%------------------------------------------------------------------------------
% 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    : Eval_FO/0005_Ailamazyan/prob_04176_174547__16124894_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1851 ( 595 unt; 599 typ;   0 def)
%            Number of atoms       : 3869 (2287 equ;   0 cnn)
%            Maximal formula atoms :   22 (   3 avg)
%            Number of connectives : 19662 ( 836   ~;  61   |; 389   &;16255   @)
%                                         (   0 <=>;2121  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  124 ( 123 usr)
%            Number of type conns  : 2337 (2337   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  479 ( 476 usr;  26 con; 0-8 aty)
%            Number of variables   : 4804 (  61   ^;4576   !; 167   ?;4804   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:12:37.427
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P6164600145584960654at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P5056861408695629236_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8373899037510109440at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_li8553678471444574443_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_s602657755630655083_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1828647624359046049st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
    set_Pr5845495582615845127_set_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_li5450038453877631591at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc7819656566062154093et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc7258583773236448510at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc5454855657571582244_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_li3402123583998030149at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_l4703314356710769291_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_li6526943997496501093_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    produc1703568184450464039_set_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4575160907756185873st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    option_Sum_sum_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_Sum_sum_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Sum_sum_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    set_list_list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    product_prod_b_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__List__Olist_It__FO__Ofo____term_Itf__a_J_J,type,
    list_fo_term_a: $tType ).

thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    sum_sum_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    set_list_a: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    list_set_a: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__FO__Ofo____term_It__Nat__Onat_J,type,
    fo_term_nat: $tType ).

thf(ty_n_t__FO__Ofo____fmla_Itf__a_Mtf__b_J,type,
    fo_fmla_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__FO__Ofo____term_Itf__a_J,type,
    fo_term_a: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (476)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
    ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_Ailamazyan_Oad__equiv__list_001tf__a_001t__Nat__Onat,type,
    ad_equiv_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_Ailamazyan_Oadd__nth_001t__Nat__Onat,type,
    add_nth_nat: nat > nat > list_nat > list_nat ).

thf(sy_c_Ailamazyan_Oadd__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    add_nt4212672348507122516_a_nat: nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Ailamazyan_Oall__tuples_001t__Nat__Onat,type,
    all_tuples_nat: set_nat > nat > set_list_nat ).

thf(sy_c_Ailamazyan_Oall__tuples_001tf__a,type,
    all_tuples_a: set_a > nat > set_list_a ).

thf(sy_c_Ailamazyan_Oeval__eq_001tf__a,type,
    eval_eq_a: fo_term_a > fo_term_a > produc8208987855206359447_a_nat ).

thf(sy_c_Ailamazyan_Oeval__eterm_001tf__a_001t__Nat__Onat,type,
    eval_eterm_a_nat: ( nat > sum_sum_a_nat ) > fo_term_a > sum_sum_a_nat ).

thf(sy_c_Ailamazyan_Oeval__eterm__rel_001tf__a_001t__Nat__Onat,type,
    eval_eterm_rel_a_nat: produc9064192874753994617term_a > produc9064192874753994617term_a > $o ).

thf(sy_c_Ailamazyan_Oeval__eterms_001tf__a_001t__Nat__Onat,type,
    eval_eterms_a_nat: ( nat > sum_sum_a_nat ) > list_fo_term_a > list_Sum_sum_a_nat ).

thf(sy_c_Ailamazyan_Oeval__neg_001tf__a,type,
    eval_neg_a: list_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat ).

thf(sy_c_Ailamazyan_Oeval__neg__rel_001tf__a,type,
    eval_neg_rel_a: produc7015024766196696536_a_nat > produc7015024766196696536_a_nat > $o ).

thf(sy_c_Ailamazyan_Oeval__table_001tf__a_001t__Nat__Onat,type,
    eval_table_a_nat: list_fo_term_a > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Ailamazyan_Ofo__adom_001tf__a_001t__Nat__Onat,type,
    fo_adom_a_nat: produc8208987855206359447_a_nat > set_a ).

thf(sy_c_Ailamazyan_Ofo__adom__rel_001tf__a_001t__Nat__Onat,type,
    fo_adom_rel_a_nat: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o ).

thf(sy_c_Ailamazyan_Ofo__fin_001tf__a,type,
    fo_fin_a: produc8208987855206359447_a_nat > $o ).

thf(sy_c_Ailamazyan_Ofo__nmlz_001tf__a,type,
    fo_nmlz_a: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Ailamazyan_Ofo__rep_001tf__a_001t__Nat__Onat,type,
    fo_rep_a_nat: produc8208987855206359447_a_nat > set_list_a ).

thf(sy_c_Ailamazyan_Ofo__wf_001tf__a_001tf__b,type,
    fo_wf_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > produc8208987855206359447_a_nat > $o ).

thf(sy_c_Ailamazyan_Onall__tuples_001t__Nat__Onat,type,
    nall_tuples_nat: set_nat > nat > set_li3402123583998030149at_nat ).

thf(sy_c_Ailamazyan_Onall__tuples_001tf__a,type,
    nall_tuples_a: set_a > nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Ailamazyan_Onall__tuples__rec_001tf__a,type,
    nall_tuples_rec_a: set_a > nat > nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Ailamazyan_Onats,type,
    nats: list_nat > $o ).

thf(sy_c_Ailamazyan_Orem__nth_001t__Nat__Onat,type,
    rem_nth_nat: nat > list_nat > list_nat ).

thf(sy_c_Ailamazyan_Orem__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    rem_nt658808235856662061_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Ailamazyan_Orem__nth__rel_001t__Nat__Onat,type,
    rem_nth_rel_nat: produc4575160907756185873st_nat > produc4575160907756185873st_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001t__Nat__Onat,type,
    sp_equiv_list_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_AssocList_Ozip__with__index__from_001t__Nat__Onat,type,
    zip_wi3407404960461264653om_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AssocList_Ozip__with__index__from_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    zip_wi4158353643603967848_a_nat: nat > list_s602657755630655083_a_nat > list_P4848832406072927130_a_nat ).

thf(sy_c_AssocList_Ozip__with__index__from_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    zip_wi5053028342928268674_a_nat: nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).

thf(sy_c_BNF__Def_Orel__sum_001tf__a_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8134084500293877003at_nat: ( a > a > $o ) > ( nat > nat > $o ) > sum_sum_a_nat > sum_sum_a_nat > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr5329103786779832013at_nat: set_li6526943997496501093_a_nat > ( list_Sum_sum_a_nat > nat ) > ( list_Sum_sum_a_nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_Itf__a_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr6181254616915012556at_nat: set_list_a > ( list_a > nat ) > ( list_a > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_Itf__a_J_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    bNF_Gr8898696621080283753_a_nat: set_list_a > ( list_a > nat ) > ( list_a > set_li6526943997496501093_a_nat ) > set_Pr1465208939327205952_a_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr9206233512786870250at_nat: set_list_a > ( list_a > product_prod_nat_nat ) > ( list_a > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr7318156773085685964at_nat: set_nat > ( nat > nat ) > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    bNF_Gr9061489936062418281_a_nat: set_nat > ( nat > nat ) > ( nat > set_li6526943997496501093_a_nat ) > set_Pr1465208939327205952_a_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr7492152087006117098at_nat: set_nat > ( nat > product_prod_nat_nat ) > ( nat > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr8954129895687480649at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    bNF_Gr6564037981669998636_a_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > set_li6526943997496501093_a_nat ) > set_Pr1465208939327205952_a_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr8808630244184418791at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum_001tf__a_001t__Nat__Onat,type,
    basic_7520187345070478634_a_nat: ( a > nat ) > ( nat > nat ) > sum_sum_a_nat > nat ).

thf(sy_c_FO_Ofo__fmla_OEqa_001tf__a_001tf__b,type,
    fo_Eqa_a_b: fo_term_a > fo_term_a > fo_fmla_a_b ).

thf(sy_c_FO_Ofo__term_OConst_001t__Nat__Onat,type,
    fo_Const_nat: nat > fo_term_nat ).

thf(sy_c_FO_Ofo__term_OConst_001tf__a,type,
    fo_Const_a: a > fo_term_a ).

thf(sy_c_FO_Ofo__term_OVar_001t__Nat__Onat,type,
    fo_Var_nat: nat > fo_term_nat ).

thf(sy_c_FO_Ofo__term_OVar_001tf__a,type,
    fo_Var_a: nat > fo_term_a ).

thf(sy_c_FO_Ofv__fo__terms__list_001tf__a,type,
    fv_fo_terms_list_a: list_fo_term_a > list_nat ).

thf(sy_c_FO_Olist__fo__term_001t__Nat__Onat,type,
    list_fo_term_nat: fo_term_nat > list_nat ).

thf(sy_c_FO_Olist__fo__term__rel_001t__Nat__Onat,type,
    list_fo_term_rel_nat: fo_term_nat > fo_term_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite5087377988160578214at_nat: set_li3402123583998030149at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    finite1487985464145237934_a_nat: set_li6526943997496501093_a_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
    finite_finite_list_a: set_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
    finite_finite_a: set_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    minus_7395159227704179404_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

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    member1995966531042493578at_nat: produc5224906263214031073at_nat > set_Pr7116486347545156417at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member2549530991182098686_a_nat: produc4511668648249480021_a_nat > set_Pr3095833754874480053_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Nat__Onat_J_J,type,
    member4164734889767531536at_nat: produc3232148987813434599at_nat > set_Pr1329147455154692551at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member1447830766745040226at_nat: produc3409968423812421561at_nat > set_Pr907209939874158617at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member2682851360909428368_a_nat: produc7350071986246684007_a_nat > set_Pr2396145695824571463_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    member7641717237037081872_a_nat: produc2659627244541738983_a_nat > set_Pr1993642681809099463_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member8751153172265411512_a_nat: produc6346555946419678479_a_nat > set_Pr757995773253584751_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    member5874747487503750032_a_nat: produc2571375213224653159_a_nat > set_Pr8343503725489915207_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    member3723442691059620112_a_nat: produc7017002724195966439_a_nat > set_Pr7343886759072863943_a_nat > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).

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

thf(sy_v_AD,type,
    ad: set_a ).

thf(sy_v_I,type,
    i: product_prod_b_nat > set_list_a ).

thf(sy_v_X,type,
    x: set_li6526943997496501093_a_nat ).

thf(sy_v__092_060phi_062,type,
    phi: fo_fmla_a_b ).

thf(sy_v__092_060tau_062____,type,
    tau: nat > a ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_vs____,type,
    vs: list_Sum_sum_a_nat ).

% Relevant facts (1246)
thf(fact_0_vs__in,axiom,
    member408289922725080238_a_nat @ vs @ x ).

% vs_in
thf(fact_1__092_060open_062vs_A_061_Afo__nmlz_AAD_A_Imap_AInl_A_Imap_A_092_060tau_062_A_0910_O_O_060n_093_J_J_092_060close_062,axiom,
    ( vs
    = ( fo_nmlz_a @ ad @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ ( map_nat_a @ tau @ ( upt @ zero_zero_nat @ n ) ) ) ) ) ).

% \<open>vs = fo_nmlz AD (map Inl (map \<tau> [0..<n]))\<close>
thf(fact_2_map___092_060tau_062__in,axiom,
    member_list_a @ ( map_nat_a @ tau @ ( upt @ zero_zero_nat @ n ) ) @ ( fo_rep_a_nat @ ( produc1626021218680649873_a_nat @ ad @ ( produc4622690550020908860_a_nat @ n @ x ) ) ) ).

% map_\<tau>_in
thf(fact_3_fo__fin_Ocases,axiom,
    ! [X: produc8208987855206359447_a_nat] :
      ~ ! [AD: set_a,N: nat,X2: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N @ X2 ) ) ) ).

% fo_fin.cases
thf(fact_4_fo__adom_Ocases,axiom,
    ! [X: produc8208987855206359447_a_nat] :
      ~ ! [AD: set_a,N: nat,X2: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N @ X2 ) ) ) ).

% fo_adom.cases
thf(fact_5_sp__equiv__pair_Ocases,axiom,
    ! [X: produc349518998152878311at_nat] :
      ~ ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( X
         != ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_6_sp__equiv__pair_Ocases,axiom,
    ! [X: produc7350071986246684007_a_nat] :
      ~ ! [A: produc5986856040925105290_a_nat,B: produc5986856040925105290_a_nat,A2: produc5986856040925105290_a_nat,B2: produc5986856040925105290_a_nat] :
          ( X
         != ( produc8287655239752403415_a_nat @ ( produc5595853527594814125_a_nat @ A @ B ) @ ( produc5595853527594814125_a_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_7_sp__equiv__pair_Ocases,axiom,
    ! [X: produc2659627244541738983_a_nat] :
      ~ ! [A: produc8208987855206359447_a_nat,B: produc8208987855206359447_a_nat,A2: produc8208987855206359447_a_nat,B2: produc8208987855206359447_a_nat] :
          ( X
         != ( produc7779878758203082071_a_nat @ ( produc5405793250563226055_a_nat @ A @ B ) @ ( produc5405793250563226055_a_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_8_sp__equiv__pair_Ocases,axiom,
    ! [X: produc6346555946419678479_a_nat] :
      ~ ! [A: set_a,B: produc5986856040925105290_a_nat,A2: set_a,B2: produc5986856040925105290_a_nat] :
          ( X
         != ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ A @ B ) @ ( produc1626021218680649873_a_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_9_sp__equiv__pair_Ocases,axiom,
    ! [X: produc5621125494677191285_a_nat] :
      ~ ! [A: nat,B: set_li6526943997496501093_a_nat,A2: nat,B2: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ A @ B ) @ ( produc4622690550020908860_a_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_10_sp__equiv__pair_Ocases,axiom,
    ! [X: produc859450856879609959at_nat] :
      ~ ! [A: nat,B: nat,A2: nat,B2: nat] :
          ( X
         != ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_11_fo__nmlz__map,axiom,
    ! [AD2: set_a,Sigma: a > sum_sum_a_nat,Ns: list_a] :
    ? [Tau: a > sum_sum_a_nat] :
      ( ( fo_nmlz_a @ AD2 @ ( map_a_Sum_sum_a_nat @ Sigma @ Ns ) )
      = ( map_a_Sum_sum_a_nat @ Tau @ Ns ) ) ).

% fo_nmlz_map
thf(fact_12_assms,axiom,
    fo_wf_a_b @ phi @ i @ ( produc1626021218680649873_a_nat @ ad @ ( produc4622690550020908860_a_nat @ n @ x ) ) ).

% assms
thf(fact_13__092_060tau_062__def,axiom,
    ad_agr_list_a_nat @ ad @ vs @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ ( map_nat_a @ tau @ ( upt @ zero_zero_nat @ n ) ) ) ).

% \<tau>_def
thf(fact_14_len__vs,axiom,
    ( ( size_s5283204784079214577_a_nat @ vs )
    = n ) ).

% len_vs
thf(fact_15__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060tau_062_O_Aad__agr__list_AAD_Avs_A_Imap_AInl_A_Imap_A_092_060tau_062_A_0910_O_O_060n_093_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Tau: nat > a] :
        ~ ( ad_agr_list_a_nat @ ad @ vs @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ ( map_nat_a @ Tau @ ( upt @ zero_zero_nat @ n ) ) ) ) ).

% \<open>\<And>thesis. (\<And>\<tau>. ad_agr_list AD vs (map Inl (map \<tau> [0..<n])) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_16_sum_Oinject_I1_J,axiom,
    ! [X1: a,Y1: a] :
      ( ( ( sum_Inl_a_nat @ X1 )
        = ( sum_Inl_a_nat @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_17_old_Osum_Oinject_I1_J,axiom,
    ! [A3: a,A4: a] :
      ( ( ( sum_Inl_a_nat @ A3 )
        = ( sum_Inl_a_nat @ A4 ) )
      = ( A3 = A4 ) ) ).

% old.sum.inject(1)
thf(fact_18_image__eqI,axiom,
    ! [B3: nat,F: nat > nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_19_image__eqI,axiom,
    ! [B3: nat,F: list_a > nat,X: list_a,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A5 )
       => ( member_nat @ B3 @ ( image_list_a_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_20_image__eqI,axiom,
    ! [B3: list_a,F: nat > list_a,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member_list_a @ B3 @ ( image_nat_list_a @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_21_image__eqI,axiom,
    ! [B3: list_a,F: list_a > list_a,X: list_a,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A5 )
       => ( member_list_a @ B3 @ ( image_list_a_list_a @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_22_image__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: nat > product_prod_nat_nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_23_image__eqI,axiom,
    ! [B3: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_24_image__eqI,axiom,
    ! [B3: nat,F: list_Sum_sum_a_nat > nat,X: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member408289922725080238_a_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_2535339886381165584at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_25_image__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: list_a > product_prod_nat_nat,X: list_a,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A5 )
       => ( member8440522571783428010at_nat @ B3 @ ( image_3288686587095672578at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_26_image__eqI,axiom,
    ! [B3: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member408289922725080238_a_nat @ B3 @ ( image_6262589752765146990_a_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_27_image__eqI,axiom,
    ! [B3: list_a,F: product_prod_nat_nat > list_a,X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( member_list_a @ B3 @ ( image_3765944151573559240list_a @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_28_prod_Oinject,axiom,
    ! [X1: set_a,X22: produc5986856040925105290_a_nat,Y1: set_a,Y2: produc5986856040925105290_a_nat] :
      ( ( ( produc1626021218680649873_a_nat @ X1 @ X22 )
        = ( produc1626021218680649873_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_29_prod_Oinject,axiom,
    ! [X1: nat,X22: set_li6526943997496501093_a_nat,Y1: nat,Y2: set_li6526943997496501093_a_nat] :
      ( ( ( produc4622690550020908860_a_nat @ X1 @ X22 )
        = ( produc4622690550020908860_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_30_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_31_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_32_prod_Oinject,axiom,
    ! [X1: produc5986856040925105290_a_nat,X22: produc5986856040925105290_a_nat,Y1: produc5986856040925105290_a_nat,Y2: produc5986856040925105290_a_nat] :
      ( ( ( produc5595853527594814125_a_nat @ X1 @ X22 )
        = ( produc5595853527594814125_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_33_prod_Oinject,axiom,
    ! [X1: produc8208987855206359447_a_nat,X22: produc8208987855206359447_a_nat,Y1: produc8208987855206359447_a_nat,Y2: produc8208987855206359447_a_nat] :
      ( ( ( produc5405793250563226055_a_nat @ X1 @ X22 )
        = ( produc5405793250563226055_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
    ! [A3: set_a,B3: produc5986856040925105290_a_nat,A4: set_a,B4: produc5986856040925105290_a_nat] :
      ( ( ( produc1626021218680649873_a_nat @ A3 @ B3 )
        = ( produc1626021218680649873_a_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: set_li6526943997496501093_a_nat,A4: nat,B4: set_li6526943997496501093_a_nat] :
      ( ( ( produc4622690550020908860_a_nat @ A3 @ B3 )
        = ( produc4622690550020908860_a_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_36_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_37_old_Oprod_Oinject,axiom,
    ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A3 @ B3 )
        = ( produc6161850002892822231at_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
    ! [A3: produc5986856040925105290_a_nat,B3: produc5986856040925105290_a_nat,A4: produc5986856040925105290_a_nat,B4: produc5986856040925105290_a_nat] :
      ( ( ( produc5595853527594814125_a_nat @ A3 @ B3 )
        = ( produc5595853527594814125_a_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_39_old_Oprod_Oinject,axiom,
    ! [A3: produc8208987855206359447_a_nat,B3: produc8208987855206359447_a_nat,A4: produc8208987855206359447_a_nat,B4: produc8208987855206359447_a_nat] :
      ( ( ( produc5405793250563226055_a_nat @ A3 @ B3 )
        = ( produc5405793250563226055_a_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_40_fo__adom_Oelims,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: set_a] :
      ( ( ( fo_adom_a_nat @ X )
        = Y )
     => ~ ! [AD: set_a] :
            ( ? [N: nat,X2: set_li6526943997496501093_a_nat] :
                ( X
                = ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N @ X2 ) ) )
           => ( Y != AD ) ) ) ).

