%------------------------------------------------------------------------------
% 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    : ResiduatedTransitionSystem/0001_LambdaCalculus/prob_07507_336179__14965660_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1454 ( 508 unt; 177 typ;   0 def)
%            Number of atoms       : 3908 (1890 equ;   0 cnn)
%            Maximal formula atoms :   38 (   3 avg)
%            Number of connectives : 14992 ( 732   ~;  38   |; 520   &;12125   @)
%                                         (   0 <=>;1577  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   6 avg)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  :  955 ( 955   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  169 ( 166 usr;  16 con; 0-6 aty)
%            Number of variables   : 3187 ( 173   ^;2880   !; 134   ?;3187   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:49:10.080
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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% Explicit typings (166)
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thf(sy_c_ResiduatedTransitionSystem_Opaths__in__confluent__rts_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    paths_3043240146134862789lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_6861597469809217359lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_366302053746739058lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    paths_2015741907369688696lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OIde_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_7717490669901197737lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid1x_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_3319247187298047121lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > list_lambda_lambda > lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_8607357831768207192lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    paths_3312059784463760222lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResidx1_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_2112734093361750239lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > lambda_lambda > list_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_2897053694880971696lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > set_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    paths_3172353640142224310lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > set_li7878742357367608272lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OTrgs_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_7594662086398683701lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > set_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__weakly__extensional__rts_OSrc_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_1651982994781619320lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Opaths__in__weakly__extensional__rts_OTrg_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    paths_2695323870737867955lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    arr_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    arr_li3691443969012319837lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    con_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    con_li79537494059641180lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    ide_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    ide_li6773656263178250516lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    trg_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    trg_li817795487149935061lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    coinit1392077014771647542lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    coinit1063259437155584316lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    compos7127019122090555255lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    compos3359172601056258045lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    coterm4142657152830665724lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    coterm81126318419254786lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    join_o1386658102433174083lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    join_o6956785509312998601lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    joinab2519788767067900224lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    joinab677607494065406278lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    seq_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    seq_li8093550137657879445lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    source5627170679743844886lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    source6119337877621565212lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    target5490931411331770124lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).

thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    target4026644918049360146lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).

thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    rts_ax7298520571313827089lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    rts_ax2283037410143324311lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    rts_wi3086218056445523938lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    rts_wi1553966559393445608lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    simula5235733974575307523lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    simula1175017789036123529lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    simula9164532646813739645lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    simula3069223088535412739lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    sub_rt3575465509250962105lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > $o ) > $o ).

thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    sub_rt3099298114314975807lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > $o ) > $o ).

thf(sy_c_Set_OCollect_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    collec621444948832755541lambda: ( lambda_lambda > $o ) > set_lambda_lambda ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    collec8830145823237851355lambda: ( list_lambda_lambda > $o ) > set_li7878742357367608272lambda ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
    collec4999033578922223073lambda: ( list_l5168835686021156086lambda > $o ) > set_li846743354098393302lambda ).

thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    image_2356535832909760517lambda: ( lambda_lambda > lambda_lambda ) > set_lambda_lambda > set_lambda_lambda ).

thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    image_3692823078739859851lambda: ( lambda_lambda > list_lambda_lambda ) > set_lambda_lambda > set_li7878742357367608272lambda ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    image_2458965899662700159lambda: ( list_lambda_lambda > lambda_lambda ) > set_li7878742357367608272lambda > set_lambda_lambda ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    image_5886195720093285125lambda: ( list_lambda_lambda > list_lambda_lambda ) > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    insert_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    insert8465602035467394112lambda: list_lambda_lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).

thf(sy_c_Set_Othe__elem_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    the_el5431214018478934239lambda: set_lambda_lambda > lambda_lambda ).

thf(sy_c_Set_Othe__elem_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    the_el4548681628688475493lambda: set_li7878742357367608272lambda > list_lambda_lambda ).

thf(sy_c_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
    member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).

thf(sy_c_member_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
    member2667661554319814297lambda: list_lambda_lambda > set_li7878742357367608272lambda > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
    member9126999148103521439lambda: list_l5168835686021156086lambda > set_li846743354098393302lambda > $o ).

thf(sy_v_M____,type,
    m: lambda_lambda ).

thf(sy_v_N____,type,
    n: lambda_lambda ).

thf(sy_v_Ua____,type,
    ua: list_lambda_lambda ).

thf(sy_v_u____,type,
    u: lambda_lambda ).

% Relevant facts (1273)
thf(fact_0_finite__developments,axiom,
    ! [T: lambda_lambda] : ( reduction_FD @ T ) ).

% finite_developments
thf(fact_1_lambda__calculus_Olambda_Oinject_I4_J,axiom,
    ! [X51: lambda_lambda,X52: lambda_lambda,Y51: lambda_lambda,Y52: lambda_lambda] :
      ( ( ( lambda_Beta @ X51 @ X52 )
        = ( lambda_Beta @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% lambda_calculus.lambda.inject(4)
thf(fact_2_Std,axiom,
    reduction_Std @ ( cons_lambda_lambda @ u @ ua ) ).

% Std
thf(fact_3_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,T2: lambda_lambda,U2: lambda_lambda] :
      ( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
        = ( lambda_Src @ ( lambda_Beta @ T2 @ U2 ) ) )
      = ( ( ( lambda_Src @ T )
          = ( lambda_Src @ T2 ) )
        & ( ( lambda_Src @ U )
          = ( lambda_Src @ U2 ) ) ) ) ).

% lambda_calculus.Src_eq_iff(3)
thf(fact_4_MN,axiom,
    ( ( lambda_Arr @ m )
    & ( lambda_Arr @ n ) ) ).

% MN
thf(fact_5_list_Oinject,axiom,
    ! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda,Y21: list_lambda_lambda,Y22: list_l5168835686021156086lambda] :
      ( ( ( cons_l4558873032622161958lambda @ X21 @ X22 )
        = ( cons_l4558873032622161958lambda @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_6_list_Oinject,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda,Y21: lambda_lambda,Y22: list_lambda_lambda] :
      ( ( ( cons_lambda_lambda @ X21 @ X22 )
        = ( cons_lambda_lambda @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_7__C2_C,axiom,
    ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ u @ ua ) ) ).

% "2"
thf(fact_8_seq,axiom,
    seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Beta @ m @ n ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ ua ) ).

% seq
thf(fact_9_not__Cons__self2,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( cons_l4558873032622161958lambda @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_10_not__Cons__self2,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( cons_lambda_lambda @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_11__C1_C,axiom,
    lambda_Ide @ ( lambda_Subst @ zero_zero_nat @ n @ m ) ).

% "1"
thf(fact_12_lambda__calculus_Ohead__strategy_Osimps_I8_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_head_strategy @ ( lambda_Beta @ T @ U ) )
      = ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).

% lambda_calculus.head_strategy.simps(8)
thf(fact_13_lambda__calculus_Ohead__redex_Osimps_I9_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_head_redex @ ( lambda_Beta @ T @ U ) )
      = ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).

% lambda_calculus.head_redex.simps(9)
thf(fact_14_lambda__calculus_Olambda_Osimps_I29_J,axiom,
    ! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X51: lambda_lambda,X52: lambda_lambda] :
      ( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_Beta @ X51 @ X52 ) )
      = ( F5 @ X51 @ X52 ) ) ).

% lambda_calculus.lambda.simps(29)
thf(fact_15__092_060Lambda_062_Oex__un__null,axiom,
    ? [X2: lambda_lambda] :
      ( ! [T3: lambda_lambda] :
          ( ( ( lambda_resid @ X2 @ T3 )
            = X2 )
          & ( ( lambda_resid @ T3 @ X2 )
            = X2 ) )
      & ! [Y: lambda_lambda] :
          ( ! [T4: lambda_lambda] :
              ( ( ( lambda_resid @ Y @ T4 )
                = Y )
              & ( ( lambda_resid @ T4 @ Y )
                = Y ) )
         => ( Y = X2 ) ) ) ).

% \<Lambda>.ex_un_null
thf(fact_16__092_060Lambda_062_Ocube,axiom,
    ! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
      = ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ).

% \<Lambda>.cube
thf(fact_17_cube,axiom,
    ! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
      = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).

% cube
thf(fact_18_ex__un__null,axiom,
    ? [X2: list_lambda_lambda] :
      ( ! [T3: list_lambda_lambda] :
          ( ( ( paths_8607357831768207192lambda @ lambda_resid @ X2 @ T3 )
            = X2 )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ X2 )
            = X2 ) )
      & ! [Y: list_lambda_lambda] :
          ( ! [T4: list_lambda_lambda] :
              ( ( ( paths_8607357831768207192lambda @ lambda_resid @ Y @ T4 )
                = Y )
              & ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ Y )
                = Y ) )
         => ( Y = X2 ) ) ) ).

% ex_un_null
thf(fact_19_Cube_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) )
       != nil_lambda_lambda )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) )
        = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V2 ) ) ) ) ).

% Cube(2)
thf(fact_20_Cube_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) )
       != nil_lambda_lambda )
      = ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V2 ) )
       != nil_lambda_lambda ) ) ).

% Cube(1)
thf(fact_21_Resid_Osimps_I1_J,axiom,
    ! [Uu: list_lambda_lambda] :
      ( ( paths_8607357831768207192lambda @ lambda_resid @ nil_lambda_lambda @ Uu )
      = nil_lambda_lambda ) ).

% Resid.simps(1)
thf(fact_22_Con__sym,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
      = ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 )
       != nil_lambda_lambda ) ) ).

% Con_sym
thf(fact_23_Ide_Osimps_I1_J,axiom,
    ~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).

% Ide.simps(1)
thf(fact_24_Con__initial__right,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
       != nil_lambda_lambda )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda ) ) ).

% Con_initial_right
thf(fact_25_Con__initial__left,axiom,
    ! [T: lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
       != nil_lambda_lambda ) ) ).

% Con_initial_left
thf(fact_26_Resid__rec_I2_J,axiom,
    ! [T5: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
          = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).

% Resid_rec(2)
thf(fact_27_Resid__rec_I3_J,axiom,
    ! [U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
          = ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 ) ) ) ) ).

% Resid_rec(3)
thf(fact_28_Resid__cons_I2_J,axiom,
    ! [U3: list_lambda_lambda,T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
          = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 ) ) ) ) ).

% Resid_cons(2)
thf(fact_29_Resid_Osimps_I2_J,axiom,
    ! [V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ nil_lambda_lambda )
      = nil_lambda_lambda ) ).

% Resid.simps(2)
thf(fact_30_Con__cons_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
           != nil_lambda_lambda )
          = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
             != nil_lambda_lambda )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
             != nil_lambda_lambda ) ) ) ) ) ).

% Con_cons(1)
thf(fact_31_Con__cons_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
           != nil_lambda_lambda )
          = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
             != nil_lambda_lambda )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 )
             != nil_lambda_lambda ) ) ) ) ) ).

% Con_cons(2)
thf(fact_32_Resid__Ide_I1_J,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T5 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ A )
          = T5 ) ) ) ).

% Resid_Ide(1)
thf(fact_33_Resid__Arr__Ide__ind,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ A )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ A )
          = T5 ) ) ) ).

% Resid_Arr_Ide_ind
thf(fact_34_Resid__Ide__Arr__ind,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T5 )
         != nil_lambda_lambda )
       => ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T5 ) ) ) ) ).

% Resid_Ide_Arr_ind
thf(fact_35_stdz__insert__Ide,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) )
     => ( ( reduct5390047737487997688insert @ T @ U3 )
        = nil_lambda_lambda ) ) ).

% stdz_insert_Ide
thf(fact_36_lambda__calculus_OSrc__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Src @ ( lambda_Src @ T ) )
        = ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Src_Src
thf(fact_37_lambda__calculus_Oresid__Arr__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
        = T ) ) ).

% lambda_calculus.resid_Arr_Src
thf(fact_38_lambda__calculus_OSrc__Subst,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,D: nat] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( lambda_Src @ ( lambda_Subst @ D @ T @ U ) )
          = ( lambda_Subst @ D @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ) ) ).

% lambda_calculus.Src_Subst
thf(fact_39_Con__consI_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
           != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 )
             != nil_lambda_lambda )
           => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
             != nil_lambda_lambda ) ) ) ) ) ).

% Con_consI(2)
thf(fact_40_Con__consI_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
           != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
             != nil_lambda_lambda )
           => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
             != nil_lambda_lambda ) ) ) ) ) ).

% Con_consI(1)
thf(fact_41_Resid__rec_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
        = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ).

% Resid_rec(1)
thf(fact_42_paths__in__confluent__rts__axioms,axiom,
    paths_9040249923598999359lambda @ lambda_resid ).

% paths_in_confluent_rts_axioms
thf(fact_43_mem__Collect__eq,axiom,
    ! [A2: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
      ( ( member9126999148103521439lambda @ A2 @ ( collec4999033578922223073lambda @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
    ! [A2: list_lambda_lambda,P: list_lambda_lambda > $o] :
      ( ( member2667661554319814297lambda @ A2 @ ( collec8830145823237851355lambda @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A2: lambda_lambda,P: lambda_lambda > $o] :
      ( ( member_lambda_lambda @ A2 @ ( collec621444948832755541lambda @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A: set_li846743354098393302lambda] :
      ( ( collec4999033578922223073lambda
        @ ^ [X3: list_l5168835686021156086lambda] : ( member9126999148103521439lambda @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A: set_lambda_lambda] :
      ( ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
    ! [P: list_l5168835686021156086lambda > $o,Q: list_l5168835686021156086lambda > $o] :
      ( ! [X2: list_l5168835686021156086lambda] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec4999033578922223073lambda @ P )
        = ( collec4999033578922223073lambda @ Q ) ) ) ).

% Collect_cong
thf(fact_50_Collect__cong,axiom,
    ! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
      ( ! [X2: list_lambda_lambda] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec8830145823237851355lambda @ P )
        = ( collec8830145823237851355lambda @ Q ) ) ) ).

% Collect_cong
thf(fact_51_Collect__cong,axiom,
    ! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
      ( ! [X2: lambda_lambda] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec621444948832755541lambda @ P )
        = ( collec621444948832755541lambda @ Q ) ) ) ).

% Collect_cong
thf(fact_52_complete__development__cons,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct7282684938351565978opment @ T @ ( cons_lambda_lambda @ U @ U3 ) )
     => ( reduct7282684938351565978opment @ ( lambda_resid @ T @ U ) @ U3 ) ) ).

% complete_development_cons
thf(fact_53_complete__development__Ide__iff,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct7282684938351565978opment @ T @ U3 )
     => ( ( lambda_Ide @ T )
        = ( U3 = nil_lambda_lambda ) ) ) ).

% complete_development_Ide_iff
thf(fact_54_is__confluent__rts,axiom,
    conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).

% is_confluent_rts
thf(fact_55_reduction__paths_OStd_Osimps_I1_J,axiom,
    reduction_Std @ nil_lambda_lambda ).

% reduction_paths.Std.simps(1)
thf(fact_56_transpose_Ocases,axiom,
    ! [X: list_l7543285377636199382lambda] :
      ( ( X != nil_li3187382615321999190lambda )
     => ( ! [Xss: list_l7543285377636199382lambda] :
            ( X
           != ( cons_l203434155165082374lambda @ nil_se6914803783601386320lambda @ Xss ) )
       => ~ ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Xss: list_l7543285377636199382lambda] :
              ( X
             != ( cons_l203434155165082374lambda @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_57_transpose_Ocases,axiom,
    ! [X: list_l4928420112573288188lambda] :
      ( ( X != nil_li6069905736445159420lambda )
     => ( ! [Xss: list_l4928420112573288188lambda] :
            ( X
           != ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ Xss ) )
       => ~ ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Xss: list_l4928420112573288188lambda] :
              ( X
             != ( cons_l4910729503725484972lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_58_transpose_Ocases,axiom,
    ! [X: list_l5168835686021156086lambda] :
      ( ( X != nil_li8379291187143545974lambda )
     => ( ! [Xss: list_l5168835686021156086lambda] :
            ( X
           != ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
       => ~ ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
              ( X
             != ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_59_lambda__calculus_OArr__Subst,axiom,
    ! [V: lambda_lambda,T: lambda_lambda,N: nat] :
      ( ( lambda_Arr @ V )
     => ( ( lambda_Arr @ T )
       => ( lambda_Arr @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).

% lambda_calculus.Arr_Subst
thf(fact_60_lambda__calculus_OIde__Subst,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,N: nat] :
      ( ( lambda_Ide @ T )
     => ( ( lambda_Ide @ V )
       => ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).

% lambda_calculus.Ide_Subst
thf(fact_61_lambda__calculus_Oresid__Arr__Ide,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( lambda_Ide @ A2 )
     => ( ( ( lambda_Arr @ T )
          & ( lambda_Arr @ A2 )
          & ( ( lambda_Src @ T )
            = ( lambda_Src @ A2 ) ) )
       => ( ( lambda_resid @ T @ A2 )
          = T ) ) ) ).

% lambda_calculus.resid_Arr_Ide
thf(fact_62_lambda__calculus_Oresid__Ide__Arr,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( lambda_Ide @ A2 )
     => ( ( ( lambda_Arr @ A2 )
          & ( lambda_Arr @ T )
          & ( ( lambda_Src @ A2 )
            = ( lambda_Src @ T ) ) )
       => ( lambda_Ide @ ( lambda_resid @ A2 @ T ) ) ) ) ).

% lambda_calculus.resid_Ide_Arr
thf(fact_63_lambda__calculus_OArr__head__redex,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_Arr @ ( lambda_head_redex @ T ) ) ) ).

% lambda_calculus.Arr_head_redex
thf(fact_64_lambda__calculus_OIde__implies__Arr,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Ide @ T )
     => ( lambda_Arr @ T ) ) ).

% lambda_calculus.Ide_implies_Arr
thf(fact_65_lambda__calculus_OArr__head__strategy,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_Arr @ ( lambda_head_strategy @ T ) ) ) ).

% lambda_calculus.Arr_head_strategy
thf(fact_66_lambda__calculus_OSrc__head__redex,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Src @ ( lambda_head_redex @ T ) )
        = ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Src_head_redex
thf(fact_67_lambda__calculus_Ohead__strategy__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_head_strategy @ ( lambda_Src @ T ) )
        = ( lambda_head_strategy @ T ) ) ) ).

% lambda_calculus.head_strategy_Src
thf(fact_68_lambda__calculus_OSrc__head__strategy,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Src @ ( lambda_head_strategy @ T ) )
        = ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Src_head_strategy
thf(fact_69_lambda__calculus_OIde__iff__Src__self,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Ide @ T )
        = ( ( lambda_Src @ T )
          = T ) ) ) ).

% lambda_calculus.Ide_iff_Src_self
thf(fact_70_lambda__calculus_OIde__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Ide_Src
thf(fact_71_list__nonempty__induct,axiom,
    ! [Xs: list_s345339197461576016lambda,P: list_s345339197461576016lambda > $o] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ( ! [X2: set_lambda_lambda] : ( P @ ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) )
       => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
              ( ( Xs2 != nil_se6914803783601386320lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_72_list__nonempty__induct,axiom,
    ! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( ! [X2: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) )
       => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
              ( ( Xs2 != nil_li8379291187143545974lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_73_list__nonempty__induct,axiom,
    ! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
      ( ( Xs != nil_lambda_lambda )
     => ( ! [X2: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) )
       => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] :
              ( ( Xs2 != nil_lambda_lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_74_list__induct2_H,axiom,
    ! [P: list_s345339197461576016lambda > list_s345339197461576016lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( P @ nil_se6914803783601386320lambda @ nil_se6914803783601386320lambda )
     => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ nil_se6914803783601386320lambda )
       => ( ! [Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_75_list__induct2_H,axiom,
    ! [P: list_s345339197461576016lambda > list_lambda_lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_lambda_lambda] :
      ( ( P @ nil_se6914803783601386320lambda @ nil_lambda_lambda )
     => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ nil_lambda_lambda )
       => ( ! [Y2: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_lambda_lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_76_list__induct2_H,axiom,
    ! [P: list_s345339197461576016lambda > list_l5168835686021156086lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_l5168835686021156086lambda] :
      ( ( P @ nil_se6914803783601386320lambda @ nil_li8379291187143545974lambda )
     => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ nil_li8379291187143545974lambda )
       => ( ! [Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_77_list__induct2_H,axiom,
    ! [P: list_lambda_lambda > list_s345339197461576016lambda > $o,Xs: list_lambda_lambda,Ys: list_s345339197461576016lambda] :
      ( ( P @ nil_lambda_lambda @ nil_se6914803783601386320lambda )
     => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ nil_se6914803783601386320lambda )
       => ( ! [Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_lambda_lambda @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_78_list__induct2_H,axiom,
    ! [P: list_lambda_lambda > list_l5168835686021156086lambda > $o,Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
      ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
     => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ nil_li8379291187143545974lambda )
       => ( ! [Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_lambda_lambda @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_79_list__induct2_H,axiom,
    ! [P: list_l5168835686021156086lambda > list_s345339197461576016lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_s345339197461576016lambda] :
      ( ( P @ nil_li8379291187143545974lambda @ nil_se6914803783601386320lambda )
     => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ nil_se6914803783601386320lambda )
       => ( ! [Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y2: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_80_list__induct2_H,axiom,
    ! [P: list_l5168835686021156086lambda > list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda] :
      ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
     => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ nil_lambda_lambda )
       => ( ! [Y2: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_lambda_lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_81_list__induct2_H,axiom,
    ! [P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
     => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ nil_li8379291187143545974lambda )
       => ( ! [Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_l4558873032622161958lambda @ X2 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_82_list__induct2_H,axiom,
    ! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
     => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ nil_lambda_lambda )
       => ( ! [Y2: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y2 @ Ys2 ) )
         => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_83_neq__Nil__conv,axiom,
    ! [Xs: list_s345339197461576016lambda] :
      ( ( Xs != nil_se6914803783601386320lambda )
      = ( ? [Y3: set_lambda_lambda,Ys3: list_s345339197461576016lambda] :
            ( Xs
            = ( cons_s3132251510596564736lambda @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_84_neq__Nil__conv,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
      = ( ? [Y3: list_lambda_lambda,Ys3: list_l5168835686021156086lambda] :
            ( Xs
            = ( cons_l4558873032622161958lambda @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_85_neq__Nil__conv,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
      = ( ? [Y3: lambda_lambda,Ys3: list_lambda_lambda] :
            ( Xs
            = ( cons_lambda_lambda @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_86_remdups__adj_Ocases,axiom,
    ! [X: list_s345339197461576016lambda] :
      ( ( X != nil_se6914803783601386320lambda )
     => ( ! [X2: set_lambda_lambda] :
            ( X
           != ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) )
       => ~ ! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
              ( X
             != ( cons_s3132251510596564736lambda @ X2 @ ( cons_s3132251510596564736lambda @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_87_remdups__adj_Ocases,axiom,
    ! [X: list_l5168835686021156086lambda] :
      ( ( X != nil_li8379291187143545974lambda )
     => ( ! [X2: list_lambda_lambda] :
            ( X
           != ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) )
       => ~ ! [X2: list_lambda_lambda,Y2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
              ( X
             != ( cons_l4558873032622161958lambda @ X2 @ ( cons_l4558873032622161958lambda @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_88_remdups__adj_Ocases,axiom,
    ! [X: list_lambda_lambda] :
      ( ( X != nil_lambda_lambda )
     => ( ! [X2: lambda_lambda] :
            ( X
           != ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) )
       => ~ ! [X2: lambda_lambda,Y2: lambda_lambda,Xs2: list_lambda_lambda] :
              ( X
             != ( cons_lambda_lambda @ X2 @ ( cons_lambda_lambda @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_89_min__list_Ocases,axiom,
    ! [X: list_s345339197461576016lambda] :
      ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
          ( X
         != ( cons_s3132251510596564736lambda @ X2 @ Xs2 ) )
     => ( X = nil_se6914803783601386320lambda ) ) ).

% min_list.cases
thf(fact_90_list_Oexhaust,axiom,
    ! [Y4: list_s345339197461576016lambda] :
      ( ( Y4 != nil_se6914803783601386320lambda )
     => ~ ! [X212: set_lambda_lambda,X222: list_s345339197461576016lambda] :
            ( Y4
           != ( cons_s3132251510596564736lambda @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_91_list_Oexhaust,axiom,
    ! [Y4: list_l5168835686021156086lambda] :
      ( ( Y4 != nil_li8379291187143545974lambda )
     => ~ ! [X212: list_lambda_lambda,X222: list_l5168835686021156086lambda] :
            ( Y4
           != ( cons_l4558873032622161958lambda @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_92_list_Oexhaust,axiom,
    ! [Y4: list_lambda_lambda] :
      ( ( Y4 != nil_lambda_lambda )
     => ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
            ( Y4
           != ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_93_list_OdiscI,axiom,
    ! [List: list_s345339197461576016lambda,X21: set_lambda_lambda,X22: list_s345339197461576016lambda] :
      ( ( List
        = ( cons_s3132251510596564736lambda @ X21 @ X22 ) )
     => ( List != nil_se6914803783601386320lambda ) ) ).

% list.discI
thf(fact_94_list_OdiscI,axiom,
    ! [List: list_l5168835686021156086lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
      ( ( List
        = ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
     => ( List != nil_li8379291187143545974lambda ) ) ).

% list.discI
thf(fact_95_list_OdiscI,axiom,
    ! [List: list_lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
      ( ( List
        = ( cons_lambda_lambda @ X21 @ X22 ) )
     => ( List != nil_lambda_lambda ) ) ).

% list.discI
thf(fact_96_list_Odistinct_I1_J,axiom,
    ! [X21: set_lambda_lambda,X22: list_s345339197461576016lambda] :
      ( nil_se6914803783601386320lambda
     != ( cons_s3132251510596564736lambda @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_97_list_Odistinct_I1_J,axiom,
    ! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
      ( nil_li8379291187143545974lambda
     != ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_98_list_Odistinct_I1_J,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda] :
      ( nil_lambda_lambda
     != ( cons_lambda_lambda @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_99_reduction__paths_Ostdz__insert_Osimps_I13_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( ( ( lambda_Arr @ T )
          & ( lambda_Arr @ U ) )
       => ( ( reduct5390047737487997688insert @ ( lambda_Beta @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
          = ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_Subst @ zero_zero_nat @ U @ T ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
      & ( ~ ( ( lambda_Arr @ T )
            & ( lambda_Arr @ U ) )
       => ( ( reduct5390047737487997688insert @ ( lambda_Beta @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
          = nil_lambda_lambda ) ) ) ).

% reduction_paths.stdz_insert.simps(13)
thf(fact_100_lambda__calculus_OArr_Osimps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ ( lambda_Beta @ T @ U ) )
      = ( ( lambda_Arr @ T )
        & ( lambda_Arr @ U ) ) ) ).

% lambda_calculus.Arr.simps(5)
thf(fact_101_lambda__calculus_OIde_Osimps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ~ ( lambda_Ide @ ( lambda_Beta @ T @ U ) ) ).

% lambda_calculus.Ide.simps(5)
thf(fact_102_lambda__calculus_OArr__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Arr_Src
thf(fact_103_reduction__paths_Ostandardize_Ocases,axiom,
    ! [X: list_lambda_lambda] :
      ( ( X != nil_lambda_lambda )
     => ~ ! [V3: lambda_lambda,Va2: list_lambda_lambda] :
            ( X
           != ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ).

% reduction_paths.standardize.cases
thf(fact_104_reduction__paths_OStd_Ocases,axiom,
    ! [X: list_lambda_lambda] :
      ( ( X != nil_lambda_lambda )
     => ( ! [T4: lambda_lambda] :
            ( X
           != ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
       => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
              ( X
             != ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ).

% reduction_paths.Std.cases
thf(fact_105_partial__magma__axioms,axiom,
    partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).

% partial_magma_axioms
thf(fact_106__092_060Lambda_062_Opartial__magma__axioms,axiom,
    partia3924075026459072775lambda @ lambda_resid ).

% \<Lambda>.partial_magma_axioms
thf(fact_107_rts__with__composites__axioms,axiom,
    rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).

% rts_with_composites_axioms
thf(fact_108_composable__iff__seq,axiom,
    ! [G: list_lambda_lambda,F: list_lambda_lambda] :
      ( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F )
      = ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F ) ) ).

% composable_iff_seq
thf(fact_109_composable__imp__seq,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% composable_imp_seq
thf(fact_110_has__composites,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% has_composites
thf(fact_111_Ide__iff__standard__development__empty,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Ide @ T )
        = ( ( reduct1603535164923377302opment @ T )
          = nil_lambda_lambda ) ) ) ).

% Ide_iff_standard_development_empty
thf(fact_112_development__Ide,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( lambda_Ide @ T )
     => ( ( reduct9208763521642324371opment @ T @ U3 )
        = ( U3 = nil_lambda_lambda ) ) ) ).

% development_Ide
thf(fact_113_Resid__cons_I1_J,axiom,
    ! [U3: list_lambda_lambda,T: lambda_lambda,T5: list_lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
          = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).

% Resid_cons(1)
thf(fact_114_Resid__rec_I4_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ U3 ) )
           != nil_lambda_lambda )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ U3 ) )
            = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).

% Resid_rec(4)
thf(fact_115_Ide_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( ide_lambda_lambda @ lambda_resid @ T ) ) ).

% Ide.simps(2)
thf(fact_116_ind,axiom,
    ( ( ( lambda_Arr @ m )
      & ( lambda_Arr @ n ) )
   => ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ ua ) )
        & ( reduction_Std @ ( cons_lambda_lambda @ u @ ua ) ) )
     => ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) )
        & ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) @ ( reduct5390047737487997688insert @ ( lambda_Subst @ zero_zero_nat @ n @ m ) @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ).

% ind
thf(fact_117__092_060Lambda_062_Oweak__extensionality,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ T )
       => ( ( ide_lambda_lambda @ lambda_resid @ U )
         => ( T = U ) ) ) ) ).

% \<Lambda>.weak_extensionality
thf(fact_118__092_060Lambda_062_Oprfx__transitive,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
       => ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) ) ) ) ).

% \<Lambda>.prfx_transitive
thf(fact_119__092_060Lambda_062_Oide__backward__stable,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ A2 ) )
       => ( ide_lambda_lambda @ lambda_resid @ T ) ) ) ).

% \<Lambda>.ide_backward_stable
thf(fact_120__092_060Lambda_062_Ocong__transitive,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
     => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) ) )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) ) ) ) ) ).

% \<Lambda>.cong_transitive
thf(fact_121__092_060Lambda_062_Ocong__symmetric,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ) ).

% \<Lambda>.cong_symmetric
thf(fact_122_development__imp__Arr,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct9208763521642324371opment @ T @ U3 )
     => ( lambda_Arr @ T ) ) ).

% development_imp_Arr
thf(fact_123_Std__standard__development,axiom,
    ! [T: lambda_lambda] : ( reduction_Std @ ( reduct1603535164923377302opment @ T ) ) ).

% Std_standard_development
thf(fact_124_prfx__transitive,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
       => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) ) ) ) ).

% prfx_transitive
thf(fact_125_ide__backward__stable,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ A2 ) )
       => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).

% ide_backward_stable
thf(fact_126_cong__transitive,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) ) ) ) ) ).

% cong_transitive
thf(fact_127_cong__symmetric,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ).

% cong_symmetric
thf(fact_128_development__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( reduct9208763521642324371opment @ T @ ( reduct1603535164923377302opment @ T ) ) ) ).

% development_standard_development
thf(fact_129_Resid__append__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( V2 != nil_lambda_lambda )
         => ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
               != nil_lambda_lambda )
              = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 )
                 != nil_lambda_lambda )
                & ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V2 ) )
                 != nil_lambda_lambda ) ) )
            & ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
               != nil_lambda_lambda )
              = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 )
                 != nil_lambda_lambda )
                & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ U3 )
                 != nil_lambda_lambda ) ) )
            & ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
               != nil_lambda_lambda )
             => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
                = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V2 ) ) ) ) )
            & ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
               != nil_lambda_lambda )
             => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
                = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ U3 ) ) ) ) ) ) ) ).

% Resid_append_ind
thf(fact_130_Con__append_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( V2 != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
             != nil_lambda_lambda )
            = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 )
               != nil_lambda_lambda )
              & ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) )
               != nil_lambda_lambda ) ) ) ) ) ) ).

% Con_append(1)
thf(fact_131_Con__append_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( V2 != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
             != nil_lambda_lambda )
            = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
               != nil_lambda_lambda )
              & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ V2 )
               != nil_lambda_lambda ) ) ) ) ) ) ).

% Con_append(2)
thf(fact_132_ide__char,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( paths_7717490669901197737lambda @ lambda_resid @ T5 ) ) ).

% ide_char
thf(fact_133_cong__respects__seq_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T6: list_lambda_lambda,U4: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T6 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T5 ) ) )
       => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U4 ) ) ) ) ).

% cong_respects_seq\<^sub>P
thf(fact_134_cong__respects__seq,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,T2: list_lambda_lambda,U2: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T2 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ T ) ) )
       => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 @ U2 ) ) ) ) ).

% cong_respects_seq
thf(fact_135_append_Oassoc,axiom,
    ! [A2: list_l5168835686021156086lambda,B: list_l5168835686021156086lambda,C: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ A2 @ B ) @ C )
      = ( append1017360404025248779lambda @ A2 @ ( append1017360404025248779lambda @ B @ C ) ) ) ).

% append.assoc
thf(fact_136_append_Oassoc,axiom,
    ! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
      ( ( append_lambda_lambda @ ( append_lambda_lambda @ A2 @ B ) @ C )
      = ( append_lambda_lambda @ A2 @ ( append_lambda_lambda @ B @ C ) ) ) ).

% append.assoc
thf(fact_137_append__assoc,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) @ Zs )
      = ( append1017360404025248779lambda @ Xs @ ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_138_append__assoc,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
      ( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs )
      = ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_139_append__same__eq,axiom,
    ! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Ys @ Xs )
        = ( append1017360404025248779lambda @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_140_append__same__eq,axiom,
    ! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Ys @ Xs )
        = ( append_lambda_lambda @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_141_same__append__eq,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Ys )
        = ( append1017360404025248779lambda @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_142_same__append__eq,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Ys )
        = ( append_lambda_lambda @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_143_complete__developments__cong,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ~ ( lambda_Ide @ T )
     => ( ( reduct7282684938351565978opment @ T @ U3 )
       => ( ( reduct7282684938351565978opment @ T @ V2 )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V2 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) ) ) ) ) ) ).

% complete_developments_cong
thf(fact_144_prfx__decomp,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).

% prfx_decomp
thf(fact_145_complete__development__cong,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct7282684938351565978opment @ T @ U3 )
     => ( ~ ( lambda_Ide @ T )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).

% complete_development_cong
thf(fact_146_head__redex__decomp,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) ) ) ) ) ).

% head_redex_decomp
thf(fact_147_cong__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( ( lambda_Arr @ T )
        & ~ ( lambda_Ide @ T ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( reduct1603535164923377302opment @ T ) ) ) ) ) ).

% cong_standard_development
thf(fact_148_append__is__Nil__conv,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Ys )
        = nil_li8379291187143545974lambda )
      = ( ( Xs = nil_li8379291187143545974lambda )
        & ( Ys = nil_li8379291187143545974lambda ) ) ) ).

% append_is_Nil_conv
thf(fact_149_append__is__Nil__conv,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( ( append2117223597850273125lambda @ Xs @ Ys )
        = nil_se6914803783601386320lambda )
      = ( ( Xs = nil_se6914803783601386320lambda )
        & ( Ys = nil_se6914803783601386320lambda ) ) ) ).

% append_is_Nil_conv
thf(fact_150_append__is__Nil__conv,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Ys )
        = nil_lambda_lambda )
      = ( ( Xs = nil_lambda_lambda )
        & ( Ys = nil_lambda_lambda ) ) ) ).

% append_is_Nil_conv
thf(fact_151_Nil__is__append__conv,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( nil_li8379291187143545974lambda
        = ( append1017360404025248779lambda @ Xs @ Ys ) )
      = ( ( Xs = nil_li8379291187143545974lambda )
        & ( Ys = nil_li8379291187143545974lambda ) ) ) ).

% Nil_is_append_conv
thf(fact_152_Nil__is__append__conv,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( nil_se6914803783601386320lambda
        = ( append2117223597850273125lambda @ Xs @ Ys ) )
      = ( ( Xs = nil_se6914803783601386320lambda )
        & ( Ys = nil_se6914803783601386320lambda ) ) ) ).

% Nil_is_append_conv
thf(fact_153_Nil__is__append__conv,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( nil_lambda_lambda
        = ( append_lambda_lambda @ Xs @ Ys ) )
      = ( ( Xs = nil_lambda_lambda )
        & ( Ys = nil_lambda_lambda ) ) ) ).

% Nil_is_append_conv
thf(fact_154_self__append__conv2,axiom,
    ! [Y4: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
      ( ( Y4
        = ( append1017360404025248779lambda @ Xs @ Y4 ) )
      = ( Xs = nil_li8379291187143545974lambda ) ) ).

% self_append_conv2
thf(fact_155_self__append__conv2,axiom,
    ! [Y4: list_s345339197461576016lambda,Xs: list_s345339197461576016lambda] :
      ( ( Y4
        = ( append2117223597850273125lambda @ Xs @ Y4 ) )
      = ( Xs = nil_se6914803783601386320lambda ) ) ).

% self_append_conv2
thf(fact_156_self__append__conv2,axiom,
    ! [Y4: list_lambda_lambda,Xs: list_lambda_lambda] :
      ( ( Y4
        = ( append_lambda_lambda @ Xs @ Y4 ) )
      = ( Xs = nil_lambda_lambda ) ) ).

% self_append_conv2
thf(fact_157_append__self__conv2,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li8379291187143545974lambda ) ) ).

% append_self_conv2
thf(fact_158_append__self__conv2,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( ( append2117223597850273125lambda @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_se6914803783601386320lambda ) ) ).

% append_self_conv2
thf(fact_159_append__self__conv2,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_lambda_lambda ) ) ).

% append_self_conv2
thf(fact_160_self__append__conv,axiom,
    ! [Y4: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( Y4
        = ( append1017360404025248779lambda @ Y4 @ Ys ) )
      = ( Ys = nil_li8379291187143545974lambda ) ) ).

% self_append_conv
thf(fact_161_self__append__conv,axiom,
    ! [Y4: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( Y4
        = ( append2117223597850273125lambda @ Y4 @ Ys ) )
      = ( Ys = nil_se6914803783601386320lambda ) ) ).

% self_append_conv
thf(fact_162_self__append__conv,axiom,
    ! [Y4: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( Y4
        = ( append_lambda_lambda @ Y4 @ Ys ) )
      = ( Ys = nil_lambda_lambda ) ) ).

% self_append_conv
thf(fact_163_append__self__conv,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li8379291187143545974lambda ) ) ).

