bot_bot_term_f,S: bot_bot_term_f,R5: set_Pr6841673932529763677term_f,S5: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ R2 @ S ) @ R5 )
     => ( ( S5 = S )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ R2 @ S5 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_936_ssubst__Pair__rhs,axiom,
    ! [R2: ground_gterm_f,S: ground_gterm_f,R5: set_Pr989862937836626183term_f,S5: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ R2 @ S ) @ R5 )
     => ( ( S5 = S )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ R2 @ S5 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_937_ta__der_H__Fun,axiom,
    ! [Q: v,A8: tree_ta_v_f,F2: f,Ts: list_term_f_v] :
      ( ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A8 @ ( fun_f_v @ F2 @ Ts ) ) )
      = ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( fun_f_v @ F2 @ Ts ) ) ) ) ).

% ta_der'_Fun
thf(fact_938_gterm__of__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [F4: f,Ts2: list_term_f_v] :
          ( X
         != ( fun_f_v @ F4 @ Ts2 ) )
     => ~ ! [V4: v] :
            ( X
           != ( var_v_f @ V4 ) ) ) ).

% gterm_of_term.cases
thf(fact_939_ta__der_H_Ocases,axiom,
    ! [X: produc5998591847903407270rm_f_v] :
      ( ! [A10: tree_ta_v_f,P3: v] :
          ( X
         != ( produc5293873249180174558rm_f_v @ A10 @ ( var_v_f @ P3 ) ) )
     => ~ ! [A10: tree_ta_v_f,F4: f,Ts2: list_term_f_v] :
            ( X
           != ( produc5293873249180174558rm_f_v @ A10 @ ( fun_f_v @ F4 @ Ts2 ) ) ) ) ).

% ta_der'.cases
thf(fact_940_term__to__bot__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [Uu2: v] :
          ( X
         != ( var_v_f @ Uu2 ) )
     => ~ ! [F4: f,Ts2: list_term_f_v] :
            ( X
           != ( fun_f_v @ F4 @ Ts2 ) ) ) ).

% term_to_bot_term.cases
thf(fact_941_Term_Oterm_Oexhaust,axiom,
    ! [Y: term_f_v] :
      ( ! [X12: v] :
          ( Y
         != ( var_v_f @ X12 ) )
     => ~ ! [X21: f,X22: list_term_f_v] :
            ( Y
           != ( fun_f_v @ X21 @ X22 ) ) ) ).

% Term.term.exhaust
thf(fact_942_term_Odistinct_I1_J,axiom,
    ! [X1: v,X212: f,X222: list_term_f_v] :
      ( ( var_v_f @ X1 )
     != ( fun_f_v @ X212 @ X222 ) ) ).

% term.distinct(1)
thf(fact_943_funas__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [X3: v] :
          ( X
         != ( var_v_f @ X3 ) )
     => ~ ! [F4: f,Ts2: list_term_f_v] :
            ( X
           != ( fun_f_v @ F4 @ Ts2 ) ) ) ).

% funas_term.cases
thf(fact_944_ground__term__of__gterm_H,axiom,
    ! [S: ground_gterm_f,F2: f,Ss: list_term_f_v] :
      ( ( ( ground1072145386804415820rm_f_v @ S )
        = ( fun_f_v @ F2 @ Ss ) )
     => ( term_ground_f_v @ ( fun_f_v @ F2 @ Ss ) ) ) ).

% ground_term_of_gterm'
thf(fact_945_term__fset__induct,axiom,
    ! [P2: term_f_v > $o,Term: term_f_v] :
      ( ! [X3: v] : ( P2 @ ( var_v_f @ X3 ) )
     => ( ! [X1a: f,X23: list_term_f_v] :
            ( ! [X2a: term_f_v] :
                ( ( fmember_term_f_v @ X2a @ ( fset_o6574051099709074461rm_f_v @ X23 ) )
               => ( P2 @ X2a ) )
           => ( P2 @ ( fun_f_v @ X1a @ X23 ) ) )
       => ( P2 @ Term ) ) ) ).

% term_fset_induct
thf(fact_946_ta__productiveI_H,axiom,
    ! [Q: v,A8: tree_ta_v_f,C2: subterm_and_ctxt_f_v,P: v,P2: fset_v] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ P ) ) ) )
     => ( ( fmember_v @ Q @ ( tree_t2678504896727164544ve_v_f @ P2 @ A8 ) )
       => ( fmember_v @ P @ ( tree_t2678504896727164544ve_v_f @ P2 @ A8 ) ) ) ) ).

% ta_productiveI'
thf(fact_947_ta__productiveI,axiom,
    ! [Q5: v,A8: tree_ta_v_f,C2: subterm_and_ctxt_f_v,Q: v,P2: fset_v] :
      ( ( fmember_v @ Q5 @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C2 @ ( var_v_f @ Q ) ) ) )
     => ( ( fmember_v @ Q5 @ P2 )
       => ( fmember_v @ Q @ ( tree_t2678504896727164544ve_v_f @ P2 @ A8 ) ) ) ) ).

% ta_productiveI
thf(fact_948_ta__productiveE,axiom,
    ! [Q: v,P2: fset_v,A8: tree_ta_v_f] :
      ( ( fmember_v @ Q @ ( tree_t2678504896727164544ve_v_f @ P2 @ A8 ) )
     => ~ ! [Q7: v] :
            ( ? [C6: subterm_and_ctxt_f_v] : ( fmember_v @ Q7 @ ( tree_ta_der_v_f @ A8 @ ( subter6886040689604212261rm_f_v @ C6 @ ( var_v_f @ Q ) ) ) )
           => ~ ( fmember_v @ Q7 @ P2 ) ) ) ).

% ta_productiveE
thf(fact_949_ta__contains__aux__state__set,axiom,
    ! [F3: set_Pr5245412377734449720_f_nat,Q3: fset_v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( tree_t3101685780840579086ux_f_v @ F3 @ Q3 @ A8 @ Q3 )
     => ( ( member5261315044688711901term_f @ T @ ( ground_T_G_f @ F3 ) )
       => ? [Q2: v] :
            ( ( fmember_v @ Q2 @ Q3 )
            & ( fmember_v @ Q2 @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) ) ) ) ) ).

% ta_contains_aux_state_set
thf(fact_950_finsert__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ ( finser1703841043590550767rm_f_v @ B @ A3 ) )
      = ( ( A = B )
        | ( fmembe774991991480784110rm_f_v @ A @ A3 ) ) ) ).

% finsert_iff
thf(fact_951_finsert__iff,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ ( finser7989854760751408709term_f @ B @ A3 ) )
      = ( ( A = B )
        | ( fmembe6950792022577883716term_f @ A @ A3 ) ) ) ).

% finsert_iff
thf(fact_952_finsert__iff,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ ( finser8320498213884029254term_f @ B @ A3 ) )
      = ( ( A = B )
        | ( fmembe1418995490368054919term_f @ A @ A3 ) ) ) ).

% finsert_iff
thf(fact_953_finsertCI,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( ~ ( fmembe774991991480784110rm_f_v @ A @ B4 )
       => ( A = B ) )
     => ( fmembe774991991480784110rm_f_v @ A @ ( finser1703841043590550767rm_f_v @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_954_finsertCI,axiom,
    ! [A: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f,B: produc250624644672406269term_f] :
      ( ( ~ ( fmembe6950792022577883716term_f @ A @ B4 )
       => ( A = B ) )
     => ( fmembe6950792022577883716term_f @ A @ ( finser7989854760751408709term_f @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_955_finsertCI,axiom,
    ! [A: bot_bot_term_f,B4: fset_Bot_bot_term_f,B: bot_bot_term_f] :
      ( ( ~ ( fmembe1418995490368054919term_f @ A @ B4 )
       => ( A = B ) )
     => ( fmembe1418995490368054919term_f @ A @ ( finser8320498213884029254term_f @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_956_fimage__finsert,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fimage84417724776064284_f_nat @ F2 @ ( finser1703841043590550767rm_f_v @ A @ B4 ) )
      = ( finser2928796382700762858_f_nat @ ( F2 @ A ) @ ( fimage84417724776064284_f_nat @ F2 @ B4 ) ) ) ).

% fimage_finsert
thf(fact_957_funion__finsert__left,axiom,
    ! [A: product_prod_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ ( finser8068648205224448314_f_nat @ A @ B4 ) @ C2 )
      = ( finser8068648205224448314_f_nat @ A @ ( sup_su8375471814421675870_f_nat @ B4 @ C2 ) ) ) ).

% funion_finsert_left
thf(fact_958_funion__finsert__right,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,A: product_prod_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( sup_su8375471814421675870_f_nat @ A3 @ ( finser8068648205224448314_f_nat @ A @ B4 ) )
      = ( finser8068648205224448314_f_nat @ A @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ) ).

% funion_finsert_right
thf(fact_959_finsert__fimage,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( finser2928796382700762858_f_nat @ ( F2 @ X ) @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) )
        = ( fimage84417724776064284_f_nat @ F2 @ A3 ) ) ) ).

% finsert_fimage
thf(fact_960_finsert__fsubset,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( ord_le4587745213494032429rm_f_v @ ( finser1703841043590550767rm_f_v @ X @ A3 ) @ B4 )
      = ( ( fmembe774991991480784110rm_f_v @ X @ B4 )
        & ( ord_le4587745213494032429rm_f_v @ A3 @ B4 ) ) ) ).

% finsert_fsubset
thf(fact_961_finsert__fsubset,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( ord_le7092741666294821507term_f @ ( finser7989854760751408709term_f @ X @ A3 ) @ B4 )
      = ( ( fmembe6950792022577883716term_f @ X @ B4 )
        & ( ord_le7092741666294821507term_f @ A3 @ B4 ) ) ) ).

% finsert_fsubset
thf(fact_962_finsert__fsubset,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ ( finser8320498213884029254term_f @ X @ A3 ) @ B4 )
      = ( ( fmembe1418995490368054919term_f @ X @ B4 )
        & ( ord_le7572167314378612510term_f @ A3 @ B4 ) ) ) ).

% finsert_fsubset
thf(fact_963_finsert__fsubset,axiom,
    ! [X: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ ( finser8068648205224448314_f_nat @ X @ A3 ) @ B4 )
      = ( ( fmembe6795308708001954811_f_nat @ X @ B4 )
        & ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ) ).

% finsert_fsubset
thf(fact_964_finter__finsert__right__ifffempty,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( inf_in6563521360087198331rm_f_v @ A3 @ ( finser1703841043590550767rm_f_v @ A @ B4 ) )
        = ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) ) ).

% finter_finsert_right_ifffempty
thf(fact_965_finter__finsert__right__ifffempty,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( inf_in2682540704405482705term_f @ A3 @ ( finser7989854760751408709term_f @ A @ B4 ) )
        = ( inf_in2682540704405482705term_f @ A3 @ B4 ) ) ) ).

% finter_finsert_right_ifffempty
thf(fact_966_finter__finsert__right__ifffempty,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ~ ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( inf_in2113687546666124112term_f @ A3 @ ( finser8320498213884029254term_f @ A @ B4 ) )
        = ( inf_in2113687546666124112term_f @ A3 @ B4 ) ) ) ).

% finter_finsert_right_ifffempty
thf(fact_967_finter__finsert__left__ifffempty,axiom,
    ! [A: produc8027700073345061287rm_f_v,C2: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ A @ C2 )
     => ( ( inf_in6563521360087198331rm_f_v @ ( finser1703841043590550767rm_f_v @ A @ B4 ) @ C2 )
        = ( inf_in6563521360087198331rm_f_v @ B4 @ C2 ) ) ) ).

% finter_finsert_left_ifffempty
thf(fact_968_finter__finsert__left__ifffempty,axiom,
    ! [A: produc250624644672406269term_f,C2: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ A @ C2 )
     => ( ( inf_in2682540704405482705term_f @ ( finser7989854760751408709term_f @ A @ B4 ) @ C2 )
        = ( inf_in2682540704405482705term_f @ B4 @ C2 ) ) ) ).

% finter_finsert_left_ifffempty
thf(fact_969_finter__finsert__left__ifffempty,axiom,
    ! [A: bot_bot_term_f,C2: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ~ ( fmembe1418995490368054919term_f @ A @ C2 )
     => ( ( inf_in2113687546666124112term_f @ ( finser8320498213884029254term_f @ A @ B4 ) @ C2 )
        = ( inf_in2113687546666124112term_f @ B4 @ C2 ) ) ) ).

% finter_finsert_left_ifffempty
thf(fact_970_finter__finsert__right__if1,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( inf_in6563521360087198331rm_f_v @ A3 @ ( finser1703841043590550767rm_f_v @ A @ B4 ) )
        = ( finser1703841043590550767rm_f_v @ A @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right_if1
thf(fact_971_finter__finsert__right__if1,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( inf_in2682540704405482705term_f @ A3 @ ( finser7989854760751408709term_f @ A @ B4 ) )
        = ( finser7989854760751408709term_f @ A @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right_if1
thf(fact_972_finter__finsert__right__if1,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( inf_in2113687546666124112term_f @ A3 @ ( finser8320498213884029254term_f @ A @ B4 ) )
        = ( finser8320498213884029254term_f @ A @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right_if1
thf(fact_973_finter__finsert__left__if1,axiom,
    ! [A: produc8027700073345061287rm_f_v,C2: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ C2 )
     => ( ( inf_in6563521360087198331rm_f_v @ ( finser1703841043590550767rm_f_v @ A @ B4 ) @ C2 )
        = ( finser1703841043590550767rm_f_v @ A @ ( inf_in6563521360087198331rm_f_v @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left_if1
thf(fact_974_finter__finsert__left__if1,axiom,
    ! [A: produc250624644672406269term_f,C2: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ C2 )
     => ( ( inf_in2682540704405482705term_f @ ( finser7989854760751408709term_f @ A @ B4 ) @ C2 )
        = ( finser7989854760751408709term_f @ A @ ( inf_in2682540704405482705term_f @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left_if1
thf(fact_975_finter__finsert__left__if1,axiom,
    ! [A: bot_bot_term_f,C2: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ C2 )
     => ( ( inf_in2113687546666124112term_f @ ( finser8320498213884029254term_f @ A @ B4 ) @ C2 )
        = ( finser8320498213884029254term_f @ A @ ( inf_in2113687546666124112term_f @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left_if1
thf(fact_976_ta__reachable__gtermI,axiom,
    ! [Q: v,A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) ) )
     => ( fmember_v @ Q @ ( tree_t1547661003257875222le_v_f @ A8 ) ) ) ).

% ta_reachable_gtermI
thf(fact_977_fvars__term__Var,axiom,
    ! [X: v] :
      ( ( fSet_fvars_term_f_v @ ( var_v_f @ X ) )
      = ( finsert_v @ X @ bot_bot_fset_v ) ) ).

% fvars_term_Var
thf(fact_978_fImage__singleton__iff,axiom,
    ! [B: term_f_v,R2: fset_P8018961893305114765rm_f_v,A: term_f_v] :
      ( ( fmember_term_f_v @ B @ ( fSet_f3701177912297302022rm_f_v @ R2 @ ( finsert_term_f_v @ A @ bot_bo9049458698243030439rm_f_v ) ) )
      = ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R2 ) ) ).

% fImage_singleton_iff
thf(fact_979_fImage__singleton__iff,axiom,
    ! [B: bot_bot_term_f,R2: fset_P2882900458636368611term_f,A: bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( fSet_f7835841554439036636term_f @ R2 @ ( finser8320498213884029254term_f @ A @ bot_bo4668628460741225810term_f ) ) )
      = ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R2 ) ) ).

% fImage_singleton_iff
thf(fact_980_fImage__singleton__iff,axiom,
    ! [B: ground_gterm_f,R2: fset_P5324236452418035597term_f,A: ground_gterm_f] :
      ( ( fmembe8264398993653236347term_f @ B @ ( fSet_f1232815550461614398term_f @ R2 @ ( finser2403200890035884156term_f @ A @ bot_bo5857025647839695366term_f ) ) )
      = ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ R2 ) ) ).