% fo_adom.elims
thf(fact_41_fo__adom_Osimps,axiom,
    ! [AD2: set_a,N2: nat,X3: set_li6526943997496501093_a_nat] :
      ( ( fo_adom_a_nat @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N2 @ X3 ) ) )
      = AD2 ) ).

% fo_adom.simps
thf(fact_42_fin__AD,axiom,
    finite_finite_a @ ad ).

% fin_AD
thf(fact_43_ad__agr__list__comm,axiom,
    ! [X3: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ad_agr_list_a_nat @ X3 @ Xs @ Ys )
     => ( ad_agr_list_a_nat @ X3 @ Ys @ Xs ) ) ).

% ad_agr_list_comm
thf(fact_44_ad__agr__list__refl,axiom,
    ! [X3: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X3 @ Xs @ Xs ) ).

% ad_agr_list_refl
thf(fact_45_ad__agr__list__trans,axiom,
    ! [X3: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
      ( ( ad_agr_list_a_nat @ X3 @ Xs @ Ys )
     => ( ( ad_agr_list_a_nat @ X3 @ Ys @ Zs )
       => ( ad_agr_list_a_nat @ X3 @ Xs @ Zs ) ) ) ).

% ad_agr_list_trans
thf(fact_46_ad__agr__list__length,axiom,
    ! [X3: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ad_agr_list_a_nat @ X3 @ Xs @ Ys )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% ad_agr_list_length
thf(fact_47_eval__ajoin_Ocases,axiom,
    ! [X: produc8481584770077217079_a_nat] :
      ~ ! [Ns_phi: list_nat,AD_phi: set_a,Uu: nat,X_phi: set_li6526943997496501093_a_nat,Ns_psi: list_nat,AD_psi: set_a,Uv: nat,X_psi: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc5266839609353528303_a_nat @ Ns_phi @ ( produc6936860417230850472_a_nat @ ( produc1626021218680649873_a_nat @ AD_phi @ ( produc4622690550020908860_a_nat @ Uu @ X_phi ) ) @ ( produc5478379421572013840_a_nat @ Ns_psi @ ( produc1626021218680649873_a_nat @ AD_psi @ ( produc4622690550020908860_a_nat @ Uv @ X_psi ) ) ) ) ) ) ).

% eval_ajoin.cases
thf(fact_48_eval__forall_Ocases,axiom,
    ! [X: produc3169862784723210109_a_nat] :
      ~ ! [I: nat,Ns2: list_nat,AD: set_a,Uu: nat,X2: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc1383156333761652911_a_nat @ I @ ( produc5478379421572013840_a_nat @ Ns2 @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X2 ) ) ) ) ) ).

% eval_forall.cases
thf(fact_49_fo__nmlz__length,axiom,
    ! [AD2: set_a,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_a @ AD2 @ Xs ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% fo_nmlz_length
thf(fact_50_fo__nmlz__ad__agr,axiom,
    ! [AD2: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ AD2 @ Xs @ ( fo_nmlz_a @ AD2 @ Xs ) ) ).

% fo_nmlz_ad_agr
thf(fact_51_fo__nmlz__eqI,axiom,
    ! [AD2: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
      ( ( ad_agr_list_a_nat @ AD2 @ Vs @ Vs2 )
     => ( ( fo_nmlz_a @ AD2 @ Vs )
        = ( fo_nmlz_a @ AD2 @ Vs2 ) ) ) ).

% fo_nmlz_eqI
thf(fact_52_fo__nmlz__eqD,axiom,
    ! [AD2: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
      ( ( ( fo_nmlz_a @ AD2 @ Vs )
        = ( fo_nmlz_a @ AD2 @ Vs2 ) )
     => ( ad_agr_list_a_nat @ AD2 @ Vs @ Vs2 ) ) ).

% fo_nmlz_eqD
thf(fact_53_fo__nmlz__eq,axiom,
    ! [AD2: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
      ( ( ( fo_nmlz_a @ AD2 @ Vs )
        = ( fo_nmlz_a @ AD2 @ Vs2 ) )
      = ( ad_agr_list_a_nat @ AD2 @ Vs @ Vs2 ) ) ).

% fo_nmlz_eq
thf(fact_54_eval__neg_Ocases,axiom,
    ! [X: produc7015024766196696536_a_nat] :
      ~ ! [Ns2: list_nat,AD: set_a,Uu: nat,X2: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc5478379421572013840_a_nat @ Ns2 @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X2 ) ) ) ) ).

% eval_neg.cases
thf(fact_55_Pair__inject,axiom,
    ! [A3: set_a,B3: produc5986856040925105290_a_nat,A4: set_a,B4: produc5986856040925105290_a_nat] :
      ( ( ( produc1626021218680649873_a_nat @ A3 @ B3 )
        = ( produc1626021218680649873_a_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_56_Pair__inject,axiom,
    ! [A3: nat,B3: set_li6526943997496501093_a_nat,A4: nat,B4: set_li6526943997496501093_a_nat] :
      ( ( ( produc4622690550020908860_a_nat @ A3 @ B3 )
        = ( produc4622690550020908860_a_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_57_Pair__inject,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_58_Pair__inject,axiom,
    ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A3 @ B3 )
        = ( produc6161850002892822231at_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_59_Pair__inject,axiom,
    ! [A3: produc5986856040925105290_a_nat,B3: produc5986856040925105290_a_nat,A4: produc5986856040925105290_a_nat,B4: produc5986856040925105290_a_nat] :
      ( ( ( produc5595853527594814125_a_nat @ A3 @ B3 )
        = ( produc5595853527594814125_a_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_60_Pair__inject,axiom,
    ! [A3: produc8208987855206359447_a_nat,B3: produc8208987855206359447_a_nat,A4: produc8208987855206359447_a_nat,B4: produc8208987855206359447_a_nat] :
      ( ( ( produc5405793250563226055_a_nat @ A3 @ B3 )
        = ( produc5405793250563226055_a_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_61_prod__cases,axiom,
    ! [P: produc8208987855206359447_a_nat > $o,P2: produc8208987855206359447_a_nat] :
      ( ! [A: set_a,B: produc5986856040925105290_a_nat] : ( P @ ( produc1626021218680649873_a_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_62_prod__cases,axiom,
    ! [P: produc5986856040925105290_a_nat > $o,P2: produc5986856040925105290_a_nat] :
      ( ! [A: nat,B: set_li6526943997496501093_a_nat] : ( P @ ( produc4622690550020908860_a_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_63_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_64_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_65_prod__cases,axiom,
    ! [P: produc5621125494677191285_a_nat > $o,P2: produc5621125494677191285_a_nat] :
      ( ! [A: produc5986856040925105290_a_nat,B: produc5986856040925105290_a_nat] : ( P @ ( produc5595853527594814125_a_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_66_prod__cases,axiom,
    ! [P: produc6346555946419678479_a_nat > $o,P2: produc6346555946419678479_a_nat] :
      ( ! [A: produc8208987855206359447_a_nat,B: produc8208987855206359447_a_nat] : ( P @ ( produc5405793250563226055_a_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_67_surj__pair,axiom,
    ! [P2: produc8208987855206359447_a_nat] :
    ? [X4: set_a,Y3: produc5986856040925105290_a_nat] :
      ( P2
      = ( produc1626021218680649873_a_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_68_surj__pair,axiom,
    ! [P2: produc5986856040925105290_a_nat] :
    ? [X4: nat,Y3: set_li6526943997496501093_a_nat] :
      ( P2
      = ( produc4622690550020908860_a_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_69_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_70_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_71_surj__pair,axiom,
    ! [P2: produc5621125494677191285_a_nat] :
    ? [X4: produc5986856040925105290_a_nat,Y3: produc5986856040925105290_a_nat] :
      ( P2
      = ( produc5595853527594814125_a_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_72_surj__pair,axiom,
    ! [P2: produc6346555946419678479_a_nat] :
    ? [X4: produc8208987855206359447_a_nat,Y3: produc8208987855206359447_a_nat] :
      ( P2
      = ( produc5405793250563226055_a_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_73_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8208987855206359447_a_nat] :
      ~ ! [A: set_a,B: produc5986856040925105290_a_nat] :
          ( Y
         != ( produc1626021218680649873_a_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_74_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5986856040925105290_a_nat] :
      ~ ! [A: nat,B: set_li6526943997496501093_a_nat] :
          ( Y
         != ( produc4622690550020908860_a_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_75_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A: nat,B: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_76_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_77_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5621125494677191285_a_nat] :
      ~ ! [A: produc5986856040925105290_a_nat,B: produc5986856040925105290_a_nat] :
          ( Y
         != ( produc5595853527594814125_a_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_78_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6346555946419678479_a_nat] :
      ~ ! [A: produc8208987855206359447_a_nat,B: produc8208987855206359447_a_nat] :
          ( Y
         != ( produc5405793250563226055_a_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_79_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: nat,F: nat > nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_80_rev__image__eqI,axiom,
    ! [X: list_a,A5: set_list_a,B3: nat,F: list_a > nat] :
      ( ( member_list_a @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_list_a_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_81_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: list_a,F: nat > list_a] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_list_a @ B3 @ ( image_nat_list_a @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_82_rev__image__eqI,axiom,
    ! [X: list_a,A5: set_list_a,B3: list_a,F: list_a > list_a] :
      ( ( member_list_a @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_list_a @ B3 @ ( image_list_a_list_a @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_83_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: product_prod_nat_nat,F: nat > product_prod_nat_nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_84_rev__image__eqI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B3: nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_85_rev__image__eqI,axiom,
    ! [X: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B3: nat,F: list_Sum_sum_a_nat > nat] :
      ( ( member408289922725080238_a_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_2535339886381165584at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_86_rev__image__eqI,axiom,
    ! [X: list_a,A5: set_list_a,B3: product_prod_nat_nat,F: list_a > product_prod_nat_nat] :
      ( ( member_list_a @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B3 @ ( image_3288686587095672578at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_87_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member408289922725080238_a_nat @ B3 @ ( image_6262589752765146990_a_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_88_rev__image__eqI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B3: list_a,F: product_prod_nat_nat > list_a] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_list_a @ B3 @ ( image_3765944151573559240list_a @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_89_mem__Collect__eq,axiom,
    ! [A3: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( member408289922725080238_a_nat @ A3 @ ( collec7555443234367654128_a_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
    ! [A3: list_a,P: list_a > $o] :
      ( ( member_list_a @ A3 @ ( collect_list_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( member_nat @ A3 @ ( collect_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
    ! [A3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A5: set_li6526943997496501093_a_nat] :
      ( ( collec7555443234367654128_a_nat
        @ ^ [X5: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X5 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A5: set_list_a] :
      ( ( collect_list_a
        @ ^ [X5: list_a] : ( member_list_a @ X5 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_97_ball__imageD,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ! [X4: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X4 @ ( image_5081948215111134021_a_nat @ F @ A5 ) )
         => ( P @ X4 ) )
     => ! [X6: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X6 @ A5 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_98_ball__imageD,axiom,
    ! [F: list_a > list_Sum_sum_a_nat,A5: set_list_a,P: list_Sum_sum_a_nat > $o] :
      ( ! [X4: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X4 @ ( image_674313660629153798_a_nat @ F @ A5 ) )
         => ( P @ X4 ) )
     => ! [X6: list_a] :
          ( ( member_list_a @ X6 @ A5 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_99_ball__imageD,axiom,
    ! [F: nat > nat,A5: set_nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A5 ) )
         => ( P @ X4 ) )
     => ! [X6: nat] :
          ( ( member_nat @ X6 @ A5 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_100_image__cong,axiom,
    ! [M: set_li6526943997496501093_a_nat,N3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( M = N3 )
     => ( ! [X4: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X4 @ N3 )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( image_5081948215111134021_a_nat @ F @ M )
          = ( image_5081948215111134021_a_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_101_image__cong,axiom,
    ! [M: set_list_a,N3: set_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
      ( ( M = N3 )
     => ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ N3 )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( image_674313660629153798_a_nat @ F @ M )
          = ( image_674313660629153798_a_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_102_image__cong,axiom,
    ! [M: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M = N3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ N3 )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( image_nat_nat @ F @ M )
          = ( image_nat_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_103_bex__imageD,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ? [X6: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X6 @ ( image_5081948215111134021_a_nat @ F @ A5 ) )
          & ( P @ X6 ) )
     => ? [X4: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X4 @ A5 )
          & ( P @ ( F @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_104_bex__imageD,axiom,
    ! [F: list_a > list_Sum_sum_a_nat,A5: set_list_a,P: list_Sum_sum_a_nat > $o] :
      ( ? [X6: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X6 @ ( image_674313660629153798_a_nat @ F @ A5 ) )
          & ( P @ X6 ) )
     => ? [X4: list_a] :
          ( ( member_list_a @ X4 @ A5 )
          & ( P @ ( F @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_105_bex__imageD,axiom,
    ! [F: nat > nat,A5: set_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( image_nat_nat @ F @ A5 ) )
          & ( P @ X6 ) )
     => ? [X4: nat] :
          ( ( member_nat @ X4 @ A5 )
          & ( P @ ( F @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_106_image__iff,axiom,
    ! [Z: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ Z @ ( image_5081948215111134021_a_nat @ F @ A5 ) )
      = ( ? [X5: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X5 @ A5 )
            & ( Z
              = ( F @ X5 ) ) ) ) ) ).

% image_iff
thf(fact_107_image__iff,axiom,
    ! [Z: list_Sum_sum_a_nat,F: list_a > list_Sum_sum_a_nat,A5: set_list_a] :
      ( ( member408289922725080238_a_nat @ Z @ ( image_674313660629153798_a_nat @ F @ A5 ) )
      = ( ? [X5: list_a] :
            ( ( member_list_a @ X5 @ A5 )
            & ( Z
              = ( F @ X5 ) ) ) ) ) ).

% image_iff
thf(fact_108_image__iff,axiom,
    ! [Z: nat,F: nat > nat,A5: set_nat] :
      ( ( member_nat @ Z @ ( image_nat_nat @ F @ A5 ) )
      = ( ? [X5: nat] :
            ( ( member_nat @ X5 @ A5 )
            & ( Z
              = ( F @ X5 ) ) ) ) ) ).

% image_iff
thf(fact_109_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_110_imageI,axiom,
    ! [X: list_a,A5: set_list_a,F: list_a > nat] :
      ( ( member_list_a @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_list_a_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_111_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > list_a] :
      ( ( member_nat @ X @ A5 )
     => ( member_list_a @ ( F @ X ) @ ( image_nat_list_a @ F @ A5 ) ) ) ).

% imageI
thf(fact_112_imageI,axiom,
    ! [X: list_a,A5: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ X @ A5 )
     => ( member_list_a @ ( F @ X ) @ ( image_list_a_list_a @ F @ A5 ) ) ) ).

% imageI
thf(fact_113_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > product_prod_nat_nat] :
      ( ( member_nat @ X @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_114_imageI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_115_imageI,axiom,
    ! [X: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
      ( ( member408289922725080238_a_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_2535339886381165584at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_116_imageI,axiom,
    ! [X: list_a,A5: set_list_a,F: list_a > product_prod_nat_nat] :
      ( ( member_list_a @ X @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_3288686587095672578at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_117_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > list_Sum_sum_a_nat] :
      ( ( member_nat @ X @ A5 )
     => ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_6262589752765146990_a_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_118_imageI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > list_a] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( member_list_a @ ( F @ X ) @ ( image_3765944151573559240list_a @ F @ A5 ) ) ) ).

% imageI
thf(fact_119_Inl__inject,axiom,
    ! [X: a,Y: a] :
      ( ( ( sum_Inl_a_nat @ X )
        = ( sum_Inl_a_nat @ Y ) )
     => ( X = Y ) ) ).