% append_self_conv
thf(fact_164_append__self__conv,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( ( append2117223597850273125lambda @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_se6914803783601386320lambda ) ) ).

% append_self_conv
thf(fact_165_append__self__conv,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_lambda_lambda ) ) ).

% append_self_conv
thf(fact_166_append__Nil2,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ Xs @ nil_li8379291187143545974lambda )
      = Xs ) ).

% append_Nil2
thf(fact_167_append__Nil2,axiom,
    ! [Xs: list_s345339197461576016lambda] :
      ( ( append2117223597850273125lambda @ Xs @ nil_se6914803783601386320lambda )
      = Xs ) ).

% append_Nil2
thf(fact_168_append__Nil2,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
      = Xs ) ).

% append_Nil2
thf(fact_169_append_Oright__neutral,axiom,
    ! [A2: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ A2 @ nil_li8379291187143545974lambda )
      = A2 ) ).

% append.right_neutral
thf(fact_170_append_Oright__neutral,axiom,
    ! [A2: list_s345339197461576016lambda] :
      ( ( append2117223597850273125lambda @ A2 @ nil_se6914803783601386320lambda )
      = A2 ) ).

% append.right_neutral
thf(fact_171_append_Oright__neutral,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( append_lambda_lambda @ A2 @ nil_lambda_lambda )
      = A2 ) ).

% append.right_neutral
thf(fact_172_complete__development__def,axiom,
    ( reduct7282684938351565978opment
    = ( ^ [T7: lambda_lambda,U5: list_lambda_lambda] :
          ( ( reduct9208763521642324371opment @ T7 @ U5 )
          & ( ( lambda_Ide @ T7 )
            | ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T7 @ nil_lambda_lambda ) @ U5 ) ) ) ) ) ) ).

% complete_development_def
thf(fact_173__092_060open_062_092_060And_062t_O_Aseq_A_091t_093_A_091_093_A_092_060and_062_AStd_A_091_093_A_092_060longrightarrow_062_AStd_A_Istdz__insert_At_A_091_093_J_A_092_060and_062_A_I_092_060not_062_AIde_A_091t_093_A_092_060longrightarrow_062_Astdz__insert_At_A_091_093_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_091t_093_J_092_060close_062,axiom,
    ! [T: lambda_lambda] :
      ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ nil_lambda_lambda )
        & ( reduction_Std @ nil_lambda_lambda ) )
     => ( ( reduction_Std @ ( reduct5390047737487997688insert @ T @ nil_lambda_lambda ) )
        & ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( reduct5390047737487997688insert @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).

% \<open>\<And>t. seq [t] [] \<and> Std [] \<longrightarrow> Std (stdz_insert t []) \<and> (\<not> Ide [t] \<longrightarrow> stdz_insert t [] \<^sup>*\<sim>\<^sup>* [t])\<close>
thf(fact_174_development__implies,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct9208763521642324371opment @ T @ U3 )
     => ( ( reduct8314398545252454212n_path @ U3 )
        & ( ( U3 != nil_lambda_lambda )
         => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).

% development_implies
thf(fact_175_append1__eq__conv,axiom,
    ! [Xs: list_s345339197461576016lambda,X: set_lambda_lambda,Ys: list_s345339197461576016lambda,Y4: set_lambda_lambda] :
      ( ( ( append2117223597850273125lambda @ Xs @ ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) )
        = ( append2117223597850273125lambda @ Ys @ ( cons_s3132251510596564736lambda @ Y4 @ nil_se6914803783601386320lambda ) ) )
      = ( ( Xs = Ys )
        & ( X = Y4 ) ) ) ).

% append1_eq_conv
thf(fact_176_append1__eq__conv,axiom,
    ! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Y4: list_lambda_lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
        = ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ Y4 @ nil_li8379291187143545974lambda ) ) )
      = ( ( Xs = Ys )
        & ( X = Y4 ) ) ) ).

% append1_eq_conv
thf(fact_177_append1__eq__conv,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda,Ys: list_lambda_lambda,Y4: lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
        = ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ Y4 @ nil_lambda_lambda ) ) )
      = ( ( Xs = Ys )
        & ( X = Y4 ) ) ) ).

% append1_eq_conv
thf(fact_178_lambda__calculus_Oprfx__char,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
      = ( lambda_Ide @ ( lambda_resid @ T @ U ) ) ) ).

% lambda_calculus.prfx_char
thf(fact_179_lambda__calculus_Oide__char,axiom,
    ! [T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ T )
      = ( lambda_Ide @ T ) ) ).

% lambda_calculus.ide_char
thf(fact_180_Resid__append2,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda,W: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( V2 != nil_lambda_lambda )
         => ( ( W != nil_lambda_lambda )
           => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ V2 @ W ) )
               != nil_lambda_lambda )
             => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ V2 @ W ) )
                = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ W ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) ) ) ) ) ) ) ) ) ) ).

% Resid_append2
thf(fact_181_Resid__append_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
          = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) ) ) ) ) ) ).

% Resid_append(1)
thf(fact_182_Resid__append_I2_J,axiom,
    ! [U3: list_lambda_lambda,V2: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( V2 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
           != nil_lambda_lambda )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
            = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ V2 ) ) ) ) ) ).

% Resid_append(2)
thf(fact_183_Con__appendI_I1_J,axiom,
    ! [T5: list_lambda_lambda,V2: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ V2 )
       != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
         != nil_lambda_lambda ) ) ) ).

% Con_appendI(1)
thf(fact_184_Con__appendI_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) @ V2 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
         != nil_lambda_lambda ) ) ) ).

% Con_appendI(2)
thf(fact_185_Resid__single__ide_I2_J,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T5 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
          = T5 ) ) ) ).

% Resid_single_ide(2)
thf(fact_186_cong__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T6: list_lambda_lambda,U4: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T6 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T5 ) ) )
       => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ T6 @ U4 ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U4 ) @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ) ) ).

% cong_append
thf(fact_187_cong__append__ideI_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ U3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ) ).

% cong_append_ideI(1)
thf(fact_188_cong__append__ideI_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ T5 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ) ).

% cong_append_ideI(2)
thf(fact_189_cong__append__ideI_I3_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( append_lambda_lambda @ T5 @ U3 ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ U3 ) ) ) ) ) ).

% cong_append_ideI(3)
thf(fact_190_cong__append__ideI_I4_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( append_lambda_lambda @ T5 @ U3 ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ T5 ) ) ) ) ) ).

% cong_append_ideI(4)
thf(fact_191_cong__cons,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda,T2: lambda_lambda,U4: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
     => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T2 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T ) ) )
       => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ ( cons_lambda_lambda @ T2 @ U4 ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ U4 ) @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ) ) ).

% cong_cons
thf(fact_192_cong__cons__ideI_I1_J,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
     => ( ( ide_lambda_lambda @ lambda_resid @ T )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ U3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ) ).

% cong_cons_ideI(1)
thf(fact_193_cong__cons__ideI_I2_J,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
     => ( ( ide_lambda_lambda @ lambda_resid @ T )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ U3 ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ U3 ) ) ) ) ) ).

% cong_cons_ideI(2)
thf(fact_194_complete__development__bottom__up__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ).

% complete_development_bottom_up_development
thf(fact_195_append__eq__appendI,axiom,
    ! [Xs: list_l5168835686021156086lambda,Xs1: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append1017360404025248779lambda @ Xs1 @ Us ) )
       => ( ( append1017360404025248779lambda @ Xs @ Ys )
          = ( append1017360404025248779lambda @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_196_append__eq__appendI,axiom,
    ! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_lambda_lambda @ Xs1 @ Us ) )
       => ( ( append_lambda_lambda @ Xs @ Ys )
          = ( append_lambda_lambda @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_197_append__eq__append__conv2,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ts: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Xs @ Ys )
        = ( append1017360404025248779lambda @ Zs @ Ts ) )
      = ( ? [Us2: list_l5168835686021156086lambda] :
            ( ( ( Xs
                = ( append1017360404025248779lambda @ Zs @ Us2 ) )
              & ( ( append1017360404025248779lambda @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append1017360404025248779lambda @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append1017360404025248779lambda @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_198_append__eq__append__conv2,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Xs @ Ys )
        = ( append_lambda_lambda @ Zs @ Ts ) )
      = ( ? [Us2: list_lambda_lambda] :
            ( ( ( Xs
                = ( append_lambda_lambda @ Zs @ Us2 ) )
              & ( ( append_lambda_lambda @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_lambda_lambda @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_lambda_lambda @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_199_lambda__calculus_Olambda_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,Lambda: lambda_lambda] :
      ( ( H @ ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ Lambda ) )
      = ( lambda_case_lambda_o @ ( H @ F1 )
        @ ^ [X3: nat] : ( H @ ( F2 @ X3 ) )
        @ ^ [X3: lambda_lambda] : ( H @ ( F3 @ X3 ) )
        @ ^ [X1: lambda_lambda,X23: lambda_lambda] : ( H @ ( F4 @ X1 @ X23 ) )
        @ ^ [X1: lambda_lambda,X23: lambda_lambda] : ( H @ ( F5 @ X1 @ X23 ) )
        @ Lambda ) ) ).

% lambda_calculus.lambda.case_distrib
thf(fact_200_append__Cons,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ Ys )
      = ( cons_l4558873032622161958lambda @ X @ ( append1017360404025248779lambda @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_201_append__Cons,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( append_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) @ Ys )
      = ( cons_lambda_lambda @ X @ ( append_lambda_lambda @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_202_Cons__eq__appendI,axiom,
    ! [X: list_lambda_lambda,Xs1: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
      ( ( ( cons_l4558873032622161958lambda @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append1017360404025248779lambda @ Xs1 @ Zs ) )
       => ( ( cons_l4558873032622161958lambda @ X @ Xs )
          = ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_203_Cons__eq__appendI,axiom,
    ! [X: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
      ( ( ( cons_lambda_lambda @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_lambda_lambda @ Xs1 @ Zs ) )
       => ( ( cons_lambda_lambda @ X @ Xs )
          = ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_204_eq__Nil__appendI,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_205_eq__Nil__appendI,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_206_eq__Nil__appendI,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_lambda_lambda @ nil_lambda_lambda @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_207_append_Oleft__neutral,axiom,
    ! [A2: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_208_append_Oleft__neutral,axiom,
    ! [A2: list_s345339197461576016lambda] :
      ( ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_209_append_Oleft__neutral,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( append_lambda_lambda @ nil_lambda_lambda @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_210_append__Nil,axiom,
    ! [Ys: list_l5168835686021156086lambda] :
      ( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_211_append__Nil,axiom,
    ! [Ys: list_s345339197461576016lambda] :
      ( ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_212_append__Nil,axiom,
    ! [Ys: list_lambda_lambda] :
      ( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_213_lambda__calculus_Oeq__Ide__are__cong,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( T = U )
     => ( ( lambda_Ide @ T )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ).

% lambda_calculus.eq_Ide_are_cong
thf(fact_214_lambda__calculus_Ocong__Ide__are__eq,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
     => ( ( lambda_Ide @ T )
       => ( ( lambda_Ide @ U )
         => ( T = U ) ) ) ) ).

% lambda_calculus.cong_Ide_are_eq
thf(fact_215_rev__nonempty__induct,axiom,
    ! [Xs: list_s345339197461576016lambda,P: list_s345339197461576016lambda > $o] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ( ! [X2: set_lambda_lambda] : ( P @ ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) )
       => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
              ( ( Xs2 != nil_se6914803783601386320lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( append2117223597850273125lambda @ Xs2 @ ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_216_rev__nonempty__induct,axiom,
    ! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( ! [X2: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) )
       => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
              ( ( Xs2 != nil_li8379291187143545974lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_217_rev__nonempty__induct,axiom,
    ! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
      ( ( Xs != nil_lambda_lambda )
     => ( ! [X2: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) )
       => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] :
              ( ( Xs2 != nil_lambda_lambda )
             => ( ( P @ Xs2 )
               => ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_218_append__eq__Cons__conv,axiom,
    ! [Ys: list_s345339197461576016lambda,Zs: list_s345339197461576016lambda,X: set_lambda_lambda,Xs: list_s345339197461576016lambda] :
      ( ( ( append2117223597850273125lambda @ Ys @ Zs )
        = ( cons_s3132251510596564736lambda @ X @ Xs ) )
      = ( ( ( Ys = nil_se6914803783601386320lambda )
          & ( Zs
            = ( cons_s3132251510596564736lambda @ X @ Xs ) ) )
        | ? [Ys4: list_s345339197461576016lambda] :
            ( ( Ys
              = ( cons_s3132251510596564736lambda @ X @ Ys4 ) )
            & ( ( append2117223597850273125lambda @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_219_append__eq__Cons__conv,axiom,
    ! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( ( append1017360404025248779lambda @ Ys @ Zs )
        = ( cons_l4558873032622161958lambda @ X @ Xs ) )
      = ( ( ( Ys = nil_li8379291187143545974lambda )
          & ( Zs
            = ( cons_l4558873032622161958lambda @ X @ Xs ) ) )
        | ? [Ys4: list_l5168835686021156086lambda] :
            ( ( Ys
              = ( cons_l4558873032622161958lambda @ X @ Ys4 ) )
            & ( ( append1017360404025248779lambda @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_220_append__eq__Cons__conv,axiom,
    ! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( ( append_lambda_lambda @ Ys @ Zs )
        = ( cons_lambda_lambda @ X @ Xs ) )
      = ( ( ( Ys = nil_lambda_lambda )
          & ( Zs
            = ( cons_lambda_lambda @ X @ Xs ) ) )
        | ? [Ys4: list_lambda_lambda] :
            ( ( Ys
              = ( cons_lambda_lambda @ X @ Ys4 ) )
            & ( ( append_lambda_lambda @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_221_Cons__eq__append__conv,axiom,
    ! [X: set_lambda_lambda,Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda,Zs: list_s345339197461576016lambda] :
      ( ( ( cons_s3132251510596564736lambda @ X @ Xs )
        = ( append2117223597850273125lambda @ Ys @ Zs ) )
      = ( ( ( Ys = nil_se6914803783601386320lambda )
          & ( ( cons_s3132251510596564736lambda @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_s345339197461576016lambda] :
            ( ( ( cons_s3132251510596564736lambda @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append2117223597850273125lambda @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_222_Cons__eq__append__conv,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
      ( ( ( cons_l4558873032622161958lambda @ X @ Xs )
        = ( append1017360404025248779lambda @ Ys @ Zs ) )
      = ( ( ( Ys = nil_li8379291187143545974lambda )
          & ( ( cons_l4558873032622161958lambda @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_l5168835686021156086lambda] :
            ( ( ( cons_l4558873032622161958lambda @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append1017360404025248779lambda @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_223_Cons__eq__append__conv,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
      ( ( ( cons_lambda_lambda @ X @ Xs )
        = ( append_lambda_lambda @ Ys @ Zs ) )
      = ( ( ( Ys = nil_lambda_lambda )
          & ( ( cons_lambda_lambda @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_lambda_lambda] :
            ( ( ( cons_lambda_lambda @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_224_rev__exhaust,axiom,
    ! [Xs: list_s345339197461576016lambda] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ~ ! [Ys2: list_s345339197461576016lambda,Y2: set_lambda_lambda] :
            ( Xs
           != ( append2117223597850273125lambda @ Ys2 @ ( cons_s3132251510596564736lambda @ Y2 @ nil_se6914803783601386320lambda ) ) ) ) ).

% rev_exhaust
thf(fact_225_rev__exhaust,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ~ ! [Ys2: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
            ( Xs
           != ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ Y2 @ nil_li8379291187143545974lambda ) ) ) ) ).

% rev_exhaust
thf(fact_226_rev__exhaust,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ~ ! [Ys2: list_lambda_lambda,Y2: lambda_lambda] :
            ( Xs
           != ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) ) ) ).

% rev_exhaust
thf(fact_227_rev__induct,axiom,
    ! [P: list_s345339197461576016lambda > $o,Xs: list_s345339197461576016lambda] :
      ( ( P @ nil_se6914803783601386320lambda )
     => ( ! [X2: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
            ( ( P @ Xs2 )
           => ( P @ ( append2117223597850273125lambda @ Xs2 @ ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_228_rev__induct,axiom,
    ! [P: list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda] :
      ( ( P @ nil_li8379291187143545974lambda )
     => ( ! [X2: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
            ( ( P @ Xs2 )
           => ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_229_rev__induct,axiom,
    ! [P: list_lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( P @ nil_lambda_lambda )
     => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda] :
            ( ( P @ Xs2 )
           => ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_230_reduction__paths_Ostdz__insert_Osimps_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( reduct5390047737487997688insert @ T @ nil_lambda_lambda )
      = ( reduct1603535164923377302opment @ T ) ) ).

% reduction_paths.stdz_insert.simps(1)
thf(fact_231_reduction__paths_Odevelopment_Osimps_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( reduct9208763521642324371opment @ T @ nil_lambda_lambda )
      = ( lambda_Arr @ T ) ) ).

% reduction_paths.development.simps(1)
thf(fact_232_reduction__paths_OBeta__decomp_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ U @ T ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).

% reduction_paths.Beta_decomp(1)
thf(fact_233_reduction__paths_Ostandard__development_Osimps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( ( lambda_Arr @ T )
          & ( lambda_Arr @ U ) )
       => ( ( reduct1603535164923377302opment @ ( lambda_Beta @ T @ U ) )
          = ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ ( reduct1603535164923377302opment @ ( lambda_Subst @ zero_zero_nat @ U @ T ) ) ) ) )
      & ( ~ ( ( lambda_Arr @ T )
            & ( lambda_Arr @ U ) )
       => ( ( reduct1603535164923377302opment @ ( lambda_Beta @ T @ U ) )
          = nil_lambda_lambda ) ) ) ).

% reduction_paths.standard_development.simps(5)
thf(fact_234__092_060open_062_092_060And_062u_AU_O_Aseq_A_091_092_060_094bold_062_092_060sharp_062_093_A_Iu_A_D_AU_J_A_092_060and_062_AStd_A_Iu_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_092_060_094bold_062_092_060sharp_062_A_Iu_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_092_060_094bold_062_092_060sharp_062_A_D_Au_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_092_060_094bold_062_092_060sharp_062_A_Iu_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_092_060_094bold_062_092_060sharp_062_A_D_Au_A_D_AU_J_092_060close_062,axiom,
    ! [U: lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ lambda_Nil @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
        & ( reduction_Std @ ( cons_lambda_lambda @ U @ U3 ) ) )
     => ( ( reduction_Std @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) )
        & ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U3 ) ) ) ) ) ) ) ) ).

% \<open>\<And>u U. seq [\<^bold>\<sharp>] (u # U) \<and> Std (u # U) \<longrightarrow> Std (stdz_insert \<^bold>\<sharp> (u # U)) \<and> (\<not> Ide (\<^bold>\<sharp> # u # U) \<longrightarrow> stdz_insert \<^bold>\<sharp> (u # U) \<^sup>*\<sim>\<^sup>* \<^bold>\<sharp> # u # U)\<close>
thf(fact_235_cong__implies__coterminal,axiom,
    ! [U: list_lambda_lambda,U2: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
     => ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U2 ) ) ).

% cong_implies_coterminal
thf(fact_236_identities__form__coherent__normal__sub__rts,axiom,
    cohere6622258358606131289lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( collec8830145823237851355lambda @ ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).

% identities_form_coherent_normal_sub_rts
thf(fact_237_paths__in__confluent__rts_Ois__confluent__rts,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ( paths_3043240146134862789lambda @ Resid )
     => ( conflu1940213901339057510lambda @ ( paths_3312059784463760222lambda @ Resid ) ) ) ).

% paths_in_confluent_rts.is_confluent_rts
thf(fact_238_paths__in__confluent__rts_Ois__confluent__rts,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ( paths_9040249923598999359lambda @ Resid )
     => ( conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ Resid ) ) ) ).

% paths_in_confluent_rts.is_confluent_rts
thf(fact_239__092_060Lambda_062_Oidentities__form__coherent__normal__sub__rts,axiom,
    cohere1690309856004283859lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ).

% \<Lambda>.identities_form_coherent_normal_sub_rts
thf(fact_240__092_060Lambda_062_Ocong__implies__coterminal,axiom,
    ! [U: lambda_lambda,U2: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
     => ( coterm4142657152830665724lambda @ lambda_resid @ U @ U2 ) ) ).

% \<Lambda>.cong_implies_coterminal
thf(fact_241_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list7041979136695072894lambda @ N @ nil_li8379291187143545974lambda )
          = ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ nil_li6069905736445159420lambda ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list7041979136695072894lambda @ N @ nil_li8379291187143545974lambda )
          = nil_li6069905736445159420lambda ) ) ) ).

% n_lists_Nil
thf(fact_242_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list5221165152849976152lambda @ N @ nil_se6914803783601386320lambda )
          = ( cons_l203434155165082374lambda @ nil_se6914803783601386320lambda @ nil_li3187382615321999190lambda ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list5221165152849976152lambda @ N @ nil_se6914803783601386320lambda )
          = nil_li3187382615321999190lambda ) ) ) ).

% n_lists_Nil
thf(fact_243_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list4163717777911954552lambda @ N @ nil_lambda_lambda )
          = ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list4163717777911954552lambda @ N @ nil_lambda_lambda )
          = nil_li8379291187143545974lambda ) ) ) ).

% n_lists_Nil
thf(fact_244_rts__with__composites_Ohas__composites,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( rts_wi3086218056445523938lambda @ Resid )
     => ( ( seq_lambda_lambda @ Resid @ T @ U )
       => ( compos7127019122090555255lambda @ Resid @ T @ U ) ) ) ).

% rts_with_composites.has_composites
thf(fact_245_rts__with__composites_Ohas__composites,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( rts_wi1553966559393445608lambda @ Resid )
     => ( ( seq_li8093550137657879445lambda @ Resid @ T @ U )
       => ( compos3359172601056258045lambda @ Resid @ T @ U ) ) ) ).

% rts_with_composites.has_composites
thf(fact_246_rts__with__composites_Ocomposable__iff__seq,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F: lambda_lambda] :
      ( ( rts_wi3086218056445523938lambda @ Resid )
     => ( ( compos7127019122090555255lambda @ Resid @ G @ F )
        = ( seq_lambda_lambda @ Resid @ G @ F ) ) ) ).

% rts_with_composites.composable_iff_seq
thf(fact_247_rts__with__composites_Ocomposable__iff__seq,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F: list_lambda_lambda] :
      ( ( rts_wi1553966559393445608lambda @ Resid )
     => ( ( compos3359172601056258045lambda @ Resid @ G @ F )
        = ( seq_li8093550137657879445lambda @ Resid @ G @ F ) ) ) ).

% rts_with_composites.composable_iff_seq
thf(fact_248_arr__append__imp__seq,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T5 @ U3 ) )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).

% arr_append_imp_seq
thf(fact_249__092_060Lambda_062_Ocomposable__imp__seq,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
     => ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.composable_imp_seq
thf(fact_250__092_060Lambda_062_Ocong__respects__seq,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,T2: lambda_lambda,U2: lambda_lambda] :
      ( ( seq_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T2 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T ) ) )
       => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
         => ( seq_lambda_lambda @ lambda_resid @ T2 @ U2 ) ) ) ) ).

% \<Lambda>.cong_respects_seq
thf(fact_251_cong__reflexive,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ).

% cong_reflexive
thf(fact_252_ide__implies__arr,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).

% ide_implies_arr
thf(fact_253_prfx__reflexive,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ).

% prfx_reflexive
thf(fact_254_composableD_I2_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ).

% composableD(2)
thf(fact_255_composableD_I1_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).

% composableD(1)
thf(fact_256_lambda__calculus_OConI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( ( lambda_Src @ T )
            = ( lambda_Src @ U ) )
         => ( ( lambda_resid @ T @ U )
           != lambda_Nil ) ) ) ) ).

% lambda_calculus.ConI
thf(fact_257_arr__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ).

% arr_append
thf(fact_258_rts_Ocoterminal_Ocong,axiom,
    coterm81126318419254786lambda = coterm81126318419254786lambda ).

% rts.coterminal.cong
thf(fact_259_rts_Ocoterminal_Ocong,axiom,
    coterm4142657152830665724lambda = coterm4142657152830665724lambda ).

% rts.coterminal.cong
thf(fact_260_residuation_Oarr_Ocong,axiom,
    arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).

% residuation.arr.cong
thf(fact_261_residuation_Oarr_Ocong,axiom,
    arr_lambda_lambda = arr_lambda_lambda ).

% residuation.arr.cong
thf(fact_262_lambda__calculus_Olambda_Odisc__eq__case_I1_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( Lambda = lambda_Nil )
      = ( lambda_case_lambda_o @ $true
        @ ^ [Uu2: nat] : $false
        @ ^ [Uu2: lambda_lambda] : $false
        @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $false
        @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $false
        @ Lambda ) ) ).

% lambda_calculus.lambda.disc_eq_case(1)
thf(fact_263_lambda__calculus_Oresid_Osimps_I26_J,axiom,
    ! [Uu: lambda_lambda] :
      ( ( lambda_resid @ Uu @ lambda_Nil )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(26)
thf(fact_264_lambda__calculus_Oresid_Osimps_I7_J,axiom,
    ! [Uv2: lambda_lambda] :
      ( ( lambda_resid @ lambda_Nil @ Uv2 )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(7)
thf(fact_265_lambda__calculus_OCube,axiom,
    ! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
       != lambda_Nil )
     => ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
        = ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).

% lambda_calculus.Cube
thf(fact_266_lambda__calculus_OCon__sym,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
      = ( ( lambda_resid @ U @ T )
       != lambda_Nil ) ) ).

% lambda_calculus.Con_sym
thf(fact_267_lambda__calculus_Olambda_Odistinct_I7_J,axiom,
    ! [X51: lambda_lambda,X52: lambda_lambda] :
      ( lambda_Nil
     != ( lambda_Beta @ X51 @ X52 ) ) ).

% lambda_calculus.lambda.distinct(7)
thf(fact_268_lambda__calculus_OIde_Osimps_I1_J,axiom,
    ~ ( lambda_Ide @ lambda_Nil ) ).

% lambda_calculus.Ide.simps(1)
thf(fact_269_lambda__calculus_OArr_Osimps_I1_J,axiom,
    ~ ( lambda_Arr @ lambda_Nil ) ).

% lambda_calculus.Arr.simps(1)
thf(fact_270_lambda__calculus_OArr__not__Nil,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( T != lambda_Nil ) ) ).

% lambda_calculus.Arr_not_Nil
thf(fact_271_lambda__calculus_OSrc_Osimps_I1_J,axiom,
    ( ( lambda_Src @ lambda_Nil )
    = lambda_Nil ) ).

% lambda_calculus.Src.simps(1)
thf(fact_272_lambda__calculus_Ohead__strategy_Osimps_I9_J,axiom,
    ( ( lambda_head_strategy @ lambda_Nil )
    = lambda_Nil ) ).

% lambda_calculus.head_strategy.simps(9)
thf(fact_273_lambda__calculus_Ohead__redex_Osimps_I1_J,axiom,
    ( ( lambda_head_redex @ lambda_Nil )
    = lambda_Nil ) ).

% lambda_calculus.head_redex.simps(1)
thf(fact_274_lambda__calculus_OSubst_Osimps_I1_J,axiom,
    ! [Uu: nat,Uv2: lambda_lambda] :
      ( ( lambda_Subst @ Uu @ Uv2 @ lambda_Nil )
      = lambda_Nil ) ).

% lambda_calculus.Subst.simps(1)
thf(fact_275_lambda__calculus_OSubst__Nil,axiom,
    ! [N: nat,V: lambda_lambda] :
      ( ( lambda_Subst @ N @ V @ lambda_Nil )
      = lambda_Nil ) ).

% lambda_calculus.Subst_Nil
thf(fact_276_lambda__calculus_OSubst__not__Nil,axiom,
    ! [V: lambda_lambda,T: lambda_lambda,N: nat] :
      ( ( V != lambda_Nil )
     => ( ( T != lambda_Nil )
       => ( ( T != lambda_Nil )
         => ( ( lambda_Subst @ N @ V @ T )
           != lambda_Nil ) ) ) ) ).

% lambda_calculus.Subst_not_Nil
thf(fact_277_lambda__calculus_Olambda_Osimps_I25_J,axiom,
    ! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o] :
      ( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ lambda_Nil )
      = F1 ) ).

% lambda_calculus.lambda.simps(25)
thf(fact_278_lambda__calculus_Oresid_Osimps_I19_J,axiom,
    ! [V: lambda_lambda,Va: lambda_lambda] :
      ( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(19)
thf(fact_279_lambda__calculus_Oresid_Osimps_I38_J,axiom,
    ! [V: lambda_lambda,Va: lambda_lambda] :
      ( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(38)
thf(fact_280_lambda__calculus_OArr__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
     => ( lambda_Arr @ ( lambda_resid @ T @ U ) ) ) ).

% lambda_calculus.Arr_resid
thf(fact_281_lambda__calculus_OCon__implies__Arr1,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
     => ( lambda_Arr @ T ) ) ).

% lambda_calculus.Con_implies_Arr1
thf(fact_282_lambda__calculus_OCon__implies__Arr2,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
     => ( lambda_Arr @ U ) ) ).

% lambda_calculus.Con_implies_Arr2
thf(fact_283_lambda__calculus_Oresid__Subst,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda,N: nat] :
      ( ( ( lambda_resid @ T @ T2 )
       != lambda_Nil )
     => ( ( ( lambda_resid @ U @ U2 )
         != lambda_Nil )
       => ( ( lambda_resid @ ( lambda_Subst @ N @ T @ U ) @ ( lambda_Subst @ N @ T2 @ U2 ) )
          = ( lambda_Subst @ N @ ( lambda_resid @ T @ T2 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).

% lambda_calculus.resid_Subst
thf(fact_284_reduction__paths_Ostandard__development_Osimps_I1_J,axiom,
    ( ( reduct1603535164923377302opment @ lambda_Nil )
    = nil_lambda_lambda ) ).

% reduction_paths.standard_development.simps(1)
thf(fact_285_paths__in__rts_OResid_Ocong,axiom,
    paths_3312059784463760222lambda = paths_3312059784463760222lambda ).

% paths_in_rts.Resid.cong
thf(fact_286_paths__in__rts_OResid_Ocong,axiom,
    paths_8607357831768207192lambda = paths_8607357831768207192lambda ).

% paths_in_rts.Resid.cong
thf(fact_287_reduction__paths_Ostdz__insert_Osimps_I14_J,axiom,
    ! [V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ V @ Va ) )
      = nil_lambda_lambda ) ).

% reduction_paths.stdz_insert.simps(14)
thf(fact_288_residuation_Oide_Ocong,axiom,
    ide_lambda_lambda = ide_lambda_lambda ).

% residuation.ide.cong
thf(fact_289_residuation_Oide_Ocong,axiom,
    ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).

% residuation.ide.cong
thf(fact_290_lambda__calculus_OCon__Arr__Src_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
       != lambda_Nil ) ) ).

% lambda_calculus.Con_Arr_Src(2)
thf(fact_291_lambda__calculus_OCon__Arr__Src_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
       != lambda_Nil ) ) ).

% lambda_calculus.Con_Arr_Src(1)
thf(fact_292_lambda__calculus_OCon__implies__Coinitial3_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
       != lambda_Nil )
     => ( ( lambda_Arr @ U )
        & ( lambda_Arr @ T )
        & ( ( lambda_Src @ U )
          = ( lambda_Src @ T ) ) ) ) ).

% lambda_calculus.Con_implies_Coinitial3(3)
thf(fact_293_lambda__calculus_OCon__implies__Coinitial3_I2_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
       != lambda_Nil )
     => ( ( lambda_Arr @ V )
        & ( lambda_Arr @ T )
        & ( ( lambda_Src @ V )
          = ( lambda_Src @ T ) ) ) ) ).

% lambda_calculus.Con_implies_Coinitial3(2)
thf(fact_294_lambda__calculus_OCon__implies__Coinitial3_I1_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
       != lambda_Nil )
     => ( ( lambda_Arr @ V )
        & ( lambda_Arr @ U )
        & ( ( lambda_Src @ V )
          = ( lambda_Src @ U ) ) ) ) ).

% lambda_calculus.Con_implies_Coinitial3(1)
thf(fact_295_lambda__calculus_OCoinitial__iff__Con,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_Arr @ T )
        & ( lambda_Arr @ U )
        & ( ( lambda_Src @ T )
          = ( lambda_Src @ U ) ) )
      = ( ( lambda_resid @ T @ U )
       != lambda_Nil ) ) ).

% lambda_calculus.Coinitial_iff_Con
thf(fact_296_lambda__calculus_OCoinitial__resid__resid,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ V )
       != lambda_Nil )
     => ( ( ( lambda_resid @ U @ V )
         != lambda_Nil )
       => ( ( lambda_Arr @ ( lambda_resid @ T @ V ) )
          & ( lambda_Arr @ ( lambda_resid @ U @ V ) )
          & ( ( lambda_Src @ ( lambda_resid @ T @ V ) )
            = ( lambda_Src @ ( lambda_resid @ U @ V ) ) ) ) ) ) ).

% lambda_calculus.Coinitial_resid_resid
thf(fact_297_lambda__calculus_OCon__head__strategy,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ T @ ( lambda_head_strategy @ T ) )
       != lambda_Nil ) ) ).

% lambda_calculus.Con_head_strategy
thf(fact_298_lambda__calculus_OCon__Arr__head__redex,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ T @ ( lambda_head_redex @ T ) )
       != lambda_Nil ) ) ).

% lambda_calculus.Con_Arr_head_redex
thf(fact_299_rts_Oseq_Ocong,axiom,
    seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).

% rts.seq.cong
thf(fact_300_rts_Oseq_Ocong,axiom,
    seq_lambda_lambda = seq_lambda_lambda ).

% rts.seq.cong
thf(fact_301_paths__in__rts_OIde_Ocong,axiom,
    paths_7717490669901197737lambda = paths_7717490669901197737lambda ).

% paths_in_rts.Ide.cong
thf(fact_302_rts_Ocomposable_Ocong,axiom,
    compos3359172601056258045lambda = compos3359172601056258045lambda ).

% rts.composable.cong
thf(fact_303_rts_Ocomposable_Ocong,axiom,
    compos7127019122090555255lambda = compos7127019122090555255lambda ).

% rts.composable.cong
thf(fact_304_lambda__calculus_Oresid_Osimps_I4_J,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
      ( ( ( ( ( lambda_resid @ T @ T2 )
            = lambda_Nil )
          | ( ( lambda_resid @ U @ U2 )
            = lambda_Nil ) )
       => ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
          = lambda_Nil ) )
      & ( ~ ( ( ( lambda_resid @ T @ T2 )
              = lambda_Nil )
            | ( ( lambda_resid @ U @ U2 )
              = lambda_Nil ) )
       => ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
          = ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U2 ) @ ( lambda_resid @ T @ T2 ) ) ) ) ) ).

% lambda_calculus.resid.simps(4)
thf(fact_305_partial__magma_Ointro,axiom,
    ! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ? [X4: list_lambda_lambda] :
          ( ! [T4: list_lambda_lambda] :
              ( ( ( OP @ X4 @ T4 )
                = X4 )
              & ( ( OP @ T4 @ X4 )
                = X4 ) )
          & ! [Y2: list_lambda_lambda] :
              ( ! [T3: list_lambda_lambda] :
                  ( ( ( OP @ Y2 @ T3 )
                    = Y2 )
                  & ( ( OP @ T3 @ Y2 )
                    = Y2 ) )
             => ( Y2 = X4 ) ) )
     => ( partia748546660539056525lambda @ OP ) ) ).

% partial_magma.intro
thf(fact_306_partial__magma_Ointro,axiom,
    ! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ? [X4: lambda_lambda] :
          ( ! [T4: lambda_lambda] :
              ( ( ( OP @ X4 @ T4 )
                = X4 )
              & ( ( OP @ T4 @ X4 )
                = X4 ) )
          & ! [Y2: lambda_lambda] :
              ( ! [T3: lambda_lambda] :
                  ( ( ( OP @ Y2 @ T3 )
                    = Y2 )
                  & ( ( OP @ T3 @ Y2 )
                    = Y2 ) )
             => ( Y2 = X4 ) ) )
     => ( partia3924075026459072775lambda @ OP ) ) ).

% partial_magma.intro
thf(fact_307_partial__magma_Oex__un__null,axiom,
    ! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ( partia748546660539056525lambda @ OP )
     => ? [X2: list_lambda_lambda] :
          ( ! [T3: list_lambda_lambda] :
              ( ( ( OP @ X2 @ T3 )
                = X2 )
              & ( ( OP @ T3 @ X2 )
                = X2 ) )
          & ! [Y: list_lambda_lambda] :
              ( ! [T4: list_lambda_lambda] :
                  ( ( ( OP @ Y @ T4 )
                    = Y )
                  & ( ( OP @ T4 @ Y )
                    = Y ) )
             => ( Y = X2 ) ) ) ) ).

% partial_magma.ex_un_null
thf(fact_308_partial__magma_Oex__un__null,axiom,
    ! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ( partia3924075026459072775lambda @ OP )
     => ? [X2: lambda_lambda] :
          ( ! [T3: lambda_lambda] :
              ( ( ( OP @ X2 @ T3 )
                = X2 )
              & ( ( OP @ T3 @ X2 )
                = X2 ) )
          & ! [Y: lambda_lambda] :
              ( ! [T4: lambda_lambda] :
                  ( ( ( OP @ Y @ T4 )
                    = Y )
                  & ( ( OP @ T4 @ Y )
                    = Y ) )
             => ( Y = X2 ) ) ) ) ).

% partial_magma.ex_un_null
thf(fact_309_partial__magma__def,axiom,
    ( partia748546660539056525lambda
    = ( ^ [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
        ? [X3: list_lambda_lambda] :
          ( ! [T7: list_lambda_lambda] :
              ( ( ( OP2 @ X3 @ T7 )
                = X3 )
              & ( ( OP2 @ T7 @ X3 )
                = X3 ) )
          & ! [Y3: list_lambda_lambda] :
              ( ! [T7: list_lambda_lambda] :
                  ( ( ( OP2 @ Y3 @ T7 )
                    = Y3 )
                  & ( ( OP2 @ T7 @ Y3 )
                    = Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% partial_magma_def
thf(fact_310_partial__magma__def,axiom,
    ( partia3924075026459072775lambda
    = ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
        ? [X3: lambda_lambda] :
          ( ! [T7: lambda_lambda] :
              ( ( ( OP2 @ X3 @ T7 )
                = X3 )
              & ( ( OP2 @ T7 @ X3 )
                = X3 ) )
          & ! [Y3: lambda_lambda] :
              ( ! [T7: lambda_lambda] :
                  ( ( ( OP2 @ Y3 @ T7 )
                    = Y3 )
                  & ( ( OP2 @ T7 @ Y3 )
                    = Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% partial_magma_def
thf(fact_311_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( n_list7041979136695072894lambda @ zero_zero_nat @ Xs )
      = ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ nil_li6069905736445159420lambda ) ) ).