% fImage_singleton_iff
thf(fact_981_finter__finsert__left,axiom,
    ! [A: produc8027700073345061287rm_f_v,C2: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( ( fmembe774991991480784110rm_f_v @ A @ C2 )
       => ( ( inf_in6563521360087198331rm_f_v @ ( finser1703841043590550767rm_f_v @ A @ B4 ) @ C2 )
          = ( finser1703841043590550767rm_f_v @ A @ ( inf_in6563521360087198331rm_f_v @ B4 @ C2 ) ) ) )
      & ( ~ ( fmembe774991991480784110rm_f_v @ A @ C2 )
       => ( ( inf_in6563521360087198331rm_f_v @ ( finser1703841043590550767rm_f_v @ A @ B4 ) @ C2 )
          = ( inf_in6563521360087198331rm_f_v @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left
thf(fact_982_finter__finsert__left,axiom,
    ! [A: produc250624644672406269term_f,C2: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( ( fmembe6950792022577883716term_f @ A @ C2 )
       => ( ( inf_in2682540704405482705term_f @ ( finser7989854760751408709term_f @ A @ B4 ) @ C2 )
          = ( finser7989854760751408709term_f @ A @ ( inf_in2682540704405482705term_f @ B4 @ C2 ) ) ) )
      & ( ~ ( fmembe6950792022577883716term_f @ A @ C2 )
       => ( ( inf_in2682540704405482705term_f @ ( finser7989854760751408709term_f @ A @ B4 ) @ C2 )
          = ( inf_in2682540704405482705term_f @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left
thf(fact_983_finter__finsert__left,axiom,
    ! [A: bot_bot_term_f,C2: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( ( fmembe1418995490368054919term_f @ A @ C2 )
       => ( ( inf_in2113687546666124112term_f @ ( finser8320498213884029254term_f @ A @ B4 ) @ C2 )
          = ( finser8320498213884029254term_f @ A @ ( inf_in2113687546666124112term_f @ B4 @ C2 ) ) ) )
      & ( ~ ( fmembe1418995490368054919term_f @ A @ C2 )
       => ( ( inf_in2113687546666124112term_f @ ( finser8320498213884029254term_f @ A @ B4 ) @ C2 )
          = ( inf_in2113687546666124112term_f @ B4 @ C2 ) ) ) ) ).

% finter_finsert_left
thf(fact_984_finter__finsert__right,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
       => ( ( inf_in6563521360087198331rm_f_v @ A3 @ ( finser1703841043590550767rm_f_v @ A @ B4 ) )
          = ( finser1703841043590550767rm_f_v @ A @ ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) ) )
      & ( ~ ( fmembe774991991480784110rm_f_v @ A @ A3 )
       => ( ( inf_in6563521360087198331rm_f_v @ A3 @ ( finser1703841043590550767rm_f_v @ A @ B4 ) )
          = ( inf_in6563521360087198331rm_f_v @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right
thf(fact_985_finter__finsert__right,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( ( inf_in2682540704405482705term_f @ A3 @ ( finser7989854760751408709term_f @ A @ B4 ) )
          = ( finser7989854760751408709term_f @ A @ ( inf_in2682540704405482705term_f @ A3 @ B4 ) ) ) )
      & ( ~ ( fmembe6950792022577883716term_f @ A @ A3 )
       => ( ( inf_in2682540704405482705term_f @ A3 @ ( finser7989854760751408709term_f @ A @ B4 ) )
          = ( inf_in2682540704405482705term_f @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right
thf(fact_986_finter__finsert__right,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( ( inf_in2113687546666124112term_f @ A3 @ ( finser8320498213884029254term_f @ A @ B4 ) )
          = ( finser8320498213884029254term_f @ A @ ( inf_in2113687546666124112term_f @ A3 @ B4 ) ) ) )
      & ( ~ ( fmembe1418995490368054919term_f @ A @ A3 )
       => ( ( inf_in2113687546666124112term_f @ A3 @ ( finser8320498213884029254term_f @ A @ B4 ) )
          = ( inf_in2113687546666124112term_f @ A3 @ B4 ) ) ) ) ).

% finter_finsert_right
thf(fact_987_fsubset__fsingletonD,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,X: product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
     => ( ( A3 = bot_bo3974441229639268550_f_nat )
        | ( A3
          = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) ) ) ) ).

% fsubset_fsingletonD
thf(fact_988_finsert__is__funion,axiom,
    ( finser8068648205224448314_f_nat
    = ( ^ [A7: product_prod_f_nat] : ( sup_su8375471814421675870_f_nat @ ( finser8068648205224448314_f_nat @ A7 @ bot_bo3974441229639268550_f_nat ) ) ) ) ).

% finsert_is_funion
thf(fact_989_fsingleton__funion__iff,axiom,
    ! [X: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat )
        = ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bo3974441229639268550_f_nat )
          & ( B4
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) ) )
        | ( ( A3
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
          & ( B4 = bot_bo3974441229639268550_f_nat ) )
        | ( ( A3
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
          & ( B4
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) ) ) ) ) ).

% fsingleton_funion_iff
thf(fact_990_funion__fsingleton__iff,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,X: product_prod_f_nat] :
      ( ( ( sup_su8375471814421675870_f_nat @ A3 @ B4 )
        = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
      = ( ( ( A3 = bot_bo3974441229639268550_f_nat )
          & ( B4
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) ) )
        | ( ( A3
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
          & ( B4 = bot_bo3974441229639268550_f_nat ) )
        | ( ( A3
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) )
          & ( B4
            = ( finser8068648205224448314_f_nat @ X @ bot_bo3974441229639268550_f_nat ) ) ) ) ) ).

% funion_fsingleton_iff
thf(fact_991_finsert__mono,axiom,
    ! [C2: fset_P6228066233360383026_f_nat,D: fset_P6228066233360383026_f_nat,A: product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C2 @ D )
     => ( ord_le1552505484586773650_f_nat @ ( finser8068648205224448314_f_nat @ A @ C2 ) @ ( finser8068648205224448314_f_nat @ A @ D ) ) ) ).

% finsert_mono
thf(fact_992_fsubset__finsertI,axiom,
    ! [B4: fset_P6228066233360383026_f_nat,A: product_prod_f_nat] : ( ord_le1552505484586773650_f_nat @ B4 @ ( finser8068648205224448314_f_nat @ A @ B4 ) ) ).

% fsubset_finsertI
thf(fact_993_fsubset__finsertI2,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,B: product_prod_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ord_le1552505484586773650_f_nat @ A3 @ ( finser8068648205224448314_f_nat @ B @ B4 ) ) ) ).

% fsubset_finsertI2
thf(fact_994_mk__disjoint__finsert,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ? [B8: fset_P8018961893305114765rm_f_v] :
          ( ( A3
            = ( finser1703841043590550767rm_f_v @ A @ B8 ) )
          & ~ ( fmembe774991991480784110rm_f_v @ A @ B8 ) ) ) ).

% mk_disjoint_finsert
thf(fact_995_mk__disjoint__finsert,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ? [B8: fset_P2882900458636368611term_f] :
          ( ( A3
            = ( finser7989854760751408709term_f @ A @ B8 ) )
          & ~ ( fmembe6950792022577883716term_f @ A @ B8 ) ) ) ).

% mk_disjoint_finsert
thf(fact_996_mk__disjoint__finsert,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ? [B8: fset_Bot_bot_term_f] :
          ( ( A3
            = ( finser8320498213884029254term_f @ A @ B8 ) )
          & ~ ( fmembe1418995490368054919term_f @ A @ B8 ) ) ) ).

% mk_disjoint_finsert
thf(fact_997_finsert__eq__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ~ ( fmembe774991991480784110rm_f_v @ B @ B4 )
       => ( ( ( finser1703841043590550767rm_f_v @ A @ A3 )
            = ( finser1703841043590550767rm_f_v @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C5: fset_P8018961893305114765rm_f_v] :
                  ( ( A3
                    = ( finser1703841043590550767rm_f_v @ B @ C5 ) )
                  & ~ ( fmembe774991991480784110rm_f_v @ B @ C5 )
                  & ( B4
                    = ( finser1703841043590550767rm_f_v @ A @ C5 ) )
                  & ~ ( fmembe774991991480784110rm_f_v @ A @ C5 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_998_finsert__eq__iff,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ~ ( fmembe6950792022577883716term_f @ B @ B4 )
       => ( ( ( finser7989854760751408709term_f @ A @ A3 )
            = ( finser7989854760751408709term_f @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C5: fset_P2882900458636368611term_f] :
                  ( ( A3
                    = ( finser7989854760751408709term_f @ B @ C5 ) )
                  & ~ ( fmembe6950792022577883716term_f @ B @ C5 )
                  & ( B4
                    = ( finser7989854760751408709term_f @ A @ C5 ) )
                  & ~ ( fmembe6950792022577883716term_f @ A @ C5 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_999_finsert__eq__iff,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ~ ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ~ ( fmembe1418995490368054919term_f @ B @ B4 )
       => ( ( ( finser8320498213884029254term_f @ A @ A3 )
            = ( finser8320498213884029254term_f @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C5: fset_Bot_bot_term_f] :
                  ( ( A3
                    = ( finser8320498213884029254term_f @ B @ C5 ) )
                  & ~ ( fmembe1418995490368054919term_f @ B @ C5 )
                  & ( B4
                    = ( finser8320498213884029254term_f @ A @ C5 ) )
                  & ~ ( fmembe1418995490368054919term_f @ A @ C5 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_1000_finsert__absorb,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( finser1703841043590550767rm_f_v @ A @ A3 )
        = A3 ) ) ).

% finsert_absorb
thf(fact_1001_finsert__absorb,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( finser7989854760751408709term_f @ A @ A3 )
        = A3 ) ) ).

% finsert_absorb
thf(fact_1002_finsert__absorb,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( finser8320498213884029254term_f @ A @ A3 )
        = A3 ) ) ).

% finsert_absorb
thf(fact_1003_finsert__ident,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ~ ( fmembe774991991480784110rm_f_v @ X @ B4 )
       => ( ( ( finser1703841043590550767rm_f_v @ X @ A3 )
            = ( finser1703841043590550767rm_f_v @ X @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_1004_finsert__ident,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( ~ ( fmembe6950792022577883716term_f @ X @ B4 )
       => ( ( ( finser7989854760751408709term_f @ X @ A3 )
            = ( finser7989854760751408709term_f @ X @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_1005_finsert__ident,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ~ ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( ~ ( fmembe1418995490368054919term_f @ X @ B4 )
       => ( ( ( finser8320498213884029254term_f @ X @ A3 )
            = ( finser8320498213884029254term_f @ X @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_1006_set__finsert,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ~ ! [B8: fset_P8018961893305114765rm_f_v] :
            ( ( A3
              = ( finser1703841043590550767rm_f_v @ X @ B8 ) )
           => ( fmembe774991991480784110rm_f_v @ X @ B8 ) ) ) ).

% set_finsert
thf(fact_1007_set__finsert,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ A3 )
     => ~ ! [B8: fset_P2882900458636368611term_f] :
            ( ( A3
              = ( finser7989854760751408709term_f @ X @ B8 ) )
           => ( fmembe6950792022577883716term_f @ X @ B8 ) ) ) ).

% set_finsert
thf(fact_1008_set__finsert,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ A3 )
     => ~ ! [B8: fset_Bot_bot_term_f] :
            ( ( A3
              = ( finser8320498213884029254term_f @ X @ B8 ) )
           => ( fmembe1418995490368054919term_f @ X @ B8 ) ) ) ).

% set_finsert
thf(fact_1009_finsertI2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v,B: produc8027700073345061287rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ B4 )
     => ( fmembe774991991480784110rm_f_v @ A @ ( finser1703841043590550767rm_f_v @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_1010_finsertI2,axiom,
    ! [A: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f,B: produc250624644672406269term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ B4 )
     => ( fmembe6950792022577883716term_f @ A @ ( finser7989854760751408709term_f @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_1011_finsertI2,axiom,
    ! [A: bot_bot_term_f,B4: fset_Bot_bot_term_f,B: bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ B4 )
     => ( fmembe1418995490368054919term_f @ A @ ( finser8320498213884029254term_f @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_1012_finsertI1,axiom,
    ! [A: produc8027700073345061287rm_f_v,B4: fset_P8018961893305114765rm_f_v] : ( fmembe774991991480784110rm_f_v @ A @ ( finser1703841043590550767rm_f_v @ A @ B4 ) ) ).

% finsertI1
thf(fact_1013_finsertI1,axiom,
    ! [A: produc250624644672406269term_f,B4: fset_P2882900458636368611term_f] : ( fmembe6950792022577883716term_f @ A @ ( finser7989854760751408709term_f @ A @ B4 ) ) ).

% finsertI1
thf(fact_1014_finsertI1,axiom,
    ! [A: bot_bot_term_f,B4: fset_Bot_bot_term_f] : ( fmembe1418995490368054919term_f @ A @ ( finser8320498213884029254term_f @ A @ B4 ) ) ).

% finsertI1
thf(fact_1015_finsertE,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ A @ ( finser1703841043590550767rm_f_v @ B @ A3 ) )
     => ( ( A != B )
       => ( fmembe774991991480784110rm_f_v @ A @ A3 ) ) ) ).

% finsertE
thf(fact_1016_finsertE,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ ( finser7989854760751408709term_f @ B @ A3 ) )
     => ( ( A != B )
       => ( fmembe6950792022577883716term_f @ A @ A3 ) ) ) ).

% finsertE
thf(fact_1017_finsertE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ ( finser8320498213884029254term_f @ B @ A3 ) )
     => ( ( A != B )
       => ( fmembe1418995490368054919term_f @ A @ A3 ) ) ) ).

% finsertE
thf(fact_1018_fsubset__finsert,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ~ ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( ord_le4587745213494032429rm_f_v @ A3 @ ( finser1703841043590550767rm_f_v @ X @ B4 ) )
        = ( ord_le4587745213494032429rm_f_v @ A3 @ B4 ) ) ) ).

% fsubset_finsert
thf(fact_1019_fsubset__finsert,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ~ ( fmembe6950792022577883716term_f @ X @ A3 )
     => ( ( ord_le7092741666294821507term_f @ A3 @ ( finser7989854760751408709term_f @ X @ B4 ) )
        = ( ord_le7092741666294821507term_f @ A3 @ B4 ) ) ) ).

% fsubset_finsert
thf(fact_1020_fsubset__finsert,axiom,
    ! [X: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ~ ( fmembe1418995490368054919term_f @ X @ A3 )
     => ( ( ord_le7572167314378612510term_f @ A3 @ ( finser8320498213884029254term_f @ X @ B4 ) )
        = ( ord_le7572167314378612510term_f @ A3 @ B4 ) ) ) ).

% fsubset_finsert
thf(fact_1021_fsubset__finsert,axiom,
    ! [X: product_prod_f_nat,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ~ ( fmembe6795308708001954811_f_nat @ X @ A3 )
     => ( ( ord_le1552505484586773650_f_nat @ A3 @ ( finser8068648205224448314_f_nat @ X @ B4 ) )
        = ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ) ).