% Inl_inject
thf(fact_120_prod__induct4,axiom,
    ! [P: produc6346555946419678479_a_nat > $o,X: produc6346555946419678479_a_nat] :
      ( ! [A: produc8208987855206359447_a_nat,B: set_a,C: nat,D: set_li6526943997496501093_a_nat] : ( P @ ( produc5405793250563226055_a_nat @ A @ ( produc1626021218680649873_a_nat @ B @ ( produc4622690550020908860_a_nat @ C @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_121_prod__induct3,axiom,
    ! [P: produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat] :
      ( ! [A: set_a,B: nat,C: set_li6526943997496501093_a_nat] : ( P @ ( produc1626021218680649873_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_122_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A: product_prod_nat_nat,B: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A @ ( product_Pair_nat_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_123_prod__induct3,axiom,
    ! [P: produc5621125494677191285_a_nat > $o,X: produc5621125494677191285_a_nat] :
      ( ! [A: produc5986856040925105290_a_nat,B: nat,C: set_li6526943997496501093_a_nat] : ( P @ ( produc5595853527594814125_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_124_prod__induct3,axiom,
    ! [P: produc6346555946419678479_a_nat > $o,X: produc6346555946419678479_a_nat] :
      ( ! [A: produc8208987855206359447_a_nat,B: set_a,C: produc5986856040925105290_a_nat] : ( P @ ( produc5405793250563226055_a_nat @ A @ ( produc1626021218680649873_a_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_125_prod__cases4,axiom,
    ! [Y: produc6346555946419678479_a_nat] :
      ~ ! [A: produc8208987855206359447_a_nat,B: set_a,C: nat,D: set_li6526943997496501093_a_nat] :
          ( Y
         != ( produc5405793250563226055_a_nat @ A @ ( produc1626021218680649873_a_nat @ B @ ( produc4622690550020908860_a_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_126_prod__cases3,axiom,
    ! [Y: produc8208987855206359447_a_nat] :
      ~ ! [A: set_a,B: nat,C: set_li6526943997496501093_a_nat] :
          ( Y
         != ( produc1626021218680649873_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_127_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A: product_prod_nat_nat,B: nat,C: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A @ ( product_Pair_nat_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_128_prod__cases3,axiom,
    ! [Y: produc5621125494677191285_a_nat] :
      ~ ! [A: produc5986856040925105290_a_nat,B: nat,C: set_li6526943997496501093_a_nat] :
          ( Y
         != ( produc5595853527594814125_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_129_prod__cases3,axiom,
    ! [Y: produc6346555946419678479_a_nat] :
      ~ ! [A: produc8208987855206359447_a_nat,B: set_a,C: produc5986856040925105290_a_nat] :
          ( Y
         != ( produc5405793250563226055_a_nat @ A @ ( produc1626021218680649873_a_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_130_length__map,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_131_length__map,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( size_s5283204784079214577_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_132_length__map,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_map
thf(fact_133_length__map,axiom,
    ! [F: nat > sum_sum_a_nat,Xs: list_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_134_length__map,axiom,
    ! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_map
thf(fact_135_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_136_finite__imageI,axiom,
    ! [F2: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( finite1487985464145237934_a_nat @ F2 )
     => ( finite1487985464145237934_a_nat @ ( image_5081948215111134021_a_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_137_finite__imageI,axiom,
    ! [F2: set_list_a,H: list_a > list_Sum_sum_a_nat] :
      ( ( finite_finite_list_a @ F2 )
     => ( finite1487985464145237934_a_nat @ ( image_674313660629153798_a_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_138_finite__imageI,axiom,
    ! [F2: set_a,H: a > a] :
      ( ( finite_finite_a @ F2 )
     => ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_139_finite__imageI,axiom,
    ! [F2: set_a,H: a > nat] :
      ( ( finite_finite_a @ F2 )
     => ( finite_finite_nat @ ( image_a_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_140_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > a] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_a @ ( image_nat_a @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_141_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_142_map__eq__imp__length__eq,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_143_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_144_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_a_Sum_sum_a_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_145_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: nat > sum_sum_a_nat,Ys: list_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_na823391071729141993_a_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_146_map__eq__imp__length__eq,axiom,
    ! [F: sum_sum_a_nat > a,Xs: list_Sum_sum_a_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_Sum_sum_a_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_147_map__eq__imp__length__eq,axiom,
    ! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_148_map__eq__imp__length__eq,axiom,
    ! [F: nat > sum_sum_a_nat,Xs: list_nat,G: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( map_na823391071729141993_a_nat @ F @ Xs )
        = ( map_a_Sum_sum_a_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_149_map__eq__imp__length__eq,axiom,
    ! [F: nat > a,Xs: list_nat,G: sum_sum_a_nat > a,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_Sum_sum_a_nat_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_150_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Su5227373005390213643at_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_151_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_152_same__fstI,axiom,
    ! [P: nat > $o,X: nat,Y4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,R: nat > set_Pr8343503725489915207_a_nat] :
      ( ( P @ X )
     => ( ( member5874747487503750032_a_nat @ ( produc7939031333681902295_a_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y4 ) @ ( produc4622690550020908860_a_nat @ X @ Y ) ) @ ( same_f6449513938715440935_a_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_153_same__fstI,axiom,
    ! [P: nat > $o,X: nat,Y4: nat,Y: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_154_same__fstI,axiom,
    ! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Y4: product_prod_nat_nat,Y: product_prod_nat_nat,R: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( P @ X )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ X @ Y4 ) @ ( produc6161850002892822231at_nat @ X @ Y ) ) @ ( same_f4956014544515070124at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_155_same__fstI,axiom,
    ! [P: produc5986856040925105290_a_nat > $o,X: produc5986856040925105290_a_nat,Y4: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,R: produc5986856040925105290_a_nat > set_Pr3551440518166979541_a_nat] :
      ( ( P @ X )
     => ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member2682851360909428368_a_nat @ ( produc8287655239752403415_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y4 ) @ ( produc5595853527594814125_a_nat @ X @ Y ) ) @ ( same_f6262323589219774978_a_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_156_same__fstI,axiom,
    ! [P: set_a > $o,X: set_a,Y4: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,R: set_a > set_Pr3551440518166979541_a_nat] :
      ( ( P @ X )
     => ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y4 ) @ ( produc1626021218680649873_a_nat @ X @ Y ) ) @ ( same_f7968148086661921148_a_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_157_same__fstI,axiom,
    ! [P: produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat,Y4: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat,R: produc8208987855206359447_a_nat > set_Pr757995773253584751_a_nat] :
      ( ( P @ X )
     => ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ Y4 @ Y ) @ ( R @ X ) )
       => ( member7641717237037081872_a_nat @ ( produc7779878758203082071_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y4 ) @ ( produc5405793250563226055_a_nat @ X @ Y ) ) @ ( same_f8229844822800482076_a_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_158_nall__tuples__rec__fo__nmlz,axiom,
    ! [Xs: list_Sum_sum_a_nat,AD2: set_a] :
      ( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD2 @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) ) )
      = ( ( fo_nmlz_a @ AD2 @ Xs )
        = Xs ) ) ).

% nall_tuples_rec_fo_nmlz
thf(fact_159_divides__aux__eq,axiom,
    ! [Q: nat,R2: nat] :
      ( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_160_nall__tuples__rec__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,AD2: set_a,I2: nat,N2: nat] :
      ( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD2 @ I2 @ N2 ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = N2 ) ) ).

% nall_tuples_rec_length
thf(fact_161_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_162_neq__if__length__neq,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
       != ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_163_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_164_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_165_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_166_nats__def,axiom,
    ( nats
    = ( ^ [Ns3: list_nat] :
          ( Ns3
          = ( upt @ zero_zero_nat @ ( size_size_list_nat @ Ns3 ) ) ) ) ) ).

% nats_def
thf(fact_167_fo__fin,axiom,
    ! [Phi: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a,T: produc8208987855206359447_a_nat] :
      ( ( fo_wf_a_b @ Phi @ I3 @ T )
     => ( ( fo_fin_a @ T )
        = ( finite_finite_list_a @ ( fo_rep_a_nat @ T ) ) ) ) ).

% fo_fin
thf(fact_168_fo__wf__eval__table,axiom,
    ! [Phi: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a,AD2: set_a,N2: nat,X3: set_li6526943997496501093_a_nat] :
      ( ( fo_wf_a_b @ Phi @ I3 @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N2 @ X3 ) ) )
     => ( X3
        = ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 ) @ ( eval_table_a_nat @ ( map_nat_fo_term_a @ fo_Var_a @ ( upt @ zero_zero_nat @ N2 ) ) @ X3 ) ) ) ) ).

% fo_wf_eval_table
thf(fact_169_length__code,axiom,
    ( size_s5283204784079214577_a_nat
    = ( gen_le1340941697924381074_a_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_170_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_171_fo__rep__fin,axiom,
    ! [Phi: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a,AD2: set_a,N2: nat,X3: set_li6526943997496501093_a_nat] :
      ( ( fo_wf_a_b @ Phi @ I3 @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N2 @ X3 ) ) )
     => ( ( fo_fin_a @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N2 @ X3 ) ) )
       => ( ( fo_rep_a_nat @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N2 @ X3 ) ) )
          = ( image_6862639887416039236list_a @ ( map_Sum_sum_a_nat_a @ sum_projl_a_nat ) @ X3 ) ) ) ) ).

% fo_rep_fin
thf(fact_172_map__eq__map__tailrec,axiom,
    map_a_Sum_sum_a_nat = map_ta822586218839268111_a_nat ).

% map_eq_map_tailrec
thf(fact_173_map__eq__map__tailrec,axiom,
    map_nat_a = map_tailrec_nat_a ).

% map_eq_map_tailrec
thf(fact_174_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_175_lists__image,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( lists_Sum_sum_a_nat @ ( image_7142520692256960453_a_nat @ F @ A5 ) )
      = ( image_5081948215111134021_a_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ).

% lists_image
thf(fact_176_lists__image,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( lists_1938969594694982521_a_nat @ ( image_5081948215111134021_a_nat @ F @ A5 ) )
      = ( image_7904961972494007493_a_nat @ ( map_li6507455427659069316_a_nat @ F ) @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ).

% lists_image
thf(fact_177_lists__image,axiom,
    ! [F: list_a > list_Sum_sum_a_nat,A5: set_list_a] :
      ( ( lists_1938969594694982521_a_nat @ ( image_674313660629153798_a_nat @ F @ A5 ) )
      = ( image_6917023031416304646_a_nat @ ( map_li6996171348340886213_a_nat @ F ) @ ( lists_list_a @ A5 ) ) ) ).

% lists_image
thf(fact_178_lists__image,axiom,
    ! [F: a > sum_sum_a_nat,A5: set_a] :
      ( ( lists_Sum_sum_a_nat @ ( image_7873763678140191238_a_nat @ F @ A5 ) )
      = ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ F ) @ ( lists_a @ A5 ) ) ) ).

% lists_image
thf(fact_179_lists__image,axiom,
    ! [F: nat > a,A5: set_nat] :
      ( ( lists_a @ ( image_nat_a @ F @ A5 ) )
      = ( image_1070249332343121117list_a @ ( map_nat_a @ F ) @ ( lists_nat @ A5 ) ) ) ).

% lists_image
thf(fact_180_lists__image,axiom,
    ! [F: nat > nat,A5: set_nat] :
      ( ( lists_nat @ ( image_nat_nat @ F @ A5 ) )
      = ( image_7976474329151083847st_nat @ ( map_nat_nat @ F ) @ ( lists_nat @ A5 ) ) ) ).

% lists_image
thf(fact_181_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc6237156593867757212_a_nat] :
      ~ ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A: nat,B: nat,Acc: set_li6526943997496501093_a_nat] :
          ( X
         != ( produc6069846434347434318_a_nat @ F3 @ ( produc4479492211097124851_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_182_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_183_sum_Osel_I1_J,axiom,
    ! [X1: a] :
      ( ( sum_projl_a_nat @ ( sum_Inl_a_nat @ X1 ) )
      = X1 ) ).

% sum.sel(1)
thf(fact_184_nall__tuples__finite,axiom,
    ! [AD2: set_a,N2: nat] :
      ( ( finite_finite_a @ AD2 )
     => ( finite1487985464145237934_a_nat @ ( nall_tuples_a @ AD2 @ N2 ) ) ) ).

% nall_tuples_finite
thf(fact_185_nall__tuples__finite,axiom,
    ! [AD2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ AD2 )
     => ( finite5087377988160578214at_nat @ ( nall_tuples_nat @ AD2 @ N2 ) ) ) ).

% nall_tuples_finite
thf(fact_186_map__projl__Inl,axiom,
    ! [Xs: list_a] :
      ( ( map_Sum_sum_a_nat_a @ sum_projl_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Xs ) )
      = Xs ) ).

% map_projl_Inl
thf(fact_187_all__tuples__finite,axiom,
    ! [Xs: set_a,N2: nat] :
      ( ( finite_finite_a @ Xs )
     => ( finite_finite_list_a @ ( all_tuples_a @ Xs @ N2 ) ) ) ).

% all_tuples_finite
thf(fact_188_all__tuples__finite,axiom,
    ! [Xs: set_nat,N2: nat] :
      ( ( finite_finite_nat @ Xs )
     => ( finite8100373058378681591st_nat @ ( all_tuples_nat @ Xs @ N2 ) ) ) ).

% all_tuples_finite
thf(fact_189_sum_Ocollapse_I1_J,axiom,
    ! [Sum: sum_sum_a_nat] :
      ( ( sum_isl_a_nat @ Sum )
     => ( ( sum_Inl_a_nat @ ( sum_projl_a_nat @ Sum ) )
        = Sum ) ) ).

% sum.collapse(1)
thf(fact_190_eval__neg_Osimps,axiom,
    ! [Ns: list_nat,AD2: set_a,Uu2: nat,X3: set_li6526943997496501093_a_nat] :
      ( ( eval_neg_a @ Ns @ ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ Uu2 @ X3 ) ) )
      = ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ ( size_size_list_nat @ Ns ) @ ( minus_7395159227704179404_a_nat @ ( nall_tuples_a @ AD2 @ ( size_size_list_nat @ Ns ) ) @ X3 ) ) ) ) ).

% eval_neg.simps
thf(fact_191_eval__neg_Oelims,axiom,
    ! [X: list_nat,Xa: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat] :
      ( ( ( eval_neg_a @ X @ Xa )
        = Y )
     => ~ ! [AD: set_a,Uu: nat,X2: set_li6526943997496501093_a_nat] :
            ( ( Xa
              = ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X2 ) ) )
           => ( Y
             != ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ ( size_size_list_nat @ X ) @ ( minus_7395159227704179404_a_nat @ ( nall_tuples_a @ AD @ ( size_size_list_nat @ X ) ) @ X2 ) ) ) ) ) ) ).

% eval_neg.elims
thf(fact_192_fo__adom_Opelims,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: set_a] :
      ( ( ( fo_adom_a_nat @ X )
        = Y )
     => ( ( accp_P3058278626556831054_a_nat @ fo_adom_rel_a_nat @ X )
       => ~ ! [AD: set_a,N: nat,X2: set_li6526943997496501093_a_nat] :
              ( ( X
                = ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N @ X2 ) ) )
             => ( ( Y = AD )
               => ~ ( accp_P3058278626556831054_a_nat @ fo_adom_rel_a_nat @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ N @ X2 ) ) ) ) ) ) ) ).

% fo_adom.pelims
thf(fact_193_size__neq__size__imp__neq,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ X )
       != ( size_s5283204784079214577_a_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_194_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_195_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_196_DiffI,axiom,
    ! [C2: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ A5 )
     => ( ~ ( member408289922725080238_a_nat @ C2 @ B5 )
       => ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_197_DiffI,axiom,
    ! [C2: list_a,A5: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C2 @ A5 )
     => ( ~ ( member_list_a @ C2 @ B5 )
       => ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_198_DiffI,axiom,
    ! [C2: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ A5 )
     => ( ~ ( member_nat @ C2 @ B5 )
       => ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_199_DiffI,axiom,
    ! [C2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A5 )
     => ( ~ ( member8440522571783428010at_nat @ C2 @ B5 )
       => ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_200_Diff__iff,axiom,
    ! [C2: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A5 @ B5 ) )
      = ( ( member408289922725080238_a_nat @ C2 @ A5 )
        & ~ ( member408289922725080238_a_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_201_Diff__iff,axiom,
    ! [C2: list_a,A5: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A5 @ B5 ) )
      = ( ( member_list_a @ C2 @ A5 )
        & ~ ( member_list_a @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_202_Diff__iff,axiom,
    ! [C2: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
      = ( ( member_nat @ C2 @ A5 )
        & ~ ( member_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_203_Diff__iff,axiom,
    ! [C2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A5 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ C2 @ A5 )
        & ~ ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_204_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ A3 @ A3 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_205_diff__zero,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% diff_zero
thf(fact_206_zero__diff,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_207_finite__Diff2,axiom,
    ! [B5: set_a,A5: set_a] :
      ( ( finite_finite_a @ B5 )
     => ( ( finite_finite_a @ ( minus_minus_set_a @ A5 @ B5 ) )
        = ( finite_finite_a @ A5 ) ) ) ).

% finite_Diff2
thf(fact_208_finite__Diff2,axiom,
    ! [B5: set_nat,A5: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ B5 ) )
        = ( finite_finite_nat @ A5 ) ) ) ).

% finite_Diff2
thf(fact_209_finite__Diff,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( finite_finite_a @ A5 )
     => ( finite_finite_a @ ( minus_minus_set_a @ A5 @ B5 ) ) ) ).

% finite_Diff
thf(fact_210_finite__Diff,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A5 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).

% finite_Diff
thf(fact_211_DiffE,axiom,
    ! [C2: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A5 @ B5 ) )
     => ~ ( ( member408289922725080238_a_nat @ C2 @ A5 )
         => ( member408289922725080238_a_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_212_DiffE,axiom,
    ! [C2: list_a,A5: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A5 @ B5 ) )
     => ~ ( ( member_list_a @ C2 @ A5 )
         => ( member_list_a @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_213_DiffE,axiom,
    ! [C2: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
     => ~ ( ( member_nat @ C2 @ A5 )
         => ( member_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_214_DiffE,axiom,
    ! [C2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A5 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ C2 @ A5 )
         => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_215_DiffD1,axiom,
    ! [C2: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A5 @ B5 ) )
     => ( member408289922725080238_a_nat @ C2 @ A5 ) ) ).

% DiffD1
thf(fact_216_DiffD1,axiom,
    ! [C2: list_a,A5: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A5 @ B5 ) )
     => ( member_list_a @ C2 @ A5 ) ) ).

% DiffD1
thf(fact_217_DiffD1,axiom,
    ! [C2: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
     => ( member_nat @ C2 @ A5 ) ) ).

% DiffD1
thf(fact_218_DiffD1,axiom,
    ! [C2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A5 @ B5 ) )
     => ( member8440522571783428010at_nat @ C2 @ A5 ) ) ).

% DiffD1
thf(fact_219_DiffD2,axiom,
    ! [C2: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A5 @ B5 ) )
     => ~ ( member408289922725080238_a_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_220_DiffD2,axiom,
    ! [C2: list_a,A5: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A5 @ B5 ) )
     => ~ ( member_list_a @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_221_DiffD2,axiom,
    ! [C2: nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
     => ~ ( member_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_222_DiffD2,axiom,
    ! [C2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A5 @ B5 ) )
     => ~ ( member8440522571783428010at_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_223_diff__right__commute,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C2 ) @ B3 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C2 ) ) ).

% diff_right_commute
thf(fact_224_Diff__infinite__finite,axiom,
    ! [T2: set_a,S: set_a] :
      ( ( finite_finite_a @ T2 )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_225_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_226_sum_Odisc_I1_J,axiom,
    ! [X1: a] : ( sum_isl_a_nat @ ( sum_Inl_a_nat @ X1 ) ) ).

% sum.disc(1)
thf(fact_227_sum_OdiscI_I1_J,axiom,
    ! [Sum: sum_sum_a_nat,X1: a] :
      ( ( Sum
        = ( sum_Inl_a_nat @ X1 ) )
     => ( sum_isl_a_nat @ Sum ) ) ).

% sum.discI(1)
thf(fact_228_isl__def,axiom,
    ( sum_isl_a_nat
    = ( ^ [Sum2: sum_sum_a_nat] :
        ? [X12: a] :
          ( Sum2
          = ( sum_Inl_a_nat @ X12 ) ) ) ) ).

% isl_def
thf(fact_229_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_230_eval__neg_Opelims,axiom,
    ! [X: list_nat,Xa: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat] :
      ( ( ( eval_neg_a @ X @ Xa )
        = Y )
     => ( ( accp_P3318534179699624161_a_nat @ eval_neg_rel_a @ ( produc5478379421572013840_a_nat @ X @ Xa ) )
       => ~ ! [AD: set_a,Uu: nat,X2: set_li6526943997496501093_a_nat] :
              ( ( Xa
                = ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X2 ) ) )
             => ( ( Y
                  = ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ ( size_size_list_nat @ X ) @ ( minus_7395159227704179404_a_nat @ ( nall_tuples_a @ AD @ ( size_size_list_nat @ X ) ) @ X2 ) ) ) )
               => ~ ( accp_P3318534179699624161_a_nat @ eval_neg_rel_a @ ( produc5478379421572013840_a_nat @ X @ ( produc1626021218680649873_a_nat @ AD @ ( produc4622690550020908860_a_nat @ Uu @ X2 ) ) ) ) ) ) ) ) ).