% n_lists.simps(1)
thf(fact_312_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_s345339197461576016lambda] :
      ( ( n_list5221165152849976152lambda @ zero_zero_nat @ Xs )
      = ( cons_l203434155165082374lambda @ nil_se6914803783601386320lambda @ nil_li3187382615321999190lambda ) ) ).

% n_lists.simps(1)
thf(fact_313_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( n_list4163717777911954552lambda @ zero_zero_nat @ Xs )
      = ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda ) ) ).

% n_lists.simps(1)
thf(fact_314_paths__in__confluent__rts_Oaxioms_I2_J,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ( paths_3043240146134862789lambda @ Resid )
     => ( conflu4036547884742428384lambda @ Resid ) ) ).

% paths_in_confluent_rts.axioms(2)
thf(fact_315_paths__in__confluent__rts_Oaxioms_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ( paths_9040249923598999359lambda @ Resid )
     => ( conflu4927472611300549594lambda @ Resid ) ) ).

% paths_in_confluent_rts.axioms(2)
thf(fact_316_Src__hd__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ~ ( lambda_Ide @ T )
       => ( ( lambda_Src @ ( hd_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
          = ( lambda_Src @ T ) ) ) ) ).

% Src_hd_standard_development
thf(fact_317_Src__hd__eqI,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) ) )
     => ( ( lambda_Src @ ( hd_lambda_lambda @ T5 ) )
        = ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) ) ).

% Src_hd_eqI
thf(fact_318_Resid__parallel,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ T ) ) )
     => ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
          = ( paths_8607357831768207192lambda @ lambda_resid @ U @ T2 ) ) ) ) ).

% Resid_parallel
thf(fact_319_coinitial__ide__are__cong,axiom,
    ! [A2: list_lambda_lambda,A3: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
       => ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 )
         => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) ) ) ) ) ).

% coinitial_ide_are_cong
thf(fact_320_cong__implies__coinitial,axiom,
    ! [U: list_lambda_lambda,U2: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
     => ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U2 ) ) ).

% cong_implies_coinitial
thf(fact_321_coterminalE,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
         => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
           => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
             != ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).

% coterminalE
thf(fact_322_coterminal__iff,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda] :
      ( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T2 )
      = ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        & ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 )
        & ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
          = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) ) ) ) ).

% coterminal_iff
thf(fact_323_filter__notIde__Ide,axiom,
    ! [U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
     => ( ( filter_lambda_lambda
          @ ^ [U6: lambda_lambda] :
              ~ ( lambda_Ide @ U6 )
          @ U3 )
        = nil_lambda_lambda ) ) ).

% filter_notIde_Ide
thf(fact_324__092_060Lambda_062_Oprfx__reflexive,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) ) ) ).

% \<Lambda>.prfx_reflexive
thf(fact_325__092_060Lambda_062_Oide__implies__arr,axiom,
    ! [A2: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( arr_lambda_lambda @ lambda_resid @ A2 ) ) ).

% \<Lambda>.ide_implies_arr
thf(fact_326__092_060Lambda_062_Ocong__reflexive,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) ) ) ) ).

% \<Lambda>.cong_reflexive
thf(fact_327__092_060Lambda_062_OcomposableD_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
     => ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% \<Lambda>.composableD(1)
thf(fact_328__092_060Lambda_062_OcomposableD_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
     => ( arr_lambda_lambda @ lambda_resid @ U ) ) ).

% \<Lambda>.composableD(2)
thf(fact_329_targets__cong__closed,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B2 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B ) ) )
       => ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).

% targets_cong_closed
thf(fact_330_targets__are__cong,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B2 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B ) ) ) ) ) ).

% targets_are_cong
thf(fact_331_target__is__ide,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).

% target_is_ide
thf(fact_332_Ide__imp__Ide__hd,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T5 ) ) ) ).

% Ide_imp_Ide_hd
thf(fact_333_targets__char,axiom,
    ! [T: list_lambda_lambda] :
      ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( collec8830145823237851355lambda
        @ ^ [B3: list_lambda_lambda] :
            ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) @ B3 ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ) ) ).

% targets_char
thf(fact_334_filter__filter,axiom,
    ! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda] :
      ( ( filter1131015166755930349lambda @ P @ ( filter1131015166755930349lambda @ Q @ Xs ) )
      = ( filter1131015166755930349lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( Q @ X3 )
            & ( P @ X3 ) )
        @ Xs ) ) ).

% filter_filter
thf(fact_335_filter__filter,axiom,
    ! [P: lambda_lambda > $o,Q: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( filter_lambda_lambda @ P @ ( filter_lambda_lambda @ Q @ Xs ) )
      = ( filter_lambda_lambda
        @ ^ [X3: lambda_lambda] :
            ( ( Q @ X3 )
            & ( P @ X3 ) )
        @ Xs ) ) ).

% filter_filter
thf(fact_336_lambda__calculus_Oarr__char,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
      = ( lambda_Arr @ T ) ) ).

% lambda_calculus.arr_char
thf(fact_337_filter__append,axiom,
    ! [P: list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( filter1131015166755930349lambda @ P @ ( append1017360404025248779lambda @ Xs @ Ys ) )
      = ( append1017360404025248779lambda @ ( filter1131015166755930349lambda @ P @ Xs ) @ ( filter1131015166755930349lambda @ P @ Ys ) ) ) ).

% filter_append
thf(fact_338_filter__append,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( filter_lambda_lambda @ P @ ( append_lambda_lambda @ Xs @ Ys ) )
      = ( append_lambda_lambda @ ( filter_lambda_lambda @ P @ Xs ) @ ( filter_lambda_lambda @ P @ Ys ) ) ) ).

% filter_append
thf(fact_339_hd__append2,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
        = ( hd_lis2183642875561147995lambda @ Xs ) ) ) ).

% hd_append2
thf(fact_340_hd__append2,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ( ( hd_set_lambda_lambda @ ( append2117223597850273125lambda @ Xs @ Ys ) )
        = ( hd_set_lambda_lambda @ Xs ) ) ) ).

% hd_append2
thf(fact_341_hd__append2,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
        = ( hd_lambda_lambda @ Xs ) ) ) ).

% hd_append2
thf(fact_342_targets__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 ) ) ) ).

% targets_append
thf(fact_343_coterminalI,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
          = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
       => ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ).

% coterminalI
thf(fact_344_rts_Otargets_Ocong,axiom,
    target4026644918049360146lambda = target4026644918049360146lambda ).

% rts.targets.cong
thf(fact_345_rts_Otargets_Ocong,axiom,
    target5490931411331770124lambda = target5490931411331770124lambda ).

% rts.targets.cong
thf(fact_346_rts_Ocoinitial_Ocong,axiom,
    coinit1063259437155584316lambda = coinit1063259437155584316lambda ).

% rts.coinitial.cong
thf(fact_347_rts_Ocoinitial_Ocong,axiom,
    coinit1392077014771647542lambda = coinit1392077014771647542lambda ).

% rts.coinitial.cong
thf(fact_348_filter_Osimps_I2_J,axiom,
    ! [P: list_lambda_lambda > $o,X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( ( P @ X )
       => ( ( filter1131015166755930349lambda @ P @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
          = ( cons_l4558873032622161958lambda @ X @ ( filter1131015166755930349lambda @ P @ Xs ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter1131015166755930349lambda @ P @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
          = ( filter1131015166755930349lambda @ P @ Xs ) ) ) ) ).

% filter.simps(2)
thf(fact_349_filter_Osimps_I2_J,axiom,
    ! [P: lambda_lambda > $o,X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( ( P @ X )
       => ( ( filter_lambda_lambda @ P @ ( cons_lambda_lambda @ X @ Xs ) )
          = ( cons_lambda_lambda @ X @ ( filter_lambda_lambda @ P @ Xs ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter_lambda_lambda @ P @ ( cons_lambda_lambda @ X @ Xs ) )
          = ( filter_lambda_lambda @ P @ Xs ) ) ) ) ).

% filter.simps(2)
thf(fact_350_filter_Osimps_I1_J,axiom,
    ! [P: list_lambda_lambda > $o] :
      ( ( filter1131015166755930349lambda @ P @ nil_li8379291187143545974lambda )
      = nil_li8379291187143545974lambda ) ).

% filter.simps(1)
thf(fact_351_filter_Osimps_I1_J,axiom,
    ! [P: set_lambda_lambda > $o] :
      ( ( filter5882915293525797703lambda @ P @ nil_se6914803783601386320lambda )
      = nil_se6914803783601386320lambda ) ).

% filter.simps(1)
thf(fact_352_filter_Osimps_I1_J,axiom,
    ! [P: lambda_lambda > $o] :
      ( ( filter_lambda_lambda @ P @ nil_lambda_lambda )
      = nil_lambda_lambda ) ).

% filter.simps(1)
thf(fact_353_list_Osel_I1_J,axiom,
    ! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
      ( ( hd_lis2183642875561147995lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_354_list_Osel_I1_J,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda] :
      ( ( hd_lambda_lambda @ ( cons_lambda_lambda @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_355_lambda__calculus_Ois__confluent__rts,axiom,
    conflu4927472611300549594lambda @ lambda_resid ).

% lambda_calculus.is_confluent_rts
thf(fact_356_longest__common__prefix,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
    ? [Ps: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys5: list_l5168835686021156086lambda] :
      ( ( Xs
        = ( append1017360404025248779lambda @ Ps @ Xs3 ) )
      & ( Ys
        = ( append1017360404025248779lambda @ Ps @ Ys5 ) )
      & ( ( Xs3 = nil_li8379291187143545974lambda )
        | ( Ys5 = nil_li8379291187143545974lambda )
        | ( ( hd_lis2183642875561147995lambda @ Xs3 )
         != ( hd_lis2183642875561147995lambda @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_357_longest__common__prefix,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
    ? [Ps: list_s345339197461576016lambda,Xs3: list_s345339197461576016lambda,Ys5: list_s345339197461576016lambda] :
      ( ( Xs
        = ( append2117223597850273125lambda @ Ps @ Xs3 ) )
      & ( Ys
        = ( append2117223597850273125lambda @ Ps @ Ys5 ) )
      & ( ( Xs3 = nil_se6914803783601386320lambda )
        | ( Ys5 = nil_se6914803783601386320lambda )
        | ( ( hd_set_lambda_lambda @ Xs3 )
         != ( hd_set_lambda_lambda @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_358_longest__common__prefix,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
    ? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
      ( ( Xs
        = ( append_lambda_lambda @ Ps @ Xs3 ) )
      & ( Ys
        = ( append_lambda_lambda @ Ps @ Ys5 ) )
      & ( ( Xs3 = nil_lambda_lambda )
        | ( Ys5 = nil_lambda_lambda )
        | ( ( hd_lambda_lambda @ Xs3 )
         != ( hd_lambda_lambda @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_359_hd__append,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( ( Xs = nil_li8379291187143545974lambda )
       => ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
          = ( hd_lis2183642875561147995lambda @ Ys ) ) )
      & ( ( Xs != nil_li8379291187143545974lambda )
       => ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
          = ( hd_lis2183642875561147995lambda @ Xs ) ) ) ) ).

% hd_append
thf(fact_360_hd__append,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( ( Xs = nil_se6914803783601386320lambda )
       => ( ( hd_set_lambda_lambda @ ( append2117223597850273125lambda @ Xs @ Ys ) )
          = ( hd_set_lambda_lambda @ Ys ) ) )
      & ( ( Xs != nil_se6914803783601386320lambda )
       => ( ( hd_set_lambda_lambda @ ( append2117223597850273125lambda @ Xs @ Ys ) )
          = ( hd_set_lambda_lambda @ Xs ) ) ) ) ).

% hd_append
thf(fact_361_hd__append,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( Xs = nil_lambda_lambda )
       => ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( hd_lambda_lambda @ Ys ) ) )
      & ( ( Xs != nil_lambda_lambda )
       => ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( hd_lambda_lambda @ Xs ) ) ) ) ).

% hd_append
thf(fact_362_stdz__insert__Beta,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( lambda_is_Beta @ T )
     => ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
       => ( lambda_is_Beta @ ( hd_lambda_lambda @ ( reduct5390047737487997688insert @ T @ U3 ) ) ) ) ) ).

% stdz_insert_Beta
thf(fact_363_stdz__insert__Ide__Std,axiom,
    ! [U: lambda_lambda,U3: list_lambda_lambda] :
      ( ( lambda_Ide @ U )
     => ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ U3 )
       => ( ( reduction_Std @ U3 )
         => ( ( reduct5390047737487997688insert @ U @ U3 )
            = ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ U3 ) @ ( tl_lambda_lambda @ U3 ) ) ) ) ) ) ).

% stdz_insert_Ide_Std
thf(fact_364_joinable__implies__coinitial,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% joinable_implies_coinitial
thf(fact_365_cong__filter__notIde,axiom,
    ! [U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
     => ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
            @ ( paths_8607357831768207192lambda @ lambda_resid
              @ ( filter_lambda_lambda
                @ ^ [U6: lambda_lambda] :
                    ~ ( lambda_Ide @ U6 )
                @ U3 )
              @ U3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
            @ ( paths_8607357831768207192lambda @ lambda_resid @ U3
              @ ( filter_lambda_lambda
                @ ^ [U6: lambda_lambda] :
                    ~ ( lambda_Ide @ U6 )
                @ U3 ) ) ) ) ) ) ).

% cong_filter_notIde
thf(fact_366_arr__iff__has__target,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
       != bot_bo1397037886704484924lambda ) ) ).

% arr_iff_has_target
thf(fact_367_Std__consE,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduction_Std @ ( cons_lambda_lambda @ T @ U3 ) )
     => ~ ( ( lambda_Arr @ T )
         => ( ( ( U3 != nil_lambda_lambda )
             => ( lambda_sseq @ T @ ( hd_lambda_lambda @ U3 ) ) )
           => ~ ( reduction_Std @ U3 ) ) ) ) ).

% Std_consE
thf(fact_368_seq__def,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        & ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
        & ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
          = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ).

% seq_def
thf(fact_369__092_060Lambda_062_Otarget__is__ide,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ).

% \<Lambda>.target_is_ide
thf(fact_370__092_060Lambda_062_Otargets__are__cong,axiom,
    ! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) ) ) ) ).

% \<Lambda>.targets_are_cong
thf(fact_371__092_060Lambda_062_Otargets__cong__closed,axiom,
    ! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) )
       => ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).

% \<Lambda>.targets_cong_closed
thf(fact_372__092_060Lambda_062_Oresid__ide_I1_J,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ T )
       => ( ( lambda_resid @ T @ A2 )
          = T ) ) ) ).

% \<Lambda>.resid_ide(1)
thf(fact_373__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
    ! [A2: lambda_lambda,A3: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ A3 )
       => ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A3 )
         => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) ) ) ) ) ).

% \<Lambda>.coinitial_ide_are_cong
thf(fact_374__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
    ! [A2: lambda_lambda,A3: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ A3 )
       => ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A3 )
         => ( A2 = A3 ) ) ) ) ).

% \<Lambda>.coinitial_ide_are_eq
thf(fact_375__092_060Lambda_062_Ocong__implies__coinitial,axiom,
    ! [U: lambda_lambda,U2: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
     => ( coinit1392077014771647542lambda @ lambda_resid @ U @ U2 ) ) ).

% \<Lambda>.cong_implies_coinitial
thf(fact_376__092_060Lambda_062_Oarr__has__un__target,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ? [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
          & ! [Y: lambda_lambda] :
              ( ( member_lambda_lambda @ Y @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
             => ( Y = X2 ) ) ) ) ).

% \<Lambda>.arr_has_un_target
thf(fact_377_Arr_Osimps_I1_J,axiom,
    ~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).

% Arr.simps(1)
thf(fact_378_Ide__implies__Arr,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( paths_366302053746739058lambda @ lambda_resid @ T5 ) ) ).

% Ide_implies_Arr
thf(fact_379__092_060Lambda_062_Ocoterminal__iff,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T2 )
      = ( ( arr_lambda_lambda @ lambda_resid @ T )
        & ( arr_lambda_lambda @ lambda_resid @ T2 )
        & ( ( target5490931411331770124lambda @ lambda_resid @ T )
          = ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ) ).

% \<Lambda>.coterminal_iff
thf(fact_380__092_060Lambda_062_OcoterminalE,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
     => ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
         => ( ( arr_lambda_lambda @ lambda_resid @ U )
           => ( ( target5490931411331770124lambda @ lambda_resid @ T )
             != ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ) ) ).

% \<Lambda>.coterminalE
thf(fact_381__092_060Lambda_062_Otargets__char,axiom,
    ! [T: lambda_lambda] :
      ( ( target5490931411331770124lambda @ lambda_resid @ T )
      = ( collec621444948832755541lambda
        @ ^ [B3: lambda_lambda] :
            ( ( arr_lambda_lambda @ lambda_resid @ T )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ T ) @ B3 ) )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ ( lambda_resid @ T @ T ) ) ) ) ) ) ).

% \<Lambda>.targets_char
thf(fact_382_Con__implies__Arr_I2_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( paths_366302053746739058lambda @ lambda_resid @ U3 ) ) ).

% Con_implies_Arr(2)
thf(fact_383_Con__implies__Arr_I1_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( paths_366302053746739058lambda @ lambda_resid @ T5 ) ) ).

% Con_implies_Arr(1)
thf(fact_384_Arr__iff__Con__self,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
      = ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T5 )
       != nil_lambda_lambda ) ) ).

% Arr_iff_Con_self
thf(fact_385_Con__Arr__self,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T5 )
       != nil_lambda_lambda ) ) ).

% Con_Arr_self
thf(fact_386_Con__imp__Arr__Resid,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) ) ) ).

% Con_imp_Arr_Resid
thf(fact_387_Arr__imp__Arr__tl,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( ( tl_lambda_lambda @ T5 )
         != nil_lambda_lambda )
       => ( paths_366302053746739058lambda @ lambda_resid @ ( tl_lambda_lambda @ T5 ) ) ) ) ).

% Arr_imp_Arr_tl
thf(fact_388_Ide__imp__Ide__tl,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ( ( tl_lambda_lambda @ T5 )
         != nil_lambda_lambda )
       => ( paths_7717490669901197737lambda @ lambda_resid @ ( tl_lambda_lambda @ T5 ) ) ) ) ).

% Ide_imp_Ide_tl
thf(fact_389_source__is__ide,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).

% source_is_ide
thf(fact_390_sources__are__cong,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) ) ) ) ).

% sources_are_cong
thf(fact_391_sources__cong__closed,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) )
       => ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).

% sources_cong_closed
thf(fact_392_Resid__Arr__self,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T5 ) ) ) ).

% Resid_Arr_self
thf(fact_393_Arr__imp__arr__hd,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( arr_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T5 ) ) ) ).

% Arr_imp_arr_hd
thf(fact_394_Std__imp__Arr,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( reduction_Std @ T5 )
     => ( ( T5 != nil_lambda_lambda )
       => ( paths_366302053746739058lambda @ lambda_resid @ T5 ) ) ) ).

% Std_imp_Arr
thf(fact_395_arr__char,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( paths_366302053746739058lambda @ lambda_resid @ T5 ) ) ).

% arr_char
thf(fact_396_resid__source__in__targets,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).

% resid_source_in_targets
thf(fact_397_Arr__Resid__single,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda )
     => ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).

% Arr_Resid_single
thf(fact_398_Arr_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% Arr.simps(2)
thf(fact_399_Ide__imp__sources__eq__targets,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).

% Ide_imp_sources_eq_targets
thf(fact_400_coinitialE,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
         => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
           => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
             != ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).

% coinitialE
thf(fact_401_coinitial__iff,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda] :
      ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T2 )
      = ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        & ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 )
        & ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
          = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) ) ) ) ).

% coinitial_iff
thf(fact_402_composableD_I3_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).

% composableD(3)
thf(fact_403_arr__iff__has__source,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
       != bot_bo1397037886704484924lambda ) ) ).

% arr_iff_has_source
thf(fact_404_sources__cons,axiom,
    ! [T: lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ T5 ) )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).

% sources_cons
thf(fact_405_seqE,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
         => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
           => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
             != ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).

% seqE
thf(fact_406_single__Src__hd__in__sources,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).

% single_Src_hd_in_sources
thf(fact_407_in__sources__iff,axiom,
    ! [T5: list_lambda_lambda,A: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
        = ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ A ) ) ) ) ) ).

% in_sources_iff
thf(fact_408_sources__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( collec8830145823237851355lambda
        @ ^ [A4: list_lambda_lambda] :
            ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A4 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ A4 ) ) ) ) ) ).

% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_409_tl__append2,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( ( tl_lis5905685823335986783lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
        = ( append1017360404025248779lambda @ ( tl_lis5905685823335986783lambda @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_410_tl__append2,axiom,
    ! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ( ( tl_set_lambda_lambda @ ( append2117223597850273125lambda @ Xs @ Ys ) )
        = ( append2117223597850273125lambda @ ( tl_set_lambda_lambda @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_411_tl__append2,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
        = ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_412_lambda__calculus_Ocoinitial__char,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
      = ( ( lambda_Arr @ T )
        & ( lambda_Arr @ U )
        & ( ( lambda_Src @ T )
          = ( lambda_Src @ U ) ) ) ) ).

% lambda_calculus.coinitial_char
thf(fact_413__092_060Lambda_062_OcoterminalI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
          = ( target5490931411331770124lambda @ lambda_resid @ U ) )
       => ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).

% \<Lambda>.coterminalI
thf(fact_414_list_Ocollapse,axiom,
    ! [List: list_s345339197461576016lambda] :
      ( ( List != nil_se6914803783601386320lambda )
     => ( ( cons_s3132251510596564736lambda @ ( hd_set_lambda_lambda @ List ) @ ( tl_set_lambda_lambda @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_415_list_Ocollapse,axiom,
    ! [List: list_l5168835686021156086lambda] :
      ( ( List != nil_li8379291187143545974lambda )
     => ( ( cons_l4558873032622161958lambda @ ( hd_lis2183642875561147995lambda @ List ) @ ( tl_lis5905685823335986783lambda @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_416_list_Ocollapse,axiom,
    ! [List: list_lambda_lambda] :
      ( ( List != nil_lambda_lambda )
     => ( ( cons_lambda_lambda @ ( hd_lambda_lambda @ List ) @ ( tl_lambda_lambda @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_417_hd__Cons__tl,axiom,
    ! [Xs: list_s345339197461576016lambda] :
      ( ( Xs != nil_se6914803783601386320lambda )
     => ( ( cons_s3132251510596564736lambda @ ( hd_set_lambda_lambda @ Xs ) @ ( tl_set_lambda_lambda @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_418_hd__Cons__tl,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( ( cons_l4558873032622161958lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( tl_lis5905685823335986783lambda @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_419_hd__Cons__tl,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ( ( cons_lambda_lambda @ ( hd_lambda_lambda @ Xs ) @ ( tl_lambda_lambda @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_420_arrI_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ).

% arrI\<^sub>P
thf(fact_421_sources__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).

% sources_append
thf(fact_422_coinitialI,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
          = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
       => ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ).

% coinitialI
thf(fact_423_seqI,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
       => ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
            = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ) ).

% seqI
thf(fact_424_sources__single__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).

% sources_single_Src
thf(fact_425_targets__single__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).

% targets_single_Src
thf(fact_426_paths__in__rts_OArr_Ocong,axiom,
    paths_2015741907369688696lambda = paths_2015741907369688696lambda ).

% paths_in_rts.Arr.cong
thf(fact_427_paths__in__rts_OArr_Ocong,axiom,
    paths_366302053746739058lambda = paths_366302053746739058lambda ).

% paths_in_rts.Arr.cong
thf(fact_428_rts_Ojoinable_Ocong,axiom,
    joinab677607494065406278lambda = joinab677607494065406278lambda ).

% rts.joinable.cong
thf(fact_429_rts_Ojoinable_Ocong,axiom,
    joinab2519788767067900224lambda = joinab2519788767067900224lambda ).

% rts.joinable.cong
thf(fact_430_rts_Osources_Ocong,axiom,
    source6119337877621565212lambda = source6119337877621565212lambda ).

% rts.sources.cong
thf(fact_431_rts_Osources_Ocong,axiom,
    source5627170679743844886lambda = source5627170679743844886lambda ).

% rts.sources.cong
thf(fact_432_lambda__calculus_Osseq_Osimps_I1_J,axiom,
    ! [Uu: lambda_lambda] :
      ~ ( lambda_sseq @ Uu @ lambda_Nil ) ).

% lambda_calculus.sseq.simps(1)
thf(fact_433_list_Osel_I3_J,axiom,
    ! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
      ( ( tl_lis5905685823335986783lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
      = X22 ) ).

% list.sel(3)
thf(fact_434_list_Osel_I3_J,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda] :
      ( ( tl_lambda_lambda @ ( cons_lambda_lambda @ X21 @ X22 ) )
      = X22 ) ).

% list.sel(3)
thf(fact_435_list_Osel_I2_J,axiom,
    ( ( tl_lambda_lambda @ nil_lambda_lambda )
    = nil_lambda_lambda ) ).

% list.sel(2)
thf(fact_436_lambda__calculus_Ois__Beta__def,axiom,
    ( lambda_is_Beta
    = ( ^ [Lambda2: lambda_lambda] :
        ? [X512: lambda_lambda,X522: lambda_lambda] :
          ( Lambda2
          = ( lambda_Beta @ X512 @ X522 ) ) ) ) ).

% lambda_calculus.is_Beta_def
thf(fact_437_lambda__calculus_Olambda_OdiscI_I4_J,axiom,
    ! [Lambda: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
      ( ( Lambda
        = ( lambda_Beta @ X51 @ X52 ) )
     => ( lambda_is_Beta @ Lambda ) ) ).

% lambda_calculus.lambda.discI(4)
thf(fact_438_lambda__calculus_Olambda_Odisc_I20_J,axiom,
    ! [X51: lambda_lambda,X52: lambda_lambda] : ( lambda_is_Beta @ ( lambda_Beta @ X51 @ X52 ) ) ).

% lambda_calculus.lambda.disc(20)
thf(fact_439_lambda__calculus_Olambda_Odisc_I16_J,axiom,
    ~ ( lambda_is_Beta @ lambda_Nil ) ).

% lambda_calculus.lambda.disc(16)
thf(fact_440_lambda__calculus_Olambda_Odistinct__disc_I8_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( lambda_is_Beta @ Lambda )
     => ( Lambda != lambda_Nil ) ) ).

% lambda_calculus.lambda.distinct_disc(8)
thf(fact_441_lambda__calculus_Olambda_Odistinct__disc_I7_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( Lambda = lambda_Nil )
     => ~ ( lambda_is_Beta @ Lambda ) ) ).

% lambda_calculus.lambda.distinct_disc(7)
thf(fact_442_lambda__calculus_Osseq_Osimps_I28_J,axiom,
    ! [V: lambda_lambda,Va: lambda_lambda] :
      ~ ( lambda_sseq @ lambda_Nil @ ( lambda_Beta @ V @ Va ) ) ).

% lambda_calculus.sseq.simps(28)
thf(fact_443_lambda__calculus_Osseq__imp__seq,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_sseq @ T @ U )
     => ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% lambda_calculus.sseq_imp_seq
thf(fact_444_Nil__tl,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( nil_lambda_lambda
        = ( tl_lambda_lambda @ Xs ) )
      = ( ( Xs = nil_lambda_lambda )
        | ? [X3: lambda_lambda] :
            ( Xs
            = ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) ) ).

% Nil_tl
thf(fact_445_tl__Nil,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( ( tl_lambda_lambda @ Xs )
        = nil_lambda_lambda )
      = ( ( Xs = nil_lambda_lambda )
        | ? [X3: lambda_lambda] :
            ( Xs
            = ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) ) ).

% tl_Nil
thf(fact_446_tl__append__if,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( Xs = nil_lambda_lambda )
       => ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( tl_lambda_lambda @ Ys ) ) )
      & ( ( Xs != nil_lambda_lambda )
       => ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys ) ) ) ) ).

% tl_append_if
thf(fact_447_list_Oexpand,axiom,
    ! [List: list_lambda_lambda,List2: list_lambda_lambda] :
      ( ( ( List = nil_lambda_lambda )
        = ( List2 = nil_lambda_lambda ) )
     => ( ( ( List != nil_lambda_lambda )
         => ( ( List2 != nil_lambda_lambda )
           => ( ( ( hd_lambda_lambda @ List )
                = ( hd_lambda_lambda @ List2 ) )
              & ( ( tl_lambda_lambda @ List )
                = ( tl_lambda_lambda @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_448_coherent__normal__sub__rts_Ocoherent_H,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V4: list_lambda_lambda,W2: list_lambda_lambda,W3: list_lambda_lambda,T: list_lambda_lambda,T2: list_lambda_lambda] :
      ( ( cohere6622258358606131289lambda @ Resid @ NN )
     => ( ( member2667661554319814297lambda @ V @ NN )
       => ( ( member2667661554319814297lambda @ V4 @ NN )
         => ( ( member2667661554319814297lambda @ W2 @ NN )
           => ( ( member2667661554319814297lambda @ W3 @ NN )
             => ( ( ( source6119337877621565212lambda @ Resid @ V )
                  = ( source6119337877621565212lambda @ Resid @ W2 ) )
               => ( ( ( source6119337877621565212lambda @ Resid @ V4 )
                    = ( source6119337877621565212lambda @ Resid @ W3 ) )
                 => ( ( ( target4026644918049360146lambda @ Resid @ W2 )
                      = ( target4026644918049360146lambda @ Resid @ W3 ) )
                   => ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T2 @ V4 ) ) @ NN )
                        & ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T2 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
                     => ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T2 @ W3 ) ) @ NN )
                        & ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T2 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).

% coherent_normal_sub_rts.coherent'
thf(fact_449_coherent__normal__sub__rts_Ocoherent_H,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V4: lambda_lambda,W2: lambda_lambda,W3: lambda_lambda,T: lambda_lambda,T2: lambda_lambda] :
      ( ( cohere1690309856004283859lambda @ Resid @ NN )
     => ( ( member_lambda_lambda @ V @ NN )
       => ( ( member_lambda_lambda @ V4 @ NN )
         => ( ( member_lambda_lambda @ W2 @ NN )
           => ( ( member_lambda_lambda @ W3 @ NN )
             => ( ( ( source5627170679743844886lambda @ Resid @ V )
                  = ( source5627170679743844886lambda @ Resid @ W2 ) )
               => ( ( ( source5627170679743844886lambda @ Resid @ V4 )
                    = ( source5627170679743844886lambda @ Resid @ W3 ) )
                 => ( ( ( target5490931411331770124lambda @ Resid @ W2 )
                      = ( target5490931411331770124lambda @ Resid @ W3 ) )
                   => ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T2 @ V4 ) ) @ NN )
                        & ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
                     => ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T2 @ W3 ) ) @ NN )
                        & ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).

% coherent_normal_sub_rts.coherent'
thf(fact_450_lambda__calculus_Olambda_Odisc__eq__case_I5_J,axiom,
    ( lambda_is_Beta
    = ( lambda_case_lambda_o @ $false
      @ ^ [Uu2: nat] : $false
      @ ^ [Uu2: lambda_lambda] : $false
      @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $false
      @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $true ) ) ).

% lambda_calculus.lambda.disc_eq_case(5)
thf(fact_451_coherent__normal__sub__rts_Ocoherent,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,U2: list_lambda_lambda] :
      ( ( cohere6622258358606131289lambda @ Resid @ NN )
     => ( ( arr_li3691443969012319837lambda @ Resid @ T )
       => ( ( member2667661554319814297lambda @ U @ NN )
         => ( ( member2667661554319814297lambda @ U2 @ NN )
           => ( ( ( source6119337877621565212lambda @ Resid @ U )
                = ( source6119337877621565212lambda @ Resid @ U2 ) )
             => ( ( ( target4026644918049360146lambda @ Resid @ U )
                  = ( target4026644918049360146lambda @ Resid @ U2 ) )
               => ( ( ( source6119337877621565212lambda @ Resid @ T )
                    = ( source6119337877621565212lambda @ Resid @ U ) )
                 => ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
                    & ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).

% coherent_normal_sub_rts.coherent
thf(fact_452_coherent__normal__sub__rts_Ocoherent,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
      ( ( cohere1690309856004283859lambda @ Resid @ NN )
     => ( ( arr_lambda_lambda @ Resid @ T )
       => ( ( member_lambda_lambda @ U @ NN )
         => ( ( member_lambda_lambda @ U2 @ NN )
           => ( ( ( source5627170679743844886lambda @ Resid @ U )
                = ( source5627170679743844886lambda @ Resid @ U2 ) )
             => ( ( ( target5490931411331770124lambda @ Resid @ U )
                  = ( target5490931411331770124lambda @ Resid @ U2 ) )
               => ( ( ( source5627170679743844886lambda @ Resid @ T )
                    = ( source5627170679743844886lambda @ Resid @ U ) )
                 => ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
                    & ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).

% coherent_normal_sub_rts.coherent
thf(fact_453_list_Oexhaust__sel,axiom,
    ! [List: list_lambda_lambda] :
      ( ( List != nil_lambda_lambda )
     => ( List
        = ( cons_lambda_lambda @ ( hd_lambda_lambda @ List ) @ ( tl_lambda_lambda @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_454_reduction__paths_OStd_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( reduction_Std @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( ( lambda_sseq @ T @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
        & ( reduction_Std @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ).

% reduction_paths.Std.simps(3)
thf(fact_455__092_060open_062_092_060And_062x_Au_AU_O_Aseq_A_091hd_A_Iu_A_D_AU_J_093_A_Itl_A_Iu_A_D_AU_J_J_A_092_060and_062_AStd_A_Itl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_Ihd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_Astdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_Ahd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_A_092_060Longrightarrow_062_Aseq_A_091_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_093_A_Iu_A_D_AU_J_A_092_060and_062_AStd_A_Iu_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_Iu_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_D_Au_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_Iu_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_D_Au_A_D_AU_J_092_060close_062,axiom,
    ! [U: lambda_lambda,U3: list_lambda_lambda,X: nat] :
      ( ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) )
          & ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) )
       => ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) )
          & ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) )
           => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) ) )
              & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U3 ) ) ) ) ) ) ) ) )
     => ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Var @ X ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
          & ( reduction_Std @ ( cons_lambda_lambda @ U @ U3 ) ) )
       => ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) )
          & ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) )
           => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( cons_lambda_lambda @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) ) )
              & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) @ ( reduct5390047737487997688insert @ ( lambda_Var @ X ) @ ( cons_lambda_lambda @ U @ U3 ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>x u U. seq [hd (u # U)] (tl (u # U)) \<and> Std (tl (u # U)) \<longrightarrow> Std (stdz_insert (hd (u # U)) (tl (u # U))) \<and> (\<not> Ide (hd (u # U) # tl (u # U)) \<longrightarrow> stdz_insert (hd (u # U)) (tl (u # U)) \<^sup>*\<sim>\<^sup>* hd (u # U) # tl (u # U)) \<Longrightarrow> seq [\<^bold>\<guillemotleft>x\<^bold>\<guillemotright>] (u # U) \<and> Std (u # U) \<longrightarrow> Std (stdz_insert \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> (u # U)) \<and> (\<not> Ide (\<^bold>\<guillemotleft>x\<^bold>\<guillemotright> # u # U) \<longrightarrow> stdz_insert \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> (u # U) \<^sup>*\<sim>\<^sup>* \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> # u # U)\<close>
thf(fact_456_coterminal__def,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
       != bot_bo1397037886704484924lambda ) ) ).

% coterminal_def
thf(fact_457_coinitial__def,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
       != bot_bo1397037886704484924lambda ) ) ).

% coinitial_def
thf(fact_458_Con__single__ide__ind,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T5 )
         != nil_lambda_lambda )
        = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
          & ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% Con_single_ide_ind
thf(fact_459_targets__eqI,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda] :
      ( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) )
       != bot_bo1397037886704484924lambda )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) ) ) ).

% targets_eqI
thf(fact_460_sources__eqI,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda] :
      ( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) )
       != bot_bo1397037886704484924lambda )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 ) ) ) ).

% sources_eqI
thf(fact_461__092_060Lambda_062_Osources__cong__closed,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) )
       => ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).

% \<Lambda>.sources_cong_closed
thf(fact_462__092_060Lambda_062_Osources__are__cong,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) ) ) ) ).

% \<Lambda>.sources_are_cong
thf(fact_463__092_060Lambda_062_Osource__is__ide,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ).

% \<Lambda>.source_is_ide
thf(fact_464__092_060Lambda_062_Oarr__has__un__source,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ? [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
          & ! [Y: lambda_lambda] :
              ( ( member_lambda_lambda @ Y @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
             => ( Y = X2 ) ) ) ) ).

% \<Lambda>.arr_has_un_source
thf(fact_465__092_060Lambda_062_Oresid__source__in__targets,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( member_lambda_lambda @ ( lambda_resid @ A2 @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).

% \<Lambda>.resid_source_in_targets
thf(fact_466__092_060Lambda_062_Ojoinable__implies__coinitial,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
     => ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.joinable_implies_coinitial
thf(fact_467_Srcs_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).

% Srcs.simps(3)
thf(fact_468_ex__un__Src,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ? [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
          & ! [Y: lambda_lambda] :
              ( ( member_lambda_lambda @ Y @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
             => ( Y = X2 ) ) ) ) ).

% ex_un_Src
thf(fact_469__092_060Lambda_062_OcoinitialE,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
     => ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
         => ( ( arr_lambda_lambda @ lambda_resid @ U )
           => ( ( source5627170679743844886lambda @ lambda_resid @ T )
             != ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).

% \<Lambda>.coinitialE
thf(fact_470__092_060Lambda_062_Ocoinitial__iff,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda] :
      ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T2 )
      = ( ( arr_lambda_lambda @ lambda_resid @ T )
        & ( arr_lambda_lambda @ lambda_resid @ T2 )
        & ( ( source5627170679743844886lambda @ lambda_resid @ T )
          = ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ) ).

% \<Lambda>.coinitial_iff
thf(fact_471__092_060Lambda_062_OcomposableD_I3_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
     => ( ( target5490931411331770124lambda @ lambda_resid @ T )
        = ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).

% \<Lambda>.composableD(3)
thf(fact_472_Srcs_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).

% Srcs.simps(2)
thf(fact_473_Con__imp__eq__Srcs,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
        = ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ).

% Con_imp_eq_Srcs
thf(fact_474_Srcs__simp_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
        = ( source5627170679743844886lambda @ lambda_resid @ ( hd_lambda_lambda @ T5 ) ) ) ) ).