% fsubset_finsert
thf(fact_1022_fset__induct2,axiom,
    ! [P2: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v > $o,Xsa: fset_P8018961893305114765rm_f_v,Ysa: fset_P8018961893305114765rm_f_v] :
      ( ( P2 @ bot_bo4614643899460950777rm_f_v @ bot_bo4614643899460950777rm_f_v )
     => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v] :
            ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
           => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ bot_bo4614643899460950777rm_f_v ) )
       => ( ! [Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
              ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
             => ( P2 @ bot_bo4614643899460950777rm_f_v @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) )
         => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v,Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
                 => ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
                   => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1023_fset__induct2,axiom,
    ! [P2: fset_P8018961893305114765rm_f_v > fset_P2882900458636368611term_f > $o,Xsa: fset_P8018961893305114765rm_f_v,Ysa: fset_P2882900458636368611term_f] :
      ( ( P2 @ bot_bo4614643899460950777rm_f_v @ bot_bo6180437703705224527term_f )
     => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v] :
            ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
           => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ bot_bo6180437703705224527term_f ) )
       => ( ! [Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
              ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo4614643899460950777rm_f_v @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v,Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
                 => ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1024_fset__induct2,axiom,
    ! [P2: fset_P8018961893305114765rm_f_v > fset_Bot_bot_term_f > $o,Xsa: fset_P8018961893305114765rm_f_v,Ysa: fset_Bot_bot_term_f] :
      ( ( P2 @ bot_bo4614643899460950777rm_f_v @ bot_bo4668628460741225810term_f )
     => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v] :
            ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
           => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ bot_bo4668628460741225810term_f ) )
       => ( ! [Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
              ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo4614643899460950777rm_f_v @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: produc8027700073345061287rm_f_v,Xs: fset_P8018961893305114765rm_f_v,Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Xs )
                 => ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ Xs ) @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1025_fset__induct2,axiom,
    ! [P2: fset_P2882900458636368611term_f > fset_P8018961893305114765rm_f_v > $o,Xsa: fset_P2882900458636368611term_f,Ysa: fset_P8018961893305114765rm_f_v] :
      ( ( P2 @ bot_bo6180437703705224527term_f @ bot_bo4614643899460950777rm_f_v )
     => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f] :
            ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
           => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ bot_bo4614643899460950777rm_f_v ) )
       => ( ! [Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
              ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
             => ( P2 @ bot_bo6180437703705224527term_f @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) )
         => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f,Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
                 => ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
                   => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1026_fset__induct2,axiom,
    ! [P2: fset_P2882900458636368611term_f > fset_P2882900458636368611term_f > $o,Xsa: fset_P2882900458636368611term_f,Ysa: fset_P2882900458636368611term_f] :
      ( ( P2 @ bot_bo6180437703705224527term_f @ bot_bo6180437703705224527term_f )
     => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f] :
            ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
           => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ bot_bo6180437703705224527term_f ) )
       => ( ! [Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
              ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo6180437703705224527term_f @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f,Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
                 => ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1027_fset__induct2,axiom,
    ! [P2: fset_P2882900458636368611term_f > fset_Bot_bot_term_f > $o,Xsa: fset_P2882900458636368611term_f,Ysa: fset_Bot_bot_term_f] :
      ( ( P2 @ bot_bo6180437703705224527term_f @ bot_bo4668628460741225810term_f )
     => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f] :
            ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
           => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ bot_bo4668628460741225810term_f ) )
       => ( ! [Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
              ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo6180437703705224527term_f @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: produc250624644672406269term_f,Xs: fset_P2882900458636368611term_f,Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe6950792022577883716term_f @ X3 @ Xs )
                 => ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser7989854760751408709term_f @ X3 @ Xs ) @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1028_fset__induct2,axiom,
    ! [P2: fset_Bot_bot_term_f > fset_P8018961893305114765rm_f_v > $o,Xsa: fset_Bot_bot_term_f,Ysa: fset_P8018961893305114765rm_f_v] :
      ( ( P2 @ bot_bo4668628460741225810term_f @ bot_bo4614643899460950777rm_f_v )
     => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f] :
            ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
           => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ bot_bo4614643899460950777rm_f_v ) )
       => ( ! [Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
              ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
             => ( P2 @ bot_bo4668628460741225810term_f @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) )
         => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f,Y3: produc8027700073345061287rm_f_v,Ys: fset_P8018961893305114765rm_f_v] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
                 => ( ~ ( fmembe774991991480784110rm_f_v @ Y3 @ Ys )
                   => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ ( finser1703841043590550767rm_f_v @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1029_fset__induct2,axiom,
    ! [P2: fset_Bot_bot_term_f > fset_P2882900458636368611term_f > $o,Xsa: fset_Bot_bot_term_f,Ysa: fset_P2882900458636368611term_f] :
      ( ( P2 @ bot_bo4668628460741225810term_f @ bot_bo6180437703705224527term_f )
     => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f] :
            ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
           => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ bot_bo6180437703705224527term_f ) )
       => ( ! [Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
              ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo4668628460741225810term_f @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f,Y3: produc250624644672406269term_f,Ys: fset_P2882900458636368611term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
                 => ( ~ ( fmembe6950792022577883716term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ ( finser7989854760751408709term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1030_fset__induct2,axiom,
    ! [P2: fset_Bot_bot_term_f > fset_Bot_bot_term_f > $o,Xsa: fset_Bot_bot_term_f,Ysa: fset_Bot_bot_term_f] :
      ( ( P2 @ bot_bo4668628460741225810term_f @ bot_bo4668628460741225810term_f )
     => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f] :
            ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
           => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ bot_bo4668628460741225810term_f ) )
       => ( ! [Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
              ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
             => ( P2 @ bot_bo4668628460741225810term_f @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) )
         => ( ! [X3: bot_bot_term_f,Xs: fset_Bot_bot_term_f,Y3: bot_bot_term_f,Ys: fset_Bot_bot_term_f] :
                ( ( P2 @ Xs @ Ys )
               => ( ~ ( fmembe1418995490368054919term_f @ X3 @ Xs )
                 => ( ~ ( fmembe1418995490368054919term_f @ Y3 @ Ys )
                   => ( P2 @ ( finser8320498213884029254term_f @ X3 @ Xs ) @ ( finser8320498213884029254term_f @ Y3 @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1031_fsingleton__iff,axiom,
    ! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ B @ ( finser1703841043590550767rm_f_v @ A @ bot_bo4614643899460950777rm_f_v ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_1032_fsingleton__iff,axiom,
    ! [B: produc250624644672406269term_f,A: produc250624644672406269term_f] :
      ( ( fmembe6950792022577883716term_f @ B @ ( finser7989854760751408709term_f @ A @ bot_bo6180437703705224527term_f ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_1033_fsingleton__iff,axiom,
    ! [B: bot_bot_term_f,A: bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ B @ ( finser8320498213884029254term_f @ A @ bot_bo4668628460741225810term_f ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_1034_fset__strong__cases,axiom,
    ! [Xs2: fset_P8018961893305114765rm_f_v] :
      ( ( Xs2 != bot_bo4614643899460950777rm_f_v )
     => ~ ! [Ys: fset_P8018961893305114765rm_f_v,X3: produc8027700073345061287rm_f_v] :
            ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ Ys )
           => ( Xs2
             != ( finser1703841043590550767rm_f_v @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_1035_fset__strong__cases,axiom,
    ! [Xs2: fset_P2882900458636368611term_f] :
      ( ( Xs2 != bot_bo6180437703705224527term_f )
     => ~ ! [Ys: fset_P2882900458636368611term_f,X3: produc250624644672406269term_f] :
            ( ~ ( fmembe6950792022577883716term_f @ X3 @ Ys )
           => ( Xs2
             != ( finser7989854760751408709term_f @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_1036_fset__strong__cases,axiom,
    ! [Xs2: fset_Bot_bot_term_f] :
      ( ( Xs2 != bot_bo4668628460741225810term_f )
     => ~ ! [Ys: fset_Bot_bot_term_f,X3: bot_bot_term_f] :
            ( ~ ( fmembe1418995490368054919term_f @ X3 @ Ys )
           => ( Xs2
             != ( finser8320498213884029254term_f @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_1037_fset__induct__stronger,axiom,
    ! [P2: fset_P8018961893305114765rm_f_v > $o,S2: fset_P8018961893305114765rm_f_v] :
      ( ( P2 @ bot_bo4614643899460950777rm_f_v )
     => ( ! [X3: produc8027700073345061287rm_f_v,S6: fset_P8018961893305114765rm_f_v] :
            ( ~ ( fmembe774991991480784110rm_f_v @ X3 @ S6 )
           => ( ( P2 @ S6 )
             => ( P2 @ ( finser1703841043590550767rm_f_v @ X3 @ S6 ) ) ) )
       => ( P2 @ S2 ) ) ) ).

% fset_induct_stronger
thf(fact_1038_fset__induct__stronger,axiom,
    ! [P2: fset_P2882900458636368611term_f > $o,S2: fset_P2882900458636368611term_f] :
      ( ( P2 @ bot_bo6180437703705224527term_f )
     => ( ! [X3: produc250624644672406269term_f,S6: fset_P2882900458636368611term_f] :
            ( ~ ( fmembe6950792022577883716term_f @ X3 @ S6 )
           => ( ( P2 @ S6 )
             => ( P2 @ ( finser7989854760751408709term_f @ X3 @ S6 ) ) ) )
       => ( P2 @ S2 ) ) ) ).

% fset_induct_stronger
thf(fact_1039_fset__induct__stronger,axiom,
    ! [P2: fset_Bot_bot_term_f > $o,S2: fset_Bot_bot_term_f] :
      ( ( P2 @ bot_bo4668628460741225810term_f )
     => ( ! [X3: bot_bot_term_f,S6: fset_Bot_bot_term_f] :
            ( ~ ( fmembe1418995490368054919term_f @ X3 @ S6 )
           => ( ( P2 @ S6 )
             => ( P2 @ ( finser8320498213884029254term_f @ X3 @ S6 ) ) ) )
       => ( P2 @ S2 ) ) ) ).

% fset_induct_stronger
thf(fact_1040_ta__reachable__gtermE,axiom,
    ! [Q: v,A8: tree_ta_v_f] :
      ( ( fmember_v @ Q @ ( tree_t1547661003257875222le_v_f @ A8 ) )
     => ~ ! [T3: ground_gterm_f] :
            ~ ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T3 ) ) ) ) ).

% ta_reachable_gtermE
thf(fact_1041_ta__der__gterm__only__reach,axiom,
    ! [A8: tree_ta_v_f,T: ground_gterm_f] :
      ( ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T ) )
      = ( tree_ta_der_v_f @ ( tree_ta_restrict_v_f @ A8 @ ( tree_t1547661003257875222le_v_f @ A8 ) ) @ ( ground1072145386804415820rm_f_v @ T ) ) ) ).

% ta_der_gterm_only_reach
thf(fact_1042_ta__reachable__rule__gtermE,axiom,
    ! [A8: tree_ta_v_f,F2: f,Qs: list_v,Q: v] :
      ( ( ord_less_eq_fset_v @ ( tree_Q_v_f @ A8 ) @ ( tree_t1547661003257875222le_v_f @ A8 ) )
     => ( ( fmembe7939848548214878551le_v_f @ ( tree_ta_TA_rule_f_v @ F2 @ Qs @ Q ) @ ( tree_rules_v_f @ A8 ) )
       => ~ ! [T3: ground_gterm_f] :
              ( ( ( ground_groot_f @ T3 )
                = ( product_Pair_f_nat @ F2 @ ( size_size_list_v @ Qs ) ) )
             => ~ ( fmember_v @ Q @ ( tree_ta_der_v_f @ A8 @ ( ground1072145386804415820rm_f_v @ T3 ) ) ) ) ) ) ).

% ta_reachable_rule_gtermE
thf(fact_1043_ffUnionI,axiom,
    ! [X5: fset_P6228066233360383026_f_nat,C2: fset_f1705081619141295970_f_nat,A3: product_prod_f_nat] :
      ( ( fmembe7677060439241084203_f_nat @ X5 @ C2 )
     => ( ( fmembe6795308708001954811_f_nat @ A3 @ X5 )
       => ( fmembe6795308708001954811_f_nat @ A3 @ ( ffUnio5887076197631975402_f_nat @ C2 ) ) ) ) ).

% ffUnionI
thf(fact_1044_ffUnionI,axiom,
    ! [X5: fset_P8018961893305114765rm_f_v,C2: fset_f2722810715853128435rm_f_v,A3: produc8027700073345061287rm_f_v] :
      ( ( fmembe767321701615834196rm_f_v @ X5 @ C2 )
     => ( ( fmembe774991991480784110rm_f_v @ A3 @ X5 )
       => ( fmembe774991991480784110rm_f_v @ A3 @ ( ffUnio2002683114454723135rm_f_v @ C2 ) ) ) ) ).

% ffUnionI
thf(fact_1045_ffUnionI,axiom,
    ! [X5: fset_P2882900458636368611term_f,C2: fset_f1447116543200314953term_f,A3: produc250624644672406269term_f] :
      ( ( fmembe7032422894501393066term_f @ X5 @ C2 )
     => ( ( fmembe6950792022577883716term_f @ A3 @ X5 )
       => ( fmembe6950792022577883716term_f @ A3 @ ( ffUnio7075245367655988117term_f @ C2 ) ) ) ) ).

% ffUnionI
thf(fact_1046_ffUnionI,axiom,
    ! [X5: fset_Bot_bot_term_f,C2: fset_f2857577307919513710term_f,A3: bot_bot_term_f] :
      ( ( fmembe1485167145268358711term_f @ X5 @ C2 )
     => ( ( fmembe1418995490368054919term_f @ A3 @ X5 )
       => ( fmembe1418995490368054919term_f @ A3 @ ( ffUnio1252369753758625782term_f @ C2 ) ) ) ) ).

% ffUnionI
thf(fact_1047_ffunion__empty,axiom,
    ( ( ffUnio5887076197631975402_f_nat @ bot_bo2413912301557604854_f_nat )
    = bot_bo3974441229639268550_f_nat ) ).

% ffunion_empty
thf(fact_1048_ffunion__insert,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B4: fset_f1705081619141295970_f_nat] :
      ( ( ffUnio5887076197631975402_f_nat @ ( finser2928796382700762858_f_nat @ A @ B4 ) )
      = ( sup_su8375471814421675870_f_nat @ A @ ( ffUnio5887076197631975402_f_nat @ B4 ) ) ) ).

% ffunion_insert
thf(fact_1049_ffUnion__funion__distrib,axiom,
    ! [A3: fset_f1705081619141295970_f_nat,B4: fset_f1705081619141295970_f_nat] :
      ( ( ffUnio5887076197631975402_f_nat @ ( sup_su1580556572264692878_f_nat @ A3 @ B4 ) )
      = ( sup_su8375471814421675870_f_nat @ ( ffUnio5887076197631975402_f_nat @ A3 ) @ ( ffUnio5887076197631975402_f_nat @ B4 ) ) ) ).

% ffUnion_funion_distrib
thf(fact_1050_ta__sig__pattern__automaton,axiom,
    ! [F3: fset_P6228066233360383026_f_nat,R5: fset_term_f_v] :
      ( ( tree_t4125170225404765961rm_f_f @ ( lV_to_2851878803089403142on_f_v @ F3 @ R5 ) )
      = ( sup_su8375471814421675870_f_nat @ F3 @ ( ffUnio5887076197631975402_f_nat @ ( fimage7553981102877535508_f_nat @ fSet_ffunas_term_f_v @ R5 ) ) ) ) ).

% ta_sig_pattern_automaton
thf(fact_1051_ffunion__mono,axiom,
    ! [A3: fset_f1705081619141295970_f_nat,B4: fset_f1705081619141295970_f_nat] :
      ( ( ord_le8062249421014450114_f_nat @ A3 @ B4 )
     => ( ord_le1552505484586773650_f_nat @ ( ffUnio5887076197631975402_f_nat @ A3 ) @ ( ffUnio5887076197631975402_f_nat @ B4 ) ) ) ).

% ffunion_mono
thf(fact_1052_fset_Omap__comp,axiom,
    ! [G2: term_f_v > fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > term_f_v,V3: fset_P8018961893305114765rm_f_v] :
      ( ( fimage7553981102877535508_f_nat @ G2 @ ( fimage4320022722461925965rm_f_v @ F2 @ V3 ) )
      = ( fimage84417724776064284_f_nat @ ( comp_t4668273383396186541rm_f_v @ G2 @ F2 ) @ V3 ) ) ).

% fset.map_comp
thf(fact_1053_fset_Omap__comp,axiom,
    ! [G2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,V3: fset_P8018961893305114765rm_f_v] :
      ( ( fimage5262077887337850211_f_nat @ G2 @ ( fimage84417724776064284_f_nat @ F2 @ V3 ) )
      = ( fimage84417724776064284_f_nat @ ( comp_f3317363480862642846rm_f_v @ G2 @ F2 ) @ V3 ) ) ).

% fset.map_comp
thf(fact_1054_fset_Omap__comp,axiom,
    ! [G2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v,V3: fset_P8018961893305114765rm_f_v] :
      ( ( fimage84417724776064284_f_nat @ G2 @ ( fimage7135986912551575357rm_f_v @ F2 @ V3 ) )
      = ( fimage84417724776064284_f_nat @ ( comp_P8210875767788461423rm_f_v @ G2 @ F2 ) @ V3 ) ) ).

% fset.map_comp
thf(fact_1055_fPow__iff,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe7677060439241084203_f_nat @ A3 @ ( fPow_P4099986270981333277_f_nat @ B4 ) )
      = ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ).

% fPow_iff
thf(fact_1056_fPowI,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( fmembe7677060439241084203_f_nat @ A3 @ ( fPow_P4099986270981333277_f_nat @ B4 ) ) ) ).

% fPowI
thf(fact_1057_fimage__fPow__surj,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v,B4: fset_f1705081619141295970_f_nat] :
      ( ( ( fimage84417724776064284_f_nat @ F2 @ A3 )
        = B4 )
     => ( ( fimage7523401898302340402_f_nat @ ( fimage84417724776064284_f_nat @ F2 ) @ ( fPow_P323116099099655500rm_f_v @ A3 ) )
        = ( fPow_f5601313220398954829_f_nat @ B4 ) ) ) ).

% fimage_fPow_surj
thf(fact_1058_fPowD,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe7677060439241084203_f_nat @ A3 @ ( fPow_P4099986270981333277_f_nat @ B4 ) )
     => ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ).

% fPowD
thf(fact_1059_fPow__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
     => ( ord_le8062249421014450114_f_nat @ ( fPow_P4099986270981333277_f_nat @ A3 ) @ ( fPow_P4099986270981333277_f_nat @ B4 ) ) ) ).

% fPow_mono
thf(fact_1060_fimage__fPow__mono,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v,B4: fset_f1705081619141295970_f_nat] :
      ( ( ord_le8062249421014450114_f_nat @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) @ B4 )
     => ( ord_le3354286811264545906_f_nat @ ( fimage7523401898302340402_f_nat @ ( fimage84417724776064284_f_nat @ F2 ) @ ( fPow_P323116099099655500rm_f_v @ A3 ) ) @ ( fPow_f5601313220398954829_f_nat @ B4 ) ) ) ).

% fimage_fPow_mono
thf(fact_1061_funion__fPow__fsubset,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] : ( ord_le8062249421014450114_f_nat @ ( sup_su1580556572264692878_f_nat @ ( fPow_P4099986270981333277_f_nat @ A3 ) @ ( fPow_P4099986270981333277_f_nat @ B4 ) ) @ ( fPow_P4099986270981333277_f_nat @ ( sup_su8375471814421675870_f_nat @ A3 @ B4 ) ) ) ).

% funion_fPow_fsubset
thf(fact_1062__092_060open_062_092_060F_062_A_124_092_060union_062_124_A_124_092_060Union_062_124_A_I_Iffunas__term_A_092_060circ_062_Asnd_J_A_124_096_124_AR_J_A_061_A_092_060F_062_092_060close_062,axiom,
    ( ( sup_su8375471814421675870_f_nat @ f2 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ ( comp_t4668273383396186541rm_f_v @ fSet_ffunas_term_f_v @ produc5682565293565518221rm_f_v ) @ r ) ) )
    = f2 ) ).