% eval_neg.pelims
thf(fact_231_ad__agr__list__fv__list,axiom,
    ! [X3: set_a,Sigma: nat > sum_sum_a_nat,Ts: list_fo_term_a,Tau2: nat > sum_sum_a_nat] :
      ( ( ad_agr_list_a_nat @ X3 @ ( eval_eterms_a_nat @ Sigma @ Ts ) @ ( eval_eterms_a_nat @ Tau2 @ Ts ) )
     => ( ad_agr_list_a_nat @ X3 @ ( map_na823391071729141993_a_nat @ Sigma @ ( fv_fo_terms_list_a @ Ts ) ) @ ( map_na823391071729141993_a_nat @ Tau2 @ ( fv_fo_terms_list_a @ Ts ) ) ) ) ).

% ad_agr_list_fv_list
thf(fact_232_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_233_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_234_length__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
      = ( minus_minus_nat @ J @ I2 ) ) ).

% length_upt
thf(fact_235_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_236_diffs0__imp__equal,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M2 )
          = zero_zero_nat )
       => ( M2 = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_237_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_238_in__lex__prod,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ A4 @ B4 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_239_in__lex__prod,axiom,
    ! [A3: nat,B3: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A3 @ B3 ) @ ( produc487386426758144856at_nat @ A4 @ B4 ) ) @ ( lex_pr8029265285556086080at_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_240_in__lex__prod,axiom,
    ! [A3: product_prod_nat_nat,B3: nat,A4: product_prod_nat_nat,B4: nat,R2: set_Pr8693737435421807431at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A3 @ B3 ) @ ( produc6350711070570205562at_nat @ A4 @ B4 ) ) @ ( lex_pr4669217892513370978at_nat @ R2 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_241_in__lex__prod,axiom,
    ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ ( produc6161850002892822231at_nat @ A4 @ B4 ) ) @ ( lex_pr8801849515957261039at_nat @ R2 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_242_in__lex__prod,axiom,
    ! [A3: nat,B3: set_li6526943997496501093_a_nat,A4: nat,B4: set_li6526943997496501093_a_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr8343503725489915207_a_nat] :
      ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( produc4622690550020908860_a_nat @ A3 @ B3 ) @ ( produc4622690550020908860_a_nat @ A4 @ B4 ) ) @ ( lex_pr5038142128417811748_a_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member5874747487503750032_a_nat @ ( produc7939031333681902295_a_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_243_in__lex__prod,axiom,
    ! [A3: nat,B3: produc5986856040925105290_a_nat,A4: nat,B4: produc5986856040925105290_a_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr3551440518166979541_a_nat] :
      ( ( member3326161157614203920_a_nat @ ( produc8601807686807425623_a_nat @ ( produc4479492211097124851_a_nat @ A3 @ B3 ) @ ( produc4479492211097124851_a_nat @ A4 @ B4 ) ) @ ( lex_pr2861876452639094795_a_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_244_in__lex__prod,axiom,
    ! [A3: produc5986856040925105290_a_nat,B3: nat,A4: produc5986856040925105290_a_nat,B4: nat,R2: set_Pr3551440518166979541_a_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member4164734889767531536at_nat @ ( produc5483442371299676759at_nat @ ( produc728247398394209011at_nat @ A3 @ B3 ) @ ( produc728247398394209011at_nat @ A4 @ B4 ) ) @ ( lex_pr8334003676790954763at_nat @ R2 @ S2 ) )
      = ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_245_in__lex__prod,axiom,
    ! [A3: set_a,B3: produc5986856040925105290_a_nat,A4: set_a,B4: produc5986856040925105290_a_nat,R2: set_Pr5845495582615845127_set_a,S2: set_Pr3551440518166979541_a_nat] :
      ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( produc1626021218680649873_a_nat @ A3 @ B3 ) @ ( produc1626021218680649873_a_nat @ A4 @ B4 ) ) @ ( lex_pr3097501987248043641_a_nat @ R2 @ S2 ) )
      = ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_246_in__lex__prod,axiom,
    ! [A3: product_prod_nat_nat,B3: produc5986856040925105290_a_nat,A4: product_prod_nat_nat,B4: produc5986856040925105290_a_nat,R2: set_Pr8693737435421807431at_nat,S2: set_Pr3551440518166979541_a_nat] :
      ( ( member2549530991182098686_a_nat @ ( produc3891878052057292941_a_nat @ ( produc8375249657177185332_a_nat @ A3 @ B3 ) @ ( produc8375249657177185332_a_nat @ A4 @ B4 ) ) @ ( lex_pr114066742002689564_a_nat @ R2 @ S2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_247_in__lex__prod,axiom,
    ! [A3: produc5986856040925105290_a_nat,B3: product_prod_nat_nat,A4: produc5986856040925105290_a_nat,B4: product_prod_nat_nat,R2: set_Pr3551440518166979541_a_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( member1447830766745040226at_nat @ ( produc2777827222414826993at_nat @ ( produc3871221139725822494at_nat @ A3 @ B3 ) @ ( produc3871221139725822494at_nat @ A4 @ B4 ) ) @ ( lex_pr4833410261406102534at_nat @ R2 @ S2 ) )
      = ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ A3 @ A4 ) @ R2 )
        | ( ( A3 = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_248_sum_Oexhaust__sel,axiom,
    ! [Sum: sum_sum_a_nat] :
      ( ( Sum
       != ( sum_Inl_a_nat @ ( sum_projl_a_nat @ Sum ) ) )
     => ( Sum
        = ( sum_Inr_nat_a @ ( sum_projr_a_nat @ Sum ) ) ) ) ).

% sum.exhaust_sel
thf(fact_249_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X7 @ X8 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_250_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_251_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_252_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_253_char_Oinject,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o,Y1: $o,Y2: $o,Y32: $o,Y42: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
      ( ( ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X7 @ X8 )
        = ( char2 @ Y1 @ Y2 @ Y32 @ Y42 @ Y5 @ Y6 @ Y7 @ Y8 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 )
        & ( X32 = Y32 )
        & ( X42 = Y42 )
        & ( X52 = Y5 )
        & ( X62 = Y6 )
        & ( X7 = Y7 )
        & ( X8 = Y8 ) ) ) ).

% char.inject
thf(fact_254_Suc__diff__diff,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_255_diff__Suc__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_256_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I2: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_257_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).

% diff_commute
thf(fact_258_char_Oexhaust,axiom,
    ! [Y: char] :
      ~ ! [X13: $o,X23: $o,X33: $o,X43: $o,X53: $o,X63: $o,X72: $o,X82: $o] :
          ( Y
         != ( char2 @ X13 @ X23 @ X33 @ X43 @ X53 @ X63 @ X72 @ X82 ) ) ).

% char.exhaust
thf(fact_259_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_260_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_261_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X7 @ X8 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_262_sum_Odistinct_I1_J,axiom,
    ! [X1: a,X22: nat] :
      ( ( sum_Inl_a_nat @ X1 )
     != ( sum_Inr_nat_a @ X22 ) ) ).

% sum.distinct(1)
thf(fact_263_old_Osum_Odistinct_I2_J,axiom,
    ! [B4: nat,A3: a] :
      ( ( sum_Inr_nat_a @ B4 )
     != ( sum_Inl_a_nat @ A3 ) ) ).

% old.sum.distinct(2)
thf(fact_264_old_Osum_Odistinct_I1_J,axiom,
    ! [A3: a,B4: nat] :
      ( ( sum_Inl_a_nat @ A3 )
     != ( sum_Inr_nat_a @ B4 ) ) ).

% old.sum.distinct(1)
thf(fact_265_old_Osum_Oexhaust,axiom,
    ! [Y: sum_sum_a_nat] :
      ( ! [A: a] :
          ( Y
         != ( sum_Inl_a_nat @ A ) )
     => ~ ! [B: nat] :
            ( Y
           != ( sum_Inr_nat_a @ B ) ) ) ).

% old.sum.exhaust
thf(fact_266_sumE,axiom,
    ! [S2: sum_sum_a_nat] :
      ( ! [X4: a] :
          ( S2
         != ( sum_Inl_a_nat @ X4 ) )
     => ~ ! [Y3: nat] :
            ( S2
           != ( sum_Inr_nat_a @ Y3 ) ) ) ).

% sumE
thf(fact_267_Inr__not__Inl,axiom,
    ! [B3: nat,A3: a] :
      ( ( sum_Inr_nat_a @ B3 )
     != ( sum_Inl_a_nat @ A3 ) ) ).

% Inr_not_Inl
thf(fact_268_split__sum__ex,axiom,
    ( ( ^ [P3: sum_sum_a_nat > $o] :
        ? [X9: sum_sum_a_nat] : ( P3 @ X9 ) )
    = ( ^ [P4: sum_sum_a_nat > $o] :
          ( ? [X5: a] : ( P4 @ ( sum_Inl_a_nat @ X5 ) )
          | ? [X5: nat] : ( P4 @ ( sum_Inr_nat_a @ X5 ) ) ) ) ) ).

% split_sum_ex
thf(fact_269_split__sum__all,axiom,
    ( ( ^ [P3: sum_sum_a_nat > $o] :
        ! [X9: sum_sum_a_nat] : ( P3 @ X9 ) )
    = ( ^ [P4: sum_sum_a_nat > $o] :
          ( ! [X5: a] : ( P4 @ ( sum_Inl_a_nat @ X5 ) )
          & ! [X5: nat] : ( P4 @ ( sum_Inr_nat_a @ X5 ) ) ) ) ) ).

% split_sum_all
thf(fact_270_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_271_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_272_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_273_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_274_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_275_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N2: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M2 @ N2 ) ) ) ) ).

% diff_induct
thf(fact_276_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_277_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_278_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_279_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_280_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_281_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_282_zero__notin__Suc__image,axiom,
    ! [A5: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A5 ) ) ).

% zero_notin_Suc_image
thf(fact_283_map__Suc__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N2 ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_284_sum_Osize_I3_J,axiom,
    ! [X1: a] :
      ( ( size_s2581288809450988139_a_nat @ ( sum_Inl_a_nat @ X1 ) )
      = ( suc @ zero_zero_nat ) ) ).

% sum.size(3)
thf(fact_285_Inl__Inr__False,axiom,
    ! [X: a,Y: nat] :
      ( ( sum_Inl_a_nat @ X )
     != ( sum_Inr_nat_a @ Y ) ) ).

% Inl_Inr_False
thf(fact_286_Inr__Inl__False,axiom,
    ! [X: nat,Y: a] :
      ( ( sum_Inr_nat_a @ X )
     != ( sum_Inl_a_nat @ Y ) ) ).

% Inr_Inl_False
thf(fact_287_obj__sumE,axiom,
    ! [S2: sum_sum_a_nat] :
      ( ! [X4: a] :
          ( S2
         != ( sum_Inl_a_nat @ X4 ) )
     => ~ ! [X4: nat] :
            ( S2
           != ( sum_Inr_nat_a @ X4 ) ) ) ).

% obj_sumE
thf(fact_288_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N: nat] :
            ( ~ ( P @ N )
            & ( P @ ( suc @ N ) ) ) ) ) ).

% exists_least_lemma
thf(fact_289_ad__agr__list__def,axiom,
    ( ad_agr_list_a_nat
    = ( ^ [X10: set_a,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
          ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
            = ( size_s5283204784079214577_a_nat @ Ys2 ) )
          & ( ad_equiv_list_a_nat @ X10 @ Xs3 @ Ys2 )
          & ( sp_equiv_list_a_nat @ Xs3 @ Ys2 ) ) ) ) ).

% ad_agr_list_def
thf(fact_290_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_291_image2__eqI,axiom,
    ! [B3: nat,F: nat > nat,X: nat,C2: nat,G: nat > nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ ( bNF_Gr7318156773085685964at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_292_image2__eqI,axiom,
    ! [B3: nat,F: list_a > nat,X: list_a,C2: nat,G: list_a > nat,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_list_a @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ ( bNF_Gr6181254616915012556at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_293_image2__eqI,axiom,
    ! [B3: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C2: nat,G: product_prod_nat_nat > nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ ( bNF_Gr8954129895687480649at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_294_image2__eqI,axiom,
    ! [B3: nat,F: list_Sum_sum_a_nat > nat,X: list_Sum_sum_a_nat,C2: nat,G: list_Sum_sum_a_nat > nat,A5: set_li6526943997496501093_a_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member408289922725080238_a_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ ( bNF_Gr5329103786779832013at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_295_image2__eqI,axiom,
    ! [B3: nat,F: nat > nat,X: nat,C2: set_li6526943997496501093_a_nat,G: nat > set_li6526943997496501093_a_nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_nat @ X @ A5 )
         => ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ B3 @ C2 ) @ ( bNF_Gr9061489936062418281_a_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_296_image2__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: nat > product_prod_nat_nat,X: nat,C2: product_prod_nat_nat,G: nat > product_prod_nat_nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ C2 ) @ ( bNF_Gr7492152087006117098at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_297_image2__eqI,axiom,
    ! [B3: nat,F: list_a > nat,X: list_a,C2: set_li6526943997496501093_a_nat,G: list_a > set_li6526943997496501093_a_nat,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_list_a @ X @ A5 )
         => ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ B3 @ C2 ) @ ( bNF_Gr8898696621080283753_a_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_298_image2__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: list_a > product_prod_nat_nat,X: list_a,C2: product_prod_nat_nat,G: list_a > product_prod_nat_nat,A5: set_list_a] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member_list_a @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ C2 ) @ ( bNF_Gr9206233512786870250at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_299_image2__eqI,axiom,
    ! [B3: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C2: set_li6526943997496501093_a_nat,G: product_prod_nat_nat > set_li6526943997496501093_a_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member3071249055146688545_a_nat @ ( produc4622690550020908860_a_nat @ B3 @ C2 ) @ ( bNF_Gr6564037981669998636_a_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_300_image2__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,C2: product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ C2 ) @ ( bNF_Gr8808630244184418791at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_301_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_302_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_303_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_304_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_305_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_306_Suc__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_307_Suc__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_eq
thf(fact_308_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_309_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_310_zero__less__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% zero_less_diff
thf(fact_311_nat__neq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 != N2 )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( ord_less_nat @ N2 @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_312_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_313_less__not__refl2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( M2 != N2 ) ) ).

% less_not_refl2
thf(fact_314_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_315_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_316_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
             => ( P @ M4 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_317_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( P @ N )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
              & ~ ( P @ M4 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_318_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_319_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M5: nat] :
        ! [X5: nat] :
          ( ( member_nat @ X5 @ N4 )
         => ( ord_less_nat @ X5 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_320_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N3 )
         => ( ord_less_nat @ X4 @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_321_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M2: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_322_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_323_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_324_gr__implies__not__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_325_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_326_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_327_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P @ N )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_328_gr__implies__not0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_329_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_330_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_331_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_332_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_333_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_334_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_335_Suc__lessD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_lessD
thf(fact_336_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_337_Suc__lessI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ( suc @ M2 )
         != N2 )
       => ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_338_less__SucE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M2 @ N2 )
       => ( M2 = N2 ) ) ) ).

% less_SucE
thf(fact_339_less__SucI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_340_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N2 )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_341_less__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) ) ) ).

% less_Suc_eq
thf(fact_342_not__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_343_Nat_OAll__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N2 )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ I4 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_344_Suc__less__eq2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
      = ( ? [M6: nat] :
            ( ( M2
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N2 @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_345_less__antisym,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
       => ( M2 = N2 ) ) ) ).

% less_antisym
thf(fact_346_Suc__less__SucD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_SucD
thf(fact_347_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_348_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
       => ( ! [I: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I @ K2 ) ) ) ) )
         => ( P @ I2 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_349_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I: nat] :
            ( ( J
              = ( suc @ I ) )
           => ( P @ I ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( ( P @ ( suc @ I ) )
               => ( P @ I ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_350_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
        = ( N2 = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_351_length__induct,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ! [Xs2: list_Sum_sum_a_nat] :
          ( ! [Ys3: list_Sum_sum_a_nat] :
              ( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ys3 ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_352_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_353_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_354_diff__less__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_355_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_356_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M5: nat] :
            ( N2
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_357_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_358_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_359_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_360_diff__less,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% diff_less
thf(fact_361_diff__less__Suc,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_362_Suc__diff__Suc,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M2 @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_363_finite__maxlen,axiom,
    ! [M: set_li6526943997496501093_a_nat] :
      ( ( finite1487985464145237934_a_nat @ M )
     => ? [N: nat] :
        ! [X6: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X6 @ M )
         => ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ X6 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_364_finite__maxlen,axiom,
    ! [M: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M )
     => ? [N: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_365_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_366_rel__sum__simps_I1_J,axiom,
    ! [R1: a > a > $o,R22: nat > nat > $o,A1: a,B1: a] :
      ( ( bNF_re8134084500293877003at_nat @ R1 @ R22 @ ( sum_Inl_a_nat @ A1 ) @ ( sum_Inl_a_nat @ B1 ) )
      = ( R1 @ A1 @ B1 ) ) ).

% rel_sum_simps(1)
thf(fact_367_in__measure,axiom,
    ! [X: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_368_in__measure,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_369_in__measure,axiom,
    ! [X: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,F: produc5986856040925105290_a_nat > nat] :
      ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) @ ( measur6643369507318379970_a_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_370_in__measure,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat,F: produc8208987855206359447_a_nat > nat] :
      ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y ) @ ( measur7337299438177415887_a_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_371_rel__sum_Ocases,axiom,
    ! [R1: a > a > $o,R22: nat > nat > $o,A1: sum_sum_a_nat,A22: sum_sum_a_nat] :
      ( ( bNF_re8134084500293877003at_nat @ R1 @ R22 @ A1 @ A22 )
     => ( ! [A: a] :
            ( ( A1
              = ( sum_Inl_a_nat @ A ) )
           => ! [C: a] :
                ( ( A22
                  = ( sum_Inl_a_nat @ C ) )
               => ~ ( R1 @ A @ C ) ) )
       => ~ ! [B: nat] :
              ( ( A1
                = ( sum_Inr_nat_a @ B ) )
             => ! [D: nat] :
                  ( ( A22
                    = ( sum_Inr_nat_a @ D ) )
                 => ~ ( R22 @ B @ D ) ) ) ) ) ).

% rel_sum.cases
thf(fact_372_rel__sum_Osimps,axiom,
    ( bNF_re8134084500293877003at_nat
    = ( ^ [R12: a > a > $o,R23: nat > nat > $o,A12: sum_sum_a_nat,A23: sum_sum_a_nat] :
          ( ? [A6: a,C3: a] :
              ( ( A12
                = ( sum_Inl_a_nat @ A6 ) )
              & ( A23
                = ( sum_Inl_a_nat @ C3 ) )
              & ( R12 @ A6 @ C3 ) )
          | ? [B6: nat,D2: nat] :
              ( ( A12
                = ( sum_Inr_nat_a @ B6 ) )
              & ( A23
                = ( sum_Inr_nat_a @ D2 ) )
              & ( R23 @ B6 @ D2 ) ) ) ) ) ).