% Srcs_simp\<^sub>P
thf(fact_475__092_060Lambda_062_Oseq__def,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( seq_lambda_lambda @ lambda_resid @ T @ U )
      = ( ( arr_lambda_lambda @ lambda_resid @ T )
        & ( arr_lambda_lambda @ lambda_resid @ U )
        & ( ( target5490931411331770124lambda @ lambda_resid @ T )
          = ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ).

% \<Lambda>.seq_def
thf(fact_476__092_060Lambda_062_OseqE,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( seq_lambda_lambda @ lambda_resid @ T @ U )
     => ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
         => ( ( arr_lambda_lambda @ lambda_resid @ U )
           => ( ( target5490931411331770124lambda @ lambda_resid @ T )
             != ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).

% \<Lambda>.seqE
thf(fact_477_lambda__calculus_Olambda_Oinject_I1_J,axiom,
    ! [X24: nat,Y23: nat] :
      ( ( ( lambda_Var @ X24 )
        = ( lambda_Var @ Y23 ) )
      = ( X24 = Y23 ) ) ).

% lambda_calculus.lambda.inject(1)
thf(fact_478_confluence__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
            = ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
           != nil_lambda_lambda ) ) ) ) ).

% confluence_ind
thf(fact_479_Resid__Arr__Src,axiom,
    ! [T5: list_lambda_lambda,A2: lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
          = T5 ) ) ) ).

% Resid_Arr_Src
thf(fact_480_Srcs__Resid__Arr__single,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
        = ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).

% Srcs_Resid_Arr_single
thf(fact_481_Con__Ide__iff,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T5 )
         != nil_lambda_lambda )
        = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
          & ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
            = ( paths_2897053694880971696lambda @ lambda_resid @ A ) ) ) ) ) ).

% Con_Ide_iff
thf(fact_482_Con__IdeI_I1_J,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
       => ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
            = ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T5 )
           != nil_lambda_lambda ) ) ) ) ).

% Con_IdeI(1)
thf(fact_483_Con__IdeI_I2_J,axiom,
    ! [A: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ A )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
       => ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
            = ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ A )
           != nil_lambda_lambda ) ) ) ) ).

% Con_IdeI(2)
thf(fact_484_sources__char_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( collec8830145823237851355lambda
        @ ^ [A4: list_lambda_lambda] :
            ( ( paths_7717490669901197737lambda @ lambda_resid @ A4 )
            & ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( ( paths_2897053694880971696lambda @ lambda_resid @ A4 )
              = ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% sources_char\<^sub>P
thf(fact_485__092_060Lambda_062_OcoinitialI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
          = ( source5627170679743844886lambda @ lambda_resid @ U ) )
       => ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).

% \<Lambda>.coinitialI
thf(fact_486_Srcs__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) ) ) ).

% Srcs_append
thf(fact_487__092_060Lambda_062_OseqI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( arr_lambda_lambda @ lambda_resid @ U )
       => ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
            = ( source5627170679743844886lambda @ lambda_resid @ U ) )
         => ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).

% \<Lambda>.seqI
thf(fact_488_Con__single__ideI_I1_J,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
       => ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T5 )
           != nil_lambda_lambda ) ) ) ) ).

% Con_single_ideI(1)
thf(fact_489_Con__single__ideI_I2_J,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
       => ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
           != nil_lambda_lambda ) ) ) ) ).

% Con_single_ideI(2)
thf(fact_490_paths__in__rts_OSrcs_Ocong,axiom,
    paths_2897053694880971696lambda = paths_2897053694880971696lambda ).

% paths_in_rts.Srcs.cong
thf(fact_491_lambda__calculus_Olambda_Odistinct_I13_J,axiom,
    ! [X24: nat,X51: lambda_lambda,X52: lambda_lambda] :
      ( ( lambda_Var @ X24 )
     != ( lambda_Beta @ X51 @ X52 ) ) ).

% lambda_calculus.lambda.distinct(13)
thf(fact_492_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
    ! [X24: nat] :
      ( lambda_Nil
     != ( lambda_Var @ X24 ) ) ).

% lambda_calculus.lambda.distinct(1)
thf(fact_493_lambda__calculus_OIde_Osimps_I2_J,axiom,
    ! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).

% lambda_calculus.Ide.simps(2)
thf(fact_494_lambda__calculus_OArr_Osimps_I2_J,axiom,
    ! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).

% lambda_calculus.Arr.simps(2)
thf(fact_495_lambda__calculus_OSrc_Osimps_I2_J,axiom,
    ! [I: nat] :
      ( ( lambda_Src @ ( lambda_Var @ I ) )
      = ( lambda_Var @ I ) ) ).

% lambda_calculus.Src.simps(2)
thf(fact_496_lambda__calculus_OSrc__eq__iff_I1_J,axiom,
    ! [I: nat,I2: nat] :
      ( ( ( lambda_Src @ ( lambda_Var @ I ) )
        = ( lambda_Src @ ( lambda_Var @ I2 ) ) )
      = ( I = I2 ) ) ).

% lambda_calculus.Src_eq_iff(1)
thf(fact_497_lambda__calculus_Ohead__strategy_Osimps_I1_J,axiom,
    ! [I: nat] :
      ( ( lambda_head_strategy @ ( lambda_Var @ I ) )
      = ( lambda_Var @ I ) ) ).

% lambda_calculus.head_strategy.simps(1)
thf(fact_498_lambda__calculus_Ohead__redex_Osimps_I2_J,axiom,
    ! [X: nat] :
      ( ( lambda_head_redex @ ( lambda_Var @ X ) )
      = ( lambda_Var @ X ) ) ).

% lambda_calculus.head_redex.simps(2)
thf(fact_499_lambda__calculus_Osseq_Osimps_I2_J,axiom,
    ! [Uv2: nat,Uw: nat] :
      ~ ( lambda_sseq @ ( lambda_Var @ Uv2 ) @ ( lambda_Var @ Uw ) ) ).

% lambda_calculus.sseq.simps(2)
thf(fact_500_lambda__calculus_Olambda_Odisc_I17_J,axiom,
    ! [X24: nat] :
      ~ ( lambda_is_Beta @ ( lambda_Var @ X24 ) ) ).

% lambda_calculus.lambda.disc(17)
thf(fact_501_lambda__calculus_Olambda_Osimps_I26_J,axiom,
    ! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X24: nat] :
      ( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_Var @ X24 ) )
      = ( F2 @ X24 ) ) ).

% lambda_calculus.lambda.simps(26)
thf(fact_502_lambda__calculus_Oresid_Osimps_I1_J,axiom,
    ! [I: nat,I2: nat] :
      ( ( ( I = I2 )
       => ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
          = ( lambda_Var @ I ) ) )
      & ( ( I != I2 )
       => ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
          = lambda_Nil ) ) ) ).

% lambda_calculus.resid.simps(1)
thf(fact_503_lambda__calculus_Oprfx__Var__iff,axiom,
    ! [U: lambda_lambda,I: nat] :
      ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Var @ I ) ) )
      = ( U
        = ( lambda_Var @ I ) ) ) ).

% lambda_calculus.prfx_Var_iff
thf(fact_504_lambda__calculus_Osseq_Osimps_I29_J,axiom,
    ! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
      ~ ( lambda_sseq @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) ) ).

% lambda_calculus.sseq.simps(29)
thf(fact_505_lambda__calculus_Osseq_Osimps_I13_J,axiom,
    ! [V: nat] :
      ~ ( lambda_sseq @ lambda_Nil @ ( lambda_Var @ V ) ) ).

% lambda_calculus.sseq.simps(13)
thf(fact_506_reduction__paths_Ostandard__development_Osimps_I2_J,axiom,
    ! [Uu: nat] :
      ( ( reduct1603535164923377302opment @ ( lambda_Var @ Uu ) )
      = nil_lambda_lambda ) ).

% reduction_paths.standard_development.simps(2)
thf(fact_507_lambda__calculus_Oresid_Osimps_I20_J,axiom,
    ! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
      ( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(20)
thf(fact_508_lambda__calculus_Oresid_Osimps_I39_J,axiom,
    ! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
      ( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) )
      = lambda_Nil ) ).

% lambda_calculus.resid.simps(39)
thf(fact_509_paths__in__confluent__rts_Oconfluence__ind,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_9040249923598999359lambda @ Resid )
     => ( ( paths_366302053746739058lambda @ Resid @ T5 )
       => ( ( paths_366302053746739058lambda @ Resid @ U3 )
         => ( ( ( paths_2897053694880971696lambda @ Resid @ T5 )
              = ( paths_2897053694880971696lambda @ Resid @ U3 ) )
           => ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
             != nil_lambda_lambda ) ) ) ) ) ).

% paths_in_confluent_rts.confluence_ind
thf(fact_510_reduction__paths_Ostdz__insert_Osimps_I2_J,axiom,
    ! [Uu: nat,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( reduct5390047737487997688insert @ ( lambda_Var @ Uu ) @ ( cons_lambda_lambda @ V @ Va ) )
      = ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ).

% reduction_paths.stdz_insert.simps(2)
thf(fact_511_Ide_Oelims_I1_J,axiom,
    ! [X: list_lambda_lambda,Y4: $o] :
      ( ( ( paths_7717490669901197737lambda @ lambda_resid @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => Y4 )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
                = ( ~ ( ide_lambda_lambda @ lambda_resid @ T4 ) ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                  = ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T4 )
                        & ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                        & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% Ide.elims(1)
thf(fact_512_Ide_Oelims_I2_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ X )
     => ( ! [T4: lambda_lambda] :
            ( ( X
              = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
           => ~ ( ide_lambda_lambda @ lambda_resid @ T4 ) )
       => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
             => ~ ( ( ide_lambda_lambda @ lambda_resid @ T4 )
                  & ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                  & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% Ide.elims(2)
thf(fact_513_Ide_Oelims_I3_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X )
     => ( ( X != nil_lambda_lambda )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( ide_lambda_lambda @ lambda_resid @ T4 ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( ( ide_lambda_lambda @ lambda_resid @ T4 )
                  & ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                  & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).

% Ide.elims(3)
thf(fact_514_Arr_Oelims_I3_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ~ ( paths_366302053746739058lambda @ lambda_resid @ X )
     => ( ( X != nil_lambda_lambda )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( arr_lambda_lambda @ lambda_resid @ T4 ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( ( arr_lambda_lambda @ lambda_resid @ T4 )
                  & ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                  & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).

% Arr.elims(3)
thf(fact_515_Arr_Oelims_I2_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ X )
     => ( ! [T4: lambda_lambda] :
            ( ( X
              = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
           => ~ ( arr_lambda_lambda @ lambda_resid @ T4 ) )
       => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
             => ~ ( ( arr_lambda_lambda @ lambda_resid @ T4 )
                  & ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                  & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% Arr.elims(2)
thf(fact_516_Srcs__are__ide,axiom,
    ! [T5: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).

% Srcs_are_ide
thf(fact_517_Arr_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( ( arr_lambda_lambda @ lambda_resid @ T )
        & ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
        & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).

% Arr.simps(3)
thf(fact_518_Ide_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( ( ide_lambda_lambda @ lambda_resid @ T )
        & ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
        & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).

% Ide.simps(3)
thf(fact_519_Arr_Oelims_I1_J,axiom,
    ! [X: list_lambda_lambda,Y4: $o] :
      ( ( ( paths_366302053746739058lambda @ lambda_resid @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => Y4 )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
                = ( ~ ( arr_lambda_lambda @ lambda_resid @ T4 ) ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                  = ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T4 )
                        & ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                        & ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% Arr.elims(1)
thf(fact_520_Arr__consI_092_060_094sub_062P,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         => ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).

% Arr_consI\<^sub>P
thf(fact_521_subset__CollectI,axiom,
    ! [B4: set_lambda_lambda,A: set_lambda_lambda,Q: lambda_lambda > $o,P: lambda_lambda > $o] :
      ( ( ord_le1926335974945931754lambda @ B4 @ A )
     => ( ! [X2: lambda_lambda] :
            ( ( member_lambda_lambda @ X2 @ B4 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le1926335974945931754lambda
          @ ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( member_lambda_lambda @ X3 @ B4 )
                & ( Q @ X3 ) ) )
          @ ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( member_lambda_lambda @ X3 @ A )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_522_subset__CollectI,axiom,
    ! [B4: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,Q: list_lambda_lambda > $o,P: list_lambda_lambda > $o] :
      ( ( ord_le3064231726521830256lambda @ B4 @ A )
     => ( ! [X2: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X2 @ B4 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le3064231726521830256lambda
          @ ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( member2667661554319814297lambda @ X3 @ B4 )
                & ( Q @ X3 ) ) )
          @ ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( member2667661554319814297lambda @ X3 @ A )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_523_subset__Collect__iff,axiom,
    ! [B4: set_lambda_lambda,A: set_lambda_lambda,P: lambda_lambda > $o] :
      ( ( ord_le1926335974945931754lambda @ B4 @ A )
     => ( ( ord_le1926335974945931754lambda @ B4
          @ ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( member_lambda_lambda @ X3 @ A )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: lambda_lambda] :
              ( ( member_lambda_lambda @ X3 @ B4 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_524_subset__Collect__iff,axiom,
    ! [B4: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
      ( ( ord_le3064231726521830256lambda @ B4 @ A )
     => ( ( ord_le3064231726521830256lambda @ B4
          @ ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( member2667661554319814297lambda @ X3 @ A )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: list_lambda_lambda] :
              ( ( member2667661554319814297lambda @ X3 @ B4 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_525_confluence__single,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ! [T4: lambda_lambda,U7: lambda_lambda] :
          ( ( coinit1392077014771647542lambda @ lambda_resid @ T4 @ U7 )
         => ( con_lambda_lambda @ lambda_resid @ T4 @ U7 ) )
     => ( ( arr_lambda_lambda @ lambda_resid @ T )
       => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
         => ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
              = ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
           => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
             != nil_lambda_lambda ) ) ) ) ) ).

% confluence_single
thf(fact_526_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ X @ bot_bo1397037886704484924lambda )
      = bot_bo1397037886704484924lambda ) ).

% boolean_algebra.conj_zero_right
thf(fact_527_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ X @ bot_bo1271416176687173302lambda )
      = bot_bo1271416176687173302lambda ) ).

% boolean_algebra.conj_zero_right
thf(fact_528_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X )
      = bot_bo1397037886704484924lambda ) ).

% boolean_algebra.conj_zero_left
thf(fact_529_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X )
      = bot_bo1271416176687173302lambda ) ).

% boolean_algebra.conj_zero_left
thf(fact_530_inf__bot__right,axiom,
    ! [X: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ X @ bot_bo1397037886704484924lambda )
      = bot_bo1397037886704484924lambda ) ).

% inf_bot_right
thf(fact_531_inf__bot__right,axiom,
    ! [X: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ X @ bot_bo1271416176687173302lambda )
      = bot_bo1271416176687173302lambda ) ).

% inf_bot_right
thf(fact_532_inf__bot__left,axiom,
    ! [X: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X )
      = bot_bo1397037886704484924lambda ) ).

% inf_bot_left
thf(fact_533_inf__bot__left,axiom,
    ! [X: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X )
      = bot_bo1271416176687173302lambda ) ).

% inf_bot_left
thf(fact_534__092_060Lambda_062_Oresid__reflects__con,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ V )
     => ( ( con_lambda_lambda @ lambda_resid @ U @ V )
       => ( ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
         => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).

% \<Lambda>.resid_reflects_con
thf(fact_535__092_060Lambda_062_Ocon__sym,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( con_lambda_lambda @ lambda_resid @ U @ T ) ) ).

% \<Lambda>.con_sym
thf(fact_536__092_060Lambda_062_Otargets__eqI,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
       != bot_bo1271416176687173302lambda )
     => ( ( target5490931411331770124lambda @ lambda_resid @ T )
        = ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ).

% \<Lambda>.targets_eqI
thf(fact_537_Srcs__eqI,axiom,
    ! [T5: list_lambda_lambda,T6: list_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
       != bot_bo1271416176687173302lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
        = ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ).

% Srcs_eqI
thf(fact_538__092_060Lambda_062_Osources__eqI,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
       != bot_bo1271416176687173302lambda )
     => ( ( source5627170679743844886lambda @ lambda_resid @ T )
        = ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ).

% \<Lambda>.sources_eqI
thf(fact_539_Srcs_Osimps_I1_J,axiom,
    ( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
    = bot_bo1271416176687173302lambda ) ).

% Srcs.simps(1)
thf(fact_540__092_060Lambda_062_Oresid__ide__arr,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( con_lambda_lambda @ lambda_resid @ A2 @ T )
       => ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ T ) ) ) ) ).

% \<Lambda>.resid_ide_arr
thf(fact_541__092_060Lambda_062_Oresid__arr__ide,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ A2 )
       => ( ( lambda_resid @ T @ A2 )
          = T ) ) ) ).

% \<Lambda>.resid_arr_ide
thf(fact_542__092_060Lambda_062_Oprfx__implies__con,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
     => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.prfx_implies_con
thf(fact_543__092_060Lambda_062_Oide__imp__con__iff__cong,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ T )
     => ( ( ide_lambda_lambda @ lambda_resid @ U )
       => ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          = ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ) ).

% \<Lambda>.ide_imp_con_iff_cong
thf(fact_544__092_060Lambda_062_Oide__def,axiom,
    ! [A2: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
      = ( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
        & ( ( lambda_resid @ A2 @ A2 )
          = A2 ) ) ) ).

% \<Lambda>.ide_def
thf(fact_545__092_060Lambda_062_OideE,axiom,
    ! [A2: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ~ ( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
         => ( ( lambda_resid @ A2 @ A2 )
           != A2 ) ) ) ).

% \<Lambda>.ideE
thf(fact_546__092_060Lambda_062_Ocon__transitive__on__ide,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ B )
       => ( ( ide_lambda_lambda @ lambda_resid @ C )
         => ( ( con_lambda_lambda @ lambda_resid @ A2 @ B )
           => ( ( con_lambda_lambda @ lambda_resid @ B @ C )
             => ( con_lambda_lambda @ lambda_resid @ A2 @ C ) ) ) ) ) ) ).

% \<Lambda>.con_transitive_on_ide
thf(fact_547__092_060Lambda_062_Ocon__target,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
     => ( ( con_lambda_lambda @ lambda_resid @ U @ V )
       => ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ V @ U ) ) ) ) ).

% \<Lambda>.con_target
thf(fact_548__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ? [A5: lambda_lambda] :
          ( ( ide_lambda_lambda @ lambda_resid @ A5 )
          & ( con_lambda_lambda @ lambda_resid @ A5 @ T )
          & ( con_lambda_lambda @ lambda_resid @ A5 @ U ) ) ) ).

% \<Lambda>.con_imp_coinitial_ax
thf(fact_549__092_060Lambda_062_Ocon__ide__are__eq,axiom,
    ! [A2: lambda_lambda,A3: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ A3 )
       => ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
         => ( A2 = A3 ) ) ) ) ).

% \<Lambda>.con_ide_are_eq
thf(fact_550__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
    ! [U: lambda_lambda,U2: lambda_lambda,T: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ) ).

% \<Lambda>.cong_subst_right(1)
thf(fact_551__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
    ! [U: lambda_lambda,U2: lambda_lambda,T: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U2 ) ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).

% \<Lambda>.cong_subst_right(2)
thf(fact_552__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T ) ) )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ) ).

% \<Lambda>.cong_subst_left(1)
thf(fact_553__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
    ! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
      ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T2 ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T ) ) )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T2 @ U ) ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T2 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).

% \<Lambda>.cong_subst_left(2)
thf(fact_554__092_060Lambda_062_Ocon__implies__arr_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( arr_lambda_lambda @ lambda_resid @ U ) ) ).

% \<Lambda>.con_implies_arr(2)
thf(fact_555__092_060Lambda_062_Ocon__implies__arr_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% \<Lambda>.con_implies_arr(1)
thf(fact_556__092_060Lambda_062_OarrE,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).

% \<Lambda>.arrE
thf(fact_557__092_060Lambda_062_Oarr__def,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
      = ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).

% \<Lambda>.arr_def
thf(fact_558__092_060Lambda_062_Oarr__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ).

% \<Lambda>.arr_resid
thf(fact_559__092_060Lambda_062_Oarr__resid__iff__con,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
      = ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.arr_resid_iff_con
thf(fact_560__092_060Lambda_062_Otargets__resid__sym,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        = ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).

% \<Lambda>.targets_resid_sym
thf(fact_561__092_060Lambda_062_Otargets__are__con,axiom,
    ! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
       => ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).

% \<Lambda>.targets_are_con
thf(fact_562_Arr__has__Src,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
       != bot_bo1271416176687173302lambda ) ) ).

% Arr_has_Src
thf(fact_563_Srcs__are__con,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
     => ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
       => ( con_lambda_lambda @ lambda_resid @ A2 @ A3 ) ) ) ).

% Srcs_are_con
thf(fact_564__092_060Lambda_062_Oarr__iff__has__target,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
      = ( ( target5490931411331770124lambda @ lambda_resid @ T )
       != bot_bo1271416176687173302lambda ) ) ).

% \<Lambda>.arr_iff_has_target
thf(fact_565__092_060Lambda_062_Ocon__imp__common__source,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
       != bot_bo1271416176687173302lambda ) ) ).

% \<Lambda>.con_imp_common_source
thf(fact_566__092_060Lambda_062_Osources__are__con,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
       => ( con_lambda_lambda @ lambda_resid @ A2 @ A3 ) ) ) ).

% \<Lambda>.sources_are_con
thf(fact_567__092_060Lambda_062_Oarr__iff__has__source,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
      = ( ( source5627170679743844886lambda @ lambda_resid @ T )
       != bot_bo1271416176687173302lambda ) ) ).

% \<Lambda>.arr_iff_has_source
thf(fact_568__092_060Lambda_062_Ocon__imp__coinitial,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.con_imp_coinitial
thf(fact_569__092_060Lambda_062_Oconfluence,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
     => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.confluence
thf(fact_570__092_060Lambda_062_Ocoinitial__def,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
      = ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
       != bot_bo1271416176687173302lambda ) ) ).

% \<Lambda>.coinitial_def
thf(fact_571__092_060Lambda_062_Ocoterminal__def,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
      = ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ U ) )
       != bot_bo1271416176687173302lambda ) ) ).

% \<Lambda>.coterminal_def
thf(fact_572__092_060Lambda_062_Ojoinable__implies__con,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
     => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.joinable_implies_con
thf(fact_573__092_060Lambda_062_Ohas__joins,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( joinab2519788767067900224lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.has_joins
thf(fact_574__092_060Lambda_062_Otargets__con__closed,axiom,
    ! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ B2 )
       => ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
         => ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).

% \<Lambda>.targets_con_closed
thf(fact_575_Srcs__con__closed,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ A3 )
       => ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
         => ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% Srcs_con_closed
thf(fact_576__092_060Lambda_062_Oin__sourcesE,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ~ ( ( ide_lambda_lambda @ lambda_resid @ A2 )
         => ~ ( con_lambda_lambda @ lambda_resid @ T @ A2 ) ) ) ).

% \<Lambda>.in_sourcesE
thf(fact_577__092_060Lambda_062_Osources__con__closed,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
      ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ A3 )
       => ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
         => ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).

% \<Lambda>.sources_con_closed
thf(fact_578__092_060Lambda_062_Osources__def,axiom,
    ! [T: lambda_lambda] :
      ( ( source5627170679743844886lambda @ lambda_resid @ T )
      = ( collec621444948832755541lambda
        @ ^ [A6: lambda_lambda] :
            ( ( ide_lambda_lambda @ lambda_resid @ A6 )
            & ( con_lambda_lambda @ lambda_resid @ T @ A6 ) ) ) ) ).

% \<Lambda>.sources_def
thf(fact_579_Resid_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
          = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
      & ( ~ ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
          = nil_lambda_lambda ) ) ) ).

% Resid.simps(3)
thf(fact_580_Con__rec_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda )
      = ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% Con_rec(1)
thf(fact_581_Con__rec_I2_J,axiom,
    ! [T5: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
         != nil_lambda_lambda )
        = ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
           != nil_lambda_lambda ) ) ) ) ).

% Con_rec(2)
thf(fact_582_Con__rec_I3_J,axiom,
    ! [U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
         != nil_lambda_lambda )
        = ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 )
           != nil_lambda_lambda ) ) ) ) ).

% Con_rec(3)
thf(fact_583_Con__rec_I4_J,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ ( cons_lambda_lambda @ U @ U3 ) )
           != nil_lambda_lambda )
          = ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
             != nil_lambda_lambda )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 )
             != nil_lambda_lambda )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
             != nil_lambda_lambda ) ) ) ) ) ).

% Con_rec(4)
thf(fact_584_Srcs_Oelims,axiom,
    ! [X: list_lambda_lambda,Y4: set_lambda_lambda] :
      ( ( ( paths_2897053694880971696lambda @ lambda_resid @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4 != bot_bo1271416176687173302lambda ) )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
               != ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) )
         => ~ ! [T4: lambda_lambda] :
                ( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
                    ( X
                    = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                 != ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) ) ) ) ) ).

% Srcs.elims
thf(fact_585_lambda__calculus_Ocon__char,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
      = ( ( lambda_resid @ T @ U )
       != lambda_Nil ) ) ).

% lambda_calculus.con_char
thf(fact_586_lambda__calculus_Otargets__simps_I1_J,axiom,
    ( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
    = bot_bo1271416176687173302lambda ) ).

% lambda_calculus.targets_simps(1)
thf(fact_587_lambda__calculus_Osources__simps_I1_J,axiom,
    ( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
    = bot_bo1271416176687173302lambda ) ).

% lambda_calculus.sources_simps(1)
thf(fact_588__092_060Lambda_062_OideI,axiom,
    ! [A2: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
     => ( ( ( lambda_resid @ A2 @ A2 )
          = A2 )
       => ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ) ).

% \<Lambda>.ideI
thf(fact_589__092_060Lambda_062_OarrI,axiom,
    ! [T: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ T )
     => ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% \<Lambda>.arrI
thf(fact_590__092_060Lambda_062_Oin__sourcesI,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( con_lambda_lambda @ lambda_resid @ T @ A2 )
       => ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).

% \<Lambda>.in_sourcesI
thf(fact_591__092_060Lambda_062_Osources__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        = ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).

% \<Lambda>.sources_resid
thf(fact_592_residuation_Ocon_Ocong,axiom,
    con_lambda_lambda = con_lambda_lambda ).

% residuation.con.cong
thf(fact_593_residuation_Ocon_Ocong,axiom,
    con_li79537494059641180lambda = con_li79537494059641180lambda ).

% residuation.con.cong
thf(fact_594_confluent__rts_Oconfluence,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( conflu4927472611300549594lambda @ Resid )
     => ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
       => ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).

% confluent_rts.confluence
thf(fact_595_confluent__rts_Oconfluence,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( conflu4036547884742428384lambda @ Resid )
     => ( ( coinit1063259437155584316lambda @ Resid @ T @ U )
       => ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).

% confluent_rts.confluence
thf(fact_596_paths__in__confluent__rts_Oconfluence__single,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U3: list_l5168835686021156086lambda] :
      ( ( paths_3043240146134862789lambda @ Resid )
     => ( ! [T4: list_lambda_lambda,U7: list_lambda_lambda] :
            ( ( coinit1063259437155584316lambda @ Resid @ T4 @ U7 )
           => ( con_li79537494059641180lambda @ Resid @ T4 @ U7 ) )
       => ( ( arr_li3691443969012319837lambda @ Resid @ T )
         => ( ( paths_2015741907369688696lambda @ Resid @ U3 )
           => ( ( ( source6119337877621565212lambda @ Resid @ T )
                = ( paths_3172353640142224310lambda @ Resid @ U3 ) )
             => ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U3 )
               != nil_li8379291187143545974lambda ) ) ) ) ) ) ).

% paths_in_confluent_rts.confluence_single
thf(fact_597_paths__in__confluent__rts_Oconfluence__single,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_9040249923598999359lambda @ Resid )
     => ( ! [T4: lambda_lambda,U7: lambda_lambda] :
            ( ( coinit1392077014771647542lambda @ Resid @ T4 @ U7 )
           => ( con_lambda_lambda @ Resid @ T4 @ U7 ) )
       => ( ( arr_lambda_lambda @ Resid @ T )
         => ( ( paths_366302053746739058lambda @ Resid @ U3 )
           => ( ( ( source5627170679743844886lambda @ Resid @ T )
                = ( paths_2897053694880971696lambda @ Resid @ U3 ) )
             => ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
               != nil_lambda_lambda ) ) ) ) ) ) ).

% paths_in_confluent_rts.confluence_single
thf(fact_598_subset__empty,axiom,
    ! [A: set_lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% subset_empty
thf(fact_599_subset__empty,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% subset_empty
thf(fact_600_empty__subsetI,axiom,
    ! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).

% empty_subsetI
thf(fact_601_empty__subsetI,axiom,
    ! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).

% empty_subsetI
thf(fact_602_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_603_Srcs__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( ( reduct1603535164923377302opment @ T )
         != nil_lambda_lambda )
       => ( ( paths_2897053694880971696lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
          = ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% Srcs_standard_development
thf(fact_604_Ide__char,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
      = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
        & ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ) ).

% Ide_char
thf(fact_605_con__sym,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T ) ) ).

% con_sym
thf(fact_606_resid__reflects__con,axiom,
    ! [T: list_lambda_lambda,V: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
       => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
         => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ) ).

% resid_reflects_con
thf(fact_607_con__char,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
      = ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda ) ) ).

% con_char
thf(fact_608_cong__subst__left_I2_J,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ T ) ) )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ U ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).

% cong_subst_left(2)
thf(fact_609_cong__subst__left_I1_J,axiom,
    ! [T: list_lambda_lambda,T2: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ T ) ) )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T2 @ U ) ) ) ).

% cong_subst_left(1)
thf(fact_610_cong__subst__right_I2_J,axiom,
    ! [U: list_lambda_lambda,U2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).

% cong_subst_right(2)
thf(fact_611_cong__subst__right_I1_J,axiom,
    ! [U: list_lambda_lambda,U2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ).

% cong_subst_right(1)
thf(fact_612_con__imp__coinitial__ax,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ? [A5: list_lambda_lambda] :
          ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 )
          & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 @ T )
          & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 @ U ) ) ) ).

% con_imp_coinitial_ax
thf(fact_613_con__target,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ).

% con_target
thf(fact_614_con__transitive__on__ide,axiom,
    ! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
         => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ B )
           => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ C )
             => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ C ) ) ) ) ) ) ).

% con_transitive_on_ide
thf(fact_615_ideE,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ~ ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
         => ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
           != A2 ) ) ) ).

% ideE
thf(fact_616_ide__def,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
      = ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
        & ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
          = A2 ) ) ) ).

% ide_def
thf(fact_617_ide__imp__con__iff__cong,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
       => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
          = ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ) ) ).

% ide_imp_con_iff_cong
thf(fact_618_prfx__implies__con,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% prfx_implies_con
thf(fact_619_resid__arr__ide,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ A2 )
          = T ) ) ) ).

% resid_arr_ide
thf(fact_620_resid__ide__arr,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T )
       => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T ) ) ) ) ).

% resid_ide_arr
thf(fact_621_sources__are__con,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 ) ) ) ).

% sources_are_con
thf(fact_622_targets__are__con,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B2 ) ) ) ).

% targets_are_con
thf(fact_623_targets__resid__sym,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ).

% targets_resid_sym
thf(fact_624_con__implies__arr_I2_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ).

% con_implies_arr(2)
thf(fact_625_con__implies__arr_I1_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).

% con_implies_arr(1)
thf(fact_626_arrE,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).

% arrE
thf(fact_627_arr__def,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).

% arr_def
thf(fact_628_arr__resid,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).

% arr_resid
thf(fact_629_arr__resid__iff__con,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
      = ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% arr_resid_iff_con
thf(fact_630_con__imp__coinitial,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% con_imp_coinitial
thf(fact_631_confluence,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% confluence
thf(fact_632_confluence_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ).

% confluence\<^sub>P
thf(fact_633_joinable__implies__con,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% joinable_implies_con
thf(fact_634_in__sourcesE,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
         => ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 ) ) ) ).

% in_sourcesE
thf(fact_635_sources__con__closed,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
       => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 )
         => ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).

% sources_con_closed
thf(fact_636_targets__con__closed,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
       => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B2 )
         => ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).

% targets_con_closed
thf(fact_637_sources__def,axiom,
    ! [T: list_lambda_lambda] :
      ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( collec8830145823237851355lambda
        @ ^ [A6: list_lambda_lambda] :
            ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A6 )
            & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A6 ) ) ) ) ).

% sources_def
thf(fact_638_empty__iff,axiom,
    ! [C: list_lambda_lambda] :
      ~ ( member2667661554319814297lambda @ C @ bot_bo1397037886704484924lambda ) ).

% empty_iff
thf(fact_639_empty__iff,axiom,
    ! [C: lambda_lambda] :
      ~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).

% empty_iff
thf(fact_640_all__not__in__conv,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ( ! [X3: list_lambda_lambda] :
            ~ ( member2667661554319814297lambda @ X3 @ A ) )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% all_not_in_conv
thf(fact_641_all__not__in__conv,axiom,
    ! [A: set_lambda_lambda] :
      ( ( ! [X3: lambda_lambda] :
            ~ ( member_lambda_lambda @ X3 @ A ) )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% all_not_in_conv
thf(fact_642_Collect__empty__eq,axiom,
    ! [P: list_lambda_lambda > $o] :
      ( ( ( collec8830145823237851355lambda @ P )
        = bot_bo1397037886704484924lambda )
      = ( ! [X3: list_lambda_lambda] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_643_Collect__empty__eq,axiom,
    ! [P: lambda_lambda > $o] :
      ( ( ( collec621444948832755541lambda @ P )
        = bot_bo1271416176687173302lambda )
      = ( ! [X3: lambda_lambda] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_644_empty__Collect__eq,axiom,
    ! [P: list_lambda_lambda > $o] :
      ( ( bot_bo1397037886704484924lambda
        = ( collec8830145823237851355lambda @ P ) )
      = ( ! [X3: list_lambda_lambda] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_645_empty__Collect__eq,axiom,
    ! [P: lambda_lambda > $o] :
      ( ( bot_bo1271416176687173302lambda
        = ( collec621444948832755541lambda @ P ) )
      = ( ! [X3: lambda_lambda] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_646_set__Arr__subset__arr,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ) ).

% set_Arr_subset_arr
thf(fact_647_set__Ide__subset__ide,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ).

% set_Ide_subset_ide
thf(fact_648_con__imp__common__source,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
       != bot_bo1397037886704484924lambda ) ) ).

% con_imp_common_source
thf(fact_649_Srcs__simp_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T )
        = ( insert_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).

% Srcs_simp\<^sub>\<Lambda>\<^sub>P
thf(fact_650_set__Ide__subset__single__hd,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T5 ) @ bot_bo1271416176687173302lambda ) ) ) ).

% set_Ide_subset_single_hd
thf(fact_651_singletonI,axiom,
    ! [A2: list_lambda_lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).

% singletonI
thf(fact_652_singletonI,axiom,
    ! [A2: lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).

% singletonI
thf(fact_653_filter__True,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ! [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
         => ( P @ X2 ) )
     => ( ( filter_lambda_lambda @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_654_singleton__conv2,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( collec8830145823237851355lambda
        @ ( ^ [Y5: list_lambda_lambda,Z: list_lambda_lambda] : ( Y5 = Z )
          @ A2 ) )
      = ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).

% singleton_conv2
thf(fact_655_singleton__conv2,axiom,
    ! [A2: lambda_lambda] :
      ( ( collec621444948832755541lambda
        @ ( ^ [Y5: lambda_lambda,Z: lambda_lambda] : ( Y5 = Z )
          @ A2 ) )
      = ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).

% singleton_conv2
thf(fact_656_singleton__conv,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] : ( X3 = A2 ) )
      = ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).

% singleton_conv
thf(fact_657_singleton__conv,axiom,
    ! [A2: lambda_lambda] :
      ( ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] : ( X3 = A2 ) )
      = ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).

% singleton_conv
thf(fact_658_set__filter,axiom,
    ! [P: list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda] :
      ( ( set_li3846194476672328965lambda @ ( filter1131015166755930349lambda @ P @ Xs ) )
      = ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_659_set__filter,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( set_lambda_lambda2 @ ( filter_lambda_lambda @ P @ Xs ) )
      = ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_660_const__ide__is__Ide,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T5 ) )
       => ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T5 ) @ bot_bo1271416176687173302lambda ) )
         => ( paths_7717490669901197737lambda @ lambda_resid @ T5 ) ) ) ) ).

% const_ide_is_Ide
thf(fact_661_conI_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ).

% conI\<^sub>P
thf(fact_662_set__empty2,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( bot_bo1397037886704484924lambda
        = ( set_li3846194476672328965lambda @ Xs ) )
      = ( Xs = nil_li8379291187143545974lambda ) ) ).

% set_empty2
thf(fact_663_set__empty2,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( bot_bo1271416176687173302lambda
        = ( set_lambda_lambda2 @ Xs ) )
      = ( Xs = nil_lambda_lambda ) ) ).

% set_empty2
thf(fact_664_set__empty,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( ( set_li3846194476672328965lambda @ Xs )
        = bot_bo1397037886704484924lambda )
      = ( Xs = nil_li8379291187143545974lambda ) ) ).

% set_empty
thf(fact_665_set__empty,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( ( set_lambda_lambda2 @ Xs )
        = bot_bo1271416176687173302lambda )
      = ( Xs = nil_lambda_lambda ) ) ).

% set_empty
thf(fact_666_list_Osimps_I15_J,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda] :
      ( ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) )
      = ( insert_lambda_lambda @ X21 @ ( set_lambda_lambda2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_667_singleton__insert__inj__eq_H,axiom,
    ! [A2: lambda_lambda,A: set_lambda_lambda,B: lambda_lambda] :
      ( ( ( insert_lambda_lambda @ A2 @ A )
        = ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
      = ( ( A2 = B )
        & ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_668_singleton__insert__inj__eq_H,axiom,
    ! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: list_lambda_lambda] :
      ( ( ( insert8465602035467394112lambda @ A2 @ A )
        = ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
      = ( ( A2 = B )
        & ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_669_singleton__insert__inj__eq,axiom,
    ! [B: lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
      ( ( ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda )
        = ( insert_lambda_lambda @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_670_singleton__insert__inj__eq,axiom,
    ! [B: list_lambda_lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda )
        = ( insert8465602035467394112lambda @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_671_ideI,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
          = A2 )
       => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ) ).

% ideI
thf(fact_672_insert__disjoint_I1_J,axiom,
    ! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
      ( ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B4 )
        = bot_bo1397037886704484924lambda )
      = ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
        & ( ( inf_in3354698378728726462lambda @ A @ B4 )
          = bot_bo1397037886704484924lambda ) ) ) ).