% \<open>\<F> |\<union>| |\<Union>| ((ffunas_term \<circ> snd) |`| R) = \<F>\<close>
thf(fact_1063__092_060open_062_092_060F_062_A_124_092_060union_062_124_A_124_092_060Union_062_124_A_I_Iffunas__term_A_092_060circ_062_Afst_J_A_124_096_124_AR_J_A_061_A_092_060F_062_092_060close_062,axiom,
    ( ( sup_su8375471814421675870_f_nat @ f2 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ ( comp_t4668273383396186541rm_f_v @ fSet_ffunas_term_f_v @ produc1199688945740118859rm_f_v ) @ r ) ) )
    = f2 ) ).

% \<open>\<F> |\<union>| |\<Union>| ((ffunas_term \<circ> fst) |`| R) = \<F>\<close>
thf(fact_1064_GTT__comp__def,axiom,
    ( gTT_GT8578867084790779380rm_f_f
    = ( ^ [G_12: produc6920895926601326823rm_f_f,G_22: produc6920895926601326823rm_f_f] : ( produc1864056441719732567rm_f_f @ ( tree_T6922284161482903311rm_f_f @ ( sup_su7157879921031797853rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ G_12 ) @ ( produc1890379913328437059rm_f_f @ G_22 ) ) ) ) @ ( tree_r5405720787295496115rm_f_f @ ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1890379913328437059rm_f_f @ G_12 ) @ ( produc1890379913328437059rm_f_f @ G_22 ) ) ) ) ) @ ( sup_su1809081536757948599term_f @ ( sup_su1809081536757948599term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_12 ) ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1890379913328437059rm_f_f @ G_22 ) ) ) @ ( gTT_De7106193943236547076rm_f_f @ ( produc1110312829679706757rm_f_f @ G_12 ) @ ( produc1890379913328437059rm_f_f @ G_22 ) ) ) ) @ ( tree_T6922284161482903311rm_f_f @ ( sup_su7157879921031797853rm_f_f @ ( tree_r5405720787295496115rm_f_f @ ( produc1890379913328437059rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1110312829679706757rm_f_f @ G_12 ) @ ( produc1110312829679706757rm_f_f @ G_22 ) ) ) ) @ ( tree_r5405720787295496115rm_f_f @ ( produc1110312829679706757rm_f_f @ ( produc1864056441719732567rm_f_f @ ( produc1110312829679706757rm_f_f @ G_12 ) @ ( produc1110312829679706757rm_f_f @ G_22 ) ) ) ) ) @ ( sup_su1809081536757948599term_f @ ( sup_su1809081536757948599term_f @ ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G_12 ) ) @ ( tree_e2214990432070132610rm_f_f @ ( produc1110312829679706757rm_f_f @ G_22 ) ) ) @ ( fSet_f5567478040060018990term_f @ ( gTT_De7106193943236547076rm_f_f @ ( produc1110312829679706757rm_f_f @ G_12 ) @ ( produc1890379913328437059rm_f_f @ G_22 ) ) ) ) ) ) ) ) ).

% GTT_comp_def
thf(fact_1065_fUn__simps_I9_J,axiom,
    ! [C2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,B4: produc8027700073345061287rm_f_v > fset_P8018961893305114765rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ C2 @ ( ffUnio2002683114454723135rm_f_v @ ( fimage4047282758338652963rm_f_v @ B4 @ A3 ) ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ C2 @ ( B4 @ X4 ) ) )
          @ A3 ) ) ) ).

% fUn_simps(9)
thf(fact_1066_fUn__simps_I9_J,axiom,
    ! [C2: product_prod_f_nat > fset_P6228066233360383026_f_nat,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage7875634609417048755_f_nat @ C2 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ A3 ) ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : ( ffUnio5887076197631975402_f_nat @ ( fimage7875634609417048755_f_nat @ C2 @ ( B4 @ X4 ) ) )
          @ A3 ) ) ) ).

% fUn_simps(9)
thf(fact_1067_fUn__simps_I10_J,axiom,
    ! [B4: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage5262077887337850211_f_nat @ B4 @ ( fimage84417724776064284_f_nat @ F2 @ A3 ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [A7: produc8027700073345061287rm_f_v] : ( B4 @ ( F2 @ A7 ) )
          @ A3 ) ) ) ).

% fUn_simps(10)
thf(fact_1068_fUn__simps_I10_J,axiom,
    ! [B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,F2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ ( fimage7135986912551575357rm_f_v @ F2 @ A3 ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [A7: produc8027700073345061287rm_f_v] : ( B4 @ ( F2 @ A7 ) )
          @ A3 ) ) ) ).

% fUn_simps(10)
thf(fact_1069_fUn__simps_I4_J,axiom,
    ! [A3: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,C2: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : ( inf_in6532419608991314884_f_nat @ ( A3 @ X4 ) @ B4 )
          @ C2 ) )
      = ( inf_in6532419608991314884_f_nat @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ A3 @ C2 ) ) @ B4 ) ) ).

% fUn_simps(4)
thf(fact_1070_fUn__simps_I5_J,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,C2: fset_P8018961893305114765rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : ( inf_in6532419608991314884_f_nat @ A3 @ ( B4 @ X4 ) )
          @ C2 ) )
      = ( inf_in6532419608991314884_f_nat @ A3 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ C2 ) ) ) ) ).

% fUn_simps(5)
thf(fact_1071_fUn__simps_I8_J,axiom,
    ! [B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_f2722810715853128435rm_f_v] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ ( ffUnio2002683114454723135rm_f_v @ A3 ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage8661250477974528514_f_nat
          @ ^ [Y5: fset_P8018961893305114765rm_f_v] : ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ Y5 ) )
          @ A3 ) ) ) ).

% fUn_simps(8)
thf(fact_1072_fUn__simps_I8_J,axiom,
    ! [B4: product_prod_f_nat > fset_P6228066233360383026_f_nat,A3: fset_f1705081619141295970_f_nat] :
      ( ( ffUnio5887076197631975402_f_nat @ ( fimage7875634609417048755_f_nat @ B4 @ ( ffUnio5887076197631975402_f_nat @ A3 ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage5262077887337850211_f_nat
          @ ^ [Y5: fset_P6228066233360383026_f_nat] : ( ffUnio5887076197631975402_f_nat @ ( fimage7875634609417048755_f_nat @ B4 @ Y5 ) )
          @ A3 ) ) ) ).

% fUn_simps(8)
thf(fact_1073_fUn__simps_I1_J,axiom,
    ! [C2: fset_P8018961893305114765rm_f_v,A: product_prod_f_nat,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( ( C2 = bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( finser8068648205224448314_f_nat @ A @ ( B4 @ X4 ) )
              @ C2 ) )
          = bot_bo3974441229639268550_f_nat ) )
      & ( ( C2 != bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( finser8068648205224448314_f_nat @ A @ ( B4 @ X4 ) )
              @ C2 ) )
          = ( finser8068648205224448314_f_nat @ A @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ C2 ) ) ) ) ) ) ).

% fUn_simps(1)
thf(fact_1074_fUn__simps_I3_J,axiom,
    ! [C2: fset_P8018961893305114765rm_f_v,A3: fset_P6228066233360383026_f_nat,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( ( C2 = bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( sup_su8375471814421675870_f_nat @ A3 @ ( B4 @ X4 ) )
              @ C2 ) )
          = bot_bo3974441229639268550_f_nat ) )
      & ( ( C2 != bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( sup_su8375471814421675870_f_nat @ A3 @ ( B4 @ X4 ) )
              @ C2 ) )
          = ( sup_su8375471814421675870_f_nat @ A3 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ C2 ) ) ) ) ) ) ).

% fUn_simps(3)
thf(fact_1075_fUn__simps_I2_J,axiom,
    ! [C2: fset_P8018961893305114765rm_f_v,A3: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( ( C2 = bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( sup_su8375471814421675870_f_nat @ ( A3 @ X4 ) @ B4 )
              @ C2 ) )
          = bot_bo3974441229639268550_f_nat ) )
      & ( ( C2 != bot_bo4614643899460950777rm_f_v )
       => ( ( ffUnio5887076197631975402_f_nat
            @ ( fimage84417724776064284_f_nat
              @ ^ [X4: produc8027700073345061287rm_f_v] : ( sup_su8375471814421675870_f_nat @ ( A3 @ X4 ) @ B4 )
              @ C2 ) )
          = ( sup_su8375471814421675870_f_nat @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% fUn_simps(2)
thf(fact_1076_fimage__constant__conv,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,C: fset_P6228066233360383026_f_nat] :
      ( ( ( A3 = bot_bo4614643899460950777rm_f_v )
       => ( ( fimage84417724776064284_f_nat
            @ ^ [X4: produc8027700073345061287rm_f_v] : C
            @ A3 )
          = bot_bo2413912301557604854_f_nat ) )
      & ( ( A3 != bot_bo4614643899460950777rm_f_v )
       => ( ( fimage84417724776064284_f_nat
            @ ^ [X4: produc8027700073345061287rm_f_v] : C
            @ A3 )
          = ( finser2928796382700762858_f_nat @ C @ bot_bo2413912301557604854_f_nat ) ) ) ) ).

% fimage_constant_conv
thf(fact_1077_fimage__constant,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,C: fset_P6228066233360383026_f_nat] :
      ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
     => ( ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : C
          @ A3 )
        = ( finser2928796382700762858_f_nat @ C @ bot_bo2413912301557604854_f_nat ) ) ) ).

% fimage_constant
thf(fact_1078_fImage__fUN,axiom,
    ! [R2: fset_P7991872372518838603_f_nat,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fSet_f7437826027993719364_f_nat @ R2 @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ A3 ) ) )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage84417724776064284_f_nat
          @ ^ [X4: produc8027700073345061287rm_f_v] : ( fSet_f7437826027993719364_f_nat @ R2 @ ( B4 @ X4 ) )
          @ A3 ) ) ) ).

% fImage_fUN
thf(fact_1079_fimage__fimage,axiom,
    ! [F2: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,G2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fimage5262077887337850211_f_nat @ F2 @ ( fimage84417724776064284_f_nat @ G2 @ A3 ) )
      = ( fimage84417724776064284_f_nat
        @ ^ [X4: produc8027700073345061287rm_f_v] : ( F2 @ ( G2 @ X4 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_1080_fimage__fimage,axiom,
    ! [F2: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,G2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fimage84417724776064284_f_nat @ F2 @ ( fimage7135986912551575357rm_f_v @ G2 @ A3 ) )
      = ( fimage84417724776064284_f_nat
        @ ^ [X4: produc8027700073345061287rm_f_v] : ( F2 @ ( G2 @ X4 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_1081_fst__diag__snd,axiom,
    ( ( comp_P449394353649552844rm_f_f @ produc1890379913328437059rm_f_f
      @ ( comp_T4692968735499750204rm_f_f
        @ ^ [X4: tree_t2044664379607036063rm_f_f] : ( produc1864056441719732567rm_f_f @ X4 @ X4 )
        @ produc1110312829679706757rm_f_f ) )
    = produc1110312829679706757rm_f_f ) ).

% fst_diag_snd
thf(fact_1082_fst__diag__snd,axiom,
    ( ( comp_P6018677580669307134rm_f_v @ produc1199688945740118859rm_f_v
      @ ( comp_t8285604959496677382rm_f_v
        @ ^ [X4: term_f_v] : ( produc2510790566102465375rm_f_v @ X4 @ X4 )
        @ produc5682565293565518221rm_f_v ) )
    = produc5682565293565518221rm_f_v ) ).

% fst_diag_snd
thf(fact_1083_snd__diag__fst,axiom,
    ( ( comp_P449394353649552844rm_f_f @ produc1110312829679706757rm_f_f
      @ ( comp_T4692968735499750204rm_f_f
        @ ^ [X4: tree_t2044664379607036063rm_f_f] : ( produc1864056441719732567rm_f_f @ X4 @ X4 )
        @ produc1890379913328437059rm_f_f ) )
    = produc1890379913328437059rm_f_f ) ).

% snd_diag_fst
thf(fact_1084_snd__diag__fst,axiom,
    ( ( comp_P6018677580669307134rm_f_v @ produc5682565293565518221rm_f_v
      @ ( comp_t8285604959496677382rm_f_v
        @ ^ [X4: term_f_v] : ( produc2510790566102465375rm_f_v @ X4 @ X4 )
        @ produc1199688945740118859rm_f_v ) )
    = produc1199688945740118859rm_f_v ) ).

% snd_diag_fst
thf(fact_1085_snd__diag__snd,axiom,
    ( ( comp_P449394353649552844rm_f_f @ produc1110312829679706757rm_f_f
      @ ( comp_T4692968735499750204rm_f_f
        @ ^ [X4: tree_t2044664379607036063rm_f_f] : ( produc1864056441719732567rm_f_f @ X4 @ X4 )
        @ produc1110312829679706757rm_f_f ) )
    = produc1110312829679706757rm_f_f ) ).

% snd_diag_snd
thf(fact_1086_snd__diag__snd,axiom,
    ( ( comp_P6018677580669307134rm_f_v @ produc5682565293565518221rm_f_v
      @ ( comp_t8285604959496677382rm_f_v
        @ ^ [X4: term_f_v] : ( produc2510790566102465375rm_f_v @ X4 @ X4 )
        @ produc5682565293565518221rm_f_v ) )
    = produc5682565293565518221rm_f_v ) ).

% snd_diag_snd
thf(fact_1087_fst__diag__fst,axiom,
    ( ( comp_P449394353649552844rm_f_f @ produc1890379913328437059rm_f_f
      @ ( comp_T4692968735499750204rm_f_f
        @ ^ [X4: tree_t2044664379607036063rm_f_f] : ( produc1864056441719732567rm_f_f @ X4 @ X4 )
        @ produc1890379913328437059rm_f_f ) )
    = produc1890379913328437059rm_f_f ) ).

% fst_diag_fst
thf(fact_1088_fst__diag__fst,axiom,
    ( ( comp_P6018677580669307134rm_f_v @ produc1199688945740118859rm_f_v
      @ ( comp_t8285604959496677382rm_f_v
        @ ^ [X4: term_f_v] : ( produc2510790566102465375rm_f_v @ X4 @ X4 )
        @ produc1199688945740118859rm_f_v ) )
    = produc1199688945740118859rm_f_v ) ).

% fst_diag_fst
thf(fact_1089_fImage__eq__fUN,axiom,
    ( fSet_f577160797525143147_f_nat
    = ( ^ [R7: fset_P7275734956710824840_f_nat,B5: fset_P8018961893305114765rm_f_v] :
          ( ffUnio5887076197631975402_f_nat
          @ ( fimage84417724776064284_f_nat
            @ ^ [Y5: produc8027700073345061287rm_f_v] : ( fSet_f577160797525143147_f_nat @ R7 @ ( finser1703841043590550767rm_f_v @ Y5 @ bot_bo4614643899460950777rm_f_v ) )
            @ B5 ) ) ) ) ).

% fImage_eq_fUN
thf(fact_1090_sup__Un__eq,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( sup_su8005028717976655084rm_f_o
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ R5 )
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ S2 ) )
      = ( ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ ( sup_su1358690925027996977term_f @ R5 @ S2 ) ) ) ) ).

% sup_Un_eq
thf(fact_1091_sup__Un__eq,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( sup_su8179383342042354114rm_f_o
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ R5 )
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ S2 ) )
      = ( ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ ( sup_su8241624573636005595term_f @ R5 @ S2 ) ) ) ) ).

% sup_Un_eq
thf(fact_1092_sup__Un__eq2,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( sup_su8963822174138643348_f_v_o
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ ( sup_su1994854688125285083rm_f_v @ R5 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_1093_sup__Un__eq2,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( sup_su3703626751955279678rm_f_o
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ ( sup_su1358690925027996977term_f @ R5 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_1094_sup__Un__eq2,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( sup_su1253082160087495618rm_f_o
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ ( sup_su8241624573636005595term_f @ R5 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_1095_term_Odisc_I1_J,axiom,
    ! [X1: v] : ( is_Var_f_v @ ( var_v_f @ X1 ) ) ).

% term.disc(1)
thf(fact_1096_term_OdiscI_I1_J,axiom,
    ! [Term: term_f_v,X1: v] :
      ( ( Term
        = ( var_v_f @ X1 ) )
     => ( is_Var_f_v @ Term ) ) ).

% term.discI(1)
thf(fact_1097_is__Var__def,axiom,
    ( is_Var_f_v
    = ( ^ [Term2: term_f_v] :
        ? [X13: v] :
          ( Term2
          = ( var_v_f @ X13 ) ) ) ) ).

% is_Var_def
thf(fact_1098_is__VarE,axiom,
    ! [T: term_f_v] :
      ( ( is_Var_f_v @ T )
     => ~ ! [X3: v] :
            ( T
           != ( var_v_f @ X3 ) ) ) ).

% is_VarE
thf(fact_1099_pred__equals__eq2,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ( ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R5 ) )
        = ( ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ S2 ) ) )
      = ( R5 = S2 ) ) ).