% rel_sum.simps
thf(fact_373_finite__psubset__induct,axiom,
    ! [A5: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ A5 )
     => ( ! [A7: set_a] :
            ( ( finite_finite_a @ A7 )
           => ( ! [B7: set_a] :
                  ( ( ord_less_set_a @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A5 ) ) ) ).

% finite_psubset_induct
thf(fact_374_finite__psubset__induct,axiom,
    ! [A5: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A5 )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A5 ) ) ) ).

% finite_psubset_induct
thf(fact_375_psubset__imp__ex__mem,axiom,
    ! [A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
      ( ( ord_le5291801191193052689_a_nat @ A5 @ B5 )
     => ? [B: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ B @ ( minus_7395159227704179404_a_nat @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_376_psubset__imp__ex__mem,axiom,
    ! [A5: set_list_a,B5: set_list_a] :
      ( ( ord_less_set_list_a @ A5 @ B5 )
     => ? [B: list_a] : ( member_list_a @ B @ ( minus_646659088055828811list_a @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_377_psubset__imp__ex__mem,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A5 @ B5 )
     => ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_378_psubset__imp__ex__mem,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A5 @ B5 )
     => ? [B: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B @ ( minus_1356011639430497352at_nat @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_379_rel__sum_Ointros_I1_J,axiom,
    ! [R1: a > a > $o,A3: a,C2: a,R22: nat > nat > $o] :
      ( ( R1 @ A3 @ C2 )
     => ( bNF_re8134084500293877003at_nat @ R1 @ R22 @ ( sum_Inl_a_nat @ A3 ) @ ( sum_Inl_a_nat @ C2 ) ) ) ).

% rel_sum.intros(1)
thf(fact_380_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X ) )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_381_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ zero_zero_nat ) )
           => ( P @ I4 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_382_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ X )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_383_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I5: nat] :
      ( ( ord_less_nat @ I5 @ zero_zero_nat )
     => ( P @ I5 ) ) ).

% forall_finite(1)
thf(fact_384_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_385_psubsetD,axiom,
    ! [A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
      ( ( ord_le5291801191193052689_a_nat @ A5 @ B5 )
     => ( ( member408289922725080238_a_nat @ C2 @ A5 )
       => ( member408289922725080238_a_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_386_psubsetD,axiom,
    ! [A5: set_list_a,B5: set_list_a,C2: list_a] :
      ( ( ord_less_set_list_a @ A5 @ B5 )
     => ( ( member_list_a @ C2 @ A5 )
       => ( member_list_a @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_387_psubsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A5 @ B5 )
     => ( ( member_nat @ C2 @ A5 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_388_psubsetD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A5 @ B5 )
     => ( ( member8440522571783428010at_nat @ C2 @ A5 )
       => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_389_in__finite__psubset,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A5 @ B5 ) @ finite_psubset_a )
      = ( ( ord_less_set_a @ A5 @ B5 )
        & ( finite_finite_a @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_390_in__finite__psubset,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A5 @ B5 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A5 @ B5 )
        & ( finite_finite_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_391_eval__eterm_Oelims,axiom,
    ! [X: nat > sum_sum_a_nat,Xa: fo_term_a,Y: sum_sum_a_nat] :
      ( ( ( eval_eterm_a_nat @ X @ Xa )
        = Y )
     => ( ! [C: a] :
            ( ( Xa
              = ( fo_Const_a @ C ) )
           => ( Y
             != ( sum_Inl_a_nat @ C ) ) )
       => ~ ! [N: nat] :
              ( ( Xa
                = ( fo_Var_a @ N ) )
             => ( Y
               != ( X @ N ) ) ) ) ) ).

% eval_eterm.elims
thf(fact_392_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_393_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_394_eval__eterm_Opelims,axiom,
    ! [X: nat > sum_sum_a_nat,Xa: fo_term_a,Y: sum_sum_a_nat] :
      ( ( ( eval_eterm_a_nat @ X @ Xa )
        = Y )
     => ( ( accp_P3881667681510128130term_a @ eval_eterm_rel_a_nat @ ( produc7272171377026902249term_a @ X @ Xa ) )
       => ( ! [C: a] :
              ( ( Xa
                = ( fo_Const_a @ C ) )
             => ( ( Y
                  = ( sum_Inl_a_nat @ C ) )
               => ~ ( accp_P3881667681510128130term_a @ eval_eterm_rel_a_nat @ ( produc7272171377026902249term_a @ X @ ( fo_Const_a @ C ) ) ) ) )
         => ~ ! [N: nat] :
                ( ( Xa
                  = ( fo_Var_a @ N ) )
               => ( ( Y
                    = ( X @ N ) )
                 => ~ ( accp_P3881667681510128130term_a @ eval_eterm_rel_a_nat @ ( produc7272171377026902249term_a @ X @ ( fo_Var_a @ N ) ) ) ) ) ) ) ) ).

% eval_eterm.pelims
thf(fact_395_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_396_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_397_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_398_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_399_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_400_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_401_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_402_eval__eterm_Osimps_I1_J,axiom,
    ! [Sigma: nat > sum_sum_a_nat,C2: a] :
      ( ( eval_eterm_a_nat @ Sigma @ ( fo_Const_a @ C2 ) )
      = ( sum_Inl_a_nat @ C2 ) ) ).

% eval_eterm.simps(1)
thf(fact_403_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_404_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_405_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_406_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_407_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_408_rem__nth__length,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( size_s5283204784079214577_a_nat @ ( rem_nt658808235856662061_a_nat @ I2 @ Xs ) )
        = ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ) ).

% rem_nth_length
thf(fact_409_rem__nth__length,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( size_size_list_nat @ ( rem_nth_nat @ I2 @ Xs ) )
        = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).

% rem_nth_length
thf(fact_410_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( last_nat @ ( upt @ I2 @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_411_length__butlast,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) )
      = ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_412_length__butlast,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_413_add__right__cancel,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B3 @ A3 )
        = ( plus_plus_nat @ C2 @ A3 ) )
      = ( B3 = C2 ) ) ).

% add_right_cancel
thf(fact_414_add__left__cancel,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A3 @ B3 )
        = ( plus_plus_nat @ A3 @ C2 ) )
      = ( B3 = C2 ) ) ).

% add_left_cancel
thf(fact_415_add_Oright__neutral,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% add.right_neutral
thf(fact_416_add__cancel__left__left,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ( plus_plus_nat @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_417_add__cancel__left__right,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( plus_plus_nat @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_418_add__cancel__right__left,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3
        = ( plus_plus_nat @ B3 @ A3 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_419_add__cancel__right__right,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3
        = ( plus_plus_nat @ A3 @ B3 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_420_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_421_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_422_add__0,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% add_0
thf(fact_423_add__less__cancel__right,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
      = ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_424_add__less__cancel__left,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
      = ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_425_add__diff__cancel__right_H,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = A3 ) ).

% add_diff_cancel_right'
thf(fact_426_add__diff__cancel__right,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
      = ( minus_minus_nat @ A3 @ B3 ) ) ).

% add_diff_cancel_right
thf(fact_427_add__diff__cancel__left_H,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ A3 )
      = B3 ) ).

% add_diff_cancel_left'
thf(fact_428_add__diff__cancel__left,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
      = ( minus_minus_nat @ A3 @ B3 ) ) ).

% add_diff_cancel_left
thf(fact_429_add__is__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_430_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_431_add__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc_right
thf(fact_432_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_433_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_434_add__less__same__cancel1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
      = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_435_add__less__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_436_less__add__same__cancel1,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_437_less__add__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_438_diff__add__zero,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_439_image__add__0,axiom,
    ! [S: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
      = S ) ).

% image_add_0
thf(fact_440_add__gr__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_441_add__right__imp__eq,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B3 @ A3 )
        = ( plus_plus_nat @ C2 @ A3 ) )
     => ( B3 = C2 ) ) ).

% add_right_imp_eq
thf(fact_442_add__left__imp__eq,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A3 @ B3 )
        = ( plus_plus_nat @ A3 @ C2 ) )
     => ( B3 = C2 ) ) ).

% add_left_imp_eq
thf(fact_443_add_Oleft__commute,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) )
      = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).

% add.left_commute
thf(fact_444_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).

% add.commute
thf(fact_445_add_Oassoc,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
      = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).

% add.assoc
thf(fact_446_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B3: nat,A3: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B3 ) )
     => ( ( plus_plus_nat @ A3 @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% group_cancel.add2
thf(fact_447_group__cancel_Oadd1,axiom,
    ! [A5: nat,K: nat,A3: nat,B3: nat] :
      ( ( A5
        = ( plus_plus_nat @ K @ A3 ) )
     => ( ( plus_plus_nat @ A5 @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% group_cancel.add1
thf(fact_448_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I2 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_449_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
      = ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_450_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_451_add_Ocomm__neutral,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% add.comm_neutral
thf(fact_452_add__less__imp__less__right,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
     => ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_453_add__less__imp__less__left,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
     => ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_454_add__strict__right__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_455_add__strict__left__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_456_add__strict__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat,D3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C2 @ D3 )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D3 ) ) ) ) ).

% add_strict_mono
thf(fact_457_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_458_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_459_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_460_diff__diff__eq,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C2 )
      = ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_461_add__implies__diff,axiom,
    ! [C2: nat,B3: nat,A3: nat] :
      ( ( ( plus_plus_nat @ C2 @ B3 )
        = A3 )
     => ( C2
        = ( minus_minus_nat @ A3 @ B3 ) ) ) ).

% add_implies_diff
thf(fact_462_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_463_add__eq__self__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = M2 )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_464_add__Suc__shift,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_465_add__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc
thf(fact_466_nat__arith_Osuc1,axiom,
    ! [A5: nat,K: nat,A3: nat] :
      ( ( A5
        = ( plus_plus_nat @ K @ A3 ) )
     => ( ( suc @ A5 )
        = ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).

% nat_arith.suc1
thf(fact_467_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_468_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_469_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_470_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_471_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_472_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_473_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_474_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_475_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_476_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_cancel2
thf(fact_477_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_478_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_479_map__butlast,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( map_a_Sum_sum_a_nat @ F @ ( butlast_a @ Xs ) )
      = ( butlas5768530507476509265_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_480_map__butlast,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( map_nat_a @ F @ ( butlast_nat @ Xs ) )
      = ( butlast_a @ ( map_nat_a @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_481_map__butlast,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
      = ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_482_add__neg__neg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_483_add__pos__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_484_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ~ ! [C: nat] :
            ( ( B3
              = ( plus_plus_nat @ A3 @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_485_pos__add__strict,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_486_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A3: nat,B3: nat] :
      ( ~ ( ord_less_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
        = A3 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_487_one__is__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_488_add__is__1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_489_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_490_less__natE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_491_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_492_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_493_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N6: nat] :
        ? [K3: nat] :
          ( N6
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_494_less__imp__Suc__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_495_diff__add__0,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_496_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N6: nat] : ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_497_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_498_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_499_add__diff__inverse__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_500_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_501_gen__length__def,axiom,
    ( gen_le1340941697924381074_a_nat
    = ( ^ [N6: nat,Xs3: list_Sum_sum_a_nat] : ( plus_plus_nat @ N6 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_502_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N6: nat,Xs3: list_nat] : ( plus_plus_nat @ N6 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_503_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_504_nat__diff__split,axiom,
    ! [P: nat > $o,A3: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
      = ( ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A3
              = ( plus_plus_nat @ B3 @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_505_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A3: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
      = ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A3
                  = ( plus_plus_nat @ B3 @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_506_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N6: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N6 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N6 ) ) ) ) ) ).

% add_eq_if
thf(fact_507_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_508_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr8693737435421807431at_nat,F: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im8721129524585065343at_nat @ R2 @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_509_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im2857804880773004637at_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_510_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im5896302226278515346at_nat @ R2 @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_511_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr3551440518166979541_a_nat,F: nat > produc5986856040925105290_a_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im5411372480267770414at_nat @ R2 @ F ) )
      = ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_512_in__inv__image,axiom,
    ! [X: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,R2: set_Pr1261947904930325089at_nat,F: produc5986856040925105290_a_nat > nat] :
      ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) @ ( inv_im9162617292970686254_a_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_513_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr3551440518166979541_a_nat,F: product_prod_nat_nat > produc5986856040925105290_a_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im7320898801310391587at_nat @ R2 @ F ) )
      = ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_514_in__inv__image,axiom,
    ! [X: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,R2: set_Pr8693737435421807431at_nat,F: produc5986856040925105290_a_nat > product_prod_nat_nat] :
      ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) @ ( inv_im2601555281906978617_a_nat @ R2 @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_515_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr757995773253584751_a_nat,F: nat > produc8208987855206359447_a_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im387455589061739579at_nat @ R2 @ F ) )
      = ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_516_in__inv__image,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat,R2: set_Pr1261947904930325089at_nat,F: produc8208987855206359447_a_nat > nat] :
      ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y ) @ ( inv_im7240894327026595899_a_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_517_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A1: nat,A22: nat,A32: set_li6526943997496501093_a_nat,P: ( nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > nat > nat > set_li6526943997496501093_a_nat > $o] :
      ( ( accp_P5385810815420935763_a_nat @ set_fo7796887916366853692_a_nat @ ( produc6069846434347434318_a_nat @ A0 @ ( produc4479492211097124851_a_nat @ A1 @ ( produc4622690550020908860_a_nat @ A22 @ A32 ) ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A: nat,B: nat,Acc: set_li6526943997496501093_a_nat] :
            ( ( accp_P5385810815420935763_a_nat @ set_fo7796887916366853692_a_nat @ ( produc6069846434347434318_a_nat @ F3 @ ( produc4479492211097124851_a_nat @ A @ ( produc4622690550020908860_a_nat @ B @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B @ A )
               => ( P @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc ) ) )
             => ( P @ F3 @ A @ B @ Acc ) ) )
       => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_518_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > nat > nat,A1: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A1 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
     => ( ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
            ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B @ A )
               => ( P @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc ) ) )
             => ( P @ F3 @ A @ B @ Acc ) ) )
       => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_519_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A3: nat,B3: nat,Acc2: set_li6526943997496501093_a_nat] :
      ( ( accp_P5385810815420935763_a_nat @ set_fo7796887916366853692_a_nat @ ( produc6069846434347434318_a_nat @ F @ ( produc4479492211097124851_a_nat @ A3 @ ( produc4622690550020908860_a_nat @ B3 @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B3 @ A3 )
         => ( ( set_fo8878618562132755411_a_nat @ F @ A3 @ B3 @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B3 @ A3 )
         => ( ( set_fo8878618562132755411_a_nat @ F @ A3 @ B3 @ Acc2 )
            = ( set_fo8878618562132755411_a_nat @ F @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_520_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B3 @ A3 )
         => ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B3 @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B3 @ A3 )
         => ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B3 @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_521_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: nat,Xb: nat,Xc: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
      ( ( ( set_fo8878618562132755411_a_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P5385810815420935763_a_nat @ set_fo7796887916366853692_a_nat @ ( produc6069846434347434318_a_nat @ X @ ( produc4479492211097124851_a_nat @ Xa @ ( produc4622690550020908860_a_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo8878618562132755411_a_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P5385810815420935763_a_nat @ set_fo7796887916366853692_a_nat @ ( produc6069846434347434318_a_nat @ X @ ( produc4479492211097124851_a_nat @ Xa @ ( produc4622690550020908860_a_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_522_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_523_sum_Osize__gen_I1_J,axiom,
    ! [Xa: a > nat,X: nat > nat,X1: a] :
      ( ( basic_7520187345070478634_a_nat @ Xa @ X @ ( sum_Inl_a_nat @ X1 ) )
      = ( plus_plus_nat @ ( Xa @ X1 ) @ ( suc @ zero_zero_nat ) ) ) ).

% sum.size_gen(1)
thf(fact_524_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A3: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( P @ A @ B )
          = ( P @ B @ A ) )
     => ( ! [A: nat] : ( P @ A @ zero_zero_nat )
       => ( ! [A: nat,B: nat] :
              ( ( P @ A @ B )
             => ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
         => ( P @ A3 @ B3 ) ) ) ) ).

% Euclid_induct
thf(fact_525_verit__sum__simplify,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% verit_sum_simplify
thf(fact_526_add__0__iff,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3
        = ( plus_plus_nat @ B3 @ A3 ) )
      = ( A3 = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_527_map__upt__eqI,axiom,
    ! [Xs: list_a,N2: nat,M2: nat,F: nat > a] :
      ( ( ( size_size_list_a @ Xs )
        = ( minus_minus_nat @ N2 @ M2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
           => ( ( nth_a @ Xs @ I )
              = ( F @ ( plus_plus_nat @ M2 @ I ) ) ) )
       => ( ( map_nat_a @ F @ ( upt @ M2 @ N2 ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_528_map__upt__eqI,axiom,
    ! [Xs: list_Sum_sum_a_nat,N2: nat,M2: nat,F: nat > sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( minus_minus_nat @ N2 @ M2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
           => ( ( nth_Sum_sum_a_nat @ Xs @ I )
              = ( F @ ( plus_plus_nat @ M2 @ I ) ) ) )
       => ( ( map_na823391071729141993_a_nat @ F @ ( upt @ M2 @ N2 ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_529_map__upt__eqI,axiom,
    ! [Xs: list_nat,N2: nat,M2: nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( minus_minus_nat @ N2 @ M2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I )
              = ( F @ ( plus_plus_nat @ M2 @ I ) ) ) )
       => ( ( map_nat_nat @ F @ ( upt @ M2 @ N2 ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_530_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_531_nth__map,axiom,
    ! [N2: nat,Xs: list_a,F: a > sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
     => ( ( nth_Sum_sum_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) @ N2 )
        = ( F @ ( nth_a @ Xs @ N2 ) ) ) ) ).

% nth_map
thf(fact_532_nth__map,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( nth_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) @ N2 )
        = ( F @ ( nth_Sum_sum_a_nat @ Xs @ N2 ) ) ) ) ).