% insert_disjoint(1)
thf(fact_673_insert__disjoint_I1_J,axiom,
    ! [A2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B4 )
        = bot_bo1271416176687173302lambda )
      = ( ~ ( member_lambda_lambda @ A2 @ B4 )
        & ( ( inf_in1766959559283430712lambda @ A @ B4 )
          = bot_bo1271416176687173302lambda ) ) ) ).

% insert_disjoint(1)
thf(fact_674_insert__disjoint_I2_J,axiom,
    ! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
      ( ( bot_bo1397037886704484924lambda
        = ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B4 ) )
      = ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
        & ( bot_bo1397037886704484924lambda
          = ( inf_in3354698378728726462lambda @ A @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_675_insert__disjoint_I2_J,axiom,
    ! [A2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
      ( ( bot_bo1271416176687173302lambda
        = ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B4 ) )
      = ( ~ ( member_lambda_lambda @ A2 @ B4 )
        & ( bot_bo1271416176687173302lambda
          = ( inf_in1766959559283430712lambda @ A @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_676_disjoint__insert_I1_J,axiom,
    ! [B4: set_li7878742357367608272lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( ( inf_in3354698378728726462lambda @ B4 @ ( insert8465602035467394112lambda @ A2 @ A ) )
        = bot_bo1397037886704484924lambda )
      = ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
        & ( ( inf_in3354698378728726462lambda @ B4 @ A )
          = bot_bo1397037886704484924lambda ) ) ) ).

% disjoint_insert(1)
thf(fact_677_disjoint__insert_I1_J,axiom,
    ! [B4: set_lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ B4 @ ( insert_lambda_lambda @ A2 @ A ) )
        = bot_bo1271416176687173302lambda )
      = ( ~ ( member_lambda_lambda @ A2 @ B4 )
        & ( ( inf_in1766959559283430712lambda @ B4 @ A )
          = bot_bo1271416176687173302lambda ) ) ) ).

% disjoint_insert(1)
thf(fact_678_disjoint__insert_I2_J,axiom,
    ! [A: set_li7878742357367608272lambda,B: list_lambda_lambda,B4: set_li7878742357367608272lambda] :
      ( ( bot_bo1397037886704484924lambda
        = ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ B @ B4 ) ) )
      = ( ~ ( member2667661554319814297lambda @ B @ A )
        & ( bot_bo1397037886704484924lambda
          = ( inf_in3354698378728726462lambda @ A @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_679_disjoint__insert_I2_J,axiom,
    ! [A: set_lambda_lambda,B: lambda_lambda,B4: set_lambda_lambda] :
      ( ( bot_bo1271416176687173302lambda
        = ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ B @ B4 ) ) )
      = ( ~ ( member_lambda_lambda @ B @ A )
        & ( bot_bo1271416176687173302lambda
          = ( inf_in1766959559283430712lambda @ A @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_680_filter__False,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ! [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
         => ~ ( P @ X2 ) )
     => ( ( filter_lambda_lambda @ P @ Xs )
        = nil_lambda_lambda ) ) ).

% filter_False
thf(fact_681_arrI,axiom,
    ! [T: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
     => ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).

% arrI
thf(fact_682_in__sourcesI,axiom,
    ! [A2: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 )
       => ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).

% in_sourcesI
thf(fact_683_sources__resid,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).

% sources_resid
thf(fact_684_conI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
     => ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
       => ( ( ( lambda_Src @ ( hd_lambda_lambda @ T5 ) )
            = ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) )
         => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).

% conI\<^sub>\<Lambda>\<^sub>P
thf(fact_685_lambda__calculus_Otargets__simps_I2_J,axiom,
    ! [X: nat] :
      ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X ) )
      = ( insert_lambda_lambda @ ( lambda_Var @ X ) @ bot_bo1271416176687173302lambda ) ) ).

% lambda_calculus.targets_simps(2)
thf(fact_686_lambda__calculus_Osources__simps_I2_J,axiom,
    ! [X: nat] :
      ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X ) )
      = ( insert_lambda_lambda @ ( lambda_Var @ X ) @ bot_bo1271416176687173302lambda ) ) ).

% lambda_calculus.sources_simps(2)
thf(fact_687_lambda__calculus_Osources__simp,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( source5627170679743844886lambda @ lambda_resid @ T )
        = ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).

% lambda_calculus.sources_simp
thf(fact_688_IdeI,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T5 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) )
       => ( paths_7717490669901197737lambda @ lambda_resid @ T5 ) ) ) ).

% IdeI
thf(fact_689_lambda__calculus_Osources__subst,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U ) )
          = ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% lambda_calculus.sources_subst
thf(fact_690_less__eq__set__def,axiom,
    ( ord_le1926335974945931754lambda
    = ( ^ [A4: set_lambda_lambda,B5: set_lambda_lambda] :
          ( ord_le5488116190248881011mbda_o
          @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A4 )
          @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_691_less__eq__set__def,axiom,
    ( ord_le3064231726521830256lambda
    = ( ^ [A4: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
          ( ord_le4030711025213088621mbda_o
          @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A4 )
          @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_692_subset__code_I1_J,axiom,
    ! [Xs: list_lambda_lambda,B4: set_lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B4 )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
           => ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_693_subset__code_I1_J,axiom,
    ! [Xs: list_l5168835686021156086lambda,B4: set_li7878742357367608272lambda] :
      ( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B4 )
      = ( ! [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
           => ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_694_Collect__conv__if2,axiom,
    ! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
      ( ( ( P @ A2 )
       => ( ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bo1397037886704484924lambda ) ) ) ).

% Collect_conv_if2
thf(fact_695_Collect__conv__if2,axiom,
    ! [P: lambda_lambda > $o,A2: lambda_lambda] :
      ( ( ( P @ A2 )
       => ( ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bo1271416176687173302lambda ) ) ) ).

% Collect_conv_if2
thf(fact_696_Collect__conv__if,axiom,
    ! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
      ( ( ( P @ A2 )
       => ( ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec8830145823237851355lambda
            @ ^ [X3: list_lambda_lambda] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bo1397037886704484924lambda ) ) ) ).

% Collect_conv_if
thf(fact_697_Collect__conv__if,axiom,
    ! [P: lambda_lambda > $o,A2: lambda_lambda] :
      ( ( ( P @ A2 )
       => ( ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec621444948832755541lambda
            @ ^ [X3: lambda_lambda] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bo1271416176687173302lambda ) ) ) ).

% Collect_conv_if
thf(fact_698_singletonD,axiom,
    ! [B: list_lambda_lambda,A2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_699_singletonD,axiom,
    ! [B: lambda_lambda,A2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_700_singleton__iff,axiom,
    ! [B: list_lambda_lambda,A2: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_701_singleton__iff,axiom,
    ! [B: lambda_lambda,A2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_702_doubleton__eq__iff,axiom,
    ! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda,D: list_lambda_lambda] :
      ( ( ( insert8465602035467394112lambda @ A2 @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
        = ( insert8465602035467394112lambda @ C @ ( insert8465602035467394112lambda @ D @ bot_bo1397037886704484924lambda ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_703_doubleton__eq__iff,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda,D: lambda_lambda] :
      ( ( ( insert_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
        = ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D @ bot_bo1271416176687173302lambda ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_704_insert__not__empty,axiom,
    ! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( insert8465602035467394112lambda @ A2 @ A )
     != bot_bo1397037886704484924lambda ) ).

% insert_not_empty
thf(fact_705_insert__not__empty,axiom,
    ! [A2: lambda_lambda,A: set_lambda_lambda] :
      ( ( insert_lambda_lambda @ A2 @ A )
     != bot_bo1271416176687173302lambda ) ).

% insert_not_empty
thf(fact_706_singleton__inject,axiom,
    ! [A2: list_lambda_lambda,B: list_lambda_lambda] :
      ( ( ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda )
        = ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_707_singleton__inject,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda] :
      ( ( ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda )
        = ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_708_insert__Collect,axiom,
    ! [A2: list_lambda_lambda,P: list_lambda_lambda > $o] :
      ( ( insert8465602035467394112lambda @ A2 @ ( collec8830145823237851355lambda @ P ) )
      = ( collec8830145823237851355lambda
        @ ^ [U6: list_lambda_lambda] :
            ( ( U6 != A2 )
           => ( P @ U6 ) ) ) ) ).

% insert_Collect
thf(fact_709_insert__Collect,axiom,
    ! [A2: lambda_lambda,P: lambda_lambda > $o] :
      ( ( insert_lambda_lambda @ A2 @ ( collec621444948832755541lambda @ P ) )
      = ( collec621444948832755541lambda
        @ ^ [U6: lambda_lambda] :
            ( ( U6 != A2 )
           => ( P @ U6 ) ) ) ) ).

% insert_Collect
thf(fact_710_insert__compr,axiom,
    ( insert8465602035467394112lambda
    = ( ^ [A6: list_lambda_lambda,B5: set_li7878742357367608272lambda] :
          ( collec8830145823237851355lambda
          @ ^ [X3: list_lambda_lambda] :
              ( ( X3 = A6 )
              | ( member2667661554319814297lambda @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_711_insert__compr,axiom,
    ( insert_lambda_lambda
    = ( ^ [A6: lambda_lambda,B5: set_lambda_lambda] :
          ( collec621444948832755541lambda
          @ ^ [X3: lambda_lambda] :
              ( ( X3 = A6 )
              | ( member_lambda_lambda @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_712_filter__cong,axiom,
    ! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys ) )
           => ( ( P @ X2 )
              = ( Q @ X2 ) ) )
       => ( ( filter1131015166755930349lambda @ P @ Xs )
          = ( filter1131015166755930349lambda @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_713_filter__cong,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: lambda_lambda] :
            ( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys ) )
           => ( ( P @ X2 )
              = ( Q @ X2 ) ) )
       => ( ( filter_lambda_lambda @ P @ Xs )
          = ( filter_lambda_lambda @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_714_filter__id__conv,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( ( filter_lambda_lambda @ P @ Xs )
        = Xs )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
           => ( P @ X3 ) ) ) ) ).

% filter_id_conv
thf(fact_715_list_Oset__intros_I2_J,axiom,
    ! [Y4: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ X22 ) )
     => ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_716_list_Oset__intros_I2_J,axiom,
    ! [Y4: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
      ( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ X22 ) )
     => ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_717_list_Oset__intros_I1_J,axiom,
    ! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] : ( member2667661554319814297lambda @ X21 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_718_list_Oset__intros_I1_J,axiom,
    ! [X21: lambda_lambda,X22: list_lambda_lambda] : ( member_lambda_lambda @ X21 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_719_list_Oset__cases,axiom,
    ! [E: list_lambda_lambda,A2: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A2 ) )
     => ( ! [Z2: list_l5168835686021156086lambda] :
            ( A2
           != ( cons_l4558873032622161958lambda @ E @ Z2 ) )
       => ~ ! [Z1: list_lambda_lambda,Z2: list_l5168835686021156086lambda] :
              ( ( A2
                = ( cons_l4558873032622161958lambda @ Z1 @ Z2 ) )
             => ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_720_list_Oset__cases,axiom,
    ! [E: lambda_lambda,A2: list_lambda_lambda] :
      ( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A2 ) )
     => ( ! [Z2: list_lambda_lambda] :
            ( A2
           != ( cons_lambda_lambda @ E @ Z2 ) )
       => ~ ! [Z1: lambda_lambda,Z2: list_lambda_lambda] :
              ( ( A2
                = ( cons_lambda_lambda @ Z1 @ Z2 ) )
             => ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_721_set__ConsD,axiom,
    ! [Y4: list_lambda_lambda,X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) ) )
     => ( ( Y4 = X )
        | ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_722_set__ConsD,axiom,
    ! [Y4: lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X @ Xs ) ) )
     => ( ( Y4 = X )
        | ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_723_subset__singleton__iff,axiom,
    ! [X5: set_lambda_lambda,A2: lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ X5 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
      = ( ( X5 = bot_bo1271416176687173302lambda )
        | ( X5
          = ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ) ) ).

% subset_singleton_iff
thf(fact_724_subset__singleton__iff,axiom,
    ! [X5: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
      ( ( ord_le3064231726521830256lambda @ X5 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
      = ( ( X5 = bot_bo1397037886704484924lambda )
        | ( X5
          = ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ) ) ).

% subset_singleton_iff
thf(fact_725_subset__singletonD,axiom,
    ! [A: set_lambda_lambda,X: lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X @ bot_bo1271416176687173302lambda ) )
     => ( ( A = bot_bo1271416176687173302lambda )
        | ( A
          = ( insert_lambda_lambda @ X @ bot_bo1271416176687173302lambda ) ) ) ) ).

% subset_singletonD
thf(fact_726_subset__singletonD,axiom,
    ! [A: set_li7878742357367608272lambda,X: list_lambda_lambda] :
      ( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X @ bot_bo1397037886704484924lambda ) )
     => ( ( A = bot_bo1397037886704484924lambda )
        | ( A
          = ( insert8465602035467394112lambda @ X @ bot_bo1397037886704484924lambda ) ) ) ) ).

% subset_singletonD
thf(fact_727_set__subset__Cons,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_728_set__subset__Cons,axiom,
    ! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_729_empty__set,axiom,
    ( bot_bo1397037886704484924lambda
    = ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).

% empty_set
thf(fact_730_empty__set,axiom,
    ( bot_bo1271416176687173302lambda
    = ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).

% empty_set
thf(fact_731_split__list,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
     => ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
          ( Xs
          = ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_732_split__list,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
     => ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
          ( Xs
          = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_733_split__list__last,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
     => ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
          ( ( Xs
            = ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) )
          & ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_734_split__list__last,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
     => ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
          ( ( Xs
            = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
          & ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_735_split__list__prop,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_lambda_lambda,X2: lambda_lambda] :
          ( ? [Zs2: list_lambda_lambda] :
              ( Xs
              = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_736_split__list__first,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
     => ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
          ( ( Xs
            = ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) )
          & ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_737_split__list__first,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
     => ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
          ( ( Xs
            = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
          & ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_738_split__list__propE,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_lambda_lambda,X2: lambda_lambda] :
            ( ? [Zs2: list_lambda_lambda] :
                ( Xs
                = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_739_append__Cons__eq__iff,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs4: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
      ( ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
     => ( ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys ) )
       => ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ Ys ) )
            = ( append1017360404025248779lambda @ Xs4 @ ( cons_l4558873032622161958lambda @ X @ Ys6 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_740_append__Cons__eq__iff,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs4: list_lambda_lambda,Ys6: list_lambda_lambda] :
      ( ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
     => ( ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
       => ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ Ys ) )
            = ( append_lambda_lambda @ Xs4 @ ( cons_lambda_lambda @ X @ Ys6 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_741_in__set__conv__decomp,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
      = ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
            ( Xs
            = ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_742_in__set__conv__decomp,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
      = ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
            ( Xs
            = ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_743_split__list__last__prop,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_lambda_lambda,X2: lambda_lambda,Zs2: list_lambda_lambda] :
          ( ( Xs
            = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa: lambda_lambda] :
              ( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_744_split__list__first__prop,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_lambda_lambda,X2: lambda_lambda] :
          ( ? [Zs2: list_lambda_lambda] :
              ( Xs
              = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa: lambda_lambda] :
              ( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_745_split__list__last__propE,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_lambda_lambda,X2: lambda_lambda,Zs2: list_lambda_lambda] :
            ( ( Xs
              = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa: lambda_lambda] :
                    ( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_746_split__list__first__propE,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ? [X4: lambda_lambda] :
          ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_lambda_lambda,X2: lambda_lambda] :
            ( ? [Zs2: list_lambda_lambda] :
                ( Xs
                = ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa: lambda_lambda] :
                    ( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_747_in__set__conv__decomp__last,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
      = ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
            ( ( Xs
              = ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) )
            & ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_748_in__set__conv__decomp__last,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
      = ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
            ( ( Xs
              = ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
            & ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_749_in__set__conv__decomp__first,axiom,
    ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
      ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
      = ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
            ( ( Xs
              = ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) )
            & ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_750_in__set__conv__decomp__first,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
      = ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
            ( ( Xs
              = ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
            & ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_751_split__list__last__prop__iff,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ( ? [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda,Zs3: list_lambda_lambda] :
            ( ( Xs
              = ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: lambda_lambda] :
                ( ( member_lambda_lambda @ Y3 @ ( set_lambda_lambda2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_752_split__list__first__prop__iff,axiom,
    ! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
      ( ( ? [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda] :
            ( ? [Zs3: list_lambda_lambda] :
                ( Xs
                = ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: lambda_lambda] :
                ( ( member_lambda_lambda @ Y3 @ ( set_lambda_lambda2 @ Ys3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_753_hd__in__set,axiom,
    ! [Xs: list_l5168835686021156086lambda] :
      ( ( Xs != nil_li8379291187143545974lambda )
     => ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).

% hd_in_set
thf(fact_754_hd__in__set,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ( member_lambda_lambda @ ( hd_lambda_lambda @ Xs ) @ ( set_lambda_lambda2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_755_list_Oset__sel_I1_J,axiom,
    ! [A2: list_l5168835686021156086lambda] :
      ( ( A2 != nil_li8379291187143545974lambda )
     => ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ A2 ) @ ( set_li3846194476672328965lambda @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_756_list_Oset__sel_I1_J,axiom,
    ! [A2: list_lambda_lambda] :
      ( ( A2 != nil_lambda_lambda )
     => ( member_lambda_lambda @ ( hd_lambda_lambda @ A2 ) @ ( set_lambda_lambda2 @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_757_list_Oset__sel_I2_J,axiom,
    ! [A2: list_l5168835686021156086lambda,X: list_lambda_lambda] :
      ( ( A2 != nil_li8379291187143545974lambda )
     => ( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ ( tl_lis5905685823335986783lambda @ A2 ) ) )
       => ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ A2 ) ) ) ) ).

% list.set_sel(2)
thf(fact_758_list_Oset__sel_I2_J,axiom,
    ! [A2: list_lambda_lambda,X: lambda_lambda] :
      ( ( A2 != nil_lambda_lambda )
     => ( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ ( tl_lambda_lambda @ A2 ) ) )
       => ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ A2 ) ) ) ) ).

% list.set_sel(2)
thf(fact_759_filter__empty__conv,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( ( filter_lambda_lambda @ P @ Xs )
        = nil_lambda_lambda )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% filter_empty_conv
thf(fact_760_empty__filter__conv,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( nil_lambda_lambda
        = ( filter_lambda_lambda @ P @ Xs ) )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% empty_filter_conv
thf(fact_761_filter__is__subset,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( filter_lambda_lambda @ P @ Xs ) ) @ ( set_lambda_lambda2 @ Xs ) ) ).

% filter_is_subset
thf(fact_762_filter__is__subset,axiom,
    ! [P: list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ ( filter1131015166755930349lambda @ P @ Xs ) ) @ ( set_li3846194476672328965lambda @ Xs ) ) ).

% filter_is_subset
thf(fact_763_inter__set__filter,axiom,
    ! [A: set_li7878742357367608272lambda,Xs: list_l5168835686021156086lambda] :
      ( ( inf_in3354698378728726462lambda @ A @ ( set_li3846194476672328965lambda @ Xs ) )
      = ( set_li3846194476672328965lambda
        @ ( filter1131015166755930349lambda
          @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A )
          @ Xs ) ) ) ).

% inter_set_filter
thf(fact_764_inter__set__filter,axiom,
    ! [A: set_lambda_lambda,Xs: list_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ A @ ( set_lambda_lambda2 @ Xs ) )
      = ( set_lambda_lambda2
        @ ( filter_lambda_lambda
          @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A )
          @ Xs ) ) ) ).

% inter_set_filter
thf(fact_765_filter__eq__Cons__iff,axiom,
    ! [P: lambda_lambda > $o,Ys: list_lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( ( filter_lambda_lambda @ P @ Ys )
        = ( cons_lambda_lambda @ X @ Xs ) )
      = ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
            ( ( Ys
              = ( append_lambda_lambda @ Us2 @ ( cons_lambda_lambda @ X @ Vs ) ) )
            & ! [X3: lambda_lambda] :
                ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Us2 ) )
               => ~ ( P @ X3 ) )
            & ( P @ X )
            & ( Xs
              = ( filter_lambda_lambda @ P @ Vs ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_766_Cons__eq__filter__iff,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda,P: lambda_lambda > $o,Ys: list_lambda_lambda] :
      ( ( ( cons_lambda_lambda @ X @ Xs )
        = ( filter_lambda_lambda @ P @ Ys ) )
      = ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
            ( ( Ys
              = ( append_lambda_lambda @ Us2 @ ( cons_lambda_lambda @ X @ Vs ) ) )
            & ! [X3: lambda_lambda] :
                ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Us2 ) )
               => ~ ( P @ X3 ) )
            & ( P @ X )
            & ( Xs
              = ( filter_lambda_lambda @ P @ Vs ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_767_filter__eq__ConsD,axiom,
    ! [P: lambda_lambda > $o,Ys: list_lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( ( filter_lambda_lambda @ P @ Ys )
        = ( cons_lambda_lambda @ X @ Xs ) )
     => ? [Us3: list_lambda_lambda,Vs2: list_lambda_lambda] :
          ( ( Ys
            = ( append_lambda_lambda @ Us3 @ ( cons_lambda_lambda @ X @ Vs2 ) ) )
          & ! [X4: lambda_lambda] :
              ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Us3 ) )
             => ~ ( P @ X4 ) )
          & ( P @ X )
          & ( Xs
            = ( filter_lambda_lambda @ P @ Vs2 ) ) ) ) ).

% filter_eq_ConsD
thf(fact_768_Cons__eq__filterD,axiom,
    ! [X: lambda_lambda,Xs: list_lambda_lambda,P: lambda_lambda > $o,Ys: list_lambda_lambda] :
      ( ( ( cons_lambda_lambda @ X @ Xs )
        = ( filter_lambda_lambda @ P @ Ys ) )
     => ? [Us3: list_lambda_lambda,Vs2: list_lambda_lambda] :
          ( ( Ys
            = ( append_lambda_lambda @ Us3 @ ( cons_lambda_lambda @ X @ Vs2 ) ) )
          & ! [X4: lambda_lambda] :
              ( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Us3 ) )
             => ~ ( P @ X4 ) )
          & ( P @ X )
          & ( Xs
            = ( filter_lambda_lambda @ P @ Vs2 ) ) ) ) ).

% Cons_eq_filterD
thf(fact_769_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_770_emptyE,axiom,
    ! [A2: list_lambda_lambda] :
      ~ ( member2667661554319814297lambda @ A2 @ bot_bo1397037886704484924lambda ) ).

% emptyE
thf(fact_771_emptyE,axiom,
    ! [A2: lambda_lambda] :
      ~ ( member_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ).

% emptyE
thf(fact_772_equals0D,axiom,
    ! [A: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
      ( ( A = bot_bo1397037886704484924lambda )
     => ~ ( member2667661554319814297lambda @ A2 @ A ) ) ).

% equals0D
thf(fact_773_equals0D,axiom,
    ! [A: set_lambda_lambda,A2: lambda_lambda] :
      ( ( A = bot_bo1271416176687173302lambda )
     => ~ ( member_lambda_lambda @ A2 @ A ) ) ).

% equals0D
thf(fact_774_equals0I,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ! [Y2: list_lambda_lambda] :
          ~ ( member2667661554319814297lambda @ Y2 @ A )
     => ( A = bot_bo1397037886704484924lambda ) ) ).

% equals0I
thf(fact_775_equals0I,axiom,
    ! [A: set_lambda_lambda] :
      ( ! [Y2: lambda_lambda] :
          ~ ( member_lambda_lambda @ Y2 @ A )
     => ( A = bot_bo1271416176687173302lambda ) ) ).

% equals0I
thf(fact_776_ex__in__conv,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ( ? [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A ) )
      = ( A != bot_bo1397037886704484924lambda ) ) ).

% ex_in_conv
thf(fact_777_ex__in__conv,axiom,
    ! [A: set_lambda_lambda] :
      ( ( ? [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A ) )
      = ( A != bot_bo1271416176687173302lambda ) ) ).

% ex_in_conv
thf(fact_778_bot__set__def,axiom,
    ( bot_bo1397037886704484924lambda
    = ( collec8830145823237851355lambda @ bot_bo430095131446760353mbda_o ) ) ).

% bot_set_def
thf(fact_779_bot__set__def,axiom,
    ( bot_bo1271416176687173302lambda
    = ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).

% bot_set_def
thf(fact_780_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_9040249923598999359lambda @ Resid )
     => ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T5 @ U3 )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T5 @ U3 ) ) ) ).

% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_781_lambda__calculus_Osources__char_092_060_094sub_062_092_060Lambda_062,axiom,
    ! [T: lambda_lambda] :
      ( ( ( lambda_Arr @ T )
       => ( ( source5627170679743844886lambda @ lambda_resid @ T )
          = ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) )
      & ( ~ ( lambda_Arr @ T )
       => ( ( source5627170679743844886lambda @ lambda_resid @ T )
          = bot_bo1271416176687173302lambda ) ) ) ).

% lambda_calculus.sources_char\<^sub>\<Lambda>
thf(fact_782_empty__def,axiom,
    ( bot_bo1397037886704484924lambda
    = ( collec8830145823237851355lambda
      @ ^ [X3: list_lambda_lambda] : $false ) ) ).

% empty_def
thf(fact_783_empty__def,axiom,
    ( bot_bo1271416176687173302lambda
    = ( collec621444948832755541lambda
      @ ^ [X3: lambda_lambda] : $false ) ) ).

% empty_def
thf(fact_784_Collect__subset,axiom,
    ! [A: set_lambda_lambda,P: lambda_lambda > $o] :
      ( ord_le1926335974945931754lambda
      @ ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_785_Collect__subset,axiom,
    ! [A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
      ( ord_le3064231726521830256lambda
      @ ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_786_Collect__conj__eq,axiom,
    ! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
      ( ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( P @ X3 )
            & ( Q @ X3 ) ) )
      = ( inf_in3354698378728726462lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_787_Collect__conj__eq,axiom,
    ! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
      ( ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] :
            ( ( P @ X3 )
            & ( Q @ X3 ) ) )
      = ( inf_in1766959559283430712lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_788_Int__Collect,axiom,
    ! [X: list_lambda_lambda,A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
      ( ( member2667661554319814297lambda @ X @ ( inf_in3354698378728726462lambda @ A @ ( collec8830145823237851355lambda @ P ) ) )
      = ( ( member2667661554319814297lambda @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_789_Int__Collect,axiom,
    ! [X: lambda_lambda,A: set_lambda_lambda,P: lambda_lambda > $o] :
      ( ( member_lambda_lambda @ X @ ( inf_in1766959559283430712lambda @ A @ ( collec621444948832755541lambda @ P ) ) )
      = ( ( member_lambda_lambda @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_790_Int__def,axiom,
    ( inf_in3354698378728726462lambda
    = ( ^ [A4: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
          ( collec8830145823237851355lambda
          @ ^ [X3: list_lambda_lambda] :
              ( ( member2667661554319814297lambda @ X3 @ A4 )
              & ( member2667661554319814297lambda @ X3 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_791_Int__def,axiom,
    ( inf_in1766959559283430712lambda
    = ( ^ [A4: set_lambda_lambda,B5: set_lambda_lambda] :
          ( collec621444948832755541lambda
          @ ^ [X3: lambda_lambda] :
              ( ( member_lambda_lambda @ X3 @ A4 )
              & ( member_lambda_lambda @ X3 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_792_inf__set__def,axiom,
    ( inf_in3354698378728726462lambda
    = ( ^ [A4: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
          ( collec8830145823237851355lambda
          @ ( inf_in6319970306715391135mbda_o
            @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A4 )
            @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_793_inf__set__def,axiom,
    ( inf_in1766959559283430712lambda
    = ( ^ [A4: set_lambda_lambda,B5: set_lambda_lambda] :
          ( collec621444948832755541lambda
          @ ( inf_in2068736301489225125mbda_o
            @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A4 )
            @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_794_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_795_disjoint__iff__not__equal,axiom,
    ! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
      ( ( ( inf_in3354698378728726462lambda @ A @ B4 )
        = bot_bo1397037886704484924lambda )
      = ( ! [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ A )
           => ! [Y3: list_lambda_lambda] :
                ( ( member2667661554319814297lambda @ Y3 @ B4 )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_796_disjoint__iff__not__equal,axiom,
    ! [A: set_lambda_lambda,B4: set_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ A @ B4 )
        = bot_bo1271416176687173302lambda )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ A )
           => ! [Y3: lambda_lambda] :
                ( ( member_lambda_lambda @ Y3 @ B4 )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_797_Int__empty__right,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ A @ bot_bo1397037886704484924lambda )
      = bot_bo1397037886704484924lambda ) ).

% Int_empty_right
thf(fact_798_Int__empty__right,axiom,
    ! [A: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ A @ bot_bo1271416176687173302lambda )
      = bot_bo1271416176687173302lambda ) ).

% Int_empty_right
thf(fact_799_Int__empty__left,axiom,
    ! [B4: set_li7878742357367608272lambda] :
      ( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ B4 )
      = bot_bo1397037886704484924lambda ) ).

% Int_empty_left
thf(fact_800_Int__empty__left,axiom,
    ! [B4: set_lambda_lambda] :
      ( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B4 )
      = bot_bo1271416176687173302lambda ) ).

% Int_empty_left
thf(fact_801_disjoint__iff,axiom,
    ! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
      ( ( ( inf_in3354698378728726462lambda @ A @ B4 )
        = bot_bo1397037886704484924lambda )
      = ( ! [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ A )
           => ~ ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_802_disjoint__iff,axiom,
    ! [A: set_lambda_lambda,B4: set_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ A @ B4 )
        = bot_bo1271416176687173302lambda )
      = ( ! [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ A )
           => ~ ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_803_Int__emptyI,axiom,
    ! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
      ( ! [X2: list_lambda_lambda] :
          ( ( member2667661554319814297lambda @ X2 @ A )
         => ~ ( member2667661554319814297lambda @ X2 @ B4 ) )
     => ( ( inf_in3354698378728726462lambda @ A @ B4 )
        = bot_bo1397037886704484924lambda ) ) ).

% Int_emptyI
thf(fact_804_Int__emptyI,axiom,
    ! [A: set_lambda_lambda,B4: set_lambda_lambda] :
      ( ! [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ A )
         => ~ ( member_lambda_lambda @ X2 @ B4 ) )
     => ( ( inf_in1766959559283430712lambda @ A @ B4 )
        = bot_bo1271416176687173302lambda ) ) ).

% Int_emptyI
thf(fact_805_seq__Lam__Arr__implies,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
     => ( ( lambda_is_Lam @ T )
       => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U3 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) ) ) ) ).

% seq_Lam_Arr_implies
thf(fact_806_Srcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ T5 )
        = ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ lambda_resid @ T5 ) @ bot_bo1271416176687173302lambda ) ) ) ).

% Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_807_elementary__reduction__path__def,axiom,
    ( reduct8314398545252454212n_path
    = ( ^ [T8: list_lambda_lambda] :
          ( ( T8 = nil_lambda_lambda )
          | ( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
            & ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T8 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ) ) ) ).

% elementary_reduction_path_def
thf(fact_808_seq__char_H,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
      = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
        & ( paths_366302053746739058lambda @ lambda_resid @ U3 )
        & ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         != bot_bo1271416176687173302lambda ) ) ) ).

% seq_char'
thf(fact_809_Trgs_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).

% Trgs.simps(3)
thf(fact_810_ex__un__Trg,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ? [X2: lambda_lambda] :
          ( ( member_lambda_lambda @ X2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) )
          & ! [Y: lambda_lambda] :
              ( ( member_lambda_lambda @ Y @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) )
             => ( Y = X2 ) ) ) ) ).

% ex_un_Trg
thf(fact_811_Trgs__are__con,axiom,
    ! [B: lambda_lambda,T5: list_lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) )
     => ( ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) )
       => ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).

% Trgs_are_con
thf(fact_812_Trgs__con__closed,axiom,
    ! [B: lambda_lambda,T5: list_lambda_lambda,B2: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) )
     => ( ( ide_lambda_lambda @ lambda_resid @ B2 )
       => ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
         => ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% Trgs_con_closed
thf(fact_813_Trgs_Osimps_I1_J,axiom,
    ( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
    = bot_bo1271416176687173302lambda ) ).

% Trgs.simps(1)
thf(fact_814_Trgs__are__ide,axiom,
    ! [T5: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).

% Trgs_are_ide
thf(fact_815_Arr__has__Trg,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
       != bot_bo1271416176687173302lambda ) ) ).

% Arr_has_Trg
thf(fact_816_Std__implies__set__subset__elementary__reduction,axiom,
    ! [U3: list_lambda_lambda] :
      ( ( reduction_Std @ U3 )
     => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U3 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).

% Std_implies_set_subset_elementary_reduction
thf(fact_817_Trgs__eqI,axiom,
    ! [T5: list_lambda_lambda,T6: list_lambda_lambda] :
      ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
       != bot_bo1271416176687173302lambda )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
        = ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ).

% Trgs_eqI
thf(fact_818_Trgs__Resid__sym__Arr__single,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
       != nil_lambda_lambda )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
        = ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T5 ) ) ) ) ).

% Trgs_Resid_sym_Arr_single
thf(fact_819_Trgs_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( target5490931411331770124lambda @ lambda_resid @ T ) ) ).

% Trgs.simps(2)
thf(fact_820_seq__implies__Trgs__eq__Srcs,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
            = ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ).

% seq_implies_Trgs_eq_Srcs
thf(fact_821_set__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( reduct1603535164923377302opment @ T ) ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).

% set_standard_development
thf(fact_822_Srcs__Resid__single__Arr,axiom,
    ! [U: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T5 )
       != nil_lambda_lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T5 ) )
        = ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) ) ) ).

% Srcs_Resid_single_Arr
thf(fact_823_Trgs_Oelims,axiom,
    ! [X: list_lambda_lambda,Y4: set_lambda_lambda] :
      ( ( ( paths_7594662086398683701lambda @ lambda_resid @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4 != bot_bo1271416176687173302lambda ) )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
               != ( target5490931411331770124lambda @ lambda_resid @ T4 ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                 != ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% Trgs.elims
thf(fact_824_Arr__appendE_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
             => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
               => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
                 != ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).

% Arr_appendE\<^sub>P
thf(fact_825_seq__char,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
      = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
        & ( paths_366302053746739058lambda @ lambda_resid @ U3 )
        & ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
          = ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ).

% seq_char
thf(fact_826_Arr__append__iff_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( paths_366302053746739058lambda @ lambda_resid @ U3 )
            & ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Arr_append_iff\<^sub>P
thf(fact_827_Ide__append__iff_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
            & ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
            & ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Ide_append_iff\<^sub>P
thf(fact_828_targets__char_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( collec8830145823237851355lambda
        @ ^ [B5: list_lambda_lambda] :
            ( ( paths_7717490669901197737lambda @ lambda_resid @ B5 )
            & ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( ( paths_2897053694880971696lambda @ lambda_resid @ B5 )
              = ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% targets_char\<^sub>P
thf(fact_829_Trgs__Resid__sym,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        = ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) ) ) ) ).

% Trgs_Resid_sym
thf(fact_830_Trgs__append,axiom,
    ! [U3: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( U3 != nil_lambda_lambda )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( paths_7594662086398683701lambda @ lambda_resid @ U3 ) ) ) ).

% Trgs_append
thf(fact_831_Srcs__Resid,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        = ( paths_7594662086398683701lambda @ lambda_resid @ U3 ) ) ) ).

% Srcs_Resid
thf(fact_832_Src__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( paths_1651982994781619320lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( paths_1651982994781619320lambda @ lambda_resid @ T5 ) ) ) ).

% Src_append
thf(fact_833_lambda__calculus_Osseq__BetaI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( lambda_Ide @ T )
     => ( ( lambda_Ide @ U )
       => ( ( lambda2670285437103676873uction @ V )
         => ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ V )
           => ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ V ) ) ) ) ) ).

% lambda_calculus.sseq_BetaI
thf(fact_834_Arr__appendI_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         => ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ).

% Arr_appendI\<^sub>P
thf(fact_835_Ide__appendI_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
       => ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
         => ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ).

% Ide_appendI\<^sub>P
thf(fact_836_seqI_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
           != bot_bo1271416176687173302lambda )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).

% seqI\<^sub>P
thf(fact_837_lambda__calculus_Olambda_Odisc_I10_J,axiom,
    ! [X51: lambda_lambda,X52: lambda_lambda] :
      ~ ( lambda_is_Lam @ ( lambda_Beta @ X51 @ X52 ) ) ).

% lambda_calculus.lambda.disc(10)
thf(fact_838_lambda__calculus_Olambda_Odisc_I6_J,axiom,
    ~ ( lambda_is_Lam @ lambda_Nil ) ).

% lambda_calculus.lambda.disc(6)
thf(fact_839_lambda__calculus_Olambda_Odistinct__disc_I4_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( lambda_is_Lam @ Lambda )
     => ( Lambda != lambda_Nil ) ) ).

% lambda_calculus.lambda.distinct_disc(4)
thf(fact_840_lambda__calculus_Olambda_Odistinct__disc_I3_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( Lambda = lambda_Nil )
     => ~ ( lambda_is_Lam @ Lambda ) ) ).

% lambda_calculus.lambda.distinct_disc(3)
thf(fact_841_lambda__calculus_Olambda_Odisc_I7_J,axiom,
    ! [X24: nat] :
      ~ ( lambda_is_Lam @ ( lambda_Var @ X24 ) ) ).

% lambda_calculus.lambda.disc(7)
thf(fact_842_lambda__calculus_Osseq__imp__elementary__reduction1,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_sseq @ T @ U )
     => ( lambda2670285437103676873uction @ T ) ) ).

% lambda_calculus.sseq_imp_elementary_reduction1
thf(fact_843_lambda__calculus_Osseq__imp__elementary__reduction2,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_sseq @ T @ U )
     => ( lambda2670285437103676873uction @ U ) ) ).

% lambda_calculus.sseq_imp_elementary_reduction2
thf(fact_844_paths__in__rts_OTrgs_Ocong,axiom,
    paths_7594662086398683701lambda = paths_7594662086398683701lambda ).

% paths_in_rts.Trgs.cong
thf(fact_845_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
    paths_1651982994781619320lambda = paths_1651982994781619320lambda ).

% paths_in_weakly_extensional_rts.Src.cong
thf(fact_846_lambda__calculus_Oelementary__reduction_Osimps_I1_J,axiom,
    ~ ( lambda2670285437103676873uction @ lambda_Nil ) ).

% lambda_calculus.elementary_reduction.simps(1)
thf(fact_847_lambda__calculus_Oelementary__reduction_Osimps_I2_J,axiom,
    ! [Uu: nat] :
      ~ ( lambda2670285437103676873uction @ ( lambda_Var @ Uu ) ) ).

% lambda_calculus.elementary_reduction.simps(2)
thf(fact_848_lambda__calculus_Olambda_Odistinct__disc_I17_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( lambda_is_Lam @ Lambda )
     => ~ ( lambda_is_Beta @ Lambda ) ) ).