% pred_equals_eq2
thf(fact_1100_pred__equals__eq2,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( ( ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ R5 ) )
        = ( ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ S2 ) ) )
      = ( R5 = S2 ) ) ).

% pred_equals_eq2
thf(fact_1101_pred__equals__eq2,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( ( ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ R5 ) )
        = ( ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ S2 ) ) )
      = ( R5 = S2 ) ) ).

% pred_equals_eq2
thf(fact_1102_bot__empty__eq2,axiom,
    ( bot_bo2204096811554796588_f_v_o
    = ( ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ bot_bo4638812505780724595rm_f_v ) ) ) ).

% bot_empty_eq2
thf(fact_1103_bot__empty__eq2,axiom,
    ( bot_bo2429867386587643862rm_f_o
    = ( ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ bot_bo5323853609601926601term_f ) ) ) ).

% bot_empty_eq2
thf(fact_1104_bot__empty__eq2,axiom,
    ( bot_bo3990891926076045402rm_f_o
    = ( ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ bot_bo9223291547480392563term_f ) ) ) ).

% bot_empty_eq2
thf(fact_1105_pred__subset__eq2,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le194975665355383648_f_v_o
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ S2 ) )
      = ( ord_le4559761987009501863rm_f_v @ R5 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1106_pred__subset__eq2,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( ord_le7920209722814570250rm_f_o
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ord_le5176047027162639613term_f @ R5 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1107_pred__subset__eq2,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( ord_le8892309489402766734rm_f_o
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ord_le263819222746101927term_f @ R5 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1108_prop__restrict,axiom,
    ! [X: produc250624644672406269term_f,Z4: set_Pr6841673932529763677term_f,X5: set_Pr6841673932529763677term_f,P2: produc250624644672406269term_f > $o] :
      ( ( member4687689947140789414term_f @ X @ Z4 )
     => ( ( ord_le5176047027162639613term_f @ Z4
          @ ( collec753421845813909992term_f
            @ ^ [X4: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_1109_prop__restrict,axiom,
    ! [X: produc7245736746747425831term_f,Z4: set_Pr989862937836626183term_f,X5: set_Pr989862937836626183term_f,P2: produc7245736746747425831term_f > $o] :
      ( ( member848276444142703440term_f @ X @ Z4 )
     => ( ( ord_le263819222746101927term_f @ Z4
          @ ( collec3209569126566747026term_f
            @ ^ [X4: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_1110_Collect__restrict,axiom,
    ! [X5: set_Pr6841673932529763677term_f,P2: produc250624644672406269term_f > $o] :
      ( ord_le5176047027162639613term_f
      @ ( collec753421845813909992term_f
        @ ^ [X4: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_1111_Collect__restrict,axiom,
    ! [X5: set_Pr989862937836626183term_f,P2: produc7245736746747425831term_f > $o] :
      ( ord_le263819222746101927term_f
      @ ( collec3209569126566747026term_f
        @ ^ [X4: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_1112_subset__Collect__iff,axiom,
    ! [B4: set_Pr6841673932529763677term_f,A3: set_Pr6841673932529763677term_f,P2: produc250624644672406269term_f > $o] :
      ( ( ord_le5176047027162639613term_f @ B4 @ A3 )
     => ( ( ord_le5176047027162639613term_f @ B4
          @ ( collec753421845813909992term_f
            @ ^ [X4: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ X4 @ A3 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc250624644672406269term_f] :
              ( ( member4687689947140789414term_f @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1113_subset__Collect__iff,axiom,
    ! [B4: set_Pr989862937836626183term_f,A3: set_Pr989862937836626183term_f,P2: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ B4 @ A3 )
     => ( ( ord_le263819222746101927term_f @ B4
          @ ( collec3209569126566747026term_f
            @ ^ [X4: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X4 @ A3 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc7245736746747425831term_f] :
              ( ( member848276444142703440term_f @ X4 @ B4 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1114_subset__CollectI,axiom,
    ! [B4: set_Pr6841673932529763677term_f,A3: set_Pr6841673932529763677term_f,Q3: produc250624644672406269term_f > $o,P2: produc250624644672406269term_f > $o] :
      ( ( ord_le5176047027162639613term_f @ B4 @ A3 )
     => ( ! [X3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X3 @ B4 )
           => ( ( Q3 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le5176047027162639613term_f
          @ ( collec753421845813909992term_f
            @ ^ [X4: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ X4 @ B4 )
                & ( Q3 @ X4 ) ) )
          @ ( collec753421845813909992term_f
            @ ^ [X4: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ X4 @ A3 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1115_subset__CollectI,axiom,
    ! [B4: set_Pr989862937836626183term_f,A3: set_Pr989862937836626183term_f,Q3: produc7245736746747425831term_f > $o,P2: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ B4 @ A3 )
     => ( ! [X3: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X3 @ B4 )
           => ( ( Q3 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le263819222746101927term_f
          @ ( collec3209569126566747026term_f
            @ ^ [X4: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X4 @ B4 )
                & ( Q3 @ X4 ) ) )
          @ ( collec3209569126566747026term_f
            @ ^ [X4: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X4 @ A3 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1116_pred__subset__eq,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( ord_le7801877831724205088rm_f_o
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ R5 )
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ S2 ) )
      = ( ord_le5176047027162639613term_f @ R5 @ S2 ) ) ).

% pred_subset_eq
thf(fact_1117_pred__subset__eq,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( ord_le1842562795780935414rm_f_o
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ R5 )
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ S2 ) )
      = ( ord_le263819222746101927term_f @ R5 @ S2 ) ) ).

% pred_subset_eq
thf(fact_1118_Collect__subset,axiom,
    ! [A3: set_Pr6841673932529763677term_f,P2: produc250624644672406269term_f > $o] :
      ( ord_le5176047027162639613term_f
      @ ( collec753421845813909992term_f
        @ ^ [X4: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X4 @ A3 )
            & ( P2 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1119_Collect__subset,axiom,
    ! [A3: set_Pr989862937836626183term_f,P2: produc7245736746747425831term_f > $o] :
      ( ord_le263819222746101927term_f
      @ ( collec3209569126566747026term_f
        @ ^ [X4: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X4 @ A3 )
            & ( P2 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1120_less__eq__set__def,axiom,
    ( ord_le5176047027162639613term_f
    = ( ^ [A5: set_Pr6841673932529763677term_f,B5: set_Pr6841673932529763677term_f] :
          ( ord_le7801877831724205088rm_f_o
          @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ A5 )
          @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_1121_less__eq__set__def,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A5: set_Pr989862937836626183term_f,B5: set_Pr989862937836626183term_f] :
          ( ord_le1842562795780935414rm_f_o
          @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ A5 )
          @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_1122_gta__lang__def,axiom,
    ( tree_g2969921145280742859rm_f_f
    = ( ^ [Q4: fset_Bot_bot_term_f,A9: tree_t2044664379607036063rm_f_f] :
          ( collec3522850625759001119term_f
          @ ^ [T2: ground_gterm_f] :
              ( ( inf_in2113687546666124112term_f @ Q4 @ ( tree_g2054044545600476092rm_f_f @ A9 @ T2 ) )
             != bot_bo4668628460741225810term_f ) ) ) ) ).

% gta_lang_def
thf(fact_1123_inf__Int__eq2,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
      ( ( inf_in2796210053413908654_f_v_o
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: term_f_v,Y5: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ ( inf_in6135112614888068085rm_f_v @ R5 @ S2 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1124_inf__Int__eq2,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( inf_in2459131613052351576rm_f_o
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: bot_bot_term_f,Y5: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y5 ) @ ( inf_in5033374358511726155term_f @ R5 @ S2 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1125_inf__Int__eq2,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( inf_in6121564803225898972rm_f_o
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ R5 )
        @ ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ S2 ) )
      = ( ^ [X4: ground_gterm_f,Y5: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X4 @ Y5 ) @ ( inf_in6246359444536921845term_f @ R5 @ S2 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1126_inf__Int__eq,axiom,
    ! [R5: set_Pr6841673932529763677term_f,S2: set_Pr6841673932529763677term_f] :
      ( ( inf_in4470949146409994322rm_f_o
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ R5 )
        @ ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ S2 ) )
      = ( ^ [X4: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X4 @ ( inf_in5033374358511726155term_f @ R5 @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_1127_inf__Int__eq,axiom,
    ! [R5: set_Pr989862937836626183term_f,S2: set_Pr989862937836626183term_f] :
      ( ( inf_in1406940722187435560rm_f_o
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ R5 )
        @ ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ S2 ) )
      = ( ^ [X4: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X4 @ ( inf_in6246359444536921845term_f @ R5 @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_1128_fUN__fImage,axiom,
    ! [X5: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,I2: fset_P8018961893305114765rm_f_v,S2: fset_f] :
      ( ( fSet_fImage_f_nat @ ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ X5 @ I2 ) ) @ S2 )
      = ( ffUnion_nat
        @ ( fimage8464632880249265770et_nat
          @ ^ [I: produc8027700073345061287rm_f_v] : ( fSet_fImage_f_nat @ ( X5 @ I ) @ S2 )
          @ I2 ) ) ) ).

% fUN_fImage
thf(fact_1129_ffmember__filter,axiom,
    ! [X: produc8027700073345061287rm_f_v,P2: produc8027700073345061287rm_f_v > $o,A3: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ ( ffilte1550897425452467152rm_f_v @ P2 @ A3 ) )
      = ( ( fmembe774991991480784110rm_f_v @ X @ A3 )
        & ( P2 @ X ) ) ) ).

% ffmember_filter
thf(fact_1130_ffmember__filter,axiom,
    ! [X: produc250624644672406269term_f,P2: produc250624644672406269term_f > $o,A3: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ ( ffilte3749186692092825894term_f @ P2 @ A3 ) )
      = ( ( fmembe6950792022577883716term_f @ X @ A3 )
        & ( P2 @ X ) ) ) ).

% ffmember_filter
thf(fact_1131_ffmember__filter,axiom,
    ! [X: bot_bot_term_f,P2: bot_bot_term_f > $o,A3: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ ( ffilte1788553570390598949term_f @ P2 @ A3 ) )
      = ( ( fmembe1418995490368054919term_f @ X @ A3 )
        & ( P2 @ X ) ) ) ).

% ffmember_filter
thf(fact_1132_filter__fset,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,Xs2: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( ffilte1550897425452467152rm_f_v @ P2 @ Xs2 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( collec7769483783292507282rm_f_v @ P2 ) @ ( fset_P4617584883882644886rm_f_v @ Xs2 ) ) ) ).

% filter_fset
thf(fact_1133_eq__ffilter,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,A3: fset_P8018961893305114765rm_f_v,Q3: produc8027700073345061287rm_f_v > $o] :
      ( ( ( ffilte1550897425452467152rm_f_v @ P2 @ A3 )
        = ( ffilte1550897425452467152rm_f_v @ Q3 @ A3 ) )
      = ( ! [X4: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X4 @ A3 )
           => ( ( P2 @ X4 )
              = ( Q3 @ X4 ) ) ) ) ) ).

% eq_ffilter
thf(fact_1134_eq__ffilter,axiom,
    ! [P2: produc250624644672406269term_f > $o,A3: fset_P2882900458636368611term_f,Q3: produc250624644672406269term_f > $o] :
      ( ( ( ffilte3749186692092825894term_f @ P2 @ A3 )
        = ( ffilte3749186692092825894term_f @ Q3 @ A3 ) )
      = ( ! [X4: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X4 @ A3 )
           => ( ( P2 @ X4 )
              = ( Q3 @ X4 ) ) ) ) ) ).

% eq_ffilter
thf(fact_1135_eq__ffilter,axiom,
    ! [P2: bot_bot_term_f > $o,A3: fset_Bot_bot_term_f,Q3: bot_bot_term_f > $o] :
      ( ( ( ffilte1788553570390598949term_f @ P2 @ A3 )
        = ( ffilte1788553570390598949term_f @ Q3 @ A3 ) )
      = ( ! [X4: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X4 @ A3 )
           => ( ( P2 @ X4 )
              = ( Q3 @ X4 ) ) ) ) ) ).

% eq_ffilter
thf(fact_1136_subset__ffilter,axiom,
    ! [P2: produc8027700073345061287rm_f_v > $o,A3: fset_P8018961893305114765rm_f_v,Q3: produc8027700073345061287rm_f_v > $o] :
      ( ( ord_le4587745213494032429rm_f_v @ ( ffilte1550897425452467152rm_f_v @ P2 @ A3 ) @ ( ffilte1550897425452467152rm_f_v @ Q3 @ A3 ) )
      = ( ! [X4: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X4 @ A3 )
           => ( ( P2 @ X4 )
             => ( Q3 @ X4 ) ) ) ) ) ).

% subset_ffilter
thf(fact_1137_subset__ffilter,axiom,
    ! [P2: produc250624644672406269term_f > $o,A3: fset_P2882900458636368611term_f,Q3: produc250624644672406269term_f > $o] :
      ( ( ord_le7092741666294821507term_f @ ( ffilte3749186692092825894term_f @ P2 @ A3 ) @ ( ffilte3749186692092825894term_f @ Q3 @ A3 ) )
      = ( ! [X4: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X4 @ A3 )
           => ( ( P2 @ X4 )
             => ( Q3 @ X4 ) ) ) ) ) ).

% subset_ffilter
thf(fact_1138_subset__ffilter,axiom,
    ! [P2: bot_bot_term_f > $o,A3: fset_Bot_bot_term_f,Q3: bot_bot_term_f > $o] :
      ( ( ord_le7572167314378612510term_f @ ( ffilte1788553570390598949term_f @ P2 @ A3 ) @ ( ffilte1788553570390598949term_f @ Q3 @ A3 ) )
      = ( ! [X4: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X4 @ A3 )
           => ( ( P2 @ X4 )
             => ( Q3 @ X4 ) ) ) ) ) ).

% subset_ffilter
thf(fact_1139_subset__ffilter,axiom,
    ! [P2: product_prod_f_nat > $o,A3: fset_P6228066233360383026_f_nat,Q3: product_prod_f_nat > $o] :
      ( ( ord_le1552505484586773650_f_nat @ ( ffilte1639387882200600473_f_nat @ P2 @ A3 ) @ ( ffilte1639387882200600473_f_nat @ Q3 @ A3 ) )
      = ( ! [X4: product_prod_f_nat] :
            ( ( fmembe6795308708001954811_f_nat @ X4 @ A3 )
           => ( ( P2 @ X4 )
             => ( Q3 @ X4 ) ) ) ) ) ).

% subset_ffilter
thf(fact_1140_agtt__lang__sig,axiom,
    ! [G5: produc6920895926601326823rm_f_f,F3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( fset_P3576968334923099475_f_nat @ ( sup_su8375471814421675870_f_nat @ ( tree_t4125170225404765961rm_f_f @ ( produc1890379913328437059rm_f_f @ G5 ) ) @ ( tree_t4125170225404765961rm_f_f @ ( produc1110312829679706757rm_f_f @ G5 ) ) ) ) @ F3 )
     => ( ord_le263819222746101927term_f @ ( agtt_l7696934014624939844rm_f_f @ G5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F3 )
          @ ^ [Uu3: ground_gterm_f] : ( ground_T_G_f @ F3 ) ) ) ) ).

% agtt_lang_sig
thf(fact_1141_pair__at__agtt__conv,axiom,
    ! [Q3: fset_P2882900458636368611term_f,G: produc6920895926601326823rm_f_f] :
      ( ( ord_le7092741666294821507term_f @ Q3
        @ ( fSet_f7914580955412209254term_f @ ( tree_Q1727860138906775108rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) )
          @ ^ [Uu3: bot_bot_term_f] : ( tree_Q1727860138906775108rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) ) )
     => ( ( ( inf_in2113687546666124112term_f @ ( tree_Q1727860138906775108rm_f_f @ ( produc1890379913328437059rm_f_f @ G ) ) @ ( tree_Q1727860138906775108rm_f_f @ ( produc1110312829679706757rm_f_f @ G ) ) )
          = bot_bo4668628460741225810term_f )
       => ( ( pair_p5686396426817080910rm_f_f @ G @ Q3 )
          = ( agtt_l7696934014624939844rm_f_f @ ( pair_p7034665317573588219rm_f_f @ G @ Q3 ) ) ) ) ) ).