% nth_map
thf(fact_533_nth__map,axiom,
    ! [N2: nat,Xs: list_nat,F: nat > a] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_a @ ( map_nat_a @ F @ Xs ) @ N2 )
        = ( F @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% nth_map
thf(fact_534_nth__map,axiom,
    ! [N2: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N2 )
        = ( F @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% nth_map
thf(fact_535_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y9: list_Sum_sum_a_nat,Z2: list_Sum_sum_a_nat] : ( Y9 = Z2 ) )
    = ( ^ [Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
          ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
            = ( size_s5283204784079214577_a_nat @ Ys2 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
             => ( ( nth_Sum_sum_a_nat @ Xs3 @ I4 )
                = ( nth_Sum_sum_a_nat @ Ys2 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_536_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y9: list_nat,Z2: list_nat] : ( Y9 = Z2 ) )
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I4 )
                = ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_537_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > sum_sum_a_nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X10: sum_sum_a_nat] : ( P @ I4 @ X10 ) ) )
      = ( ? [Xs3: list_Sum_sum_a_nat] :
            ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_Sum_sum_a_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_538_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X10: nat] : ( P @ I4 @ X10 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_539_nth__equalityI,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
           => ( ( nth_Sum_sum_a_nat @ Xs @ I )
              = ( nth_Sum_sum_a_nat @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_540_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_541_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_542_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_543_map__equality__iff,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_a_Sum_sum_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_544_map__equality__iff,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: nat > sum_sum_a_nat,Ys: list_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_na823391071729141993_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_545_map__equality__iff,axiom,
    ! [F: sum_sum_a_nat > a,Xs: list_Sum_sum_a_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_Sum_sum_a_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_s5283204784079214577_a_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Sum_sum_a_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_546_map__equality__iff,axiom,
    ! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s5283204784079214577_a_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Sum_sum_a_nat @ Xs @ I4 ) )
              = ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_547_map__equality__iff,axiom,
    ! [F: nat > sum_sum_a_nat,Xs: list_nat,G: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( map_na823391071729141993_a_nat @ F @ Xs )
        = ( map_a_Sum_sum_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_548_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: sum_sum_a_nat > a,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_Sum_sum_a_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_Sum_sum_a_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_549_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Su5227373005390213643at_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I4 ) )
              = ( G @ ( nth_Sum_sum_a_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_550_map__equality__iff,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I4 ) )
              = ( G @ ( nth_Sum_sum_a_nat @ Ys @ I4 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_551_nth__butlast,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) ) )
     => ( ( nth_Sum_sum_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) @ N2 )
        = ( nth_Sum_sum_a_nat @ Xs @ N2 ) ) ) ).

% nth_butlast
thf(fact_552_nth__butlast,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
     => ( ( nth_nat @ ( butlast_nat @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ N2 ) ) ) ).

% nth_butlast
thf(fact_553_nth__map__upt,axiom,
    ! [I2: nat,N2: nat,M2: nat,F: nat > a] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N2 @ M2 ) )
     => ( ( nth_a @ ( map_nat_a @ F @ ( upt @ M2 @ N2 ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_554_nth__map__upt,axiom,
    ! [I2: nat,N2: nat,M2: nat,F: nat > nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N2 @ M2 ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N2 ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_555_eval__eq,axiom,
    ! [T: fo_term_a,T3: fo_term_a,I3: product_prod_b_nat > set_list_a] : ( fo_wf_a_b @ ( fo_Eqa_a_b @ T @ T3 ) @ I3 @ ( eval_eq_a @ T @ T3 ) ) ).

% eval_eq
thf(fact_556_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_s602657755630655083_a_nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s5225532995356599511_a_nat @ Xs ) )
     => ( ( nth_Pr4499681824854399811_a_nat @ ( enumer2400619181270887242_a_nat @ N2 @ Xs ) @ M2 )
        = ( produc4622690550020908860_a_nat @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_se3280133865147330540_a_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_557_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_Sum_sum_a_nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( nth_Pr4195520319383970909_a_nat @ ( enumer3164015132978342500_a_nat @ N2 @ Xs ) @ M2 )
        = ( produc3265382261054541654_a_nat @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_Sum_sum_a_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_558_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_559_add__nth__rem__nth__self,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( add_nt4212672348507122516_a_nat @ I2 @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( rem_nt658808235856662061_a_nat @ I2 @ Xs ) )
        = Xs ) ) ).

% add_nth_rem_nth_self
thf(fact_560_add__nth__rem__nth__self,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( add_nth_nat @ I2 @ ( nth_nat @ Xs @ I2 ) @ ( rem_nth_nat @ I2 @ Xs ) )
        = Xs ) ) ).

% add_nth_rem_nth_self
thf(fact_561_length__enumerate,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s4076174644546656840_a_nat @ ( enumer3164015132978342500_a_nat @ N2 @ Xs ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_enumerate
thf(fact_562_length__enumerate,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_563_nth__Cons__pos,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_564_nth__zip,axiom,
    ! [I2: nat,Xs: list_set_a,Ys: list_P4848832406072927130_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3272666785928748590_a_nat @ Ys ) )
       => ( ( nth_Pr7304172183617465168_a_nat @ ( zip_se7211208273872771441_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc1626021218680649873_a_nat @ ( nth_set_a @ Xs @ I2 ) @ ( nth_Pr4499681824854399811_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_565_nth__zip,axiom,
    ! [I2: nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I2 )
          = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_566_nth__zip,axiom,
    ! [I2: nat,Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s3272666785928748590_a_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3272666785928748590_a_nat @ Ys ) )
       => ( ( nth_Pr4836238651267048316_a_nat @ ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc5595853527594814125_a_nat @ ( nth_Pr4499681824854399811_a_nat @ Xs @ I2 ) @ ( nth_Pr4499681824854399811_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_567_nth__zip,axiom,
    ! [I2: nat,Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s3428558496044824507_a_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3428558496044824507_a_nat @ Ys ) )
       => ( ( nth_Pr6500259550886273046_a_nat @ ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc5405793250563226055_a_nat @ ( nth_Pr7304172183617465168_a_nat @ Xs @ I2 ) @ ( nth_Pr7304172183617465168_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_568_nth__zip,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Ys ) )
       => ( ( nth_Pr7458973636520993902_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_569_nth__zip,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr5999248435048837175at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) @ I2 )
          = ( produc7669364194715613304at_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_570_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_s602657755630655083_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s5225532995356599511_a_nat @ Ys ) )
       => ( ( nth_Pr4499681824854399811_a_nat @ ( zip_na7417237922546100252_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc4622690550020908860_a_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_se3280133865147330540_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_571_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Ys ) )
       => ( ( nth_Pr4195520319383970909_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) @ I2 )
          = ( produc3265382261054541654_a_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_572_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I2 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_573_last__conv__nth,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( last_Sum_sum_a_nat @ Xs )
        = ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_574_last__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ Xs )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_575_map__fst__enumerate,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( enumer3164015132978342500_a_nat @ N2 @ Xs ) )
      = ( upt @ N2 @ ( plus_plus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ).

% map_fst_enumerate
thf(fact_576_map__fst__enumerate,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( enumerate_nat @ N2 @ Xs ) )
      = ( upt @ N2 @ ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ).

% map_fst_enumerate
thf(fact_577_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_578_fo__nmlz__Nil,axiom,
    ! [AD2: set_a] :
      ( ( fo_nmlz_a @ AD2 @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% fo_nmlz_Nil
thf(fact_579_map__is__Nil__conv,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = nil_Sum_sum_a_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_580_map__is__Nil__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_581_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_582_Nil__is__map__conv,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( nil_Sum_sum_a_nat
        = ( map_a_Sum_sum_a_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_583_Nil__is__map__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( nil_a
        = ( map_nat_a @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_584_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_585_list_Omap__disc__iff,axiom,
    ! [F: a > sum_sum_a_nat,A3: list_a] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ A3 )
        = nil_Sum_sum_a_nat )
      = ( A3 = nil_a ) ) ).

% list.map_disc_iff
thf(fact_586_list_Omap__disc__iff,axiom,
    ! [F: nat > a,A3: list_nat] :
      ( ( ( map_nat_a @ F @ A3 )
        = nil_a )
      = ( A3 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_587_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A3: list_nat] :
      ( ( ( map_nat_nat @ F @ A3 )
        = nil_nat )
      = ( A3 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_588_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_589_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_590_Cons__in__lists__iff,axiom,
    ! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( member6656483030743143220_a_nat @ ( cons_l6604326339930385211_a_nat @ X @ Xs ) @ ( lists_1938969594694982521_a_nat @ A5 ) )
      = ( ( member408289922725080238_a_nat @ X @ A5 )
        & ( member6656483030743143220_a_nat @ Xs @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_591_Cons__in__lists__iff,axiom,
    ! [X: list_a,Xs: list_list_a,A5: set_list_a] :
      ( ( member_list_list_a @ ( cons_list_a @ X @ Xs ) @ ( lists_list_a @ A5 ) )
      = ( ( member_list_a @ X @ A5 )
        & ( member_list_list_a @ Xs @ ( lists_list_a @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_592_Cons__in__lists__iff,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( lists_3066315782749251317at_nat @ A5 ) )
      = ( ( member8440522571783428010at_nat @ X @ A5 )
        & ( member3067507820990806192at_nat @ Xs @ ( lists_3066315782749251317at_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_593_Cons__in__lists__iff,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( lists_Sum_sum_a_nat @ A5 ) )
      = ( ( member_Sum_sum_a_nat @ X @ A5 )
        & ( member408289922725080238_a_nat @ Xs @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_594_Cons__in__lists__iff,axiom,
    ! [X: a,Xs: list_a,A5: set_a] :
      ( ( member_list_a @ ( cons_a @ X @ Xs ) @ ( lists_a @ A5 ) )
      = ( ( member_a @ X @ A5 )
        & ( member_list_a @ Xs @ ( lists_a @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_595_Cons__in__lists__iff,axiom,
    ! [X: nat,Xs: list_nat,A5: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X @ Xs ) @ ( lists_nat @ A5 ) )
      = ( ( member_nat @ X @ A5 )
        & ( member_list_nat @ Xs @ ( lists_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_596_enumerate__simps_I1_J,axiom,
    ! [N2: nat] :
      ( ( enumerate_nat @ N2 @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_597_length__0__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% length_0_conv
thf(fact_598_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_599_zip__Cons__Cons,axiom,
    ! [X: set_a,Xs: list_set_a,Y: produc5986856040925105290_a_nat,Ys: list_P4848832406072927130_a_nat] :
      ( ( zip_se7211208273872771441_a_nat @ ( cons_set_a @ X @ Xs ) @ ( cons_P681071152627790164_a_nat @ Y @ Ys ) )
      = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y ) @ ( zip_se7211208273872771441_a_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_600_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: set_li6526943997496501093_a_nat,Ys: list_s602657755630655083_a_nat] :
      ( ( zip_na7417237922546100252_a_nat @ ( cons_nat @ X @ Xs ) @ ( cons_s2546857569980105499_a_nat @ Y @ Ys ) )
      = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y ) @ ( zip_na7417237922546100252_a_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_601_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_602_zip__Cons__Cons,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_603_zip__Cons__Cons,axiom,
    ! [X: produc5986856040925105290_a_nat,Xs: list_P4848832406072927130_a_nat,Y: produc5986856040925105290_a_nat,Ys: list_P4848832406072927130_a_nat] :
      ( ( zip_Pr1216249236586582477_a_nat @ ( cons_P681071152627790164_a_nat @ X @ Xs ) @ ( cons_P681071152627790164_a_nat @ Y @ Ys ) )
      = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) @ ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_604_zip__Cons__Cons,axiom,
    ! [X: produc8208987855206359447_a_nat,Xs: list_P2897000184160895271_a_nat,Y: produc8208987855206359447_a_nat,Ys: list_P2897000184160895271_a_nat] :
      ( ( zip_Pr8897620143892512999_a_nat @ ( cons_P4489214174023894113_a_nat @ X @ Xs ) @ ( cons_P4489214174023894113_a_nat @ Y @ Ys ) )
      = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y ) @ ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_605_length__nth__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% length_nth_simps(3)
thf(fact_606_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_607_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N2: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N2 ) )
      = ( nth_nat @ Xs @ N2 ) ) ).

% nth_Cons_Suc
thf(fact_608_length__greater__0__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) )
      = ( Xs != nil_Sum_sum_a_nat ) ) ).

% length_greater_0_conv
thf(fact_609_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_610_map__fst__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_611_map__fst__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_612_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_613_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_614_enumerate__simps_I2_J,axiom,
    ! [N2: nat,X: set_li6526943997496501093_a_nat,Xs: list_s602657755630655083_a_nat] :
      ( ( enumer2400619181270887242_a_nat @ N2 @ ( cons_s2546857569980105499_a_nat @ X @ Xs ) )
      = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ N2 @ X ) @ ( enumer2400619181270887242_a_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_615_enumerate__simps_I2_J,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N2 @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N2 @ X ) @ ( enumerate_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_616_lists_ONil,axiom,
    ! [A5: set_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ nil_Sum_sum_a_nat @ ( lists_Sum_sum_a_nat @ A5 ) ) ).

% lists.Nil
thf(fact_617_lists_ONil,axiom,
    ! [A5: set_a] : ( member_list_a @ nil_a @ ( lists_a @ A5 ) ) ).

% lists.Nil
thf(fact_618_lists_ONil,axiom,
    ! [A5: set_nat] : ( member_list_nat @ nil_nat @ ( lists_nat @ A5 ) ) ).

% lists.Nil
thf(fact_619_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_620_eq__fst__iff,axiom,
    ! [A3: set_a,P2: produc8208987855206359447_a_nat] :
      ( ( A3
        = ( produc4817319330785305509_a_nat @ P2 ) )
      = ( ? [B6: produc5986856040925105290_a_nat] :
            ( P2
            = ( produc1626021218680649873_a_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_621_eq__fst__iff,axiom,
    ! [A3: nat,P2: produc5986856040925105290_a_nat] :
      ( ( A3
        = ( produc8331164115582522960_a_nat @ P2 ) )
      = ( ? [B6: set_li6526943997496501093_a_nat] :
            ( P2
            = ( produc4622690550020908860_a_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_622_eq__fst__iff,axiom,
    ! [A3: nat,P2: product_prod_nat_nat] :
      ( ( A3
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B6: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_623_eq__fst__iff,axiom,
    ! [A3: product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( A3
        = ( produc3213797794245857475at_nat @ P2 ) )
      = ( ? [B6: product_prod_nat_nat] :
            ( P2
            = ( produc6161850002892822231at_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_624_eq__fst__iff,axiom,
    ! [A3: produc5986856040925105290_a_nat,P2: produc5621125494677191285_a_nat] :
      ( ( A3
        = ( produc7907315211926687385_a_nat @ P2 ) )
      = ( ? [B6: produc5986856040925105290_a_nat] :
            ( P2
            = ( produc5595853527594814125_a_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_625_eq__fst__iff,axiom,
    ! [A3: produc8208987855206359447_a_nat,P2: produc6346555946419678479_a_nat] :
      ( ( A3
        = ( produc7126340784653108659_a_nat @ P2 ) )
      = ( ? [B6: produc8208987855206359447_a_nat] :
            ( P2
            = ( produc5405793250563226055_a_nat @ A3 @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_626_fst__eqD,axiom,
    ! [X: set_a,Y: produc5986856040925105290_a_nat,A3: set_a] :
      ( ( ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_627_fst__eqD,axiom,
    ! [X: nat,Y: set_li6526943997496501093_a_nat,A3: nat] :
      ( ( ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_628_fst__eqD,axiom,
    ! [X: nat,Y: nat,A3: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_629_fst__eqD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A3: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_630_fst__eqD,axiom,
    ! [X: produc5986856040925105290_a_nat,Y: produc5986856040925105290_a_nat,A3: produc5986856040925105290_a_nat] :
      ( ( ( produc7907315211926687385_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_631_fst__eqD,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: produc8208987855206359447_a_nat,A3: produc8208987855206359447_a_nat] :
      ( ( ( produc7126340784653108659_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y ) )
        = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
thf(fact_632_fst__conv,axiom,
    ! [X1: set_a,X22: produc5986856040925105290_a_nat] :
      ( ( produc4817319330785305509_a_nat @ ( produc1626021218680649873_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_633_fst__conv,axiom,
    ! [X1: nat,X22: set_li6526943997496501093_a_nat] :
      ( ( produc8331164115582522960_a_nat @ ( produc4622690550020908860_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_634_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_635_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_636_fst__conv,axiom,
    ! [X1: produc5986856040925105290_a_nat,X22: produc5986856040925105290_a_nat] :
      ( ( produc7907315211926687385_a_nat @ ( produc5595853527594814125_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_637_fst__conv,axiom,
    ! [X1: produc8208987855206359447_a_nat,X22: produc8208987855206359447_a_nat] :
      ( ( produc7126340784653108659_a_nat @ ( produc5405793250563226055_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_638_fstI,axiom,
    ! [X: produc8208987855206359447_a_nat,Y: set_a,Z: produc5986856040925105290_a_nat] :
      ( ( X
        = ( produc1626021218680649873_a_nat @ Y @ Z ) )
     => ( ( produc4817319330785305509_a_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_639_fstI,axiom,
    ! [X: produc5986856040925105290_a_nat,Y: nat,Z: set_li6526943997496501093_a_nat] :
      ( ( X
        = ( produc4622690550020908860_a_nat @ Y @ Z ) )
     => ( ( produc8331164115582522960_a_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_640_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z ) )
     => ( ( product_fst_nat_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_641_fstI,axiom,
    ! [X: produc859450856879609959at_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( X
        = ( produc6161850002892822231at_nat @ Y @ Z ) )
     => ( ( produc3213797794245857475at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_642_fstI,axiom,
    ! [X: produc5621125494677191285_a_nat,Y: produc5986856040925105290_a_nat,Z: produc5986856040925105290_a_nat] :
      ( ( X
        = ( produc5595853527594814125_a_nat @ Y @ Z ) )
     => ( ( produc7907315211926687385_a_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_643_fstI,axiom,
    ! [X: produc6346555946419678479_a_nat,Y: produc8208987855206359447_a_nat,Z: produc8208987855206359447_a_nat] :
      ( ( X
        = ( produc5405793250563226055_a_nat @ Y @ Z ) )
     => ( ( produc7126340784653108659_a_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_644_lists_OCons,axiom,
    ! [A3: sum_sum_a_nat,A5: set_Sum_sum_a_nat,L: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ A3 @ A5 )
     => ( ( member408289922725080238_a_nat @ L @ ( lists_Sum_sum_a_nat @ A5 ) )
       => ( member408289922725080238_a_nat @ ( cons_Sum_sum_a_nat @ A3 @ L ) @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_645_lists_OCons,axiom,
    ! [A3: a,A5: set_a,L: list_a] :
      ( ( member_a @ A3 @ A5 )
     => ( ( member_list_a @ L @ ( lists_a @ A5 ) )
       => ( member_list_a @ ( cons_a @ A3 @ L ) @ ( lists_a @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_646_lists_OCons,axiom,
    ! [A3: list_Sum_sum_a_nat,A5: set_li6526943997496501093_a_nat,L: list_l4703314356710769291_a_nat] :
      ( ( member408289922725080238_a_nat @ A3 @ A5 )
     => ( ( member6656483030743143220_a_nat @ L @ ( lists_1938969594694982521_a_nat @ A5 ) )
       => ( member6656483030743143220_a_nat @ ( cons_l6604326339930385211_a_nat @ A3 @ L ) @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_647_lists_OCons,axiom,
    ! [A3: list_a,A5: set_list_a,L: list_list_a] :
      ( ( member_list_a @ A3 @ A5 )
     => ( ( member_list_list_a @ L @ ( lists_list_a @ A5 ) )
       => ( member_list_list_a @ ( cons_list_a @ A3 @ L ) @ ( lists_list_a @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_648_lists_OCons,axiom,
    ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,L: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ A3 @ A5 )
     => ( ( member3067507820990806192at_nat @ L @ ( lists_3066315782749251317at_nat @ A5 ) )
       => ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ A3 @ L ) @ ( lists_3066315782749251317at_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_649_lists_OCons,axiom,
    ! [A3: nat,A5: set_nat,L: list_nat] :
      ( ( member_nat @ A3 @ A5 )
     => ( ( member_list_nat @ L @ ( lists_nat @ A5 ) )
       => ( member_list_nat @ ( cons_nat @ A3 @ L ) @ ( lists_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_650_listsE,axiom,
    ! [X: list_Sum_sum_a_nat,L: list_l4703314356710769291_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( member6656483030743143220_a_nat @ ( cons_l6604326339930385211_a_nat @ X @ L ) @ ( lists_1938969594694982521_a_nat @ A5 ) )
     => ~ ( ( member408289922725080238_a_nat @ X @ A5 )
         => ~ ( member6656483030743143220_a_nat @ L @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_651_listsE,axiom,
    ! [X: list_a,L: list_list_a,A5: set_list_a] :
      ( ( member_list_list_a @ ( cons_list_a @ X @ L ) @ ( lists_list_a @ A5 ) )
     => ~ ( ( member_list_a @ X @ A5 )
         => ~ ( member_list_list_a @ L @ ( lists_list_a @ A5 ) ) ) ) ).