% lambda_calculus.lambda.distinct_disc(17)
thf(fact_849_lambda__calculus_Olambda_Odistinct__disc_I18_J,axiom,
    ! [Lambda: lambda_lambda] :
      ( ( lambda_is_Beta @ Lambda )
     => ~ ( lambda_is_Lam @ Lambda ) ) ).

% lambda_calculus.lambda.distinct_disc(18)
thf(fact_850_lambda__calculus_Oelementary__reduction_Osimps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda2670285437103676873uction @ ( lambda_Beta @ T @ U ) )
      = ( ( lambda_Ide @ T )
        & ( lambda_Ide @ U ) ) ) ).

% lambda_calculus.elementary_reduction.simps(5)
thf(fact_851_lambda__calculus_Ocong__elementary__reductions__are__equal,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda2670285437103676873uction @ T )
     => ( ( lambda2670285437103676873uction @ U )
       => ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
            & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
         => ( T = U ) ) ) ) ).

% lambda_calculus.cong_elementary_reductions_are_equal
thf(fact_852_lambda__calculus_Oelementary__reduction__not__ide,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda2670285437103676873uction @ T )
     => ~ ( ide_lambda_lambda @ lambda_resid @ T ) ) ).

% lambda_calculus.elementary_reduction_not_ide
thf(fact_853_lambda__calculus_Oelementary__reduction__is__arr,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda2670285437103676873uction @ T )
     => ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% lambda_calculus.elementary_reduction_is_arr
thf(fact_854_lambda__calculus_OCon__implies__is__Lam__iff__is__Lam,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
     => ( ( lambda_is_Lam @ T )
        = ( lambda_is_Lam @ U ) ) ) ).

% lambda_calculus.Con_implies_is_Lam_iff_is_Lam
thf(fact_855_lambda__calculus_Olambda_Odisc__eq__case_I3_J,axiom,
    ( lambda_is_Lam
    = ( lambda_case_lambda_o @ $false
      @ ^ [Uu2: nat] : $false
      @ ^ [Uu2: lambda_lambda] : $true
      @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $false
      @ ^ [Uu2: lambda_lambda,Uv: lambda_lambda] : $false ) ) ).

% lambda_calculus.lambda.disc_eq_case(3)
thf(fact_856_lambda__calculus_Ohead__strategy__is__elementary,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ~ ( lambda_Ide @ ( lambda_head_strategy @ T ) )
       => ( lambda2670285437103676873uction @ ( lambda_head_strategy @ T ) ) ) ) ).

% lambda_calculus.head_strategy_is_elementary
thf(fact_857_lambda__calculus_Oelementary__reduction__head__redex,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ~ ( lambda_Ide @ ( lambda_head_redex @ T ) )
       => ( lambda2670285437103676873uction @ ( lambda_head_redex @ T ) ) ) ) ).

% lambda_calculus.elementary_reduction_head_redex
thf(fact_858_reduction__paths_OStd_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( reduction_Std @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( lambda2670285437103676873uction @ T ) ) ).

% reduction_paths.Std.simps(2)
thf(fact_859_lambda__calculus_Osseq__Beta,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ V )
      = ( ( lambda_Ide @ T )
        & ( lambda_Ide @ U )
        & ( lambda2670285437103676873uction @ V )
        & ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ V ) ) ) ).

% lambda_calculus.sseq_Beta
thf(fact_860_lambda__calculus_Osseq_Osimps_I12_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
      ( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ Va @ Vb ) )
      = ( ( lambda_Ide @ T )
        & ( lambda_Ide @ U )
        & ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb ) )
        & ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ Va @ Vb ) ) ) ) ).

% lambda_calculus.sseq.simps(12)
thf(fact_861_reduction__paths_OStd_Oelims_I1_J,axiom,
    ! [X: list_lambda_lambda,Y4: $o] :
      ( ( ( reduction_Std @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ~ Y4 )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
                = ( ~ ( lambda2670285437103676873uction @ T4 ) ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                  = ( ~ ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                        & ( reduction_Std @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ).

% reduction_paths.Std.elims(1)
thf(fact_862_reduction__paths_OStd_Oelims_I2_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ( reduction_Std @ X )
     => ( ( X != nil_lambda_lambda )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ~ ( lambda2670285437103676873uction @ T4 ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ~ ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                    & ( reduction_Std @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% reduction_paths.Std.elims(2)
thf(fact_863_reduction__paths_OStd_Oelims_I3_J,axiom,
    ! [X: list_lambda_lambda] :
      ( ~ ( reduction_Std @ X )
     => ( ! [T4: lambda_lambda] :
            ( ( X
              = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
           => ( lambda2670285437103676873uction @ T4 ) )
       => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
             => ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                & ( reduction_Std @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ).

% reduction_paths.Std.elims(3)
thf(fact_864_lambda__calculus_Osseq_Osimps_I9_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,Va: nat] :
      ( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_Var @ Va ) )
      = ( ( lambda_Ide @ T )
        & ( lambda_Ide @ U )
        & ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
        & ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Var @ Va ) ) ) ) ).

% lambda_calculus.sseq.simps(9)
thf(fact_865_path__from__NF__is__Ide,axiom,
    ! [A2: lambda_lambda,U3: list_lambda_lambda] :
      ( ( lambda_NF @ A2 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U3 )
            = A2 )
         => ( paths_7717490669901197737lambda @ lambda_resid @ U3 ) ) ) ) ).

% path_from_NF_is_Ide
thf(fact_866_img__Resid__Srcs,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ord_le3064231726521830256lambda
        @ ( image_3692823078739859851lambda
          @ ^ [A6: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A6 @ nil_lambda_lambda ) @ T5 )
          @ ( paths_2897053694880971696lambda @ lambda_resid @ T5 ) )
        @ ( image_3692823078739859851lambda
          @ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
          @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) ) ) ) ).

% img_Resid_Srcs
thf(fact_867_image__empty,axiom,
    ! [F: list_lambda_lambda > list_lambda_lambda] :
      ( ( image_5886195720093285125lambda @ F @ bot_bo1397037886704484924lambda )
      = bot_bo1397037886704484924lambda ) ).

% image_empty
thf(fact_868_image__empty,axiom,
    ! [F: list_lambda_lambda > lambda_lambda] :
      ( ( image_2458965899662700159lambda @ F @ bot_bo1397037886704484924lambda )
      = bot_bo1271416176687173302lambda ) ).

% image_empty
thf(fact_869_image__empty,axiom,
    ! [F: lambda_lambda > list_lambda_lambda] :
      ( ( image_3692823078739859851lambda @ F @ bot_bo1271416176687173302lambda )
      = bot_bo1397037886704484924lambda ) ).

% image_empty
thf(fact_870_image__empty,axiom,
    ! [F: lambda_lambda > lambda_lambda] :
      ( ( image_2356535832909760517lambda @ F @ bot_bo1271416176687173302lambda )
      = bot_bo1271416176687173302lambda ) ).

% image_empty
thf(fact_871_empty__is__image,axiom,
    ! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( bot_bo1397037886704484924lambda
        = ( image_5886195720093285125lambda @ F @ A ) )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% empty_is_image
thf(fact_872_empty__is__image,axiom,
    ! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
      ( ( bot_bo1397037886704484924lambda
        = ( image_3692823078739859851lambda @ F @ A ) )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% empty_is_image
thf(fact_873_empty__is__image,axiom,
    ! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( bot_bo1271416176687173302lambda
        = ( image_2458965899662700159lambda @ F @ A ) )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% empty_is_image
thf(fact_874_empty__is__image,axiom,
    ! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
      ( ( bot_bo1271416176687173302lambda
        = ( image_2356535832909760517lambda @ F @ A ) )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% empty_is_image
thf(fact_875_image__is__empty,axiom,
    ! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( ( image_5886195720093285125lambda @ F @ A )
        = bot_bo1397037886704484924lambda )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% image_is_empty
thf(fact_876_image__is__empty,axiom,
    ! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
      ( ( ( image_3692823078739859851lambda @ F @ A )
        = bot_bo1397037886704484924lambda )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% image_is_empty
thf(fact_877_image__is__empty,axiom,
    ! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( ( image_2458965899662700159lambda @ F @ A )
        = bot_bo1271416176687173302lambda )
      = ( A = bot_bo1397037886704484924lambda ) ) ).

% image_is_empty
thf(fact_878_image__is__empty,axiom,
    ! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
      ( ( ( image_2356535832909760517lambda @ F @ A )
        = bot_bo1271416176687173302lambda )
      = ( A = bot_bo1271416176687173302lambda ) ) ).

% image_is_empty
thf(fact_879_Resid__single__ide_I1_J,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T5 )
         != nil_lambda_lambda )
       => ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T5 )
          @ ( image_3692823078739859851lambda
            @ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
            @ ( paths_7594662086398683701lambda @ lambda_resid @ T5 ) ) ) ) ) ).

% Resid_single_ide(1)
thf(fact_880_imageE,axiom,
    ! [B: list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( member2667661554319814297lambda @ B @ ( image_5886195720093285125lambda @ F @ A ) )
     => ~ ! [X2: list_lambda_lambda] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member2667661554319814297lambda @ X2 @ A ) ) ) ).

% imageE
thf(fact_881_imageE,axiom,
    ! [B: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( image_3692823078739859851lambda @ F @ A ) )
     => ~ ! [X2: lambda_lambda] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_lambda_lambda @ X2 @ A ) ) ) ).

% imageE
thf(fact_882_imageE,axiom,
    ! [B: lambda_lambda,F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
      ( ( member_lambda_lambda @ B @ ( image_2458965899662700159lambda @ F @ A ) )
     => ~ ! [X2: list_lambda_lambda] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member2667661554319814297lambda @ X2 @ A ) ) ) ).

% imageE
thf(fact_883_imageE,axiom,
    ! [B: lambda_lambda,F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( image_2356535832909760517lambda @ F @ A ) )
     => ~ ! [X2: lambda_lambda] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_lambda_lambda @ X2 @ A ) ) ) ).

% imageE
thf(fact_884_image__image,axiom,
    ! [F: list_lambda_lambda > list_lambda_lambda,G: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
      ( ( image_5886195720093285125lambda @ F @ ( image_3692823078739859851lambda @ G @ A ) )
      = ( image_3692823078739859851lambda
        @ ^ [X3: lambda_lambda] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_885_image__image,axiom,
    ! [F: lambda_lambda > list_lambda_lambda,G: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
      ( ( image_3692823078739859851lambda @ F @ ( image_2356535832909760517lambda @ G @ A ) )
      = ( image_3692823078739859851lambda
        @ ^ [X3: lambda_lambda] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_886_Compr__image__eq,axiom,
    ! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
      ( ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ ( image_5886195720093285125lambda @ F @ A ) )
            & ( P @ X3 ) ) )
      = ( image_5886195720093285125lambda @ F
        @ ( collec8830145823237851355lambda
          @ ^ [X3: list_lambda_lambda] :
              ( ( member2667661554319814297lambda @ X3 @ A )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_887_Compr__image__eq,axiom,
    ! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,P: list_lambda_lambda > $o] :
      ( ( collec8830145823237851355lambda
        @ ^ [X3: list_lambda_lambda] :
            ( ( member2667661554319814297lambda @ X3 @ ( image_3692823078739859851lambda @ F @ A ) )
            & ( P @ X3 ) ) )
      = ( image_3692823078739859851lambda @ F
        @ ( collec621444948832755541lambda
          @ ^ [X3: lambda_lambda] :
              ( ( member_lambda_lambda @ X3 @ A )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_888_Compr__image__eq,axiom,
    ! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda,P: lambda_lambda > $o] :
      ( ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( image_2458965899662700159lambda @ F @ A ) )
            & ( P @ X3 ) ) )
      = ( image_2458965899662700159lambda @ F
        @ ( collec8830145823237851355lambda
          @ ^ [X3: list_lambda_lambda] :
              ( ( member2667661554319814297lambda @ X3 @ A )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_889_Compr__image__eq,axiom,
    ! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda,P: lambda_lambda > $o] :
      ( ( collec621444948832755541lambda
        @ ^ [X3: lambda_lambda] :
            ( ( member_lambda_lambda @ X3 @ ( image_2356535832909760517lambda @ F @ A ) )
            & ( P @ X3 ) ) )
      = ( image_2356535832909760517lambda @ F
        @ ( collec621444948832755541lambda
          @ ^ [X3: lambda_lambda] :
              ( ( member_lambda_lambda @ X3 @ A )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_890_image__constant__conv,axiom,
    ! [A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
      ( ( ( A = bot_bo1397037886704484924lambda )
       => ( ( image_5886195720093285125lambda
            @ ^ [X3: list_lambda_lambda] : C
            @ A )
          = bot_bo1397037886704484924lambda ) )
      & ( ( A != bot_bo1397037886704484924lambda )
       => ( ( image_5886195720093285125lambda
            @ ^ [X3: list_lambda_lambda] : C
            @ A )
          = ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).

% image_constant_conv
thf(fact_891_image__constant__conv,axiom,
    ! [A: set_li7878742357367608272lambda,C: lambda_lambda] :
      ( ( ( A = bot_bo1397037886704484924lambda )
       => ( ( image_2458965899662700159lambda
            @ ^ [X3: list_lambda_lambda] : C
            @ A )
          = bot_bo1271416176687173302lambda ) )
      & ( ( A != bot_bo1397037886704484924lambda )
       => ( ( image_2458965899662700159lambda
            @ ^ [X3: list_lambda_lambda] : C
            @ A )
          = ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).

% image_constant_conv
thf(fact_892_image__constant__conv,axiom,
    ! [A: set_lambda_lambda,C: list_lambda_lambda] :
      ( ( ( A = bot_bo1271416176687173302lambda )
       => ( ( image_3692823078739859851lambda
            @ ^ [X3: lambda_lambda] : C
            @ A )
          = bot_bo1397037886704484924lambda ) )
      & ( ( A != bot_bo1271416176687173302lambda )
       => ( ( image_3692823078739859851lambda
            @ ^ [X3: lambda_lambda] : C
            @ A )
          = ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).

% image_constant_conv
thf(fact_893_image__constant__conv,axiom,
    ! [A: set_lambda_lambda,C: lambda_lambda] :
      ( ( ( A = bot_bo1271416176687173302lambda )
       => ( ( image_2356535832909760517lambda
            @ ^ [X3: lambda_lambda] : C
            @ A )
          = bot_bo1271416176687173302lambda ) )
      & ( ( A != bot_bo1271416176687173302lambda )
       => ( ( image_2356535832909760517lambda
            @ ^ [X3: lambda_lambda] : C
            @ A )
          = ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).

% image_constant_conv
thf(fact_894_image__constant,axiom,
    ! [X: list_lambda_lambda,A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ X @ A )
     => ( ( image_5886195720093285125lambda
          @ ^ [X3: list_lambda_lambda] : C
          @ A )
        = ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).

% image_constant
thf(fact_895_image__constant,axiom,
    ! [X: lambda_lambda,A: set_lambda_lambda,C: list_lambda_lambda] :
      ( ( member_lambda_lambda @ X @ A )
     => ( ( image_3692823078739859851lambda
          @ ^ [X3: lambda_lambda] : C
          @ A )
        = ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).

% image_constant
thf(fact_896_image__constant,axiom,
    ! [X: list_lambda_lambda,A: set_li7878742357367608272lambda,C: lambda_lambda] :
      ( ( member2667661554319814297lambda @ X @ A )
     => ( ( image_2458965899662700159lambda
          @ ^ [X3: list_lambda_lambda] : C
          @ A )
        = ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).

% image_constant
thf(fact_897_image__constant,axiom,
    ! [X: lambda_lambda,A: set_lambda_lambda,C: lambda_lambda] :
      ( ( member_lambda_lambda @ X @ A )
     => ( ( image_2356535832909760517lambda
          @ ^ [X3: lambda_lambda] : C
          @ A )
        = ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).

% image_constant
thf(fact_898_lambda__calculus_ONF__def,axiom,
    ( lambda_NF
    = ( ^ [A6: lambda_lambda] :
          ( ( lambda_Ide @ A6 )
          & ! [T7: lambda_lambda] :
              ( ( ( lambda_Arr @ T7 )
                & ( ( lambda_Src @ T7 )
                  = A6 ) )
             => ( lambda_Ide @ T7 ) ) ) ) ) ).

% lambda_calculus.NF_def
thf(fact_899_lambda__calculus_Onot__NF__elim,axiom,
    ! [T: lambda_lambda] :
      ( ~ ( lambda_NF @ T )
     => ( ( lambda_Ide @ T )
       => ~ ! [U7: lambda_lambda] :
              ~ ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U7 )
                & ~ ( lambda_Ide @ U7 ) ) ) ) ).

% lambda_calculus.not_NF_elim
thf(fact_900_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
     => ( A2 = bot_bo1271416176687173302lambda ) ) ).

% bot.extremum_uniqueI
thf(fact_901_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_li7878742357367608272lambda] :
      ( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
     => ( A2 = bot_bo1397037886704484924lambda ) ) ).

% bot.extremum_uniqueI
thf(fact_902_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_903_bot_Oextremum__unique,axiom,
    ! [A2: set_lambda_lambda] :
      ( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
      = ( A2 = bot_bo1271416176687173302lambda ) ) ).

% bot.extremum_unique
thf(fact_904_bot_Oextremum__unique,axiom,
    ! [A2: set_li7878742357367608272lambda] :
      ( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
      = ( A2 = bot_bo1397037886704484924lambda ) ) ).

% bot.extremum_unique
thf(fact_905_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_906_bot_Oextremum,axiom,
    ! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A2 ) ).

% bot.extremum
thf(fact_907_bot_Oextremum,axiom,
    ! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A2 ) ).

% bot.extremum
thf(fact_908_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_909_set__Cons__sing__Nil,axiom,
    ! [A: set_lambda_lambda] :
      ( ( set_Co4812318673604925728lambda @ A @ ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) )
      = ( image_3692823078739859851lambda
        @ ^ [X3: lambda_lambda] : ( cons_lambda_lambda @ X3 @ nil_lambda_lambda )
        @ A ) ) ).

% set_Cons_sing_Nil
thf(fact_910_Trgs__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( ( reduct1603535164923377302opment @ T )
         != nil_lambda_lambda )
       => ( ( paths_7594662086398683701lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
          = ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% Trgs_standard_development
thf(fact_911_Trgs__complete__development,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct7282684938351565978opment @ T @ U3 )
     => ( ~ ( lambda_Ide @ T )
       => ( ( paths_7594662086398683701lambda @ lambda_resid @ U3 )
          = ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% Trgs_complete_development
thf(fact_912_lambda__calculus_OTrg__Trg,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Trg @ ( lambda_Trg @ T ) )
        = ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.Trg_Trg
thf(fact_913_lambda__calculus_OSrc__Trg,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Src @ ( lambda_Trg @ T ) )
        = ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.Src_Trg
thf(fact_914_lambda__calculus_OTrg__Src,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Trg @ ( lambda_Src @ T ) )
        = ( lambda_Src @ T ) ) ) ).

% lambda_calculus.Trg_Src
thf(fact_915_lambda__calculus_OTrg__Subst,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,D: nat] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( lambda_Trg @ ( lambda_Subst @ D @ T @ U ) )
          = ( lambda_Subst @ D @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) ) ) ) ).

% lambda_calculus.Trg_Subst
thf(fact_916_lambda__calculus_Oresid__Src__Arr,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
        = ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.resid_Src_Arr
thf(fact_917_lambda__calculus_OseqI_092_060_094sub_062_092_060Lambda_062,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( ( lambda_Trg @ T )
            = ( lambda_Src @ U ) )
         => ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).

% lambda_calculus.seqI\<^sub>\<Lambda>
thf(fact_918_lambda__calculus_Otargets__simp,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( target5490931411331770124lambda @ lambda_resid @ T )
        = ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).

% lambda_calculus.targets_simp
thf(fact_919_targets__single__Trg,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).

% targets_single_Trg
thf(fact_920_sources__single__Trg,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).

% sources_single_Trg
thf(fact_921_lambda__calculus_Otargets__subst,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Arr @ U )
       => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U ) )
          = ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% lambda_calculus.targets_subst
thf(fact_922_lambda__calculus_Otargets__simps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( arr_lambda_lambda @ lambda_resid @ U )
       => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) )
          = ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% lambda_calculus.targets_simps(5)
thf(fact_923_lambda__calculus_Oresid__Arr__self,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_resid @ T @ T )
        = ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.resid_Arr_self
thf(fact_924_lambda__calculus_OIde__Trg,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.Ide_Trg
thf(fact_925_lambda__calculus_OIde__iff__Trg__self,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ( lambda_Ide @ T )
        = ( ( lambda_Trg @ T )
          = T ) ) ) ).

% lambda_calculus.Ide_iff_Trg_self
thf(fact_926_lambda__calculus_OArr__Trg,axiom,
    ! [X5: lambda_lambda] :
      ( ( lambda_Arr @ X5 )
     => ( lambda_Arr @ ( lambda_Trg @ X5 ) ) ) ).

% lambda_calculus.Arr_Trg
thf(fact_927_lambda__calculus_OTrg_Osimps_I5_J,axiom,
    ( ( lambda_Trg @ lambda_Nil )
    = lambda_Nil ) ).

% lambda_calculus.Trg.simps(5)
thf(fact_928_lambda__calculus_OTrg_Osimps_I1_J,axiom,
    ! [I: nat] :
      ( ( lambda_Trg @ ( lambda_Var @ I ) )
      = ( lambda_Var @ I ) ) ).

% lambda_calculus.Trg.simps(1)
thf(fact_929_lambda__calculus_OSrc__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != lambda_Nil )
     => ( ( lambda_Src @ ( lambda_resid @ T @ U ) )
        = ( lambda_Trg @ U ) ) ) ).

% lambda_calculus.Src_resid
thf(fact_930_lambda__calculus_OTrg_Osimps_I4_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda_Trg @ ( lambda_Beta @ T @ U ) )
      = ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.Trg.simps(4)
thf(fact_931_lambda__calculus_OseqE_092_060_094sub_062_092_060Lambda_062,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( seq_lambda_lambda @ lambda_resid @ T @ U )
     => ~ ( ( lambda_Arr @ T )
         => ( ( lambda_Arr @ U )
           => ( ( lambda_Trg @ T )
             != ( lambda_Src @ U ) ) ) ) ) ).

% lambda_calculus.seqE\<^sub>\<Lambda>
thf(fact_932_lambda__calculus_Oseq__char,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( seq_lambda_lambda @ lambda_resid @ T @ U )
      = ( ( lambda_Arr @ T )
        & ( lambda_Arr @ U )
        & ( ( lambda_Trg @ T )
          = ( lambda_Src @ U ) ) ) ) ).

% lambda_calculus.seq_char
thf(fact_933_lambda__calculus_Otargets__char_092_060_094sub_062_092_060Lambda_062,axiom,
    ! [T: lambda_lambda] :
      ( ( ( lambda_Arr @ T )
       => ( ( target5490931411331770124lambda @ lambda_resid @ T )
          = ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) )
      & ( ~ ( lambda_Arr @ T )
       => ( ( target5490931411331770124lambda @ lambda_resid @ T )
          = bot_bo1271416176687173302lambda ) ) ) ).

% lambda_calculus.targets_char\<^sub>\<Lambda>
thf(fact_934_targets__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
      = ( collec8830145823237851355lambda
        @ ^ [B5: list_lambda_lambda] :
            ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) ) )
            & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ B5 ) ) ) ) ) ).

% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_935_in__targets__iff,axiom,
    ! [T5: list_lambda_lambda,B4: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
        = ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ B4 ) ) ) ) ) ).

% in_targets_iff
thf(fact_936_seq__imp__cong__Trg__last__Src__hd,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) )
        & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) ) ) ) ) ).

% seq_imp_cong_Trg_last_Src_hd
thf(fact_937_Arr__imp__arr__last,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( arr_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T5 ) ) ) ).

% Arr_imp_arr_last
thf(fact_938_Ide__imp__Ide__last,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ide_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T5 ) ) ) ).

% Ide_imp_Ide_last
thf(fact_939_Trg__last__eqI,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) ) )
     => ( ( lambda_Trg @ ( last_lambda_lambda @ T5 ) )
        = ( lambda_Trg @ ( last_lambda_lambda @ U3 ) ) ) ) ).

% Trg_last_eqI
thf(fact_940_Trg__last__standard__development,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( ~ ( lambda_Ide @ T )
       => ( ( lambda_Trg @ ( last_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
          = ( lambda_Trg @ T ) ) ) ) ).

% Trg_last_standard_development
thf(fact_941_Trgs__simp_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
        = ( target5490931411331770124lambda @ lambda_resid @ ( last_lambda_lambda @ T5 ) ) ) ) ).

% Trgs_simp\<^sub>P
thf(fact_942_Std__append__single,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( reduction_Std @ T5 )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( lambda_sseq @ ( last_lambda_lambda @ T5 ) @ U )
         => ( reduction_Std @ ( append_lambda_lambda @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ) ) ).

% Std_append_single
thf(fact_943_Std__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduction_Std @ T5 )
     => ( ( reduction_Std @ U3 )
       => ( ( ( T5 = nil_lambda_lambda )
            | ( U3 = nil_lambda_lambda )
            | ( lambda_sseq @ ( last_lambda_lambda @ T5 ) @ ( hd_lambda_lambda @ U3 ) ) )
         => ( reduction_Std @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ).

% Std_append
thf(fact_944_Std__imp__sseq__last__hd,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduction_Std @ ( append_lambda_lambda @ T5 @ U3 ) )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( lambda_sseq @ ( last_lambda_lambda @ T5 ) @ ( hd_lambda_lambda @ U3 ) ) ) ) ) ).

% Std_imp_sseq_last_hd
thf(fact_945_Trg__last__Src__hd__eqI,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( lambda_Trg @ ( last_lambda_lambda @ T5 ) )
        = ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) ) ).

% Trg_last_Src_hd_eqI
thf(fact_946_Trgs__simp_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ T )
        = ( insert_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).

% Trgs_simp\<^sub>\<Lambda>\<^sub>P
thf(fact_947_single__Trg__last__in__targets,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T5 ) ) @ nil_lambda_lambda ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).

% single_Trg_last_in_targets
thf(fact_948_last__appendL,axiom,
    ! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
      ( ( Ys = nil_lambda_lambda )
     => ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
        = ( last_lambda_lambda @ Xs ) ) ) ).

% last_appendL
thf(fact_949_last__appendR,axiom,
    ! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
      ( ( Ys != nil_lambda_lambda )
     => ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
        = ( last_lambda_lambda @ Ys ) ) ) ).

% last_appendR
thf(fact_950_last__snoc,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda] :
      ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) )
      = X ) ).

% last_snoc
thf(fact_951_seqI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( lambda_Trg @ ( last_lambda_lambda @ T5 ) )
            = ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) )
         => ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).

% seqI\<^sub>\<Lambda>\<^sub>P
thf(fact_952_last_Osimps,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda] :
      ( ( ( Xs = nil_lambda_lambda )
       => ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_lambda_lambda )
       => ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
          = ( last_lambda_lambda @ Xs ) ) ) ) ).

% last.simps
thf(fact_953_last__ConsL,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda] :
      ( ( Xs = nil_lambda_lambda )
     => ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_954_last__ConsR,axiom,
    ! [Xs: list_lambda_lambda,X: lambda_lambda] :
      ( ( Xs != nil_lambda_lambda )
     => ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
        = ( last_lambda_lambda @ Xs ) ) ) ).

% last_ConsR
thf(fact_955_last__in__set,axiom,
    ! [As: list_l5168835686021156086lambda] :
      ( ( As != nil_li8379291187143545974lambda )
     => ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).

% last_in_set
thf(fact_956_last__in__set,axiom,
    ! [As: list_lambda_lambda] :
      ( ( As != nil_lambda_lambda )
     => ( member_lambda_lambda @ ( last_lambda_lambda @ As ) @ ( set_lambda_lambda2 @ As ) ) ) ).

% last_in_set
thf(fact_957_last__append,axiom,
    ! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
      ( ( ( Ys = nil_lambda_lambda )
       => ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( last_lambda_lambda @ Xs ) ) )
      & ( ( Ys != nil_lambda_lambda )
       => ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
          = ( last_lambda_lambda @ Ys ) ) ) ) ).

% last_append
thf(fact_958_longest__common__suffix,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
    ? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
      ( ( Xs
        = ( append_lambda_lambda @ Xs3 @ Ss ) )
      & ( Ys
        = ( append_lambda_lambda @ Ys5 @ Ss ) )
      & ( ( Xs3 = nil_lambda_lambda )
        | ( Ys5 = nil_lambda_lambda )
        | ( ( last_lambda_lambda @ Xs3 )
         != ( last_lambda_lambda @ Ys5 ) ) ) ) ).

% longest_common_suffix
thf(fact_959_hd__Nil__eq__last,axiom,
    ( ( hd_lambda_lambda @ nil_lambda_lambda )
    = ( last_lambda_lambda @ nil_lambda_lambda ) ) ).

% hd_Nil_eq_last
thf(fact_960_last__tl,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( ( Xs = nil_lambda_lambda )
        | ( ( tl_lambda_lambda @ Xs )
         != nil_lambda_lambda ) )
     => ( ( last_lambda_lambda @ ( tl_lambda_lambda @ Xs ) )
        = ( last_lambda_lambda @ Xs ) ) ) ).

% last_tl
thf(fact_961_Ide__appendE,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ~ ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
             => ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
               => ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
                 != ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).

% Ide_appendE
thf(fact_962_Ide__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
            & ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
            & ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
              = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_963_Arr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
             => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
               => ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
                 != ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).

% Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_964_Trg_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).

% Trg.simps(3)
thf(fact_965_Trg__resid__sym,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        = ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) ) ) ) ).

% Trg_resid_sym
thf(fact_966_Ide__imp__Src__eq__Trg,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ( paths_1651982994781619320lambda @ lambda_resid @ T5 )
        = ( paths_2695323870737867955lambda @ lambda_resid @ T5 ) ) ) ).

% Ide_imp_Src_eq_Trg
thf(fact_967_Arr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
            & ( paths_366302053746739058lambda @ lambda_resid @ U3 )
            & ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
              = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_968_Trg__append,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 )
     => ( ( paths_2695323870737867955lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) )
        = ( paths_2695323870737867955lambda @ lambda_resid @ U3 ) ) ) ).

% Trg_append
thf(fact_969_Src__resid,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_1651982994781619320lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        = ( paths_2695323870737867955lambda @ lambda_resid @ U3 ) ) ) ).

% Src_resid
thf(fact_970_Arr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
            = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
         => ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ).

% Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_971_Ide__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ T5 )
     => ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
       => ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
            = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
         => ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) ) ) ) ) ).

% Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_972_Trgs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ T5 )
     => ( ( paths_7594662086398683701lambda @ lambda_resid @ T5 )
        = ( insert_lambda_lambda @ ( paths_2695323870737867955lambda @ lambda_resid @ T5 ) @ bot_bo1271416176687173302lambda ) ) ) ).

% Trgs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_973_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
    paths_2695323870737867955lambda = paths_2695323870737867955lambda ).

% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_974_red__iff,axiom,
    ( lambda_red
    = ( ^ [A6: lambda_lambda,B3: lambda_lambda] :
        ? [T8: list_lambda_lambda] :
          ( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
          & ( ( paths_1651982994781619320lambda @ lambda_resid @ T8 )
            = A6 )
          & ( ( paths_2695323870737867955lambda @ lambda_resid @ T8 )
            = B3 ) ) ) ) ).

% red_iff
thf(fact_975_Ide__consE,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) )
     => ~ ( ( ide_lambda_lambda @ lambda_resid @ T )
         => ( ( ( U3 != nil_lambda_lambda )
             => ( paths_7717490669901197737lambda @ lambda_resid @ U3 ) )
           => ~ ( ( U3 != nil_lambda_lambda )
               => ( ( trg_lambda_lambda @ lambda_resid @ T )
                  = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Ide_consE
thf(fact_976__092_060Lambda_062_Otrg__def,axiom,
    ! [T: lambda_lambda] :
      ( ( trg_lambda_lambda @ lambda_resid @ T )
      = ( lambda_resid @ T @ T ) ) ).

% \<Lambda>.trg_def
thf(fact_977__092_060Lambda_062_Oapex__sym,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
      = ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ).

% \<Lambda>.apex_sym
thf(fact_978__092_060Lambda_062_Otrg__resid__sym,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        = ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).

% \<Lambda>.trg_resid_sym
thf(fact_979__092_060Lambda_062_Oide__trg,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ide_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).

% \<Lambda>.ide_trg
thf(fact_980__092_060Lambda_062_Oide__iff__trg__self,axiom,
    ! [A2: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ide_lambda_lambda @ lambda_resid @ A2 )
        = ( ( trg_lambda_lambda @ lambda_resid @ A2 )
          = A2 ) ) ) ).

% \<Lambda>.ide_iff_trg_self
thf(fact_981__092_060Lambda_062_Otrg__in__targets,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( member_lambda_lambda @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).

% \<Lambda>.trg_in_targets
thf(fact_982__092_060Lambda_062_Oresid__ide_I2_J,axiom,
    ! [A2: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ T )
       => ( ( lambda_resid @ A2 @ T )
          = ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ) ).

% \<Lambda>.resid_ide(2)
thf(fact_983__092_060Lambda_062_Ocoterminal__iff__con__trg,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
      = ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).

% \<Lambda>.coterminal_iff_con_trg
thf(fact_984__092_060Lambda_062_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
      = ( ( arr_lambda_lambda @ lambda_resid @ T )
        & ( arr_lambda_lambda @ lambda_resid @ U )
        & ( ( trg_lambda_lambda @ lambda_resid @ T )
          = ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ).

% \<Lambda>.coterminal_iff\<^sub>W\<^sub>E
thf(fact_985__092_060Lambda_062_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
     => ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
         => ( ( arr_lambda_lambda @ lambda_resid @ U )
           => ( ( trg_lambda_lambda @ lambda_resid @ T )
             != ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ) ).

% \<Lambda>.coterminalE\<^sub>W\<^sub>E
thf(fact_986__092_060Lambda_062_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T: lambda_lambda] :
      ( ( target5490931411331770124lambda @ lambda_resid @ T )
      = ( collec621444948832755541lambda
        @ ^ [B3: lambda_lambda] :
            ( ( arr_lambda_lambda @ lambda_resid @ T )
            & ( ( trg_lambda_lambda @ lambda_resid @ T )
              = B3 ) ) ) ) ).

% \<Lambda>.targets_char\<^sub>W\<^sub>E
thf(fact_987_Trg_Osimps_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
      = ( trg_lambda_lambda @ lambda_resid @ T ) ) ).

% Trg.simps(2)
thf(fact_988__092_060Lambda_062_Oin__targetsE,axiom,
    ! [B: lambda_lambda,T: lambda_lambda] :
      ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
     => ~ ( ( ide_lambda_lambda @ lambda_resid @ B )
         => ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B ) ) ) ).

% \<Lambda>.in_targetsE
thf(fact_989__092_060Lambda_062_Otargets__def,axiom,
    ! [T: lambda_lambda] :
      ( ( target5490931411331770124lambda @ lambda_resid @ T )
      = ( collec621444948832755541lambda
        @ ^ [B3: lambda_lambda] :
            ( ( ide_lambda_lambda @ lambda_resid @ B3 )
            & ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B3 ) ) ) ) ).

% \<Lambda>.targets_def
thf(fact_990_Arr__consE,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) )
     => ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
         => ( ( ( U3 != nil_lambda_lambda )
             => ( paths_366302053746739058lambda @ lambda_resid @ U3 ) )
           => ~ ( ( U3 != nil_lambda_lambda )
               => ( ( trg_lambda_lambda @ lambda_resid @ T )
                  = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).

% Arr_consE
thf(fact_991__092_060Lambda_062_Otrg__trg,axiom,
    ! [T: lambda_lambda] :
      ( ( trg_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
      = ( trg_lambda_lambda @ lambda_resid @ T ) ) ).

% \<Lambda>.trg_trg
thf(fact_992__092_060Lambda_062_Otrg__ide,axiom,
    ! [A2: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( trg_lambda_lambda @ lambda_resid @ A2 )
        = A2 ) ) ).

% \<Lambda>.trg_ide
thf(fact_993__092_060Lambda_062_Oarr__trg__iff__arr,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
      = ( arr_lambda_lambda @ lambda_resid @ T ) ) ).

% \<Lambda>.arr_trg_iff_arr
thf(fact_994__092_060Lambda_062_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( ( trg_lambda_lambda @ lambda_resid @ T )
          = ( trg_lambda_lambda @ lambda_resid @ U ) )
       => ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).

% \<Lambda>.coterminalI\<^sub>W\<^sub>E
thf(fact_995_lambda__calculus_Otrg__char,axiom,
    ! [T: lambda_lambda] :
      ( ( ( lambda_Arr @ T )
       => ( ( trg_lambda_lambda @ lambda_resid @ T )
          = ( lambda_Trg @ T ) ) )
      & ( ~ ( lambda_Arr @ T )
       => ( ( trg_lambda_lambda @ lambda_resid @ T )
          = lambda_Nil ) ) ) ).

% lambda_calculus.trg_char
thf(fact_996__092_060Lambda_062_Oin__targetsI,axiom,
    ! [B: lambda_lambda,T: lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ B )
     => ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B )
       => ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).

% \<Lambda>.in_targetsI
thf(fact_997_Trg__simp,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( paths_2695323870737867955lambda @ lambda_resid @ T5 )
        = ( trg_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T5 ) ) ) ) ).

% Trg_simp
thf(fact_998_Arr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
       => ( ( ( trg_lambda_lambda @ lambda_resid @ T )
            = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
         => ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).

% Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_999_Ide__consI,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ T )
     => ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
       => ( ( ( trg_lambda_lambda @ lambda_resid @ T )
            = ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
         => ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).

% Ide_consI
thf(fact_1000_lambda__calculus_Ored_Ointros_I2_J,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
      ( ( lambda_red @ A2 @ B )
     => ( ( lambda_red @ B @ C )
       => ( lambda_red @ A2 @ C ) ) ) ).

% lambda_calculus.red.intros(2)
thf(fact_1001_lambda__calculus_Oweak__diamond,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda,B2: lambda_lambda] :
      ( ( lambda_red @ A2 @ B )
     => ( ( lambda_red @ A2 @ B2 )
       => ~ ! [C2: lambda_lambda] :
              ( ( lambda_red @ B @ C2 )
             => ~ ( lambda_red @ B2 @ C2 ) ) ) ) ).

% lambda_calculus.weak_diamond
thf(fact_1002_residuation_Otrg_Ocong,axiom,
    trg_lambda_lambda = trg_lambda_lambda ).

% residuation.trg.cong
thf(fact_1003_residuation_Otrg_Ocong,axiom,
    trg_li817795487149935061lambda = trg_li817795487149935061lambda ).