% pair_at_agtt_conv
thf(fact_1142_mem__Sigma__iff,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( produc8672414958890659976term_f @ A3 @ B4 ) )
      = ( ( member4687689947140789414term_f @ A @ A3 )
        & ( member4687689947140789414term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1143_mem__Sigma__iff,axiom,
    ! [A: produc250624644672406269term_f,B: produc7245736746747425831term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr989862937836626183term_f] :
      ( ( member6995771964778216122term_f @ ( produc6538764457636710913term_f @ A @ B ) @ ( produc666742139920228914term_f @ A3 @ B4 ) )
      = ( ( member4687689947140789414term_f @ A @ A3 )
        & ( member848276444142703440term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1144_mem__Sigma__iff,axiom,
    ! [A: produc7245736746747425831term_f,B: produc250624644672406269term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr6841673932529763677term_f] :
      ( ( member1040703446680842726term_f @ ( produc7271451049524994605term_f @ A @ B ) @ ( produc1399428731808512606term_f @ A3 @ B4 ) )
      = ( ( member848276444142703440term_f @ A @ A3 )
        & ( member4687689947140789414term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1145_mem__Sigma__iff,axiom,
    ! [A: produc7245736746747425831term_f,B: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ B ) @ ( produc4509621232857779464term_f @ A3 @ B4 ) )
      = ( ( member848276444142703440term_f @ A @ A3 )
        & ( member848276444142703440term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1146_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_1147_mem__Sigma__iff,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( produc2005054426516035430term_f @ A3 @ B4 ) )
      = ( ( member2723211850833494437term_f @ A @ A3 )
        & ( member2723211850833494437term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1148_mem__Sigma__iff,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( produc6043514909933191368term_f @ A3 @ B4 ) )
      = ( ( member5261315044688711901term_f @ A @ A3 )
        & ( member5261315044688711901term_f @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1149_SigmaI,axiom,
    ! [A: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f,B: produc250624644672406269term_f,B4: produc250624644672406269term_f > set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ A @ A3 )
     => ( ( member4687689947140789414term_f @ B @ ( B4 @ A ) )
       => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( produc8672414958890659976term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1150_SigmaI,axiom,
    ! [A: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f,B: produc7245736746747425831term_f,B4: produc250624644672406269term_f > set_Pr989862937836626183term_f] :
      ( ( member4687689947140789414term_f @ A @ A3 )
     => ( ( member848276444142703440term_f @ B @ ( B4 @ A ) )
       => ( member6995771964778216122term_f @ ( produc6538764457636710913term_f @ A @ B ) @ ( produc666742139920228914term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1151_SigmaI,axiom,
    ! [A: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f,B: produc250624644672406269term_f,B4: produc7245736746747425831term_f > set_Pr6841673932529763677term_f] :
      ( ( member848276444142703440term_f @ A @ A3 )
     => ( ( member4687689947140789414term_f @ B @ ( B4 @ A ) )
       => ( member1040703446680842726term_f @ ( produc7271451049524994605term_f @ A @ B ) @ ( produc1399428731808512606term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1152_SigmaI,axiom,
    ! [A: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f,B: produc7245736746747425831term_f,B4: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ A @ A3 )
     => ( ( member848276444142703440term_f @ B @ ( B4 @ A ) )
       => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ B ) @ ( produc4509621232857779464term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1153_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_1154_SigmaI,axiom,
    ! [A: bot_bot_term_f,A3: set_Bot_bot_term_f,B: bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member2723211850833494437term_f @ A @ A3 )
     => ( ( member2723211850833494437term_f @ B @ ( B4 @ A ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( produc2005054426516035430term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1155_SigmaI,axiom,
    ! [A: ground_gterm_f,A3: set_Ground_gterm_f,B: ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ A @ A3 )
     => ( ( member5261315044688711901term_f @ B @ ( B4 @ A ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( produc6043514909933191368term_f @ A3 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_1156_fmem__Sigma__iff,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) )
      = ( ( fmembe1418995490368054919term_f @ A @ A3 )
        & ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1157_fmem__Sigma__iff,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: fset_Ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) )
      = ( ( fmembe8264398993653236347term_f @ A @ A3 )
        & ( fmembe8264398993653236347term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1158_fmem__Sigma__iff,axiom,
    ! [A: term_f_v,B: term_f_v,A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
      = ( ( fmember_term_f_v @ A @ A3 )
        & ( fmember_term_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1159_fmem__Sigma__iff,axiom,
    ! [A: produc250624644672406269term_f,B: bot_bot_term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_Bot_bot_term_f] :
      ( ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ A @ B ) @ ( fSet_f2646000941442319427term_f @ A3 @ B4 ) )
      = ( ( fmembe6950792022577883716term_f @ A @ A3 )
        & ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1160_fmem__Sigma__iff,axiom,
    ! [A: bot_bot_term_f,B: produc250624644672406269term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ A @ B ) @ ( fSet_f2583267287733066021term_f @ A3 @ B4 ) )
      = ( ( fmembe1418995490368054919term_f @ A @ A3 )
        & ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1161_fmem__Sigma__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: bot_bot_term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_Bot_bot_term_f] :
      ( ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ A @ B ) @ ( fSet_f4943302497394398701term_f @ A3 @ B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
        & ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1162_fmem__Sigma__iff,axiom,
    ! [A: bot_bot_term_f,B: produc8027700073345061287rm_f_v,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ A @ B ) @ ( fSet_f8638597300464633807rm_f_v @ A3 @ B4 ) )
      = ( ( fmembe1418995490368054919term_f @ A @ A3 )
        & ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1163_fmem__Sigma__iff,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( fSet_f4469069203941308296term_f @ A3 @ B4 ) )
      = ( ( fmembe6950792022577883716term_f @ A @ A3 )
        & ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1164_fmem__Sigma__iff,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc250624644672406269term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P2882900458636368611term_f] :
      ( ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ A @ B ) @ ( fSet_f5744403325833042526term_f @ A3 @ B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
        & ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1165_fmem__Sigma__iff,axiom,
    ! [A: produc250624644672406269term_f,B: produc8027700073345061287rm_f_v,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ A @ B ) @ ( fSet_f5536414834681360946rm_f_v @ A3 @ B4 ) )
      = ( ( fmembe6950792022577883716term_f @ A @ A3 )
        & ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_1166_fSigmaI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1167_fSigmaI,axiom,
    ! [A: ground_gterm_f,A3: fset_Ground_gterm_f,B: ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe8264398993653236347term_f @ A @ A3 )
     => ( ( fmembe8264398993653236347term_f @ B @ ( B4 @ A ) )
       => ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1168_fSigmaI,axiom,
    ! [A: term_f_v,A3: fset_term_f_v,B: term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmember_term_f_v @ A @ A3 )
     => ( ( fmember_term_f_v @ B @ ( B4 @ A ) )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1169_fSigmaI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: bot_bot_term_f,B4: produc250624644672406269term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) )
       => ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ A @ B ) @ ( fSet_f2646000941442319427term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1170_fSigmaI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc250624644672406269term_f,B4: bot_bot_term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) )
       => ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ A @ B ) @ ( fSet_f2583267287733066021term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1171_fSigmaI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: bot_bot_term_f,B4: produc8027700073345061287rm_f_v > fset_Bot_bot_term_f] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) )
       => ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ A @ B ) @ ( fSet_f4943302497394398701term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1172_fSigmaI,axiom,
    ! [A: bot_bot_term_f,A3: fset_Bot_bot_term_f,B: produc8027700073345061287rm_f_v,B4: bot_bot_term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe1418995490368054919term_f @ A @ A3 )
     => ( ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) )
       => ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ A @ B ) @ ( fSet_f8638597300464633807rm_f_v @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1173_fSigmaI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc250624644672406269term_f,B4: produc250624644672406269term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) )
       => ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( fSet_f4469069203941308296term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1174_fSigmaI,axiom,
    ! [A: produc8027700073345061287rm_f_v,A3: fset_P8018961893305114765rm_f_v,B: produc250624644672406269term_f,B4: produc8027700073345061287rm_f_v > fset_P2882900458636368611term_f] :
      ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
     => ( ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) )
       => ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ A @ B ) @ ( fSet_f5744403325833042526term_f @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1175_fSigmaI,axiom,
    ! [A: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B: produc8027700073345061287rm_f_v,B4: produc250624644672406269term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe6950792022577883716term_f @ A @ A3 )
     => ( ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) )
       => ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ A @ B ) @ ( fSet_f5536414834681360946rm_f_v @ A3 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_1176_ffst__image__times,axiom,
    ! [B4: fset_T2564841199962184965rm_f_f,A3: fset_T2564841199962184965rm_f_f] :
      ( ( ( B4 = bot_bo1434247796688174449rm_f_f )
       => ( ( fimage7671923670337682677rm_f_f @ produc1890379913328437059rm_f_f
            @ ( fSet_f5320715149535191048rm_f_f @ A3
              @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
          = bot_bo1434247796688174449rm_f_f ) )
      & ( ( B4 != bot_bo1434247796688174449rm_f_f )
       => ( ( fimage7671923670337682677rm_f_f @ produc1890379913328437059rm_f_f
            @ ( fSet_f5320715149535191048rm_f_f @ A3
              @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
          = A3 ) ) ) ).

% ffst_image_times
thf(fact_1177_ffst__image__times,axiom,
    ! [B4: fset_term_f_v,A3: fset_term_f_v] :
      ( ( ( B4 = bot_bo9049458698243030439rm_f_v )
       => ( ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v
            @ ( fSet_f5986999499850583952rm_f_v @ A3
              @ ^ [Uu3: term_f_v] : B4 ) )
          = bot_bo9049458698243030439rm_f_v ) )
      & ( ( B4 != bot_bo9049458698243030439rm_f_v )
       => ( ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v
            @ ( fSet_f5986999499850583952rm_f_v @ A3
              @ ^ [Uu3: term_f_v] : B4 ) )
          = A3 ) ) ) ).

% ffst_image_times
thf(fact_1178_fsnd__image__times,axiom,
    ! [A3: fset_T2564841199962184965rm_f_f,B4: fset_T2564841199962184965rm_f_f] :
      ( ( ( A3 = bot_bo1434247796688174449rm_f_f )
       => ( ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f
            @ ( fSet_f5320715149535191048rm_f_f @ A3
              @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
          = bot_bo1434247796688174449rm_f_f ) )
      & ( ( A3 != bot_bo1434247796688174449rm_f_f )
       => ( ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f
            @ ( fSet_f5320715149535191048rm_f_f @ A3
              @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
          = B4 ) ) ) ).

% fsnd_image_times
thf(fact_1179_fsnd__image__times,axiom,
    ! [A3: fset_term_f_v,B4: fset_term_f_v] :
      ( ( ( A3 = bot_bo9049458698243030439rm_f_v )
       => ( ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v
            @ ( fSet_f5986999499850583952rm_f_v @ A3
              @ ^ [Uu3: term_f_v] : B4 ) )
          = bot_bo9049458698243030439rm_f_v ) )
      & ( ( A3 != bot_bo9049458698243030439rm_f_v )
       => ( ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v
            @ ( fSet_f5986999499850583952rm_f_v @ A3
              @ ^ [Uu3: term_f_v] : B4 ) )
          = B4 ) ) ) ).

% fsnd_image_times
thf(fact_1180_fTimes__Un__distrib1,axiom,
    ! [A3: fset_f,B4: fset_f,C2: fset_nat] :
      ( ( fSet_fSigma_f_nat @ ( sup_sup_fset_f @ A3 @ B4 )
        @ ^ [Uu3: f] : C2 )
      = ( sup_su8375471814421675870_f_nat
        @ ( fSet_fSigma_f_nat @ A3
          @ ^ [Uu3: f] : C2 )
        @ ( fSet_fSigma_f_nat @ B4
          @ ^ [Uu3: f] : C2 ) ) ) ).

% fTimes_Un_distrib1
thf(fact_1181_fSigma__Un__distrib2,axiom,
    ! [I2: fset_f,A3: f > fset_nat,B4: f > fset_nat] :
      ( ( fSet_fSigma_f_nat @ I2
        @ ^ [I: f] : ( sup_sup_fset_nat @ ( A3 @ I ) @ ( B4 @ I ) ) )
      = ( sup_su8375471814421675870_f_nat @ ( fSet_fSigma_f_nat @ I2 @ A3 ) @ ( fSet_fSigma_f_nat @ I2 @ B4 ) ) ) ).

% fSigma_Un_distrib2
thf(fact_1182_fSigma__Un__distrib1,axiom,
    ! [I2: fset_f,J2: fset_f,C2: f > fset_nat] :
      ( ( fSet_fSigma_f_nat @ ( sup_sup_fset_f @ I2 @ J2 ) @ C2 )
      = ( sup_su8375471814421675870_f_nat @ ( fSet_fSigma_f_nat @ I2 @ C2 ) @ ( fSet_fSigma_f_nat @ J2 @ C2 ) ) ) ).

% fSigma_Un_distrib1
thf(fact_1183_fSigma__Union,axiom,
    ! [X5: fset_fset_f,B4: f > fset_nat] :
      ( ( fSet_fSigma_f_nat @ ( ffUnion_f @ X5 ) @ B4 )
      = ( ffUnio5887076197631975402_f_nat
        @ ( fimage726068271183321822_f_nat
          @ ^ [A5: fset_f] : ( fSet_fSigma_f_nat @ A5 @ B4 )
          @ X5 ) ) ) ).

% fSigma_Union
thf(fact_1184_fSigmaE,axiom,
    ! [C: produc250624644672406269term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
           => ! [Y3: bot_bot_term_f] :
                ( ( fmembe1418995490368054919term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2865564700782054965term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1185_fSigmaE,axiom,
    ! [C: produc7245736746747425831term_f,A3: fset_Ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ C @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) )
     => ~ ! [X3: ground_gterm_f] :
            ( ( fmembe8264398993653236347term_f @ X3 @ A3 )
           => ! [Y3: ground_gterm_f] :
                ( ( fmembe8264398993653236347term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc3560254623552331287term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1186_fSigmaE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
     => ~ ! [X3: term_f_v] :
            ( ( fmember_term_f_v @ X3 @ A3 )
           => ! [Y3: term_f_v] :
                ( ( fmember_term_f_v @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1187_fSigmaE,axiom,
    ! [C: produc3989578180274711866term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_Bot_bot_term_f] :
      ( ( fmembe2966320103993061043term_f @ C @ ( fSet_f2646000941442319427term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
           => ! [Y3: bot_bot_term_f] :
                ( ( fmembe1418995490368054919term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7216648687831246644term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1188_fSigmaE,axiom,
    ! [C: produc7402450736791287140term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe6379192660509636317term_f @ C @ ( fSet_f2583267287733066021term_f @ A3 @ B4 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
           => ! [Y3: produc250624644672406269term_f] :
                ( ( fmembe6950792022577883716term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7153915034121993238term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1189_fSigmaE,axiom,
    ! [C: produc6737520623449210596term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_Bot_bot_term_f] :
      ( ( fmembe8032078482862777437term_f @ C @ ( fSet_f4943302497394398701term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
           => ! [Y3: bot_bot_term_f] :
                ( ( fmembe1418995490368054919term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc294555725933214942term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1190_fSigmaE,axiom,
    ! [C: produc1667017993794354958rm_f_v,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe2961575853207921799rm_f_v @ C @ ( fSet_f8638597300464633807rm_f_v @ A3 @ B4 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
           => ! [Y3: produc8027700073345061287rm_f_v] :
                ( ( fmembe774991991480784110rm_f_v @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc3989850529003450048rm_f_v @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1191_fSigmaE,axiom,
    ! [C: produc6853483567480260455term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ C @ ( fSet_f4469069203941308296term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
           => ! [Y3: produc250624644672406269term_f] :
                ( ( fmembe6950792022577883716term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7121717381804241367term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1192_fSigmaE,axiom,
    ! [C: produc3700083184670943805term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P2882900458636368611term_f] :
      ( ( fmembe1382477799658998788term_f @ C @ ( fSet_f5744403325833042526term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
           => ! [Y3: produc250624644672406269term_f] :
                ( ( fmembe6950792022577883716term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2498570894422446253term_f @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1193_fSigmaE,axiom,
    ! [C: produc474063087423118609rm_f_v,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe7379829739265949400rm_f_v @ C @ ( fSet_f5536414834681360946rm_f_v @ A3 @ B4 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
           => ! [Y3: produc8027700073345061287rm_f_v] :
                ( ( fmembe774991991480784110rm_f_v @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2290582403270764673rm_f_v @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_1194_fSigmaD1,axiom,
    ! [A: term_f_v,B: term_f_v,A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
     => ( fmember_term_f_v @ A @ A3 ) ) ).

% fSigmaD1
thf(fact_1195_fSigmaD1,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) )
     => ( fmembe1418995490368054919term_f @ A @ A3 ) ) ).

% fSigmaD1
thf(fact_1196_fSigmaD1,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: fset_Ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) )
     => ( fmembe8264398993653236347term_f @ A @ A3 ) ) ).