% listsE
thf(fact_652_listsE,axiom,
    ! [X: product_prod_nat_nat,L: list_P6011104703257516679at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ X @ L ) @ ( lists_3066315782749251317at_nat @ A5 ) )
     => ~ ( ( member8440522571783428010at_nat @ X @ A5 )
         => ~ ( member3067507820990806192at_nat @ L @ ( lists_3066315782749251317at_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_653_listsE,axiom,
    ! [X: sum_sum_a_nat,L: list_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ ( cons_Sum_sum_a_nat @ X @ L ) @ ( lists_Sum_sum_a_nat @ A5 ) )
     => ~ ( ( member_Sum_sum_a_nat @ X @ A5 )
         => ~ ( member408289922725080238_a_nat @ L @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_654_listsE,axiom,
    ! [X: a,L: list_a,A5: set_a] :
      ( ( member_list_a @ ( cons_a @ X @ L ) @ ( lists_a @ A5 ) )
     => ~ ( ( member_a @ X @ A5 )
         => ~ ( member_list_a @ L @ ( lists_a @ A5 ) ) ) ) ).

% listsE
thf(fact_655_listsE,axiom,
    ! [X: nat,L: list_nat,A5: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X @ L ) @ ( lists_nat @ A5 ) )
     => ~ ( ( member_nat @ X @ A5 )
         => ~ ( member_list_nat @ L @ ( lists_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_656_last_Osimps,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = ( last_nat @ Xs ) ) ) ) ).

% last.simps
thf(fact_657_last__ConsL,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_658_last__ConsR,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = ( last_nat @ Xs ) ) ) ).

% last_ConsR
thf(fact_659_list__induct2,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
       => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat] :
              ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                = ( size_s5283204784079214577_a_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_660_list__induct2,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,P: list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_Sum_sum_a_nat @ nil_nat )
       => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat] :
              ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_661_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_Sum_sum_a_nat )
       => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s5283204784079214577_a_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_662_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_663_list__induct3,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
         => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
                ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                  = ( size_s5283204784079214577_a_nat @ Ys4 ) )
               => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                    = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_664_list__induct3,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat )
         => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                  = ( size_s5283204784079214577_a_nat @ Ys4 ) )
               => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_665_list__induct3,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat )
         => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
                ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_666_list__induct3,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
         => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_667_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s5283204784079214577_a_nat @ Ys4 ) )
               => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                    = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_668_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s5283204784079214577_a_nat @ Ys4 ) )
               => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_669_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,P: list_nat > list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_670_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_671_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_672_list__induct4,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_673_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5283204784079214577_a_nat @ Ys4 ) )
                 => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_674_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_nat > list_nat > list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( ( size_s5283204784079214577_a_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                   => ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_675_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_nat > list_nat > list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5283204784079214577_a_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5283204784079214577_a_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_676_list__induct4,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
           => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                    = ( size_s5283204784079214577_a_nat @ Ys4 ) )
                 => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_677_list__induct4,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( ( size_s5283204784079214577_a_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
           => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                   => ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_678_list__induct4,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5283204784079214577_a_nat @ Ws ) )
         => ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
           => ( ! [X4: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
                  ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5283204784079214577_a_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_Sum_sum_a_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_679_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs ) )
       => ( ( ( size_s5283204784079214577_a_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5283204784079214577_a_nat @ Ys4 ) )
                 => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                      = ( size_s5283204784079214577_a_nat @ Zs2 ) )
                   => ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_Sum_sum_a_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_680_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5283204784079214577_a_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Z3: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5283204784079214577_a_nat @ Ys4 ) )
                 => ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5283204784079214577_a_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_681_lists_Ocases,axiom,
    ! [A3: list_l4703314356710769291_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( member6656483030743143220_a_nat @ A3 @ ( lists_1938969594694982521_a_nat @ A5 ) )
     => ( ( A3 != nil_li1906260230833442699_a_nat )
       => ~ ! [A: list_Sum_sum_a_nat,L2: list_l4703314356710769291_a_nat] :
              ( ( A3
                = ( cons_l6604326339930385211_a_nat @ A @ L2 ) )
             => ( ( member408289922725080238_a_nat @ A @ A5 )
               => ~ ( member6656483030743143220_a_nat @ L2 @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_682_lists_Ocases,axiom,
    ! [A3: list_list_a,A5: set_list_a] :
      ( ( member_list_list_a @ A3 @ ( lists_list_a @ A5 ) )
     => ( ( A3 != nil_list_a )
       => ~ ! [A: list_a,L2: list_list_a] :
              ( ( A3
                = ( cons_list_a @ A @ L2 ) )
             => ( ( member_list_a @ A @ A5 )
               => ~ ( member_list_list_a @ L2 @ ( lists_list_a @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_683_lists_Ocases,axiom,
    ! [A3: list_P6011104703257516679at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ A3 @ ( lists_3066315782749251317at_nat @ A5 ) )
     => ( ( A3 != nil_Pr5478986624290739719at_nat )
       => ~ ! [A: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
              ( ( A3
                = ( cons_P6512896166579812791at_nat @ A @ L2 ) )
             => ( ( member8440522571783428010at_nat @ A @ A5 )
               => ~ ( member3067507820990806192at_nat @ L2 @ ( lists_3066315782749251317at_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_684_lists_Ocases,axiom,
    ! [A3: list_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ A3 @ ( lists_Sum_sum_a_nat @ A5 ) )
     => ( ( A3 != nil_Sum_sum_a_nat )
       => ~ ! [A: sum_sum_a_nat,L2: list_Sum_sum_a_nat] :
              ( ( A3
                = ( cons_Sum_sum_a_nat @ A @ L2 ) )
             => ( ( member_Sum_sum_a_nat @ A @ A5 )
               => ~ ( member408289922725080238_a_nat @ L2 @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_685_lists_Ocases,axiom,
    ! [A3: list_a,A5: set_a] :
      ( ( member_list_a @ A3 @ ( lists_a @ A5 ) )
     => ( ( A3 != nil_a )
       => ~ ! [A: a,L2: list_a] :
              ( ( A3
                = ( cons_a @ A @ L2 ) )
             => ( ( member_a @ A @ A5 )
               => ~ ( member_list_a @ L2 @ ( lists_a @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_686_lists_Ocases,axiom,
    ! [A3: list_nat,A5: set_nat] :
      ( ( member_list_nat @ A3 @ ( lists_nat @ A5 ) )
     => ( ( A3 != nil_nat )
       => ~ ! [A: nat,L2: list_nat] :
              ( ( A3
                = ( cons_nat @ A @ L2 ) )
             => ( ( member_nat @ A @ A5 )
               => ~ ( member_list_nat @ L2 @ ( lists_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_687_lists_Osimps,axiom,
    ! [A3: list_l4703314356710769291_a_nat,A5: set_li6526943997496501093_a_nat] :
      ( ( member6656483030743143220_a_nat @ A3 @ ( lists_1938969594694982521_a_nat @ A5 ) )
      = ( ( A3 = nil_li1906260230833442699_a_nat )
        | ? [A6: list_Sum_sum_a_nat,L3: list_l4703314356710769291_a_nat] :
            ( ( A3
              = ( cons_l6604326339930385211_a_nat @ A6 @ L3 ) )
            & ( member408289922725080238_a_nat @ A6 @ A5 )
            & ( member6656483030743143220_a_nat @ L3 @ ( lists_1938969594694982521_a_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_688_lists_Osimps,axiom,
    ! [A3: list_list_a,A5: set_list_a] :
      ( ( member_list_list_a @ A3 @ ( lists_list_a @ A5 ) )
      = ( ( A3 = nil_list_a )
        | ? [A6: list_a,L3: list_list_a] :
            ( ( A3
              = ( cons_list_a @ A6 @ L3 ) )
            & ( member_list_a @ A6 @ A5 )
            & ( member_list_list_a @ L3 @ ( lists_list_a @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_689_lists_Osimps,axiom,
    ! [A3: list_P6011104703257516679at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ A3 @ ( lists_3066315782749251317at_nat @ A5 ) )
      = ( ( A3 = nil_Pr5478986624290739719at_nat )
        | ? [A6: product_prod_nat_nat,L3: list_P6011104703257516679at_nat] :
            ( ( A3
              = ( cons_P6512896166579812791at_nat @ A6 @ L3 ) )
            & ( member8440522571783428010at_nat @ A6 @ A5 )
            & ( member3067507820990806192at_nat @ L3 @ ( lists_3066315782749251317at_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_690_lists_Osimps,axiom,
    ! [A3: list_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ A3 @ ( lists_Sum_sum_a_nat @ A5 ) )
      = ( ( A3 = nil_Sum_sum_a_nat )
        | ? [A6: sum_sum_a_nat,L3: list_Sum_sum_a_nat] :
            ( ( A3
              = ( cons_Sum_sum_a_nat @ A6 @ L3 ) )
            & ( member_Sum_sum_a_nat @ A6 @ A5 )
            & ( member408289922725080238_a_nat @ L3 @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_691_lists_Osimps,axiom,
    ! [A3: list_a,A5: set_a] :
      ( ( member_list_a @ A3 @ ( lists_a @ A5 ) )
      = ( ( A3 = nil_a )
        | ? [A6: a,L3: list_a] :
            ( ( A3
              = ( cons_a @ A6 @ L3 ) )
            & ( member_a @ A6 @ A5 )
            & ( member_list_a @ L3 @ ( lists_a @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_692_lists_Osimps,axiom,
    ! [A3: list_nat,A5: set_nat] :
      ( ( member_list_nat @ A3 @ ( lists_nat @ A5 ) )
      = ( ( A3 = nil_nat )
        | ? [A6: nat,L3: list_nat] :
            ( ( A3
              = ( cons_nat @ A6 @ L3 ) )
            & ( member_nat @ A6 @ A5 )
            & ( member_list_nat @ L3 @ ( lists_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_693_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_694_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_695_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X4: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X4 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_696_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y10: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y10 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_697_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X4: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_698_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X4: nat,Xs2: list_nat] :
          ( X
         != ( cons_nat @ X4 @ Xs2 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_699_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_700_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_701_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_702_rem__nth_Ocases,axiom,
    ! [X: produc4575160907756185873st_nat] :
      ( ! [Uu: nat] :
          ( X
         != ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
     => ( ! [X4: nat,Xs2: list_nat] :
            ( X
           != ( produc8282810413953273033st_nat @ zero_zero_nat @ ( cons_nat @ X4 @ Xs2 ) ) )
       => ~ ! [N: nat,X4: nat,Xs2: list_nat] :
              ( X
             != ( produc8282810413953273033st_nat @ ( suc @ N ) @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ).

% rem_nth.cases
thf(fact_703_merge_Ocases,axiom,
    ! [X: produc6461136567658903573_a_nat] :
      ( ! [Mys: list_P4848832406072927130_a_nat] :
          ( X
         != ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Mys ) )
     => ( ! [V: produc5986856040925105290_a_nat,Va: list_P4848832406072927130_a_nat] :
            ( X
           != ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ V @ Va ) @ nil_Pr6508300198058928388_a_nat ) )
       => ~ ! [N: nat,X4: set_li6526943997496501093_a_nat,Nxs: list_P4848832406072927130_a_nat,M3: nat,Y3: set_li6526943997496501093_a_nat,Mys: list_P4848832406072927130_a_nat] :
              ( X
             != ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ N @ X4 ) @ Nxs ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ M3 @ Y3 ) @ Mys ) ) ) ) ) ).

% merge.cases
thf(fact_704_merge_Ocases,axiom,
    ! [X: produc6392793444374437607at_nat] :
      ( ! [Mys: list_P6011104703257516679at_nat] :
          ( X
         != ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Mys ) )
     => ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
            ( X
           != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) @ nil_Pr5478986624290739719at_nat ) )
       => ~ ! [N: nat,X4: nat,Nxs: list_P6011104703257516679at_nat,M3: nat,Y3: nat,Mys: list_P6011104703257516679at_nat] :
              ( X
             != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X4 ) @ Nxs ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ M3 @ Y3 ) @ Mys ) ) ) ) ) ).

% merge.cases
thf(fact_705_splice_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ~ ! [X4: nat,Xs2: list_nat,Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 ) ) ) ).

% splice.cases
thf(fact_706_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ( ! [Xs2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_707_sorted__wrt_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ~ ! [P5: nat > nat > $o,X4: nat,Ys4: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_708_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X4: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X4: nat,Y3: nat,Xs2: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_709_zip__eq__ConsE,axiom,
    ! [Xs: list_set_a,Ys: list_P4848832406072927130_a_nat,Xy: produc8208987855206359447_a_nat,Xys: list_P2897000184160895271_a_nat] :
      ( ( ( zip_se7211208273872771441_a_nat @ Xs @ Ys )
        = ( cons_P4489214174023894113_a_nat @ Xy @ Xys ) )
     => ~ ! [X4: set_a,Xs4: list_set_a] :
            ( ( Xs
              = ( cons_set_a @ X4 @ Xs4 ) )
           => ! [Y3: produc5986856040925105290_a_nat,Ys5: list_P4848832406072927130_a_nat] :
                ( ( Ys
                  = ( cons_P681071152627790164_a_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( produc1626021218680649873_a_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_se7211208273872771441_a_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_710_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_s602657755630655083_a_nat,Xy: produc5986856040925105290_a_nat,Xys: list_P4848832406072927130_a_nat] :
      ( ( ( zip_na7417237922546100252_a_nat @ Xs @ Ys )
        = ( cons_P681071152627790164_a_nat @ Xy @ Xys ) )
     => ~ ! [X4: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs4 ) )
           => ! [Y3: set_li6526943997496501093_a_nat,Ys5: list_s602657755630655083_a_nat] :
                ( ( Ys
                  = ( cons_s2546857569980105499_a_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( produc4622690550020908860_a_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_na7417237922546100252_a_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_711_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X4: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs4 ) )
           => ! [Y3: nat,Ys5: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_712_zip__eq__ConsE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
      ( ( ( zip_Pr4664179122662387191at_nat @ Xs @ Ys )
        = ( cons_P8732206157123786781at_nat @ Xy @ Xys ) )
     => ~ ! [X4: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X4 @ Xs4 ) )
           => ! [Y3: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
                ( ( Ys
                  = ( cons_P6512896166579812791at_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( produc6161850002892822231at_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr4664179122662387191at_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_713_zip__eq__ConsE,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat,Xy: produc5621125494677191285_a_nat,Xys: list_P1596508216499738619_a_nat] :
      ( ( ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys )
        = ( cons_P6300126698356719403_a_nat @ Xy @ Xys ) )
     => ~ ! [X4: produc5986856040925105290_a_nat,Xs4: list_P4848832406072927130_a_nat] :
            ( ( Xs
              = ( cons_P681071152627790164_a_nat @ X4 @ Xs4 ) )
           => ! [Y3: produc5986856040925105290_a_nat,Ys5: list_P4848832406072927130_a_nat] :
                ( ( Ys
                  = ( cons_P681071152627790164_a_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( produc5595853527594814125_a_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr1216249236586582477_a_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_714_zip__eq__ConsE,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat,Xy: produc6346555946419678479_a_nat,Xys: list_P4631832503587151765_a_nat] :
      ( ( ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys )
        = ( cons_P8856943777243231685_a_nat @ Xy @ Xys ) )
     => ~ ! [X4: produc8208987855206359447_a_nat,Xs4: list_P2897000184160895271_a_nat] :
            ( ( Xs
              = ( cons_P4489214174023894113_a_nat @ X4 @ Xs4 ) )
           => ! [Y3: produc8208987855206359447_a_nat,Ys5: list_P2897000184160895271_a_nat] :
                ( ( Ys
                  = ( cons_P4489214174023894113_a_nat @ Y3 @ Ys5 ) )
               => ( ( Xy
                    = ( produc5405793250563226055_a_nat @ X4 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr8897620143892512999_a_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_715_ad__agr__close__rec_Ocases,axiom,
    ! [X: produc5047129986930123118_a_nat] :
      ( ! [I: nat,M3: nat > option_Sum_sum_a_nat,AD: set_a] :
          ( X
         != ( produc2762060399932752288_a_nat @ I @ ( produc4871871856725027001_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ nil_Sum_sum_a_nat ) ) ) )
     => ( ! [I: nat,M3: nat > option_Sum_sum_a_nat,AD: set_a,X4: a,Xs2: list_Sum_sum_a_nat] :
            ( X
           != ( produc2762060399932752288_a_nat @ I @ ( produc4871871856725027001_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ X4 ) @ Xs2 ) ) ) ) )
       => ~ ! [I: nat,M3: nat > option_Sum_sum_a_nat,AD: set_a,N: nat,Xs2: list_Sum_sum_a_nat] :
              ( X
             != ( produc2762060399932752288_a_nat @ I @ ( produc4871871856725027001_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ ( cons_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N ) @ Xs2 ) ) ) ) ) ) ) ).

% ad_agr_close_rec.cases
thf(fact_716_fo__nmlz__rec_Ocases,axiom,
    ! [X: produc7400749417448455946_a_nat] :
      ( ! [I: nat,M3: sum_sum_a_nat > option_nat,AD: set_a] :
          ( X
         != ( produc9161022033385942332_a_nat @ I @ ( produc2047461453323441237_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ nil_Sum_sum_a_nat ) ) ) )
     => ( ! [I: nat,M3: sum_sum_a_nat > option_nat,AD: set_a,X4: a,Xs2: list_Sum_sum_a_nat] :
            ( X
           != ( produc9161022033385942332_a_nat @ I @ ( produc2047461453323441237_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ X4 ) @ Xs2 ) ) ) ) )
       => ~ ! [I: nat,M3: sum_sum_a_nat > option_nat,AD: set_a,N: nat,Xs2: list_Sum_sum_a_nat] :
              ( X
             != ( produc9161022033385942332_a_nat @ I @ ( produc2047461453323441237_a_nat @ M3 @ ( produc5501371206643777598_a_nat @ AD @ ( cons_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N ) @ Xs2 ) ) ) ) ) ) ) ).