% residuation.trg.cong
thf(fact_1004_lambda__calculus_ONF__reduct__is__trivial,axiom,
    ! [A2: lambda_lambda,B: lambda_lambda] :
      ( ( lambda_NF @ A2 )
     => ( ( lambda_red @ A2 @ B )
       => ( A2 = B ) ) ) ).

% lambda_calculus.NF_reduct_is_trivial
thf(fact_1005_lambda__calculus_ONF__unique,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
      ( ( lambda_red @ T @ U )
     => ( ( lambda_red @ T @ U2 )
       => ( ( lambda_NF @ U )
         => ( ( lambda_NF @ U2 )
           => ( U = U2 ) ) ) ) ) ).

% lambda_calculus.NF_unique
thf(fact_1006_lambda__calculus_Ored_Osimps,axiom,
    ( lambda_red
    = ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
          ( ? [T7: lambda_lambda] :
              ( ( A1
                = ( lambda_Src @ T7 ) )
              & ( A22
                = ( lambda_Trg @ T7 ) )
              & ( lambda_Arr @ T7 ) )
          | ? [A6: lambda_lambda,B3: lambda_lambda,C3: lambda_lambda] :
              ( ( A1 = A6 )
              & ( A22 = C3 )
              & ( lambda_red @ A6 @ B3 )
              & ( lambda_red @ B3 @ C3 ) ) ) ) ) ).

% lambda_calculus.red.simps
thf(fact_1007_lambda__calculus_Ored_Ocases,axiom,
    ! [A12: lambda_lambda,A23: lambda_lambda] :
      ( ( lambda_red @ A12 @ A23 )
     => ( ! [T4: lambda_lambda] :
            ( ( A12
              = ( lambda_Src @ T4 ) )
           => ( ( A23
                = ( lambda_Trg @ T4 ) )
             => ~ ( lambda_Arr @ T4 ) ) )
       => ~ ! [B6: lambda_lambda] :
              ( ( lambda_red @ A12 @ B6 )
             => ~ ( lambda_red @ B6 @ A23 ) ) ) ) ).

% lambda_calculus.red.cases
thf(fact_1008_lambda__calculus_Ored_Ointros_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_Arr @ T )
     => ( lambda_red @ ( lambda_Src @ T ) @ ( lambda_Trg @ T ) ) ) ).

% lambda_calculus.red.intros(1)
thf(fact_1009_lambda__calculus_Onormalizable__def,axiom,
    ( lambda_normalizable
    = ( ^ [A6: lambda_lambda] :
          ( ( lambda_Ide @ A6 )
          & ? [B3: lambda_lambda] :
              ( ( lambda_red @ A6 @ B3 )
              & ( lambda_NF @ B3 ) ) ) ) ) ).

% lambda_calculus.normalizable_def
thf(fact_1010_reflects__con,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) @ ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ U ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) )
       != nil_lambda_lambda )
     => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% reflects_con
thf(fact_1011_trg__def,axiom,
    ! [T: list_lambda_lambda] :
      ( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).

% trg_def
thf(fact_1012_null__eqI,axiom,
    ! [N: list_lambda_lambda] :
      ( ! [T4: list_lambda_lambda] :
          ( ( ( paths_8607357831768207192lambda @ lambda_resid @ N @ T4 )
            = N )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ N )
            = N ) )
     => ( N
        = ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% null_eqI
thf(fact_1013_cube__ax,axiom,
    ! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
        = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ).

% cube_ax
thf(fact_1014_con__sym__ax,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% con_sym_ax
thf(fact_1015_con__imp__arr__resid,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% con_imp_arr_resid
thf(fact_1016_ide__trg,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).

% ide_trg
thf(fact_1017_null__char,axiom,
    ( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
    = nil_lambda_lambda ) ).

% null_char
thf(fact_1018_trg__in__targets,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( member2667661554319814297lambda @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).

% trg_in_targets
thf(fact_1019_con__def,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% con_def
thf(fact_1020_conE,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% conE
thf(fact_1021_not__arr__null,axiom,
    ~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).

% not_arr_null
thf(fact_1022_coterminal__iff__con__trg,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).

% coterminal_iff_con_trg
thf(fact_1023_in__targetsE,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
     => ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
         => ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B ) ) ) ).

% in_targetsE
thf(fact_1024_targets__def,axiom,
    ! [T: list_lambda_lambda] :
      ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
      = ( collec8830145823237851355lambda
        @ ^ [B3: list_lambda_lambda] :
            ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
            & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B3 ) ) ) ) ).

% targets_def
thf(fact_1025_null__is__zero_I2_J,axiom,
    ! [T: list_lambda_lambda] :
      ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
      = ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).

% null_is_zero(2)
thf(fact_1026_null__is__zero_I1_J,axiom,
    ! [T: list_lambda_lambda] :
      ( ( paths_8607357831768207192lambda @ lambda_resid @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) @ T )
      = ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).

% null_is_zero(1)
thf(fact_1027_conI,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
       != ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
     => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).

% conI
thf(fact_1028_in__targetsI,axiom,
    ! [B: list_lambda_lambda,T: list_lambda_lambda] :
      ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B )
       => ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).

% in_targetsI
thf(fact_1029_partial__magma_Onull__is__zero_I2_J,axiom,
    ! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( partia748546660539056525lambda @ OP )
     => ( ( OP @ T @ ( partia6873046715171720854lambda @ OP ) )
        = ( partia6873046715171720854lambda @ OP ) ) ) ).

% partial_magma.null_is_zero(2)
thf(fact_1030_partial__magma_Onull__is__zero_I2_J,axiom,
    ! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( partia3924075026459072775lambda @ OP )
     => ( ( OP @ T @ ( partia1461432607296028304lambda @ OP ) )
        = ( partia1461432607296028304lambda @ OP ) ) ) ).

% partial_magma.null_is_zero(2)
thf(fact_1031_partial__magma_Onull__is__zero_I1_J,axiom,
    ! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( partia748546660539056525lambda @ OP )
     => ( ( OP @ ( partia6873046715171720854lambda @ OP ) @ T )
        = ( partia6873046715171720854lambda @ OP ) ) ) ).

% partial_magma.null_is_zero(1)
thf(fact_1032_partial__magma_Onull__is__zero_I1_J,axiom,
    ! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( partia3924075026459072775lambda @ OP )
     => ( ( OP @ ( partia1461432607296028304lambda @ OP ) @ T )
        = ( partia1461432607296028304lambda @ OP ) ) ) ).

% partial_magma.null_is_zero(1)
thf(fact_1033_partial__magma_Onull__eqI,axiom,
    ! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
      ( ( partia748546660539056525lambda @ OP )
     => ( ! [T4: list_lambda_lambda] :
            ( ( ( OP @ N @ T4 )
              = N )
            & ( ( OP @ T4 @ N )
              = N ) )
       => ( N
          = ( partia6873046715171720854lambda @ OP ) ) ) ) ).

% partial_magma.null_eqI
thf(fact_1034_partial__magma_Onull__eqI,axiom,
    ! [OP: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
      ( ( partia3924075026459072775lambda @ OP )
     => ( ! [T4: lambda_lambda] :
            ( ( ( OP @ N @ T4 )
              = N )
            & ( ( OP @ T4 @ N )
              = N ) )
       => ( N
          = ( partia1461432607296028304lambda @ OP ) ) ) ) ).

% partial_magma.null_eqI
thf(fact_1035_partial__magma_Onull_Ocong,axiom,
    partia6873046715171720854lambda = partia6873046715171720854lambda ).

% partial_magma.null.cong
thf(fact_1036_partial__magma_Onull_Ocong,axiom,
    partia1461432607296028304lambda = partia1461432607296028304lambda ).

% partial_magma.null.cong
thf(fact_1037_incl__is__simulation,axiom,
    ( simula1175017789036123529lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid )
    @ ^ [T7: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( cons_lambda_lambda @ T7 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).

% incl_is_simulation
thf(fact_1038_incl__is__injective,axiom,
    ( inj_on3897920697455699703lambda
    @ ^ [T7: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( cons_lambda_lambda @ T7 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
    @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ).

% incl_is_injective
thf(fact_1039__092_060Lambda_062_Ocon__imp__arr__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != ( partia1461432607296028304lambda @ lambda_resid ) )
     => ( ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U ) )
       != ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% \<Lambda>.con_imp_arr_resid
thf(fact_1040__092_060Lambda_062_Ocon__sym__ax,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != ( partia1461432607296028304lambda @ lambda_resid ) )
     => ( ( lambda_resid @ U @ T )
       != ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% \<Lambda>.con_sym_ax
thf(fact_1041__092_060Lambda_062_Ocube__ax,axiom,
    ! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
       != ( partia1461432607296028304lambda @ lambda_resid ) )
     => ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
        = ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).

% \<Lambda>.cube_ax
thf(fact_1042__092_060Lambda_062_Onull__eqI,axiom,
    ! [N: lambda_lambda] :
      ( ! [T4: lambda_lambda] :
          ( ( ( lambda_resid @ N @ T4 )
            = N )
          & ( ( lambda_resid @ T4 @ N )
            = N ) )
     => ( N
        = ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% \<Lambda>.null_eqI
thf(fact_1043__092_060Lambda_062_Ocon__def,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
      = ( ( lambda_resid @ T @ U )
       != ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% \<Lambda>.con_def
thf(fact_1044__092_060Lambda_062_OconE,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( con_lambda_lambda @ lambda_resid @ T @ U )
     => ( ( lambda_resid @ T @ U )
       != ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% \<Lambda>.conE
thf(fact_1045__092_060Lambda_062_Onot__arr__null,axiom,
    ~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% \<Lambda>.not_arr_null
thf(fact_1046_Trg_Osimps_I1_J,axiom,
    ( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
    = ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% Trg.simps(1)
thf(fact_1047__092_060Lambda_062_Onull__is__zero_I2_J,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_resid @ T @ ( partia1461432607296028304lambda @ lambda_resid ) )
      = ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% \<Lambda>.null_is_zero(2)
thf(fact_1048__092_060Lambda_062_Onull__is__zero_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T )
      = ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% \<Lambda>.null_is_zero(1)
thf(fact_1049_Trg_Oelims,axiom,
    ! [X: list_lambda_lambda,Y4: lambda_lambda] :
      ( ( ( paths_2695323870737867955lambda @ lambda_resid @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4
           != ( partia1461432607296028304lambda @ lambda_resid ) ) )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ( Y4
               != ( trg_lambda_lambda @ lambda_resid @ T4 ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                 != ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% Trg.elims
thf(fact_1050_lambda__calculus_Onull__char,axiom,
    ( ( partia1461432607296028304lambda @ lambda_resid )
    = lambda_Nil ) ).

% lambda_calculus.null_char
thf(fact_1051__092_060Lambda_062_OconI,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( lambda_resid @ T @ U )
       != ( partia1461432607296028304lambda @ lambda_resid ) )
     => ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).

% \<Lambda>.conI
thf(fact_1052_simulation_Opreserves__trg,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( arr_lambda_lambda @ A @ T )
       => ( ( F6 @ ( trg_lambda_lambda @ A @ T ) )
          = ( trg_lambda_lambda @ B4 @ ( F6 @ T ) ) ) ) ) ).

% simulation.preserves_trg
thf(fact_1053_simulation_Opreserves__trg,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( arr_lambda_lambda @ A @ T )
       => ( ( F6 @ ( trg_lambda_lambda @ A @ T ) )
          = ( trg_li817795487149935061lambda @ B4 @ ( F6 @ T ) ) ) ) ) ).

% simulation.preserves_trg
thf(fact_1054_simulation_Opreserves__trg,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( arr_li3691443969012319837lambda @ A @ T )
       => ( ( F6 @ ( trg_li817795487149935061lambda @ A @ T ) )
          = ( trg_lambda_lambda @ B4 @ ( F6 @ T ) ) ) ) ) ).

% simulation.preserves_trg
thf(fact_1055_simulation_Opreserves__trg,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( arr_li3691443969012319837lambda @ A @ T )
       => ( ( F6 @ ( trg_li817795487149935061lambda @ A @ T ) )
          = ( trg_li817795487149935061lambda @ B4 @ ( F6 @ T ) ) ) ) ) ).

% simulation.preserves_trg
thf(fact_1056_simulation_Oextensional,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ~ ( arr_li3691443969012319837lambda @ A @ T )
       => ( ( F6 @ T )
          = ( partia6873046715171720854lambda @ B4 ) ) ) ) ).

% simulation.extensional
thf(fact_1057_simulation_Oextensional,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ~ ( arr_lambda_lambda @ A @ T )
       => ( ( F6 @ T )
          = ( partia6873046715171720854lambda @ B4 ) ) ) ) ).

% simulation.extensional
thf(fact_1058_simulation_Oextensional,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ~ ( arr_li3691443969012319837lambda @ A @ T )
       => ( ( F6 @ T )
          = ( partia1461432607296028304lambda @ B4 ) ) ) ) ).

% simulation.extensional
thf(fact_1059_simulation_Oextensional,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ~ ( arr_lambda_lambda @ A @ T )
       => ( ( F6 @ T )
          = ( partia1461432607296028304lambda @ B4 ) ) ) ) ).

% simulation.extensional
thf(fact_1060_simulation_Opreserves__con,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( con_lambda_lambda @ A @ T @ U )
       => ( con_lambda_lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ).

% simulation.preserves_con
thf(fact_1061_simulation_Opreserves__con,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( con_lambda_lambda @ A @ T @ U )
       => ( con_li79537494059641180lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ).

% simulation.preserves_con
thf(fact_1062_simulation_Opreserves__con,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( con_li79537494059641180lambda @ A @ T @ U )
       => ( con_lambda_lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ).

% simulation.preserves_con
thf(fact_1063_simulation_Opreserves__con,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( con_li79537494059641180lambda @ A @ T @ U )
       => ( con_li79537494059641180lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ).

% simulation.preserves_con
thf(fact_1064_simulation_Opreserves__resid,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( con_lambda_lambda @ A @ T @ U )
       => ( ( F6 @ ( A @ T @ U ) )
          = ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ) ).

% simulation.preserves_resid
thf(fact_1065_simulation_Opreserves__reflects__arr,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( arr_li3691443969012319837lambda @ B4 @ ( F6 @ T ) )
        = ( arr_li3691443969012319837lambda @ A @ T ) ) ) ).

% simulation.preserves_reflects_arr
thf(fact_1066_simulation_Opreserves__reflects__arr,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( arr_li3691443969012319837lambda @ B4 @ ( F6 @ T ) )
        = ( arr_lambda_lambda @ A @ T ) ) ) ).

% simulation.preserves_reflects_arr
thf(fact_1067_simulation_Opreserves__reflects__arr,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( arr_lambda_lambda @ B4 @ ( F6 @ T ) )
        = ( arr_li3691443969012319837lambda @ A @ T ) ) ) ).

% simulation.preserves_reflects_arr
thf(fact_1068_simulation_Opreserves__reflects__arr,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( arr_lambda_lambda @ B4 @ ( F6 @ T ) )
        = ( arr_lambda_lambda @ A @ T ) ) ) ).

% simulation.preserves_reflects_arr
thf(fact_1069_simulation_Opreserves__prfx,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
       => ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ) ).

% simulation.preserves_prfx
thf(fact_1070_simulation_Opreserves__prfx,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
       => ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ) ).

% simulation.preserves_prfx
thf(fact_1071_simulation_Opreserves__prfx,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
       => ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ) ).

% simulation.preserves_prfx
thf(fact_1072_simulation_Opreserves__prfx,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
       => ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) ) ) ) ).

% simulation.preserves_prfx
thf(fact_1073_simulation_Opreserves__cong,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
          & ( ide_lambda_lambda @ A @ ( A @ U @ T ) ) )
       => ( ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) )
          & ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ U ) @ ( F6 @ T ) ) ) ) ) ) ).

% simulation.preserves_cong
thf(fact_1074_simulation_Opreserves__cong,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
          & ( ide_lambda_lambda @ A @ ( A @ U @ T ) ) )
       => ( ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) )
          & ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ U ) @ ( F6 @ T ) ) ) ) ) ) ).

% simulation.preserves_cong
thf(fact_1075_simulation_Opreserves__cong,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
          & ( ide_li6773656263178250516lambda @ A @ ( A @ U @ T ) ) )
       => ( ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) )
          & ( ide_lambda_lambda @ B4 @ ( B4 @ ( F6 @ U ) @ ( F6 @ T ) ) ) ) ) ) ).

% simulation.preserves_cong
thf(fact_1076_simulation_Opreserves__cong,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
          & ( ide_li6773656263178250516lambda @ A @ ( A @ U @ T ) ) )
       => ( ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ T ) @ ( F6 @ U ) ) )
          & ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F6 @ U ) @ ( F6 @ T ) ) ) ) ) ) ).

% simulation.preserves_cong
thf(fact_1077_simulation_Opreserves__ide,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,A2: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( ide_lambda_lambda @ A @ A2 )
       => ( ide_lambda_lambda @ B4 @ ( F6 @ A2 ) ) ) ) ).

% simulation.preserves_ide
thf(fact_1078_simulation_Opreserves__ide,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,A2: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( ide_lambda_lambda @ A @ A2 )
       => ( ide_li6773656263178250516lambda @ B4 @ ( F6 @ A2 ) ) ) ) ).

% simulation.preserves_ide
thf(fact_1079_simulation_Opreserves__ide,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,A2: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( ide_li6773656263178250516lambda @ A @ A2 )
       => ( ide_lambda_lambda @ B4 @ ( F6 @ A2 ) ) ) ) ).

% simulation.preserves_ide
thf(fact_1080_simulation_Opreserves__ide,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( ide_li6773656263178250516lambda @ A @ A2 )
       => ( ide_li6773656263178250516lambda @ B4 @ ( F6 @ A2 ) ) ) ) ).

% simulation.preserves_ide
thf(fact_1081_simulation_Opreserves__sources,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F6 @ ( source6119337877621565212lambda @ A @ T ) ) @ ( source6119337877621565212lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_sources
thf(fact_1082_simulation_Opreserves__sources,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F6 @ ( source6119337877621565212lambda @ A @ T ) ) @ ( source5627170679743844886lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_sources
thf(fact_1083_simulation_Opreserves__sources,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F6 @ ( source5627170679743844886lambda @ A @ T ) ) @ ( source6119337877621565212lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_sources
thf(fact_1084_simulation_Opreserves__sources,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F6 @ ( source5627170679743844886lambda @ A @ T ) ) @ ( source5627170679743844886lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_sources
thf(fact_1085_simulation_Opreserves__targets,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F6 @ ( target4026644918049360146lambda @ A @ T ) ) @ ( target4026644918049360146lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_targets
thf(fact_1086_simulation_Opreserves__targets,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F6 @ ( target4026644918049360146lambda @ A @ T ) ) @ ( target5490931411331770124lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_targets
thf(fact_1087_simulation_Opreserves__targets,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F6 @ ( target5490931411331770124lambda @ A @ T ) ) @ ( target4026644918049360146lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_targets
thf(fact_1088_simulation_Opreserves__targets,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F6 @ ( target5490931411331770124lambda @ A @ T ) ) @ ( target5490931411331770124lambda @ B4 @ ( F6 @ T ) ) ) ) ).

% simulation.preserves_targets
thf(fact_1089_inj__on__filter__key__eq,axiom,
    ! [F: lambda_lambda > list_lambda_lambda,Y4: lambda_lambda,Xs: list_lambda_lambda] :
      ( ( inj_on3897920697455699703lambda @ F @ ( insert_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Xs ) ) )
     => ( ( filter_lambda_lambda
          @ ^ [X3: lambda_lambda] :
              ( ( F @ Y4 )
              = ( F @ X3 ) )
          @ Xs )
        = ( filter_lambda_lambda
          @ ( ^ [Y5: lambda_lambda,Z: lambda_lambda] : ( Y5 = Z )
            @ Y4 )
          @ Xs ) ) ) ).

% inj_on_filter_key_eq
thf(fact_1090_inj__on__empty,axiom,
    ! [F: lambda_lambda > list_lambda_lambda] : ( inj_on3897920697455699703lambda @ F @ bot_bo1271416176687173302lambda ) ).

% inj_on_empty
thf(fact_1091_inj__on__iff__surj,axiom,
    ! [A: set_li7878742357367608272lambda,A7: set_lambda_lambda] :
      ( ( A != bot_bo1397037886704484924lambda )
     => ( ( ? [F7: list_lambda_lambda > lambda_lambda] :
              ( ( inj_on2664063518378540011lambda @ F7 @ A )
              & ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F7 @ A ) @ A7 ) ) )
        = ( ? [G2: lambda_lambda > list_lambda_lambda] :
              ( ( image_3692823078739859851lambda @ G2 @ A7 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1092_inj__on__iff__surj,axiom,
    ! [A: set_lambda_lambda,A7: set_lambda_lambda] :
      ( ( A != bot_bo1271416176687173302lambda )
     => ( ( ? [F7: lambda_lambda > lambda_lambda] :
              ( ( inj_on287184477777394545lambda @ F7 @ A )
              & ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F7 @ A ) @ A7 ) ) )
        = ( ? [G2: lambda_lambda > lambda_lambda] :
              ( ( image_2356535832909760517lambda @ G2 @ A7 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1093_inj__on__iff__surj,axiom,
    ! [A: set_li7878742357367608272lambda,A7: set_li7878742357367608272lambda] :
      ( ( A != bot_bo1397037886704484924lambda )
     => ( ( ? [F7: list_lambda_lambda > list_lambda_lambda] :
              ( ( inj_on8159307203718685297lambda @ F7 @ A )
              & ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F7 @ A ) @ A7 ) ) )
        = ( ? [G2: list_lambda_lambda > list_lambda_lambda] :
              ( ( image_5886195720093285125lambda @ G2 @ A7 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1094_inj__on__iff__surj,axiom,
    ! [A: set_lambda_lambda,A7: set_li7878742357367608272lambda] :
      ( ( A != bot_bo1271416176687173302lambda )
     => ( ( ? [F7: lambda_lambda > list_lambda_lambda] :
              ( ( inj_on3897920697455699703lambda @ F7 @ A )
              & ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F7 @ A ) @ A7 ) ) )
        = ( ? [G2: list_lambda_lambda > lambda_lambda] :
              ( ( image_2458965899662700159lambda @ G2 @ A7 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1095_inj__singleton,axiom,
    ! [A: set_li7878742357367608272lambda] :
      ( inj_on4344131335932687313lambda
      @ ^ [X3: list_lambda_lambda] : ( insert8465602035467394112lambda @ X3 @ bot_bo1397037886704484924lambda )
      @ A ) ).

% inj_singleton
thf(fact_1096_inj__singleton,axiom,
    ! [A: set_lambda_lambda] :
      ( inj_on1689892504205702737lambda
      @ ^ [X3: lambda_lambda] : ( insert_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda )
      @ A ) ).

% inj_singleton
thf(fact_1097_inj__on__Cons1,axiom,
    ! [X: lambda_lambda,A: set_li7878742357367608272lambda] : ( inj_on8159307203718685297lambda @ ( cons_lambda_lambda @ X ) @ A ) ).

% inj_on_Cons1
thf(fact_1098_Resid_Osimps_I4_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
           != ( partia1461432607296028304lambda @ lambda_resid ) ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
          = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ nil_lambda_lambda ) ) )
      & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
             != ( partia1461432607296028304lambda @ lambda_resid ) ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
          = nil_lambda_lambda ) ) ) ).

% Resid.simps(4)
thf(fact_1099_Resid1x__as__Resid_H,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
         != nil_lambda_lambda )
       => ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 )
          = ( hd_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 ) ) ) )
      & ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
          = nil_lambda_lambda )
       => ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 )
          = ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).

% Resid1x_as_Resid'
thf(fact_1100_Resid1x_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
      = ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ).

% Resid1x.simps(3)
thf(fact_1101_Resid1x__null,axiom,
    ! [T5: list_lambda_lambda] :
      ( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T5 )
      = ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% Resid1x_null
thf(fact_1102_Resid1x_Osimps_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
      = ( lambda_resid @ T @ U ) ) ).

% Resid1x.simps(2)
thf(fact_1103_Resid1x_Osimps_I1_J,axiom,
    ! [T: lambda_lambda] :
      ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
      = ( partia1461432607296028304lambda @ lambda_resid ) ) ).

% Resid1x.simps(1)
thf(fact_1104_Resid1x__ide,axiom,
    ! [A2: lambda_lambda,T5: list_lambda_lambda] :
      ( ( ide_lambda_lambda @ lambda_resid @ A2 )
     => ( ( ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T5 )
         != ( partia1461432607296028304lambda @ lambda_resid ) )
       => ( ide_lambda_lambda @ lambda_resid @ ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T5 ) ) ) ) ).

% Resid1x_ide
thf(fact_1105_development__append,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( reduct9208763521642324371opment @ T @ U3 )
     => ( ( reduct9208763521642324371opment @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 ) @ V2 )
       => ( reduct9208763521642324371opment @ T @ ( append_lambda_lambda @ U3 @ V2 ) ) ) ) ).

% development_append
thf(fact_1106_Resid1x__as__Resid,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
       != nil_lambda_lambda )
     => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
        = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 ) @ nil_lambda_lambda ) ) ) ).

% Resid1x_as_Resid
thf(fact_1107_Resid__cons_H,axiom,
    ! [T5: list_lambda_lambda,T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
          = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).

% Resid_cons'
thf(fact_1108_Resid1x_Oelims,axiom,
    ! [X: lambda_lambda,Xa2: list_lambda_lambda,Y4: lambda_lambda] :
      ( ( ( paths_3319247187298047121lambda @ lambda_resid @ X @ Xa2 )
        = Y4 )
     => ( ( ( Xa2 = nil_lambda_lambda )
         => ( Y4
           != ( partia1461432607296028304lambda @ lambda_resid ) ) )
       => ( ! [U7: lambda_lambda] :
              ( ( Xa2
                = ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) )
             => ( Y4
               != ( lambda_resid @ X @ U7 ) ) )
         => ~ ! [U7: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( Xa2
                  = ( cons_lambda_lambda @ U7 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ( Y4
                 != ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X @ U7 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).

% Resid1x.elims
thf(fact_1109_paths__in__rts_OResid1x_Ocong,axiom,
    paths_3319247187298047121lambda = paths_3319247187298047121lambda ).

% paths_in_rts.Resid1x.cong
thf(fact_1110_Resid_Oelims,axiom,
    ! [X: list_lambda_lambda,Xa2: list_lambda_lambda,Y4: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ X @ Xa2 )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4 != nil_lambda_lambda ) )
       => ( ( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( X
                = ( cons_lambda_lambda @ V3 @ Va2 ) )
           => ( ( Xa2 = nil_lambda_lambda )
             => ( Y4 != nil_lambda_lambda ) ) )
         => ( ! [T4: lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
               => ! [U7: lambda_lambda] :
                    ( ( Xa2
                      = ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) )
                   => ~ ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                         => ( Y4
                            = ( cons_lambda_lambda @ ( lambda_resid @ T4 @ U7 ) @ nil_lambda_lambda ) ) )
                        & ( ~ ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                         => ( Y4 = nil_lambda_lambda ) ) ) ) )
           => ( ! [T4: lambda_lambda] :
                  ( ( X
                    = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
                 => ! [U7: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                      ( ( Xa2
                        = ( cons_lambda_lambda @ U7 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                     => ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                              & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                               != ( partia1461432607296028304lambda @ lambda_resid ) ) )
                           => ( Y4
                              = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) @ nil_lambda_lambda ) ) )
                          & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                                & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
                                 != ( partia1461432607296028304lambda @ lambda_resid ) ) )
                           => ( Y4 = nil_lambda_lambda ) ) ) ) )
             => ( ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                    ( ( X
                      = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                   => ! [U7: lambda_lambda] :
                        ( ( Xa2
                          = ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) )
                       => ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                                & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) )
                                 != nil_lambda_lambda ) )
                             => ( Y4
                                = ( cons_lambda_lambda @ ( lambda_resid @ T4 @ U7 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) ) ) ) )
                            & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                                  & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) )
                                   != nil_lambda_lambda ) )
                             => ( Y4 = nil_lambda_lambda ) ) ) ) )
               => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                      ( ( X
                        = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
                     => ! [U7: lambda_lambda,Vb2: lambda_lambda,Vc: list_lambda_lambda] :
                          ( ( Xa2
                            = ( cons_lambda_lambda @ U7 @ ( cons_lambda_lambda @ Vb2 @ Vc ) ) )
                         => ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                                  & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) )
                                   != ( partia1461432607296028304lambda @ lambda_resid ) )
                                  & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T4 @ U7 ) ) )
                                   != nil_lambda_lambda ) )
                               => ( Y4
                                  = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T4 @ U7 ) ) ) ) ) )
                              & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T4 @ U7 )
                                    & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T4 @ U7 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) )
                                     != ( partia1461432607296028304lambda @ lambda_resid ) )
                                    & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U7 @ T4 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T4 @ U7 ) ) )
                                     != nil_lambda_lambda ) )
                               => ( Y4 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).

% Resid.elims
thf(fact_1111_Residx1_Osimps_I1_J,axiom,
    ! [U: lambda_lambda] :
      ( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
      = nil_lambda_lambda ) ).

% Residx1.simps(1)
thf(fact_1112_Residx1__as__Resid,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( paths_2112734093361750239lambda @ lambda_resid @ T5 @ U )
      = ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ).

% Residx1_as_Resid
thf(fact_1113_Residx1_Oelims,axiom,
    ! [X: list_lambda_lambda,Xa2: lambda_lambda,Y4: list_lambda_lambda] :
      ( ( ( paths_2112734093361750239lambda @ lambda_resid @ X @ Xa2 )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4 != nil_lambda_lambda ) )
       => ( ! [T4: lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
             => ~ ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ Xa2 )
                   => ( Y4
                      = ( cons_lambda_lambda @ ( lambda_resid @ T4 @ Xa2 ) @ nil_lambda_lambda ) ) )
                  & ( ~ ( con_lambda_lambda @ lambda_resid @ T4 @ Xa2 )
                   => ( Y4 = nil_lambda_lambda ) ) ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( ( X
                  = ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
               => ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T4 @ Xa2 )
                        & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa2 @ T4 ) )
                         != nil_lambda_lambda ) )
                     => ( Y4
                        = ( cons_lambda_lambda @ ( lambda_resid @ T4 @ Xa2 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa2 @ T4 ) ) ) ) )
                    & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T4 @ Xa2 )
                          & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa2 @ T4 ) )
                           != nil_lambda_lambda ) )
                     => ( Y4 = nil_lambda_lambda ) ) ) ) ) ) ) ).

% Residx1.elims
thf(fact_1114_Residx1_Osimps_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
          = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
      & ( ~ ( con_lambda_lambda @ lambda_resid @ T @ U )
       => ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
          = nil_lambda_lambda ) ) ) ).

% Residx1.simps(2)
thf(fact_1115_Residx1_Osimps_I3_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
           != nil_lambda_lambda ) )
       => ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
          = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
      & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
             != nil_lambda_lambda ) )
       => ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
          = nil_lambda_lambda ) ) ) ).

% Residx1.simps(3)
thf(fact_1116_Con__sym1,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] :
      ( ( ( paths_2112734093361750239lambda @ lambda_resid @ T5 @ U )
       != nil_lambda_lambda )
      = ( ( paths_3319247187298047121lambda @ lambda_resid @ U @ T5 )
       != ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).

% Con_sym1
thf(fact_1117_Resid_Osimps_I5_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
           != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
          = ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
      & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
             != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
          = nil_lambda_lambda ) ) ) ).

% Resid.simps(5)
thf(fact_1118_Resid_Osimps_I7_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda,Vb: lambda_lambda,Vc2: list_lambda_lambda] :
      ( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
           != ( partia1461432607296028304lambda @ lambda_resid ) )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
           != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
          = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
      & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
             != ( partia1461432607296028304lambda @ lambda_resid ) )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
             != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
          = nil_lambda_lambda ) ) ) ).

% Resid.simps(7)
thf(fact_1119_Resid_Osimps_I6_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc2: list_lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
          & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) )
           != ( partia1461432607296028304lambda @ lambda_resid ) )
          & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
           != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) )
          = ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
      & ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
            & ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) )
             != ( partia1461432607296028304lambda @ lambda_resid ) )
            & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
             != nil_lambda_lambda ) )
       => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) )
          = nil_lambda_lambda ) ) ) ).

% Resid.simps(6)
thf(fact_1120_paths__in__rts_OResidx1_Ocong,axiom,
    paths_2112734093361750239lambda = paths_2112734093361750239lambda ).

% paths_in_rts.Residx1.cong
thf(fact_1121_join__of__arr__src_I1_J,axiom,
    ! [T: list_lambda_lambda,A2: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T @ T ) ) ) ).

% join_of_arr_src(1)
thf(fact_1122_join__of__arr__src_I2_J,axiom,
    ! [T: list_lambda_lambda,A2: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
       => ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 @ T ) ) ) ).

% join_of_arr_src(2)
thf(fact_1123_join__of__symmetric,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V ) ) ).

% join_of_symmetric
thf(fact_1124_join__of__un__upto__cong,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V4 )
       => ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V4 ) )
          & ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ V ) ) ) ) ) ).

% join_of_un_upto_cong
thf(fact_1125_con__with__join__of__iff_I2_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
       => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
          & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ) ).

% con_with_join_of_iff(2)
thf(fact_1126_con__with__join__of__iff_I1_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
     => ( ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
          & ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V ) ) ) ).

% con_with_join_of_iff(1)
thf(fact_1127_join__of__resid,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V @ W2 )
       => ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ V ) ) ) ) ).

% join_of_resid
thf(fact_1128_sources__join__of_I2_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).

% sources_join_of(2)
thf(fact_1129_sources__join__of_I1_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
        = ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).

% sources_join_of(1)
thf(fact_1130_targets__join__of_I2_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).

% targets_join_of(2)
thf(fact_1131_targets__join__of_I1_J,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
     => ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
        = ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).

% targets_join_of(1)
thf(fact_1132_join__of__arr__self,axiom,
    ! [T: list_lambda_lambda] :
      ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
     => ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T @ T ) ) ).

% join_of_arr_self
thf(fact_1133_joinable__def,axiom,
    ! [T: list_lambda_lambda,U: list_lambda_lambda] :
      ( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
      = ( ? [X6: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X6 ) ) ) ).

% joinable_def
thf(fact_1134_simulation_Opreserves__joins,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( simula3069223088535412739lambda @ A @ B4 @ F6 )
     => ( ( join_o6956785509312998601lambda @ A @ T @ U @ V )
       => ( join_o6956785509312998601lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) @ ( F6 @ V ) ) ) ) ).

% simulation.preserves_joins
thf(fact_1135_simulation_Opreserves__joins,axiom,
    ! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
      ( ( simula9164532646813739645lambda @ A @ B4 @ F6 )
     => ( ( join_o6956785509312998601lambda @ A @ T @ U @ V )
       => ( join_o1386658102433174083lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) @ ( F6 @ V ) ) ) ) ).

% simulation.preserves_joins
thf(fact_1136_simulation_Opreserves__joins,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F6: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( simula1175017789036123529lambda @ A @ B4 @ F6 )
     => ( ( join_o1386658102433174083lambda @ A @ T @ U @ V )
       => ( join_o6956785509312998601lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) @ ( F6 @ V ) ) ) ) ).

% simulation.preserves_joins
thf(fact_1137_simulation_Opreserves__joins,axiom,
    ! [A: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F6: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( simula5235733974575307523lambda @ A @ B4 @ F6 )
     => ( ( join_o1386658102433174083lambda @ A @ T @ U @ V )
       => ( join_o1386658102433174083lambda @ B4 @ ( F6 @ T ) @ ( F6 @ U ) @ ( F6 @ V ) ) ) ) ).

% simulation.preserves_joins
thf(fact_1138_rts_Ojoin__of_Ocong,axiom,
    join_o6956785509312998601lambda = join_o6956785509312998601lambda ).

% rts.join_of.cong
thf(fact_1139_rts_Ojoin__of_Ocong,axiom,
    join_o1386658102433174083lambda = join_o1386658102433174083lambda ).

% rts.join_of.cong
thf(fact_1140_reduction__paths_Ostandardize_Oelims,axiom,
    ! [X: list_lambda_lambda,Y4: list_lambda_lambda] :
      ( ( ( reduct6308743955541908015ardize @ X )
        = Y4 )
     => ( ( ( X = nil_lambda_lambda )
         => ( Y4 != nil_lambda_lambda ) )
       => ~ ! [V3: lambda_lambda,Va2: list_lambda_lambda] :
              ( ( X
                = ( cons_lambda_lambda @ V3 @ Va2 ) )
             => ( Y4
               != ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) @ ( reduct6308743955541908015ardize @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).

% reduction_paths.standardize.elims
thf(fact_1141_complete__development__bottom__up__development__ind,axiom,
    ! [T: lambda_lambda,N: nat] :
      ( ( lambda_Arr @ T )
     => ( ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( reduct6613843825294780173opment @ T ) ) @ N )
       => ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ) ).

% complete_development_bottom_up_development_ind
thf(fact_1142__092_060Lambda_062_Ojoin__of__symmetric,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( join_o1386658102433174083lambda @ lambda_resid @ U @ T @ V ) ) ).

% \<Lambda>.join_of_symmetric
thf(fact_1143__092_060Lambda_062_Ojoin__of__un__upto__cong,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V4 )
       => ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V4 ) )
          & ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V4 @ V ) ) ) ) ) ).

% \<Lambda>.join_of_un_upto_cong
thf(fact_1144__092_060Lambda_062_Ocon__with__join__of__iff_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
     => ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
       => ( ( con_lambda_lambda @ lambda_resid @ T @ V )
          & ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) ) ) ) ) ).

% \<Lambda>.con_with_join_of_iff(2)
thf(fact_1145__092_060Lambda_062_Ocon__with__join__of__iff_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
     => ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
          & ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) )
       => ( con_lambda_lambda @ lambda_resid @ W2 @ V ) ) ) ).

% \<Lambda>.con_with_join_of_iff(1)
thf(fact_1146__092_060Lambda_062_Ojoin__of__resid,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
     => ( ( con_lambda_lambda @ lambda_resid @ V @ W2 )
       => ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W2 @ V ) ) ) ) ).

% \<Lambda>.join_of_resid
thf(fact_1147__092_060Lambda_062_Ojoin__of__arr__self,axiom,
    ! [T: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( join_o1386658102433174083lambda @ lambda_resid @ T @ T @ T ) ) ).