% fSigmaD1
thf(fact_1197_fSigmaD2,axiom,
    ! [A: term_f_v,B: term_f_v,A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
     => ( fmember_term_f_v @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_1198_fSigmaD2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) )
     => ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_1199_fSigmaD2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: fset_Ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) )
     => ( fmembe8264398993653236347term_f @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_1200_fSigmaE2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f7914580955412209254term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe1418995490368054919term_f @ A @ A3 )
         => ~ ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1201_fSigmaE2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: fset_Ground_gterm_f,B4: ground_gterm_f > fset_Ground_gterm_f] :
      ( ( fmembe2803290242953082350term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( fSet_f6236091190789097416term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe8264398993653236347term_f @ A @ A3 )
         => ~ ( fmembe8264398993653236347term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1202_fSigmaE2,axiom,
    ! [A: term_f_v,B: term_f_v,A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
     => ~ ( ( fmember_term_f_v @ A @ A3 )
         => ~ ( fmember_term_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1203_fSigmaE2,axiom,
    ! [A: produc250624644672406269term_f,B: bot_bot_term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_Bot_bot_term_f] :
      ( ( fmembe2966320103993061043term_f @ ( produc7216648687831246644term_f @ A @ B ) @ ( fSet_f2646000941442319427term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe6950792022577883716term_f @ A @ A3 )
         => ~ ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1204_fSigmaE2,axiom,
    ! [A: bot_bot_term_f,B: produc250624644672406269term_f,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe6379192660509636317term_f @ ( produc7153915034121993238term_f @ A @ B ) @ ( fSet_f2583267287733066021term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe1418995490368054919term_f @ A @ A3 )
         => ~ ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1205_fSigmaE2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: bot_bot_term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_Bot_bot_term_f] :
      ( ( fmembe8032078482862777437term_f @ ( produc294555725933214942term_f @ A @ B ) @ ( fSet_f4943302497394398701term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
         => ~ ( fmembe1418995490368054919term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1206_fSigmaE2,axiom,
    ! [A: bot_bot_term_f,B: produc8027700073345061287rm_f_v,A3: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe2961575853207921799rm_f_v @ ( produc3989850529003450048rm_f_v @ A @ B ) @ ( fSet_f8638597300464633807rm_f_v @ A3 @ B4 ) )
     => ~ ( ( fmembe1418995490368054919term_f @ A @ A3 )
         => ~ ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1207_fSigmaE2,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( fSet_f4469069203941308296term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe6950792022577883716term_f @ A @ A3 )
         => ~ ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1208_fSigmaE2,axiom,
    ! [A: produc8027700073345061287rm_f_v,B: produc250624644672406269term_f,A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P2882900458636368611term_f] :
      ( ( fmembe1382477799658998788term_f @ ( produc2498570894422446253term_f @ A @ B ) @ ( fSet_f5744403325833042526term_f @ A3 @ B4 ) )
     => ~ ( ( fmembe774991991480784110rm_f_v @ A @ A3 )
         => ~ ( fmembe6950792022577883716term_f @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1209_fSigmaE2,axiom,
    ! [A: produc250624644672406269term_f,B: produc8027700073345061287rm_f_v,A3: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P8018961893305114765rm_f_v] :
      ( ( fmembe7379829739265949400rm_f_v @ ( produc2290582403270764673rm_f_v @ A @ B ) @ ( fSet_f5536414834681360946rm_f_v @ A3 @ B4 ) )
     => ~ ( ( fmembe6950792022577883716term_f @ A @ A3 )
         => ~ ( fmembe774991991480784110rm_f_v @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_1210_fSigma__mono,axiom,
    ! [A3: fset_f,C2: fset_f,B4: f > fset_nat,D: f > fset_nat] :
      ( ( ord_less_eq_fset_f @ A3 @ C2 )
     => ( ! [X3: f] :
            ( ( fmember_f @ X3 @ A3 )
           => ( ord_less_eq_fset_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le1552505484586773650_f_nat @ ( fSet_fSigma_f_nat @ A3 @ B4 ) @ ( fSet_fSigma_f_nat @ C2 @ D ) ) ) ) ).

% fSigma_mono
thf(fact_1211_fSigma__mono,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,C2: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,D: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( ord_le4587745213494032429rm_f_v @ A3 @ C2 )
     => ( ! [X3: produc8027700073345061287rm_f_v] :
            ( ( fmembe774991991480784110rm_f_v @ X3 @ A3 )
           => ( ord_le1552505484586773650_f_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le6966081965213977576_f_nat @ ( fSet_f5147502566550621281_f_nat @ A3 @ B4 ) @ ( fSet_f5147502566550621281_f_nat @ C2 @ D ) ) ) ) ).

% fSigma_mono
thf(fact_1212_fSigma__mono,axiom,
    ! [A3: fset_P2882900458636368611term_f,C2: fset_P2882900458636368611term_f,B4: produc250624644672406269term_f > fset_P6228066233360383026_f_nat,D: produc250624644672406269term_f > fset_P6228066233360383026_f_nat] :
      ( ( ord_le7092741666294821507term_f @ A3 @ C2 )
     => ( ! [X3: produc250624644672406269term_f] :
            ( ( fmembe6950792022577883716term_f @ X3 @ A3 )
           => ( ord_le1552505484586773650_f_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le9058847183658067006_f_nat @ ( fSet_f8854351057391813559_f_nat @ A3 @ B4 ) @ ( fSet_f8854351057391813559_f_nat @ C2 @ D ) ) ) ) ).

% fSigma_mono
thf(fact_1213_fSigma__mono,axiom,
    ! [A3: fset_Bot_bot_term_f,C2: fset_Bot_bot_term_f,B4: bot_bot_term_f > fset_P6228066233360383026_f_nat,D: bot_bot_term_f > fset_P6228066233360383026_f_nat] :
      ( ( ord_le7572167314378612510term_f @ A3 @ C2 )
     => ( ! [X3: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X3 @ A3 )
           => ( ord_le1552505484586773650_f_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le899626389396378871_f_nat @ ( fSet_f2794117901059782234_f_nat @ A3 @ B4 ) @ ( fSet_f2794117901059782234_f_nat @ C2 @ D ) ) ) ) ).

% fSigma_mono
thf(fact_1214_fSigma__mono,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B4: product_prod_f_nat > fset_P6228066233360383026_f_nat,D: product_prod_f_nat > fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A3 @ C2 )
     => ( ! [X3: product_prod_f_nat] :
            ( ( fmembe6795308708001954811_f_nat @ X3 @ A3 )
           => ( ord_le1552505484586773650_f_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le6425377310842345707_f_nat @ ( fSet_f1026950353325567950_f_nat @ A3 @ B4 ) @ ( fSet_f1026950353325567950_f_nat @ C2 @ D ) ) ) ) ).

% fSigma_mono
thf(fact_1215_SigmaE2,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( produc8672414958890659976term_f @ A3 @ B4 ) )
     => ~ ( ( member4687689947140789414term_f @ A @ A3 )
         => ~ ( member4687689947140789414term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1216_SigmaE2,axiom,
    ! [A: produc250624644672406269term_f,B: produc7245736746747425831term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr989862937836626183term_f] :
      ( ( member6995771964778216122term_f @ ( produc6538764457636710913term_f @ A @ B ) @ ( produc666742139920228914term_f @ A3 @ B4 ) )
     => ~ ( ( member4687689947140789414term_f @ A @ A3 )
         => ~ ( member848276444142703440term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1217_SigmaE2,axiom,
    ! [A: produc7245736746747425831term_f,B: produc250624644672406269term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr6841673932529763677term_f] :
      ( ( member1040703446680842726term_f @ ( produc7271451049524994605term_f @ A @ B ) @ ( produc1399428731808512606term_f @ A3 @ B4 ) )
     => ~ ( ( member848276444142703440term_f @ A @ A3 )
         => ~ ( member4687689947140789414term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1218_SigmaE2,axiom,
    ! [A: produc7245736746747425831term_f,B: produc7245736746747425831term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ B ) @ ( produc4509621232857779464term_f @ A3 @ B4 ) )
     => ~ ( ( member848276444142703440term_f @ A @ A3 )
         => ~ ( member848276444142703440term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1219_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_1220_SigmaE2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( produc2005054426516035430term_f @ A3 @ B4 ) )
     => ~ ( ( member2723211850833494437term_f @ A @ A3 )
         => ~ ( member2723211850833494437term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1221_SigmaE2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( produc6043514909933191368term_f @ A3 @ B4 ) )
     => ~ ( ( member5261315044688711901term_f @ A @ A3 )
         => ~ ( member5261315044688711901term_f @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1222_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_1223_SigmaD2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( produc2005054426516035430term_f @ A3 @ B4 ) )
     => ( member2723211850833494437term_f @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_1224_SigmaD2,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( produc6043514909933191368term_f @ A3 @ B4 ) )
     => ( member5261315044688711901term_f @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_1225_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_1226_SigmaD1,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A3: set_Bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( produc2005054426516035430term_f @ A3 @ B4 ) )
     => ( member2723211850833494437term_f @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1227_SigmaD1,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,A3: set_Ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( produc6043514909933191368term_f @ A3 @ B4 ) )
     => ( member5261315044688711901term_f @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1228_SigmaE,axiom,
    ! [C: produc6853483567480260455term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ C @ ( produc8672414958890659976term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X3 @ A3 )
           => ! [Y3: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7121717381804241367term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1229_SigmaE,axiom,
    ! [C: produc4208813338463212049term_f,A3: set_Pr6841673932529763677term_f,B4: produc250624644672406269term_f > set_Pr989862937836626183term_f] :
      ( ( member6995771964778216122term_f @ C @ ( produc666742139920228914term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X3 @ A3 )
           => ! [Y3: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc6538764457636710913term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1230_SigmaE,axiom,
    ! [C: produc7477116857220614461term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr6841673932529763677term_f] :
      ( ( member1040703446680842726term_f @ C @ ( produc1399428731808512606term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X3 @ A3 )
           => ! [Y3: produc250624644672406269term_f] :
                ( ( member4687689947140789414term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7271451049524994605term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1231_SigmaE,axiom,
    ! [C: produc8979644124618241255term_f,A3: set_Pr989862937836626183term_f,B4: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ C @ ( produc4509621232857779464term_f @ A3 @ B4 ) )
     => ~ ! [X3: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X3 @ A3 )
           => ! [Y3: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc7348823797596265303term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1232_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 ) )
     => ~ ! [X3: term_f_v] :
            ( ( member_term_f_v @ X3 @ A3 )
           => ! [Y3: term_f_v] :
                ( ( member_term_f_v @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1233_SigmaE,axiom,
    ! [C: produc250624644672406269term_f,A3: set_Bot_bot_term_f,B4: bot_bot_term_f > set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ C @ ( produc2005054426516035430term_f @ A3 @ B4 ) )
     => ~ ! [X3: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ X3 @ A3 )
           => ! [Y3: bot_bot_term_f] :
                ( ( member2723211850833494437term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc2865564700782054965term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1234_SigmaE,axiom,
    ! [C: produc7245736746747425831term_f,A3: set_Ground_gterm_f,B4: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ C @ ( produc6043514909933191368term_f @ A3 @ B4 ) )
     => ~ ! [X3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X3 @ A3 )
           => ! [Y3: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ Y3 @ ( B4 @ X3 ) )
               => ( C
                 != ( produc3560254623552331287term_f @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1235_fSigma__repeq,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P8224949580060463702rm_f_v
        @ ( fSet_f1217649319540403464rm_f_v @ A3
          @ ^ [Uu3: produc8027700073345061287rm_f_v] : B4 ) )
      = ( produc7856489909025154568rm_f_v @ ( fset_P4617584883882644886rm_f_v @ A3 )
        @ ^ [Uu3: produc8027700073345061287rm_f_v] : ( fset_P4617584883882644886rm_f_v @ B4 ) ) ) ).

% fSigma_repeq
thf(fact_1236_fSigma__repeq,axiom,
    ! [A3: fset_term_f_v,B4: fset_term_f_v] :
      ( ( fset_P4617584883882644886rm_f_v
        @ ( fSet_f5986999499850583952rm_f_v @ A3
          @ ^ [Uu3: term_f_v] : B4 ) )
      = ( produc6667182599969081488rm_f_v @ ( fset_term_f_v2 @ A3 )
        @ ^ [Uu3: term_f_v] : ( fset_term_f_v2 @ B4 ) ) ) ).

% fSigma_repeq
thf(fact_1237_mem__Times__iff,axiom,
    ! [X: produc6853483567480260455term_f,A3: set_Pr6841673932529763677term_f,B4: set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ X
        @ ( produc8672414958890659976term_f @ A3
          @ ^ [Uu3: produc250624644672406269term_f] : B4 ) )
      = ( ( member4687689947140789414term_f @ ( produc529750796343735235term_f @ X ) @ A3 )
        & ( member4687689947140789414term_f @ ( produc6080605871244463365term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1238_mem__Times__iff,axiom,
    ! [X: produc4208813338463212049term_f,A3: set_Pr6841673932529763677term_f,B4: set_Pr989862937836626183term_f] :
      ( ( member6995771964778216122term_f @ X
        @ ( produc666742139920228914term_f @ A3
          @ ^ [Uu3: produc250624644672406269term_f] : B4 ) )
      = ( ( member4687689947140789414term_f @ ( produc3748741038034202093term_f @ X ) @ A3 )
        & ( member848276444142703440term_f @ ( produc3831370311506299439term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1239_mem__Times__iff,axiom,
    ! [X: produc7477116857220614461term_f,A3: set_Pr989862937836626183term_f,B4: set_Pr6841673932529763677term_f] :
      ( ( member1040703446680842726term_f @ X
        @ ( produc1399428731808512606term_f @ A3
          @ ^ [Uu3: produc7245736746747425831term_f] : B4 ) )
      = ( ( member848276444142703440term_f @ ( produc4481427629922485785term_f @ X ) @ A3 )
        & ( member4687689947140789414term_f @ ( produc4564056903394583131term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1240_mem__Times__iff,axiom,
    ! [X: produc8979644124618241255term_f,A3: set_Pr989862937836626183term_f,B4: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ X
        @ ( produc4509621232857779464term_f @ A3
          @ ^ [Uu3: produc7245736746747425831term_f] : B4 ) )
      = ( ( member848276444142703440term_f @ ( produc4728556686632001859term_f @ X ) @ A3 )
        & ( member848276444142703440term_f @ ( produc5507147304211513477term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1241_mem__Times__iff,axiom,
    ! [X: produc250624644672406269term_f,A3: set_Bot_bot_term_f,B4: set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ X
        @ ( produc2005054426516035430term_f @ A3
          @ ^ [Uu3: bot_bot_term_f] : B4 ) )
      = ( ( member2723211850833494437term_f @ ( produc5789359262401207841term_f @ X ) @ A3 )
        & ( member2723211850833494437term_f @ ( produc656023363694804067term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1242_mem__Times__iff,axiom,
    ! [X: produc7245736746747425831term_f,A3: set_Ground_gterm_f,B4: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ X
        @ ( produc6043514909933191368term_f @ A3
          @ ^ [Uu3: ground_gterm_f] : B4 ) )
      = ( ( member5261315044688711901term_f @ ( produc1239122367099457539term_f @ X ) @ A3 )
        & ( member5261315044688711901term_f @ ( produc1464490519189811525term_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1243_mem__Times__iff,axiom,
    ! [X: produc6920895926601326823rm_f_f,A3: set_Tr6091377851431777407rm_f_f,B4: set_Tr6091377851431777407rm_f_f] :
      ( ( member7222014889447216400rm_f_f @ X
        @ ( produc6018639678417508616rm_f_f @ A3
          @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
      = ( ( member8324439534195101384rm_f_f @ ( produc1890379913328437059rm_f_f @ X ) @ A3 )
        & ( member8324439534195101384rm_f_f @ ( produc1110312829679706757rm_f_f @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1244_mem__Times__iff,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: set_term_f_v,B4: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ X
        @ ( produc6667182599969081488rm_f_v @ A3
          @ ^ [Uu3: term_f_v] : B4 ) )
      = ( ( member_term_f_v @ ( produc1199688945740118859rm_f_v @ X ) @ A3 )
        & ( member_term_f_v @ ( produc5682565293565518221rm_f_v @ X ) @ B4 ) ) ) ).

% mem_Times_iff
thf(fact_1245_fTimes__subset__cancel2,axiom,
    ! [X: produc8027700073345061287rm_f_v,C2: fset_P8018961893305114765rm_f_v,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe774991991480784110rm_f_v @ X @ C2 )
     => ( ( ord_le5784093408781744682rm_f_v
          @ ( fSet_f1351405651625405787rm_f_v @ A3
            @ ^ [Uu3: product_prod_f_nat] : C2 )
          @ ( fSet_f1351405651625405787rm_f_v @ B4
            @ ^ [Uu3: product_prod_f_nat] : C2 ) )
        = ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ) ).

% fTimes_subset_cancel2
thf(fact_1246_fTimes__subset__cancel2,axiom,
    ! [X: produc250624644672406269term_f,C2: fset_P2882900458636368611term_f,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe6950792022577883716term_f @ X @ C2 )
     => ( ( ord_le4829002653170868608term_f
          @ ( fSet_f6358559406895517361term_f @ A3
            @ ^ [Uu3: product_prod_f_nat] : C2 )
          @ ( fSet_f6358559406895517361term_f @ B4
            @ ^ [Uu3: product_prod_f_nat] : C2 ) )
        = ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ) ).

% fTimes_subset_cancel2
thf(fact_1247_fTimes__subset__cancel2,axiom,
    ! [X: bot_bot_term_f,C2: fset_Bot_bot_term_f,A3: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
      ( ( fmembe1418995490368054919term_f @ X @ C2 )
     => ( ( ord_le4805696398359599223term_f
          @ ( fSet_f4524753253907642842term_f @ A3
            @ ^ [Uu3: product_prod_f_nat] : C2 )
          @ ( fSet_f4524753253907642842term_f @ B4
            @ ^ [Uu3: product_prod_f_nat] : C2 ) )
        = ( ord_le1552505484586773650_f_nat @ A3 @ B4 ) ) ) ).