% \<Lambda>.join_of_arr_self
thf(fact_1148__092_060Lambda_062_Otargets__join__of_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        = ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.targets_join_of(1)
thf(fact_1149__092_060Lambda_062_Otargets__join__of_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
        = ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.targets_join_of(2)
thf(fact_1150__092_060Lambda_062_Osources__join__of_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( source5627170679743844886lambda @ lambda_resid @ T )
        = ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.sources_join_of(1)
thf(fact_1151__092_060Lambda_062_Osources__join__of_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( source5627170679743844886lambda @ lambda_resid @ U )
        = ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.sources_join_of(2)
thf(fact_1152__092_060Lambda_062_Otrg__join__of_I2_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
        = ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.trg_join_of(2)
thf(fact_1153__092_060Lambda_062_Otrg__join__of_I1_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
      ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
     => ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
        = ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).

% \<Lambda>.trg_join_of(1)
thf(fact_1154__092_060Lambda_062_Ojoinable__def,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
      = ( ? [X6: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X6 ) ) ) ).

% \<Lambda>.joinable_def
thf(fact_1155__092_060Lambda_062_Ojoin__of__arr__src_I2_J,axiom,
    ! [T: lambda_lambda,A2: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
       => ( join_o1386658102433174083lambda @ lambda_resid @ T @ A2 @ T ) ) ) ).

% \<Lambda>.join_of_arr_src(2)
thf(fact_1156__092_060Lambda_062_Ojoin__of__arr__src_I1_J,axiom,
    ! [T: lambda_lambda,A2: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
       => ( join_o1386658102433174083lambda @ lambda_resid @ A2 @ T @ T ) ) ) ).

% \<Lambda>.join_of_arr_src(1)
thf(fact_1157_length__Resid,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
        = ( size_s4208335251162658524lambda @ T5 ) ) ) ).

% length_Resid
thf(fact_1158_length__Residx1,axiom,
    ! [T5: list_lambda_lambda,U: lambda_lambda] : ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( paths_2112734093361750239lambda @ lambda_resid @ T5 @ U ) ) @ ( size_s4208335251162658524lambda @ T5 ) ) ).

% length_Residx1
thf(fact_1159_FD__def,axiom,
    ( reduction_FD
    = ( ^ [T7: lambda_lambda] :
        ? [N2: nat] :
        ! [U5: list_lambda_lambda] :
          ( ( reduct9208763521642324371opment @ T7 @ U5 )
         => ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ U5 ) @ N2 ) ) ) ) ).

% FD_def
thf(fact_1160_append__eq__append__conv,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda,Vs3: list_lambda_lambda] :
      ( ( ( ( size_s4208335251162658524lambda @ Xs )
          = ( size_s4208335251162658524lambda @ Ys ) )
        | ( ( size_s4208335251162658524lambda @ Us )
          = ( size_s4208335251162658524lambda @ Vs3 ) ) )
     => ( ( ( append_lambda_lambda @ Xs @ Us )
          = ( append_lambda_lambda @ Ys @ Vs3 ) )
        = ( ( Xs = Ys )
          & ( Us = Vs3 ) ) ) ) ).

% append_eq_append_conv
thf(fact_1161_length__0__conv,axiom,
    ! [Xs: list_lambda_lambda] :
      ( ( ( size_s4208335251162658524lambda @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_lambda_lambda ) ) ).

% length_0_conv
thf(fact_1162_reduction__paths_Ostandardize_Osimps_I1_J,axiom,
    ( ( reduct6308743955541908015ardize @ nil_lambda_lambda )
    = nil_lambda_lambda ) ).

% reduction_paths.standardize.simps(1)
thf(fact_1163_impossible__Cons,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,X: lambda_lambda] :
      ( ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) )
     => ( Xs
       != ( cons_lambda_lambda @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_1164_length__filter__le,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] : ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( filter_lambda_lambda @ P @ Xs ) ) @ ( size_s4208335251162658524lambda @ Xs ) ) ).

% length_filter_le
thf(fact_1165_list__induct2,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > $o] :
      ( ( ( size_s4208335251162658524lambda @ Xs )
        = ( size_s4208335251162658524lambda @ Ys ) )
     => ( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
       => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
              ( ( ( size_s4208335251162658524lambda @ Xs2 )
                = ( size_s4208335251162658524lambda @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1166_list__induct3,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
      ( ( ( size_s4208335251162658524lambda @ Xs )
        = ( size_s4208335251162658524lambda @ Ys ) )
     => ( ( ( size_s4208335251162658524lambda @ Ys )
          = ( size_s4208335251162658524lambda @ Zs ) )
       => ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
         => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda,Z3: lambda_lambda,Zs2: list_lambda_lambda] :
                ( ( ( size_s4208335251162658524lambda @ Xs2 )
                  = ( size_s4208335251162658524lambda @ Ys2 ) )
               => ( ( ( size_s4208335251162658524lambda @ Ys2 )
                    = ( size_s4208335251162658524lambda @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs2 )
                   => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) @ ( cons_lambda_lambda @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_1167_list__induct4,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
      ( ( ( size_s4208335251162658524lambda @ Xs )
        = ( size_s4208335251162658524lambda @ Ys ) )
     => ( ( ( size_s4208335251162658524lambda @ Ys )
          = ( size_s4208335251162658524lambda @ Zs ) )
       => ( ( ( size_s4208335251162658524lambda @ Zs )
            = ( size_s4208335251162658524lambda @ Ws ) )
         => ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
           => ( ! [X2: lambda_lambda,Xs2: list_lambda_lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda,Z3: lambda_lambda,Zs2: list_lambda_lambda,W4: lambda_lambda,Ws2: list_lambda_lambda] :
                  ( ( ( size_s4208335251162658524lambda @ Xs2 )
                    = ( size_s4208335251162658524lambda @ Ys2 ) )
                 => ( ( ( size_s4208335251162658524lambda @ Ys2 )
                      = ( size_s4208335251162658524lambda @ Zs2 ) )
                   => ( ( ( size_s4208335251162658524lambda @ Zs2 )
                        = ( size_s4208335251162658524lambda @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_lambda_lambda @ X2 @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) @ ( cons_lambda_lambda @ Z3 @ Zs2 ) @ ( cons_lambda_lambda @ W4 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1168_list_Osize_I3_J,axiom,
    ( ( size_s4208335251162658524lambda @ nil_lambda_lambda )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1169_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_lambda_lambda] :
      ( ( size_s4208335251162658524lambda @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_1170_neq__if__length__neq,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( ( size_s4208335251162658524lambda @ Xs )
       != ( size_s4208335251162658524lambda @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1171_length__n__lists__elem,axiom,
    ! [Ys: list_lambda_lambda,N: nat,Xs: list_lambda_lambda] :
      ( ( member2667661554319814297lambda @ Ys @ ( set_li3846194476672328965lambda @ ( n_list4163717777911954552lambda @ N @ Xs ) ) )
     => ( ( size_s4208335251162658524lambda @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1172_same__length__different,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( Xs != Ys )
     => ( ( ( size_s4208335251162658524lambda @ Xs )
          = ( size_s4208335251162658524lambda @ Ys ) )
       => ? [Pre: list_lambda_lambda,X2: lambda_lambda,Xs3: list_lambda_lambda,Y2: lambda_lambda,Ys5: list_lambda_lambda] :
            ( ( X2 != Y2 )
            & ( Xs
              = ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) @ Xs3 ) ) )
            & ( Ys
              = ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1173_set__n__lists,axiom,
    ! [N: nat,Xs: list_lambda_lambda] :
      ( ( set_li3846194476672328965lambda @ ( n_list4163717777911954552lambda @ N @ Xs ) )
      = ( collec8830145823237851355lambda
        @ ^ [Ys3: list_lambda_lambda] :
            ( ( ( size_s4208335251162658524lambda @ Ys3 )
              = N )
            & ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Ys3 ) @ ( set_lambda_lambda2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1174_set__n__lists,axiom,
    ! [N: nat,Xs: list_l5168835686021156086lambda] :
      ( ( set_li6257665199064048907lambda @ ( n_list7041979136695072894lambda @ N @ Xs ) )
      = ( collec4999033578922223073lambda
        @ ^ [Ys3: list_l5168835686021156086lambda] :
            ( ( ( size_s6298197441921507170lambda @ Ys3 )
              = N )
            & ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Ys3 ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1175_reduction__paths_Ostandardize_Osimps_I2_J,axiom,
    ! [V: lambda_lambda,Va: list_lambda_lambda] :
      ( ( reduct6308743955541908015ardize @ ( cons_lambda_lambda @ V @ Va ) )
      = ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( reduct6308743955541908015ardize @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).

% reduction_paths.standardize.simps(2)
thf(fact_1176_Resid__cons__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,N: nat] :
      ( ( T5 != nil_lambda_lambda )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T5 ) @ ( size_s4208335251162658524lambda @ U3 ) ) @ N )
         => ( ! [T3: lambda_lambda] :
                ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ T5 ) @ U3 )
                 != nil_lambda_lambda )
                = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) @ U3 )
                   != nil_lambda_lambda )
                  & ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) ) )
                   != nil_lambda_lambda ) ) )
            & ! [U8: lambda_lambda] :
                ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ U3 ) )
                 != nil_lambda_lambda )
                = ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ nil_lambda_lambda ) )
                   != nil_lambda_lambda )
                  & ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ nil_lambda_lambda ) ) @ U3 )
                   != nil_lambda_lambda ) ) )
            & ! [T3: lambda_lambda] :
                ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ T5 ) @ U3 )
                 != nil_lambda_lambda )
               => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ T5 ) @ U3 )
                  = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) ) ) ) ) )
            & ! [U8: lambda_lambda] :
                ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ U3 ) )
                 != nil_lambda_lambda )
               => ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ U3 ) )
                  = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ ( cons_lambda_lambda @ U8 @ nil_lambda_lambda ) ) @ U3 ) ) ) ) ) ) ) ).

% Resid_cons_ind
thf(fact_1177_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_1178_length__Resid__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T5 ) @ ( size_s4208335251162658524lambda @ U3 ) ) @ N )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
         != nil_lambda_lambda )
       => ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
          = ( size_s4208335251162658524lambda @ T5 ) ) ) ) ).

% length_Resid_ind
thf(fact_1179_Cube__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda,N: nat] :
      ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
       != nil_lambda_lambda )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 )
         != nil_lambda_lambda )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T5 ) @ ( size_s4208335251162658524lambda @ U3 ) ) @ ( size_s4208335251162658524lambda @ V2 ) ) @ N )
         => ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) )
               != nil_lambda_lambda )
              = ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) )
               != nil_lambda_lambda ) )
            & ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) )
               != nil_lambda_lambda )
             => ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T5 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 ) )
                = ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 ) ) ) ) ) ) ) ) ).

% Cube_ind
thf(fact_1180_Con__sym__ind,axiom,
    ! [T5: list_lambda_lambda,U3: list_lambda_lambda,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T5 ) @ ( size_s4208335251162658524lambda @ U3 ) ) @ N )
     => ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U3 )
         != nil_lambda_lambda )
        = ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T5 )
         != nil_lambda_lambda ) ) ) ).

% Con_sym_ind
thf(fact_1181_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_1182_add__cancel__left__left,axiom,
    ! [B: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B @ A2 )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1183_add__cancel__left__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( plus_plus_nat @ A2 @ B )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1184_add__cancel__right__left,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ B @ A2 ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1185_add__cancel__right__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1186_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y4: nat] :
      ( ( ( plus_plus_nat @ X @ Y4 )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y4 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1187_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y4: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y4 ) )
      = ( ( X = zero_zero_nat )
        & ( Y4 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1188_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_1189_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1190_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1191_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1192_le__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1193_le__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1194_add__le__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1195_add__le__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1196_image__add__0,axiom,
    ! [S: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
      = S ) ).

% image_add_0
thf(fact_1197_length__append,axiom,
    ! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
      ( ( size_s4208335251162658524lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) ) ) ).

% length_append
thf(fact_1198_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y4 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y4 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1199_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
       => ( ( ( plus_plus_nat @ X @ Y4 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y4 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1200_add__nonpos__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1201_add__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1202_add__increasing2,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1203_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1204_add__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_1205_add__decreasing,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1206_sum__length__filter__compl,axiom,
    ! [P: lambda_lambda > $o,Xs: list_lambda_lambda] :
      ( ( plus_plus_nat @ ( size_s4208335251162658524lambda @ ( filter_lambda_lambda @ P @ Xs ) )
        @ ( size_s4208335251162658524lambda
          @ ( filter_lambda_lambda
            @ ^ [X3: lambda_lambda] :
                ~ ( P @ X3 )
            @ Xs ) ) )
      = ( size_s4208335251162658524lambda @ Xs ) ) ).

% sum_length_filter_compl
thf(fact_1207_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_1208_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_1209_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1210_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1211_inj__on__add_H,axiom,
    ! [A2: nat,A: set_nat] :
      ( inj_on_nat_nat
      @ ^ [B3: nat] : ( plus_plus_nat @ B3 @ A2 )
      @ A ) ).

% inj_on_add'
thf(fact_1212_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1213_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1214_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1215_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1216_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1217_stdz__insert__Beta__ind,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( lambda_hgt @ T ) @ ( size_s4208335251162658524lambda @ U3 ) ) @ N )
     => ( ( lambda_is_Beta @ T )
       => ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
         => ( lambda_is_Beta @ ( hd_lambda_lambda @ ( reduct5390047737487997688insert @ T @ U3 ) ) ) ) ) ) ).

% stdz_insert_Beta_ind
thf(fact_1218_length__devel__le__hgt,axiom,
    ! [T: lambda_lambda,U3: list_lambda_lambda] :
      ( ( reduct9208763521642324371opment @ T @ U3 )
     => ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ U3 ) @ ( lambda_hgt @ T ) ) ) ).

% length_devel_le_hgt
thf(fact_1219_lambda__calculus_Ohgt_Osimps_I2_J,axiom,
    ! [Uu: nat] :
      ( ( lambda_hgt @ ( lambda_Var @ Uu ) )
      = zero_zero_nat ) ).

% lambda_calculus.hgt.simps(2)
thf(fact_1220_lambda__calculus_Ohgt_Osimps_I1_J,axiom,
    ( ( lambda_hgt @ lambda_Nil )
    = zero_zero_nat ) ).

% lambda_calculus.hgt.simps(1)
thf(fact_1221_rts__axioms__def,axiom,
    ( rts_ax7298520571313827089lambda
    = ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
          ( ! [T7: lambda_lambda] :
              ( ( arr_lambda_lambda @ Resid2 @ T7 )
             => ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T7 ) ) )
          & ! [A6: lambda_lambda,T7: lambda_lambda] :
              ( ( ide_lambda_lambda @ Resid2 @ A6 )
             => ( ( con_lambda_lambda @ Resid2 @ T7 @ A6 )
               => ( ( Resid2 @ T7 @ A6 )
                  = T7 ) ) )
          & ! [A6: lambda_lambda,T7: lambda_lambda] :
              ( ( ide_lambda_lambda @ Resid2 @ A6 )
             => ( ( con_lambda_lambda @ Resid2 @ A6 @ T7 )
               => ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A6 @ T7 ) ) ) )
          & ! [T7: lambda_lambda,U6: lambda_lambda] :
              ( ( con_lambda_lambda @ Resid2 @ T7 @ U6 )
             => ? [A6: lambda_lambda] :
                  ( ( ide_lambda_lambda @ Resid2 @ A6 )
                  & ( con_lambda_lambda @ Resid2 @ A6 @ T7 )
                  & ( con_lambda_lambda @ Resid2 @ A6 @ U6 ) ) )
          & ! [T7: lambda_lambda,U6: lambda_lambda,V5: lambda_lambda] :
              ( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T7 @ U6 ) )
             => ( ( con_lambda_lambda @ Resid2 @ U6 @ V5 )
               => ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T7 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).

% rts_axioms_def
thf(fact_1222_rts__axioms__def,axiom,
    ( rts_ax2283037410143324311lambda
    = ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
          ( ! [T7: list_lambda_lambda] :
              ( ( arr_li3691443969012319837lambda @ Resid2 @ T7 )
             => ( ide_li6773656263178250516lambda @ Resid2 @ ( trg_li817795487149935061lambda @ Resid2 @ T7 ) ) )
          & ! [A6: list_lambda_lambda,T7: list_lambda_lambda] :
              ( ( ide_li6773656263178250516lambda @ Resid2 @ A6 )
             => ( ( con_li79537494059641180lambda @ Resid2 @ T7 @ A6 )
               => ( ( Resid2 @ T7 @ A6 )
                  = T7 ) ) )
          & ! [A6: list_lambda_lambda,T7: list_lambda_lambda] :
              ( ( ide_li6773656263178250516lambda @ Resid2 @ A6 )
             => ( ( con_li79537494059641180lambda @ Resid2 @ A6 @ T7 )
               => ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ A6 @ T7 ) ) ) )
          & ! [T7: list_lambda_lambda,U6: list_lambda_lambda] :
              ( ( con_li79537494059641180lambda @ Resid2 @ T7 @ U6 )
             => ? [A6: list_lambda_lambda] :
                  ( ( ide_li6773656263178250516lambda @ Resid2 @ A6 )
                  & ( con_li79537494059641180lambda @ Resid2 @ A6 @ T7 )
                  & ( con_li79537494059641180lambda @ Resid2 @ A6 @ U6 ) ) )
          & ! [T7: list_lambda_lambda,U6: list_lambda_lambda,V5: list_lambda_lambda] :
              ( ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ T7 @ U6 ) )
             => ( ( con_li79537494059641180lambda @ Resid2 @ U6 @ V5 )
               => ( con_li79537494059641180lambda @ Resid2 @ ( Resid2 @ T7 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).

% rts_axioms_def
thf(fact_1223_rts__axioms_Ointro,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ! [T4: lambda_lambda] :
          ( ( arr_lambda_lambda @ Resid @ T4 )
         => ( ide_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T4 ) ) )
     => ( ! [A5: lambda_lambda,T4: lambda_lambda] :
            ( ( ide_lambda_lambda @ Resid @ A5 )
           => ( ( con_lambda_lambda @ Resid @ T4 @ A5 )
             => ( ( Resid @ T4 @ A5 )
                = T4 ) ) )
       => ( ! [A5: lambda_lambda,T4: lambda_lambda] :
              ( ( ide_lambda_lambda @ Resid @ A5 )
             => ( ( con_lambda_lambda @ Resid @ A5 @ T4 )
               => ( ide_lambda_lambda @ Resid @ ( Resid @ A5 @ T4 ) ) ) )
         => ( ! [T4: lambda_lambda,U7: lambda_lambda] :
                ( ( con_lambda_lambda @ Resid @ T4 @ U7 )
               => ? [A8: lambda_lambda] :
                    ( ( ide_lambda_lambda @ Resid @ A8 )
                    & ( con_lambda_lambda @ Resid @ A8 @ T4 )
                    & ( con_lambda_lambda @ Resid @ A8 @ U7 ) ) )
           => ( ! [T4: lambda_lambda,U7: lambda_lambda,V3: lambda_lambda] :
                  ( ( ide_lambda_lambda @ Resid @ ( Resid @ T4 @ U7 ) )
                 => ( ( con_lambda_lambda @ Resid @ U7 @ V3 )
                   => ( con_lambda_lambda @ Resid @ ( Resid @ T4 @ U7 ) @ ( Resid @ V3 @ U7 ) ) ) )
             => ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).

% rts_axioms.intro
thf(fact_1224_rts__axioms_Ointro,axiom,
    ! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ! [T4: list_lambda_lambda] :
          ( ( arr_li3691443969012319837lambda @ Resid @ T4 )
         => ( ide_li6773656263178250516lambda @ Resid @ ( trg_li817795487149935061lambda @ Resid @ T4 ) ) )
     => ( ! [A5: list_lambda_lambda,T4: list_lambda_lambda] :
            ( ( ide_li6773656263178250516lambda @ Resid @ A5 )
           => ( ( con_li79537494059641180lambda @ Resid @ T4 @ A5 )
             => ( ( Resid @ T4 @ A5 )
                = T4 ) ) )
       => ( ! [A5: list_lambda_lambda,T4: list_lambda_lambda] :
              ( ( ide_li6773656263178250516lambda @ Resid @ A5 )
             => ( ( con_li79537494059641180lambda @ Resid @ A5 @ T4 )
               => ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ A5 @ T4 ) ) ) )
         => ( ! [T4: list_lambda_lambda,U7: list_lambda_lambda] :
                ( ( con_li79537494059641180lambda @ Resid @ T4 @ U7 )
               => ? [A8: list_lambda_lambda] :
                    ( ( ide_li6773656263178250516lambda @ Resid @ A8 )
                    & ( con_li79537494059641180lambda @ Resid @ A8 @ T4 )
                    & ( con_li79537494059641180lambda @ Resid @ A8 @ U7 ) ) )
           => ( ! [T4: list_lambda_lambda,U7: list_lambda_lambda,V3: list_lambda_lambda] :
                  ( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ T4 @ U7 ) )
                 => ( ( con_li79537494059641180lambda @ Resid @ U7 @ V3 )
                   => ( con_li79537494059641180lambda @ Resid @ ( Resid @ T4 @ U7 ) @ ( Resid @ V3 @ U7 ) ) ) )
             => ( rts_ax2283037410143324311lambda @ Resid ) ) ) ) ) ) ).

% rts_axioms.intro
thf(fact_1225_sub__rts__axioms_Ointro,axiom,
    ! [Arr: lambda_lambda > $o,R: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ! [T4: lambda_lambda] :
          ( ( Arr @ T4 )
         => ( arr_lambda_lambda @ R @ T4 ) )
     => ( ! [T4: lambda_lambda] :
            ( ( Arr @ T4 )
           => ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R @ T4 ) @ ( collec621444948832755541lambda @ Arr ) ) )
       => ( ! [T4: lambda_lambda,U7: lambda_lambda] :
              ( ( Arr @ T4 )
             => ( ( Arr @ U7 )
               => ( ( con_lambda_lambda @ R @ T4 @ U7 )
                 => ( Arr @ ( R @ T4 @ U7 ) ) ) ) )
         => ( sub_rt3575465509250962105lambda @ R @ Arr ) ) ) ) ).

% sub_rts_axioms.intro
thf(fact_1226_sub__rts__axioms_Ointro,axiom,
    ! [Arr: list_lambda_lambda > $o,R: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
      ( ! [T4: list_lambda_lambda] :
          ( ( Arr @ T4 )
         => ( arr_li3691443969012319837lambda @ R @ T4 ) )
     => ( ! [T4: list_lambda_lambda] :
            ( ( Arr @ T4 )
           => ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R @ T4 ) @ ( collec8830145823237851355lambda @ Arr ) ) )
       => ( ! [T4: list_lambda_lambda,U7: list_lambda_lambda] :
              ( ( Arr @ T4 )
             => ( ( Arr @ U7 )
               => ( ( con_li79537494059641180lambda @ R @ T4 @ U7 )
                 => ( Arr @ ( R @ T4 @ U7 ) ) ) ) )
         => ( sub_rt3099298114314975807lambda @ R @ Arr ) ) ) ) ).

% sub_rts_axioms.intro
thf(fact_1227_sub__rts__axioms__def,axiom,
    ( sub_rt3575465509250962105lambda
    = ( ^ [R2: lambda_lambda > lambda_lambda > lambda_lambda,Arr2: lambda_lambda > $o] :
          ( ! [T7: lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( arr_lambda_lambda @ R2 @ T7 ) )
          & ! [T7: lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R2 @ T7 ) @ ( collec621444948832755541lambda @ Arr2 ) ) )
          & ! [T7: lambda_lambda,U6: lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( ( Arr2 @ U6 )
               => ( ( con_lambda_lambda @ R2 @ T7 @ U6 )
                 => ( Arr2 @ ( R2 @ T7 @ U6 ) ) ) ) ) ) ) ) ).

% sub_rts_axioms_def
thf(fact_1228_sub__rts__axioms__def,axiom,
    ( sub_rt3099298114314975807lambda
    = ( ^ [R2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Arr2: list_lambda_lambda > $o] :
          ( ! [T7: list_lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( arr_li3691443969012319837lambda @ R2 @ T7 ) )
          & ! [T7: list_lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R2 @ T7 ) @ ( collec8830145823237851355lambda @ Arr2 ) ) )
          & ! [T7: list_lambda_lambda,U6: list_lambda_lambda] :
              ( ( Arr2 @ T7 )
             => ( ( Arr2 @ U6 )
               => ( ( con_li79537494059641180lambda @ R2 @ T7 @ U6 )
                 => ( Arr2 @ ( R2 @ T7 @ U6 ) ) ) ) ) ) ) ) ).

% sub_rts_axioms_def
thf(fact_1229_inf__Int__eq,axiom,
    ! [R: set_li7878742357367608272lambda,S: set_li7878742357367608272lambda] :
      ( ( inf_in6319970306715391135mbda_o
        @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ R )
        @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ S ) )
      = ( ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ ( inf_in3354698378728726462lambda @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1230_inf__Int__eq,axiom,
    ! [R: set_lambda_lambda,S: set_lambda_lambda] :
      ( ( inf_in2068736301489225125mbda_o
        @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ R )
        @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ S ) )
      = ( ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ ( inf_in1766959559283430712lambda @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1231_lambda__calculus_Oleftmost__reduction__preservation,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( lambda5425420467215166849uction @ T )
     => ( ( lambda2670285437103676873uction @ U )
       => ( ~ ( lambda5425420467215166849uction @ U )
         => ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
           => ( lambda5425420467215166849uction @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).

% lambda_calculus.leftmost_reduction_preservation
thf(fact_1232_bot__empty__eq,axiom,
    ( bot_bo430095131446760353mbda_o
    = ( ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ bot_bo1397037886704484924lambda ) ) ) ).

% bot_empty_eq
thf(fact_1233_bot__empty__eq,axiom,
    ( bot_bo3555140721347798439mbda_o
    = ( ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda ) ) ) ).

% bot_empty_eq
thf(fact_1234_pred__subset__eq,axiom,
    ! [R: set_lambda_lambda,S: set_lambda_lambda] :
      ( ( ord_le5488116190248881011mbda_o
        @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ R )
        @ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ S ) )
      = ( ord_le1926335974945931754lambda @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1235_pred__subset__eq,axiom,
    ! [R: set_li7878742357367608272lambda,S: set_li7878742357367608272lambda] :
      ( ( ord_le4030711025213088621mbda_o
        @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ R )
        @ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ S ) )
      = ( ord_le3064231726521830256lambda @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1236_listset_Osimps_I1_J,axiom,
    ( ( listse1500948262344051919lambda @ nil_se6914803783601386320lambda )
    = ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) ) ).

% listset.simps(1)
thf(fact_1237_the__elem__eq,axiom,
    ! [X: list_lambda_lambda] :
      ( ( the_el4548681628688475493lambda @ ( insert8465602035467394112lambda @ X @ bot_bo1397037886704484924lambda ) )
      = X ) ).

% the_elem_eq
thf(fact_1238_the__elem__eq,axiom,
    ! [X: lambda_lambda] :
      ( ( the_el5431214018478934239lambda @ ( insert_lambda_lambda @ X @ bot_bo1271416176687173302lambda ) )
      = X ) ).

% the_elem_eq
thf(fact_1239_the__elem__image__unique,axiom,
    ! [A: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda,X: lambda_lambda] :
      ( ( A != bot_bo1271416176687173302lambda )
     => ( ! [Y2: lambda_lambda] :
            ( ( member_lambda_lambda @ Y2 @ A )
           => ( ( F @ Y2 )
              = ( F @ X ) ) )
       => ( ( the_el4548681628688475493lambda @ ( image_3692823078739859851lambda @ F @ A ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_1240_the__elem__set,axiom,
    ! [X: lambda_lambda] :
      ( ( the_el5431214018478934239lambda @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) )
      = X ) ).

% the_elem_set
thf(fact_1241_lambda__calculus_Otargets__simps_I4_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( arr_lambda_lambda @ lambda_resid @ U )
       => ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
          = ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% lambda_calculus.targets_simps(4)
thf(fact_1242_paths__in__rts_Oimg__Resid__Srcs,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_366302053746739058lambda @ Resid @ T5 )
       => ( ord_le3064231726521830256lambda
          @ ( image_3692823078739859851lambda
            @ ^ [A6: lambda_lambda] : ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ A6 @ nil_lambda_lambda ) @ T5 )
            @ ( paths_2897053694880971696lambda @ Resid @ T5 ) )
          @ ( image_3692823078739859851lambda
            @ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
            @ ( paths_7594662086398683701lambda @ Resid @ T5 ) ) ) ) ) ).

% paths_in_rts.img_Resid_Srcs
thf(fact_1243_paths__in__rts__axioms,axiom,
    paths_6861597469809217359lambda @ lambda_resid ).

% paths_in_rts_axioms
thf(fact_1244_lambda__calculus_Olambda_Oinject_I3_J,axiom,
    ! [X41: lambda_lambda,X42: lambda_lambda,Y41: lambda_lambda,Y42: lambda_lambda] :
      ( ( ( lambda_App @ X41 @ X42 )
        = ( lambda_App @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% lambda_calculus.lambda.inject(3)
thf(fact_1245_lambda__calculus_Osources__simps_I4_J,axiom,
    ! [T: lambda_lambda,U: lambda_lambda] :
      ( ( arr_lambda_lambda @ lambda_resid @ T )
     => ( ( arr_lambda_lambda @ lambda_resid @ U )
       => ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
          = ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).

% lambda_calculus.sources_simps(4)
thf(fact_1246_paths__in__rts_Olength__Resid,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda )
       => ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) )
          = ( size_s4208335251162658524lambda @ T5 ) ) ) ) ).

% paths_in_rts.length_Resid
thf(fact_1247_paths__in__rts_OResidx1_Osimps_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_2112734093361750239lambda @ Resid @ nil_lambda_lambda @ U )
        = nil_lambda_lambda ) ) ).

% paths_in_rts.Residx1.simps(1)
thf(fact_1248_paths__in__rts_OResid1x_Osimps_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_3319247187298047121lambda @ Resid @ T @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
        = ( Resid @ T @ U ) ) ) ).

% paths_in_rts.Resid1x.simps(2)
thf(fact_1249_paths__in__rts_OTrgs__Resid__sym,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda )
       => ( ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) )
          = ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ U3 @ T5 ) ) ) ) ) ).

% paths_in_rts.Trgs_Resid_sym
thf(fact_1250_paths__in__rts_OSrcs__append,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( paths_2897053694880971696lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( paths_2897053694880971696lambda @ Resid @ T5 ) ) ) ) ).

% paths_in_rts.Srcs_append
thf(fact_1251_paths__in__rts_Ocon__char,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T5 @ U3 )
        = ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda ) ) ) ).

% paths_in_rts.con_char
thf(fact_1252_paths__in__rts_OconI_092_060_094sub_062P,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda )
       => ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T5 @ U3 ) ) ) ).

% paths_in_rts.conI\<^sub>P
thf(fact_1253_paths__in__rts_OArr__imp__Arr__tl,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_366302053746739058lambda @ Resid @ T5 )
       => ( ( ( tl_lambda_lambda @ T5 )
           != nil_lambda_lambda )
         => ( paths_366302053746739058lambda @ Resid @ ( tl_lambda_lambda @ T5 ) ) ) ) ) ).

% paths_in_rts.Arr_imp_Arr_tl
thf(fact_1254_paths__in__rts_OTrgs__append,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U3: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( paths_7594662086398683701lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) )
          = ( paths_7594662086398683701lambda @ Resid @ U3 ) ) ) ) ).

% paths_in_rts.Trgs_append
thf(fact_1255_paths__in__rts_OIde__imp__Ide__tl,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_7717490669901197737lambda @ Resid @ T5 )
       => ( ( ( tl_lambda_lambda @ T5 )
           != nil_lambda_lambda )
         => ( paths_7717490669901197737lambda @ Resid @ ( tl_lambda_lambda @ T5 ) ) ) ) ) ).

% paths_in_rts.Ide_imp_Ide_tl
thf(fact_1256_paths__in__rts_OCon__append_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( V2 != nil_lambda_lambda )
           => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
               != nil_lambda_lambda )
              = ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
                 != nil_lambda_lambda )
                & ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ V2 )
                 != nil_lambda_lambda ) ) ) ) ) ) ) ).

% paths_in_rts.Con_append(2)
thf(fact_1257_paths__in__rts_OCon__append_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( V2 != nil_lambda_lambda )
           => ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
               != nil_lambda_lambda )
              = ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 )
                 != nil_lambda_lambda )
                & ( ( paths_8607357831768207192lambda @ Resid @ U3 @ ( paths_8607357831768207192lambda @ Resid @ V2 @ T5 ) )
                 != nil_lambda_lambda ) ) ) ) ) ) ) ).

% paths_in_rts.Con_append(1)
thf(fact_1258_paths__in__rts_OCon__appendI_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ V2 )
           != nil_lambda_lambda )
         => ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
           != nil_lambda_lambda ) ) ) ) ).

% paths_in_rts.Con_appendI(2)
thf(fact_1259_paths__in__rts_OCon__appendI_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,V2: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 )
         != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ Resid @ U3 @ ( paths_8607357831768207192lambda @ Resid @ V2 @ T5 ) )
           != nil_lambda_lambda )
         => ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
           != nil_lambda_lambda ) ) ) ) ).

% paths_in_rts.Con_appendI(1)
thf(fact_1260_paths__in__rts_OResid__append_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda,T5: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( U3 != nil_lambda_lambda )
       => ( ( V2 != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
             != nil_lambda_lambda )
           => ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ U3 @ V2 ) )
              = ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ V2 ) ) ) ) ) ) ).

% paths_in_rts.Resid_append(2)
thf(fact_1261_paths__in__rts_OResid__append_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
           != nil_lambda_lambda )
         => ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ V2 )
            = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ Resid @ U3 @ ( paths_8607357831768207192lambda @ Resid @ V2 @ T5 ) ) ) ) ) ) ) ).

% paths_in_rts.Resid_append(1)
thf(fact_1262_paths__in__rts_OResid__append2,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda,W: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( V2 != nil_lambda_lambda )
           => ( ( W != nil_lambda_lambda )
             => ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ V2 @ W ) )
                 != nil_lambda_lambda )
               => ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ V2 @ W ) )
                  = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ W ) @ ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ U3 @ ( paths_8607357831768207192lambda @ Resid @ V2 @ T5 ) ) @ ( paths_8607357831768207192lambda @ Resid @ W @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) ) ) ) ) ) ) ) ) ) ) ).

% paths_in_rts.Resid_append2
thf(fact_1263_paths__in__rts_OResid__append__ind,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( V2 != nil_lambda_lambda )
           => ( ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
                 != nil_lambda_lambda )
                = ( ( ( paths_8607357831768207192lambda @ Resid @ V2 @ U3 )
                   != nil_lambda_lambda )
                  & ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( paths_8607357831768207192lambda @ Resid @ U3 @ V2 ) )
                   != nil_lambda_lambda ) ) )
              & ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
                 != nil_lambda_lambda )
                = ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 )
                   != nil_lambda_lambda )
                  & ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ U3 )
                   != nil_lambda_lambda ) ) )
              & ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
                 != nil_lambda_lambda )
               => ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V2 @ T5 ) @ U3 )
                  = ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ V2 @ U3 ) @ ( paths_8607357831768207192lambda @ Resid @ T5 @ ( paths_8607357831768207192lambda @ Resid @ U3 @ V2 ) ) ) ) )
              & ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
                 != nil_lambda_lambda )
               => ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( append_lambda_lambda @ V2 @ U3 ) )
                  = ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ U3 ) ) ) ) ) ) ) ) ).

% paths_in_rts.Resid_append_ind
thf(fact_1264_paths__in__rts_OCube_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ Resid @ V2 @ U3 ) )
         != nil_lambda_lambda )
       => ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ Resid @ V2 @ U3 ) )
          = ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ Resid @ U3 @ V2 ) ) ) ) ) ).

% paths_in_rts.Cube(2)
thf(fact_1265_paths__in__rts_OCube_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,V2: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 ) @ ( paths_8607357831768207192lambda @ Resid @ V2 @ U3 ) )
         != nil_lambda_lambda )
        = ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ V2 ) @ ( paths_8607357831768207192lambda @ Resid @ U3 @ V2 ) )
         != nil_lambda_lambda ) ) ) ).

% paths_in_rts.Cube(1)
thf(fact_1266_paths__in__rts_OResid_Osimps_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,Uu: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( paths_8607357831768207192lambda @ Resid @ nil_lambda_lambda @ Uu )
        = nil_lambda_lambda ) ) ).

% paths_in_rts.Resid.simps(1)
thf(fact_1267_paths__in__rts_OCon__sym,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ U3 )
         != nil_lambda_lambda )
        = ( ( paths_8607357831768207192lambda @ Resid @ U3 @ T5 )
         != nil_lambda_lambda ) ) ) ).

% paths_in_rts.Con_sym
thf(fact_1268_paths__in__rts_OArr_Osimps_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ~ ( paths_366302053746739058lambda @ Resid @ nil_lambda_lambda ) ) ).

% paths_in_rts.Arr.simps(1)
thf(fact_1269_paths__in__rts_OIde_Osimps_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ~ ( paths_7717490669901197737lambda @ Resid @ nil_lambda_lambda ) ) ).

% paths_in_rts.Ide.simps(1)
thf(fact_1270_paths__in__rts_OTrgs_Ocases,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X: list_lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( X != nil_lambda_lambda )
       => ( ! [T4: lambda_lambda] :
              ( X
             != ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
         => ~ ! [T4: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
                ( X
               != ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ).

% paths_in_rts.Trgs.cases
thf(fact_1271_paths__in__rts_OCon__cons_I2_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,U: lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( cons_lambda_lambda @ U @ U3 ) )
             != nil_lambda_lambda )
            = ( ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
               != nil_lambda_lambda )
              & ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T5 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 )
               != nil_lambda_lambda ) ) ) ) ) ) ).

% paths_in_rts.Con_cons(2)
thf(fact_1272_paths__in__rts_OCon__cons_I1_J,axiom,
    ! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda] :
      ( ( paths_6861597469809217359lambda @ Resid )
     => ( ( T5 != nil_lambda_lambda )
       => ( ( U3 != nil_lambda_lambda )
         => ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T5 ) @ U3 )
             != nil_lambda_lambda )
            = ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
               != nil_lambda_lambda )
              & ( ( paths_8607357831768207192lambda @ Resid @ T5 @ ( paths_8607357831768207192lambda @ Resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
               != nil_lambda_lambda ) ) ) ) ) ) ).

% paths_in_rts.Con_cons(1)

% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
    ! [X: list_lambda_lambda,Y4: list_lambda_lambda] :
      ( ( if_lis7195930416592267574lambda @ $false @ X @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
    ! [X: list_lambda_lambda,Y4: list_lambda_lambda] :
      ( ( if_lis7195930416592267574lambda @ $true @ X @ Y4 )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( reduct5390047737487997688insert @ ( lambda_Beta @ m @ n ) @ ( cons_lambda_lambda @ u @ ua ) )
    = ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ m ) @ ( lambda_Src @ n ) ) @ ( reduct5390047737487997688insert @ u @ ua ) ) ) ).

%------------------------------------------------------------------------------