% fTimes_subset_cancel2
thf(fact_1248_fTimes__subset__cancel2,axiom,
    ! [X: nat,C2: fset_nat,A3: fset_f,B4: fset_f] :
      ( ( fmember_nat @ X @ C2 )
     => ( ( ord_le1552505484586773650_f_nat
          @ ( fSet_fSigma_f_nat @ A3
            @ ^ [Uu3: f] : C2 )
          @ ( fSet_fSigma_f_nat @ B4
            @ ^ [Uu3: f] : C2 ) )
        = ( ord_less_eq_fset_f @ A3 @ B4 ) ) ) ).

% fTimes_subset_cancel2
thf(fact_1249_trancl__subset__Sigma__aux,axiom,
    ! [A: produc250624644672406269term_f,B: produc250624644672406269term_f,R2: set_Pr4896805198020829255term_f,A3: set_Pr6841673932529763677term_f] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ A @ B ) @ ( transi4676799229066866816term_f @ R2 ) )
     => ( ( ord_le1984979917675083751term_f @ R2
          @ ( produc8672414958890659976term_f @ A3
            @ ^ [Uu3: produc250624644672406269term_f] : A3 ) )
       => ( ( A = B )
          | ( member4687689947140789414term_f @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1250_trancl__subset__Sigma__aux,axiom,
    ! [A: produc7245736746747425831term_f,B: produc7245736746747425831term_f,R2: set_Pr1766941801908210887term_f,A3: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A @ B ) @ ( transi834741307384780330term_f @ R2 ) )
     => ( ( ord_le2215455632037200999term_f @ R2
          @ ( produc4509621232857779464term_f @ A3
            @ ^ [Uu3: produc7245736746747425831term_f] : A3 ) )
       => ( ( A = B )
          | ( member848276444142703440term_f @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1251_trancl__subset__Sigma__aux,axiom,
    ! [A: term_f_v,B: term_f_v,R2: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R2 ) )
     => ( ( ord_le4559761987009501863rm_f_v @ R2
          @ ( produc6667182599969081488rm_f_v @ A3
            @ ^ [Uu3: term_f_v] : A3 ) )
       => ( ( A = B )
          | ( member_term_f_v @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1252_trancl__subset__Sigma__aux,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,A3: set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi6299148271647642955term_f @ R2 ) )
     => ( ( ord_le5176047027162639613term_f @ R2
          @ ( produc2005054426516035430term_f @ A3
            @ ^ [Uu3: bot_bot_term_f] : A3 ) )
       => ( ( A = B )
          | ( member2723211850833494437term_f @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1253_trancl__subset__Sigma__aux,axiom,
    ! [A: ground_gterm_f,B: ground_gterm_f,R2: set_Pr989862937836626183term_f,A3: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A @ B ) @ ( transi7074228675819176119term_f @ R2 ) )
     => ( ( ord_le263819222746101927term_f @ R2
          @ ( produc6043514909933191368term_f @ A3
            @ ^ [Uu3: ground_gterm_f] : A3 ) )
       => ( ( A = B )
          | ( member5261315044688711901term_f @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1254_ftimes__subset__iff,axiom,
    ! [A3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat,D: fset_P6228066233360383026_f_nat] :
      ( ( ord_le6425377310842345707_f_nat
        @ ( fSet_f1026950353325567950_f_nat @ A3
          @ ^ [Uu3: product_prod_f_nat] : C2 )
        @ ( fSet_f1026950353325567950_f_nat @ B4
          @ ^ [Uu3: product_prod_f_nat] : D ) )
      = ( ( A3 = bot_bo3974441229639268550_f_nat )
        | ( C2 = bot_bo3974441229639268550_f_nat )
        | ( ( ord_le1552505484586773650_f_nat @ A3 @ B4 )
          & ( ord_le1552505484586773650_f_nat @ C2 @ D ) ) ) ) ).

% ftimes_subset_iff
thf(fact_1255_ftimes__subset__iff,axiom,
    ! [A3: fset_f,C2: fset_nat,B4: fset_f,D: fset_nat] :
      ( ( ord_le1552505484586773650_f_nat
        @ ( fSet_fSigma_f_nat @ A3
          @ ^ [Uu3: f] : C2 )
        @ ( fSet_fSigma_f_nat @ B4
          @ ^ [Uu3: f] : D ) )
      = ( ( A3 = bot_bot_fset_f )
        | ( C2 = bot_bot_fset_nat )
        | ( ( ord_less_eq_fset_f @ A3 @ B4 )
          & ( ord_less_eq_fset_nat @ C2 @ D ) ) ) ) ).

% ftimes_subset_iff
thf(fact_1256_rtrancl__Restr,axiom,
    ! [X: term_f_v,Y: term_f_v,R2: set_Pr8827868859434726151rm_f_v,A3: set_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y )
        @ ( transi3028809021613410336rm_f_v
          @ ( inf_in6135112614888068085rm_f_v @ R2
            @ ( produc6667182599969081488rm_f_v @ A3
              @ ^ [Uu3: term_f_v] : A3 ) ) ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3028809021613410336rm_f_v @ R2 ) ) ) ).

% rtrancl_Restr
thf(fact_1257_rtrancl__Restr,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R2: set_Pr6841673932529763677term_f,A3: set_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y )
        @ ( transi6299148271647642955term_f
          @ ( inf_in5033374358511726155term_f @ R2
            @ ( produc2005054426516035430term_f @ A3
              @ ^ [Uu3: bot_bot_term_f] : A3 ) ) ) )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi6299148271647642955term_f @ R2 ) ) ) ).

% rtrancl_Restr
thf(fact_1258_rtrancl__Restr,axiom,
    ! [X: ground_gterm_f,Y: ground_gterm_f,R2: set_Pr989862937836626183term_f,A3: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y )
        @ ( transi7074228675819176119term_f
          @ ( inf_in6246359444536921845term_f @ R2
            @ ( produc6043514909933191368term_f @ A3
              @ ^ [Uu3: ground_gterm_f] : A3 ) ) ) )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y ) @ ( transi7074228675819176119term_f @ R2 ) ) ) ).

% rtrancl_Restr
thf(fact_1259_fImage__fsubset,axiom,
    ! [R2: fset_P6228066233360383026_f_nat,A3: fset_f,B4: fset_nat,C2: fset_f] :
      ( ( ord_le1552505484586773650_f_nat @ R2
        @ ( fSet_fSigma_f_nat @ A3
          @ ^ [Uu3: f] : B4 ) )
     => ( ord_less_eq_fset_nat @ ( fSet_fImage_f_nat @ R2 @ C2 ) @ B4 ) ) ).

% fImage_fsubset
thf(fact_1260_fmem__Times__iff,axiom,
    ! [X: produc250624644672406269term_f,A3: fset_Bot_bot_term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ X
        @ ( fSet_f7914580955412209254term_f @ A3
          @ ^ [Uu3: bot_bot_term_f] : B4 ) )
      = ( ( fmembe1418995490368054919term_f @ ( produc5789359262401207841term_f @ X ) @ A3 )
        & ( fmembe1418995490368054919term_f @ ( produc656023363694804067term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1261_fmem__Times__iff,axiom,
    ! [X: produc8027700073345061287rm_f_v,A3: fset_term_f_v,B4: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X
        @ ( fSet_f5986999499850583952rm_f_v @ A3
          @ ^ [Uu3: term_f_v] : B4 ) )
      = ( ( fmember_term_f_v @ ( produc1199688945740118859rm_f_v @ X ) @ A3 )
        & ( fmember_term_f_v @ ( produc5682565293565518221rm_f_v @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1262_fmem__Times__iff,axiom,
    ! [X: produc3989578180274711866term_f,A3: fset_P2882900458636368611term_f,B4: fset_Bot_bot_term_f] :
      ( ( fmembe2966320103993061043term_f @ X
        @ ( fSet_f2646000941442319427term_f @ A3
          @ ^ [Uu3: produc250624644672406269term_f] : B4 ) )
      = ( ( fmembe6950792022577883716term_f @ ( produc1987789157120097864term_f @ X ) @ A3 )
        & ( fmembe1418995490368054919term_f @ ( produc1545723550160310918term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1263_fmem__Times__iff,axiom,
    ! [X: produc7402450736791287140term_f,A3: fset_Bot_bot_term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe6379192660509636317term_f @ X
        @ ( fSet_f2583267287733066021term_f @ A3
          @ ^ [Uu3: bot_bot_term_f] : B4 ) )
      = ( ( fmembe1418995490368054919term_f @ ( produc1925055503410844458term_f @ X ) @ A3 )
        & ( fmembe6950792022577883716term_f @ ( produc1482989896451057512term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1264_fmem__Times__iff,axiom,
    ! [X: produc6920895926601326823rm_f_f,A3: fset_T2564841199962184965rm_f_f,B4: fset_T2564841199962184965rm_f_f] :
      ( ( fmembe692662118792543662rm_f_f @ X
        @ ( fSet_f5320715149535191048rm_f_f @ A3
          @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : B4 ) )
      = ( ( fmembe6944376671386649446rm_f_f @ ( produc1890379913328437059rm_f_f @ X ) @ A3 )
        & ( fmembe6944376671386649446rm_f_f @ ( produc1110312829679706757rm_f_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1265_fmem__Times__iff,axiom,
    ! [X: produc6737520623449210596term_f,A3: fset_P8018961893305114765rm_f_v,B4: fset_Bot_bot_term_f] :
      ( ( fmembe8032078482862777437term_f @ X
        @ ( fSet_f4943302497394398701term_f @ A3
          @ ^ [Uu3: produc8027700073345061287rm_f_v] : B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ ( produc4149221720254882802term_f @ X ) @ A3 )
        & ( fmembe1418995490368054919term_f @ ( produc5727801712419221552term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1266_fmem__Times__iff,axiom,
    ! [X: produc1667017993794354958rm_f_v,A3: fset_Bot_bot_term_f,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe2961575853207921799rm_f_v @ X
        @ ( fSet_f8638597300464633807rm_f_v @ A3
          @ ^ [Uu3: bot_bot_term_f] : B4 ) )
      = ( ( fmembe1418995490368054919term_f @ ( produc7844516523325117908rm_f_v @ X ) @ A3 )
        & ( fmembe774991991480784110rm_f_v @ ( produc199724478634680850rm_f_v @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1267_fmem__Times__iff,axiom,
    ! [X: produc6853483567480260455term_f,A3: fset_P2882900458636368611term_f,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe5277907931857505582term_f @ X
        @ ( fSet_f4469069203941308296term_f @ A3
          @ ^ [Uu3: produc250624644672406269term_f] : B4 ) )
      = ( ( fmembe6950792022577883716term_f @ ( produc529750796343735235term_f @ X ) @ A3 )
        & ( fmembe6950792022577883716term_f @ ( produc6080605871244463365term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1268_fmem__Times__iff,axiom,
    ! [X: produc3700083184670943805term_f,A3: fset_P8018961893305114765rm_f_v,B4: fset_P2882900458636368611term_f] :
      ( ( fmembe1382477799658998788term_f @ X
        @ ( fSet_f5744403325833042526term_f @ A3
          @ ^ [Uu3: produc8027700073345061287rm_f_v] : B4 ) )
      = ( ( fmembe774991991480784110rm_f_v @ ( produc7104483540190724761term_f @ X ) @ A3 )
        & ( fmembe6950792022577883716term_f @ ( produc1676325126087730139term_f @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1269_fmem__Times__iff,axiom,
    ! [X: produc474063087423118609rm_f_v,A3: fset_P2882900458636368611term_f,B4: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe7379829739265949400rm_f_v @ X
        @ ( fSet_f5536414834681360946rm_f_v @ A3
          @ ^ [Uu3: produc250624644672406269term_f] : B4 ) )
      = ( ( fmembe6950792022577883716term_f @ ( produc6896495049039043181rm_f_v @ X ) @ A3 )
        & ( fmembe774991991480784110rm_f_v @ ( produc1468336634936048559rm_f_v @ X ) @ B4 ) ) ) ).

% fmem_Times_iff
thf(fact_1270_fsnd__image__Sigma,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat] :
      ( ( fimage7082879873702729561_f_nat @ produc6446878481558449956_f_nat @ ( fSet_f5147502566550621281_f_nat @ A3 @ B4 ) )
      = ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ A3 ) ) ) ).

% fsnd_image_Sigma
thf(fact_1271_fsnd__image__Sigma,axiom,
    ! [A3: fset_T2564841199962184965rm_f_f,B4: tree_t2044664379607036063rm_f_f > fset_T2564841199962184965rm_f_f] :
      ( ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f @ ( fSet_f5320715149535191048rm_f_f @ A3 @ B4 ) )
      = ( ffUnio8036915103833617719rm_f_f @ ( fimage2103382486912029859rm_f_f @ B4 @ A3 ) ) ) ).

% fsnd_image_Sigma
thf(fact_1272_fsnd__image__Sigma,axiom,
    ! [A3: fset_term_f_v,B4: term_f_v > fset_term_f_v] :
      ( ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v @ ( fSet_f5986999499850583952rm_f_v @ A3 @ B4 ) )
      = ( ffUnion_term_f_v @ ( fimage352304360108585205rm_f_v @ B4 @ A3 ) ) ) ).

% fsnd_image_Sigma
thf(fact_1273_fSigma__fImage,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v,B4: produc8027700073345061287rm_f_v > fset_P6228066233360383026_f_nat,X5: fset_P8018961893305114765rm_f_v] :
      ( ( fSet_f577160797525143147_f_nat @ ( fSet_f5147502566550621281_f_nat @ A3 @ B4 ) @ X5 )
      = ( ffUnio5887076197631975402_f_nat @ ( fimage84417724776064284_f_nat @ B4 @ ( inf_in6563521360087198331rm_f_v @ X5 @ A3 ) ) ) ) ).

% fSigma_fImage
thf(fact_1274_fsubset__fst__snd,axiom,
    ! [A3: fset_P4253400765599247181rm_f_f] :
      ( ord_le619118439696175853rm_f_f @ A3
      @ ( fSet_f5320715149535191048rm_f_f @ ( fimage7671923670337682677rm_f_f @ produc1890379913328437059rm_f_f @ A3 )
        @ ^ [Uu3: tree_t2044664379607036063rm_f_f] : ( fimage7671923670337682677rm_f_f @ produc1110312829679706757rm_f_f @ A3 ) ) ) ).

% fsubset_fst_snd
thf(fact_1275_fsubset__fst__snd,axiom,
    ! [A3: fset_P8018961893305114765rm_f_v] :
      ( ord_le4587745213494032429rm_f_v @ A3
      @ ( fSet_f5986999499850583952rm_f_v @ ( fimage4320022722461925965rm_f_v @ produc1199688945740118859rm_f_v @ A3 )
        @ ^ [Uu3: term_f_v] : ( fimage4320022722461925965rm_f_v @ produc5682565293565518221rm_f_v @ A3 ) ) ) ).

% fsubset_fst_snd
thf(fact_1276_fsubset__fst__snd,axiom,
    ! [A3: fset_P6228066233360383026_f_nat] :
      ( ord_le1552505484586773650_f_nat @ A3
      @ ( fSet_fSigma_f_nat @ ( fimage5271517554409401666_nat_f @ product_fst_f_nat @ A3 )
        @ ^ [Uu3: f] : ( fimage8897140481647806417at_nat @ product_snd_f_nat @ A3 ) ) ) ).

% fsubset_fst_snd

% Helper facts (15)
thf(help_If_2_1_If_001t__Term__Oterm_Itf__f_Mtf__v_J_T,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( if_term_f_v @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__f_Mtf__v_J_T,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( if_term_f_v @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Bot____Terms__Obot____term_Itf__f_J_T,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( if_Bot_bot_term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Bot____Terms__Obot____term_Itf__f_J_T,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( if_Bot_bot_term_f @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( if_fse3524882727628502648term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( if_fse3524882727628502648term_f @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_T,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
      ( ( if_Pro8994587301352985965rm_f_v @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_T,axiom,
    ! [X: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
      ( ( if_Pro8994587301352985965rm_f_v @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: produc250624644672406269term_f,Y: produc250624644672406269term_f] :
      ( ( if_Pro1782623486134218435term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: produc250624644672406269term_f,Y: produc250624644672406269term_f] :
      ( ( if_Pro1782623486134218435term_f @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_T,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( if_fse5575177004205410643rm_f_v @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_T,axiom,
    ! [X: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( if_fse5575177004205410643rm_f_v @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_T,axiom,
    ! [X: fset_P2882900458636368611term_f,Y: fset_P2882900458636368611term_f] :
      ( ( if_fse2792004000006630313term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_T,axiom,
    ! [X: fset_P2882900458636368611term_f,Y: fset_P2882900458636368611term_f] :
      ( ( if_fse2792004000006630313term_f @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    linear_term_f_v @ l ).

%------------------------------------------------------------------------------