% fo_nmlz_rec.cases
thf(fact_717_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = nil_nat ) )
      & ( ( Xs != nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_718_list_Osimps_I8_J,axiom,
    ! [F: a > sum_sum_a_nat] :
      ( ( map_a_Sum_sum_a_nat @ F @ nil_a )
      = nil_Sum_sum_a_nat ) ).

% list.simps(8)
thf(fact_719_list_Osimps_I8_J,axiom,
    ! [F: nat > a] :
      ( ( map_nat_a @ F @ nil_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_720_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_721_map__eq__Cons__conv,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( cons_Sum_sum_a_nat @ Y @ Ys ) )
      = ( ? [Z4: a,Zs3: list_a] :
            ( ( Xs
              = ( cons_a @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y )
            & ( ( map_a_Sum_sum_a_nat @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_722_map__eq__Cons__conv,axiom,
    ! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z4: nat,Zs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y )
            & ( ( map_nat_a @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_723_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z4: nat,Zs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y )
            & ( ( map_nat_nat @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_724_Cons__eq__map__conv,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( cons_Sum_sum_a_nat @ X @ Xs )
        = ( map_a_Sum_sum_a_nat @ F @ Ys ) )
      = ( ? [Z4: a,Zs3: list_a] :
            ( ( Ys
              = ( cons_a @ Z4 @ Zs3 ) )
            & ( X
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_a_Sum_sum_a_nat @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_725_Cons__eq__map__conv,axiom,
    ! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_nat_a @ F @ Ys ) )
      = ( ? [Z4: nat,Zs3: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z4 @ Zs3 ) )
            & ( X
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_nat_a @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_726_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
      = ( ? [Z4: nat,Zs3: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z4 @ Zs3 ) )
            & ( X
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_727_map__eq__Cons__D,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( cons_Sum_sum_a_nat @ Y @ Ys ) )
     => ? [Z3: a,Zs2: list_a] :
          ( ( Xs
            = ( cons_a @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_a_Sum_sum_a_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_728_map__eq__Cons__D,axiom,
    ! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_nat_a @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_729_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_730_Cons__eq__map__D,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: a > sum_sum_a_nat,Ys: list_a] :
      ( ( ( cons_Sum_sum_a_nat @ X @ Xs )
        = ( map_a_Sum_sum_a_nat @ F @ Ys ) )
     => ? [Z3: a,Zs2: list_a] :
          ( ( Ys
            = ( cons_a @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_a_Sum_sum_a_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_731_Cons__eq__map__D,axiom,
    ! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_nat_a @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_a @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_732_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_733_list_Osimps_I9_J,axiom,
    ! [F: a > sum_sum_a_nat,X21: a,X222: list_a] :
      ( ( map_a_Sum_sum_a_nat @ F @ ( cons_a @ X21 @ X222 ) )
      = ( cons_Sum_sum_a_nat @ ( F @ X21 ) @ ( map_a_Sum_sum_a_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_734_list_Osimps_I9_J,axiom,
    ! [F: nat > a,X21: nat,X222: list_nat] :
      ( ( map_nat_a @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_735_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X222: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_736_last__zip,axiom,
    ! [Xs: list_set_a,Ys: list_P4848832406072927130_a_nat] :
      ( ( Xs != nil_set_a )
     => ( ( Ys != nil_Pr6508300198058928388_a_nat )
       => ( ( ( size_size_list_set_a @ Xs )
            = ( size_s3272666785928748590_a_nat @ Ys ) )
         => ( ( last_P2345141186260599352_a_nat @ ( zip_se7211208273872771441_a_nat @ Xs @ Ys ) )
            = ( produc1626021218680649873_a_nat @ ( last_set_a @ Xs ) @ ( last_P7175624747751160363_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_737_last__zip,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ( ( Ys != nil_Pr5478986624290739719at_nat )
       => ( ( ( size_s5460976970255530739at_nat @ Xs )
            = ( size_s5460976970255530739at_nat @ Ys ) )
         => ( ( last_P5835599730086418438at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) )
            = ( produc6161850002892822231at_nat @ ( last_P6484183829340986144at_nat @ Xs ) @ ( last_P6484183829340986144at_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_738_last__zip,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat] :
      ( ( Xs != nil_Pr6508300198058928388_a_nat )
     => ( ( Ys != nil_Pr6508300198058928388_a_nat )
       => ( ( ( size_s3272666785928748590_a_nat @ Xs )
            = ( size_s3272666785928748590_a_nat @ Ys ) )
         => ( ( last_P4193942428839852692_a_nat @ ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys ) )
            = ( produc5595853527594814125_a_nat @ ( last_P7175624747751160363_a_nat @ Xs ) @ ( last_P7175624747751160363_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_739_last__zip,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat] :
      ( ( Xs != nil_Pr236201474699349009_a_nat )
     => ( ( Ys != nil_Pr236201474699349009_a_nat )
       => ( ( ( size_s3428558496044824507_a_nat @ Xs )
            = ( size_s3428558496044824507_a_nat @ Ys ) )
         => ( ( last_P3610546745493882670_a_nat @ ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys ) )
            = ( produc5405793250563226055_a_nat @ ( last_P2345141186260599352_a_nat @ Xs ) @ ( last_P2345141186260599352_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_740_last__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( Ys != nil_Sum_sum_a_nat )
       => ( ( ( size_s5283204784079214577_a_nat @ Xs )
            = ( size_s5283204784079214577_a_nat @ Ys ) )
         => ( ( last_P17124104574654342_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
            = ( produc1212125651291703639_a_nat @ ( last_Sum_sum_a_nat @ Xs ) @ ( last_Sum_sum_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_741_last__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( Ys != nil_nat )
       => ( ( ( size_s5283204784079214577_a_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ( last_P7114463934449255199at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
            = ( produc7669364194715613304at_nat @ ( last_Sum_sum_a_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_742_last__zip,axiom,
    ! [Xs: list_nat,Ys: list_s602657755630655083_a_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_se550621812924229483_a_nat )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_s5225532995356599511_a_nat @ Ys ) )
         => ( ( last_P7175624747751160363_a_nat @ ( zip_na7417237922546100252_a_nat @ Xs @ Ys ) )
            = ( produc4622690550020908860_a_nat @ ( last_nat @ Xs ) @ ( last_s9202520926264054532_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_743_last__zip,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_Sum_sum_a_nat )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_s5283204784079214577_a_nat @ Ys ) )
         => ( ( last_P5310735818784388933_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
            = ( produc3265382261054541654_a_nat @ ( last_nat @ Xs ) @ ( last_Sum_sum_a_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_744_last__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
            = ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_745_rem__nth_Osimps_I1_J,axiom,
    ! [Uu2: nat] :
      ( ( rem_nth_nat @ Uu2 @ nil_nat )
      = nil_nat ) ).

% rem_nth.simps(1)
thf(fact_746_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_747_rem__nth_Oelims,axiom,
    ! [X: nat,Xa: list_nat,Y: list_nat] :
      ( ( ( rem_nth_nat @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != nil_nat ) )
       => ( ( ( X = zero_zero_nat )
           => ! [X4: nat,Xs2: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ( Y != Xs2 ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ N ) )
               => ! [X4: nat,Xs2: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ X4 @ Xs2 ) )
                   => ( Y
                     != ( cons_nat @ X4 @ ( rem_nth_nat @ N @ Xs2 ) ) ) ) ) ) ) ) ).

% rem_nth.elims
thf(fact_748_gen__length__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( gen_length_nat @ N2 @ nil_nat )
      = N2 ) ).

% gen_length_code(1)
thf(fact_749_length__Cons,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_750_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_751_Suc__length__conv,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( suc @ N2 )
        = ( size_s5283204784079214577_a_nat @ Xs ) )
      = ( ? [Y10: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( cons_Sum_sum_a_nat @ Y10 @ Ys2 ) )
            & ( ( size_s5283204784079214577_a_nat @ Ys2 )
              = N2 ) ) ) ) ).

% Suc_length_conv
thf(fact_752_Suc__length__conv,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ( suc @ N2 )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y10: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y10 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N2 ) ) ) ) ).

% Suc_length_conv
thf(fact_753_length__Suc__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,N2: nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( suc @ N2 ) )
      = ( ? [Y10: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( cons_Sum_sum_a_nat @ Y10 @ Ys2 ) )
            & ( ( size_s5283204784079214577_a_nat @ Ys2 )
              = N2 ) ) ) ) ).

% length_Suc_conv
thf(fact_754_length__Suc__conv,axiom,
    ! [Xs: list_nat,N2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N2 ) )
      = ( ? [Y10: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y10 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N2 ) ) ) ) ).

% length_Suc_conv
thf(fact_755_length__nth__simps_I2_J,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_756_length__nth__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_757_length__nth__simps_I1_J,axiom,
    ( ( size_s5283204784079214577_a_nat @ nil_Sum_sum_a_nat )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_758_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_759_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N2: nat,Ns: list_nat,Q: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M2 @ Q ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_760_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_761_rem__nth_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( rem_nth_nat @ zero_zero_nat @ ( cons_nat @ X @ Xs ) )
      = Xs ) ).

% rem_nth.simps(2)
thf(fact_762_rem__nth_Osimps_I3_J,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( rem_nth_nat @ ( suc @ N2 ) @ ( cons_nat @ X @ Xs ) )
      = ( cons_nat @ X @ ( rem_nth_nat @ N2 @ Xs ) ) ) ).

% rem_nth.simps(3)
thf(fact_763_last__map,axiom,
    ! [Xs: list_a,F: a > sum_sum_a_nat] :
      ( ( Xs != nil_a )
     => ( ( last_Sum_sum_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
        = ( F @ ( last_a @ Xs ) ) ) ) ).

% last_map
thf(fact_764_last__map,axiom,
    ! [Xs: list_nat,F: nat > a] :
      ( ( Xs != nil_nat )
     => ( ( last_a @ ( map_nat_a @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_765_last__map,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_766_add__nth_Osimps_I1_J,axiom,
    ! [A3: nat,Xs: list_nat] :
      ( ( add_nth_nat @ zero_zero_nat @ A3 @ Xs )
      = ( cons_nat @ A3 @ Xs ) ) ).

% add_nth.simps(1)
thf(fact_767_gen__length__code_I2_J,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( gen_length_nat @ N2 @ ( cons_nat @ X @ Xs ) )
      = ( gen_length_nat @ ( suc @ N2 ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_768_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_769_enumerate__eq__zip,axiom,
    ( enumer3164015132978342500_a_nat
    = ( ^ [N6: nat,Xs3: list_Sum_sum_a_nat] : ( zip_na2013496608136855606_a_nat @ ( upt @ N6 @ ( plus_plus_nat @ N6 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).

% enumerate_eq_zip
thf(fact_770_enumerate__eq__zip,axiom,
    ( enumerate_nat
    = ( ^ [N6: nat,Xs3: list_nat] : ( zip_nat_nat @ ( upt @ N6 @ ( plus_plus_nat @ N6 @ ( size_size_list_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).

% enumerate_eq_zip
thf(fact_771_list_Osize_I4_J,axiom,
    ! [X21: sum_sum_a_nat,X222: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_772_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_773_nth__Cons_H,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
          = X ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_774_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_775_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N2: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_776_rem__nth_Opelims,axiom,
    ! [X: nat,Xa: list_nat,Y: list_nat] :
      ( ( ( rem_nth_nat @ X @ Xa )
        = Y )
     => ( ( accp_P8032398731279750170st_nat @ rem_nth_rel_nat @ ( produc8282810413953273033st_nat @ X @ Xa ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_P8032398731279750170st_nat @ rem_nth_rel_nat @ ( produc8282810413953273033st_nat @ X @ nil_nat ) ) ) )
         => ( ( ( X = zero_zero_nat )
             => ! [X4: nat,Xs2: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X4 @ Xs2 ) )
                 => ( ( Y = Xs2 )
                   => ~ ( accp_P8032398731279750170st_nat @ rem_nth_rel_nat @ ( produc8282810413953273033st_nat @ zero_zero_nat @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ N ) )
                 => ! [X4: nat,Xs2: list_nat] :
                      ( ( Xa
                        = ( cons_nat @ X4 @ Xs2 ) )
                     => ( ( Y
                          = ( cons_nat @ X4 @ ( rem_nth_nat @ N @ Xs2 ) ) )
                       => ~ ( accp_P8032398731279750170st_nat @ rem_nth_rel_nat @ ( produc8282810413953273033st_nat @ ( suc @ N ) @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% rem_nth.pelims
thf(fact_777_list__fo__term_Oelims,axiom,
    ! [X: fo_term_nat,Y: list_nat] :
      ( ( ( list_fo_term_nat @ X )
        = Y )
     => ( ! [C: nat] :
            ( ( X
              = ( fo_Const_nat @ C ) )
           => ( Y
             != ( cons_nat @ C @ nil_nat ) ) )
       => ~ ( ? [V: nat] :
                ( X
                = ( fo_Var_nat @ V ) )
           => ( Y != nil_nat ) ) ) ) ).

% list_fo_term.elims
thf(fact_778_list__fo__term_Osimps_I1_J,axiom,
    ! [C2: nat] :
      ( ( list_fo_term_nat @ ( fo_Const_nat @ C2 ) )
      = ( cons_nat @ C2 @ nil_nat ) ) ).

% list_fo_term.simps(1)
thf(fact_779_sunion__with_Ocases,axiom,
    ! [X: produc4870497388681062763_a_nat] :
      ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
          ( X
         != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( X
           != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Bs ) ) )
       => ~ ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( X
             != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ As @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_780_sunion__with_Ocases,axiom,
    ! [X: produc4583671771502781146_a_nat] :
      ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
          ( X
         != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( X
           != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Bs ) ) )
       => ~ ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( X
             != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ As @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_781_sunion__with_Ocases,axiom,
    ! [X: produc4961808859980711423at_nat] :
      ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
          ( X
         != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X
           != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F3: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
              ( X
             != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_782_list__fo__term_Osimps_I2_J,axiom,
    ! [V3: nat] :
      ( ( list_fo_term_nat @ ( fo_Var_nat @ V3 ) )
      = nil_nat ) ).

% list_fo_term.simps(2)
thf(fact_783_list__fo__term_Opelims,axiom,
    ! [X: fo_term_nat,Y: list_nat] :
      ( ( ( list_fo_term_nat @ X )
        = Y )
     => ( ( accp_fo_term_nat @ list_fo_term_rel_nat @ X )
       => ( ! [C: nat] :
              ( ( X
                = ( fo_Const_nat @ C ) )
             => ( ( Y
                  = ( cons_nat @ C @ nil_nat ) )
               => ~ ( accp_fo_term_nat @ list_fo_term_rel_nat @ ( fo_Const_nat @ C ) ) ) )
         => ~ ! [V: nat] :
                ( ( X
                  = ( fo_Var_nat @ V ) )
               => ( ( Y = nil_nat )
                 => ~ ( accp_fo_term_nat @ list_fo_term_rel_nat @ ( fo_Var_nat @ V ) ) ) ) ) ) ) ).

% list_fo_term.pelims
thf(fact_784_comp__sunion__with_Ocases,axiom,
    ! [X: produc4870497388681062763_a_nat] :
      ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
          ( X
         != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( X
           != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Bs ) ) )
       => ~ ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( X
             != ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ As @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_785_comp__sunion__with_Ocases,axiom,
    ! [X: produc4583671771502781146_a_nat] :
      ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
          ( X
         != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( X
           != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Bs ) ) )
       => ~ ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( X
             != ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ As @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_786_comp__sunion__with_Ocases,axiom,
    ! [X: produc4961808859980711423at_nat] :
      ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
          ( X
         != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X
           != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F3: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
              ( X
             != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_787_comp__sunion__with_Ocases,axiom,
    ! [X: produc2780847497663810273at_nat] :
      ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
          ( X
         != ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( X
           != ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( X
             != ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_788_comp__sunion__with_Ocases,axiom,
    ! [X: produc3088563587264749079_a_nat] :
      ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat,K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
          ( X
         != ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
            ( X
           != ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ nil_Pr8135642835551910779_a_nat @ Bs ) ) )
       => ~ ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
              ( X
             != ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ As @ nil_Pr8135642835551910779_a_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_789_comp__sunion__with_Ocases,axiom,
    ! [X: produc1520877820366851299_a_nat] :
      ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat,K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
          ( X
         != ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
     => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
            ( X
           != ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ nil_Pr6086116330685108757_a_nat @ Bs ) ) )
       => ~ ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
              ( X
             != ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ As @ nil_Pr6086116330685108757_a_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_790_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc3992752218919044276_a_nat] :
      ( ! [K2: set_a] :
          ( X
         != ( produc428680061221208750_a_nat @ K2 @ nil_Pr236201474699349009_a_nat ) )
     => ~ ! [K2: set_a,K4: set_a,V: produc5986856040925105290_a_nat,Xs2: list_P2897000184160895271_a_nat] :
            ( X
           != ( produc428680061221208750_a_nat @ K2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_791_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc7011930558909267659_a_nat] :
      ( ! [K2: nat] :
          ( X
         != ( produc3870440650713894531_a_nat @ K2 @ nil_Pr6508300198058928388_a_nat ) )
     => ~ ! [K2: nat,K4: nat,V: set_li6526943997496501093_a_nat,Xs2: list_P4848832406072927130_a_nat] :
            ( X
           != ( produc3870440650713894531_a_nat @ K2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_792_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc8472197452120411308at_nat] :
      ( ! [K2: nat] :
          ( X
         != ( produc6109913384486294878at_nat @ K2 @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [K2: nat,K4: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
            ( X
           != ( produc6109913384486294878at_nat @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_793_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc5065989759464263123at_nat] :
      ( ! [K2: product_prod_nat_nat] :
          ( X
         != ( produc4832291263848940611at_nat @ K2 @ nil_Pr2582115297535392877at_nat ) )
     => ~ ! [K2: product_prod_nat_nat,K4: product_prod_nat_nat,V: product_prod_nat_nat,Xs2: list_P8469869581646625389at_nat] :
            ( X
           != ( produc4832291263848940611at_nat @ K2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_794_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc5080830201832994534_a_nat] :
      ( ! [K2: produc5986856040925105290_a_nat] :
          ( X
         != ( produc3076400587846869144_a_nat @ K2 @ nil_Pr8135642835551910779_a_nat ) )
     => ~ ! [K2: produc5986856040925105290_a_nat,K4: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,Xs2: list_P1596508216499738619_a_nat] :
            ( X
           != ( produc3076400587846869144_a_nat @ K2 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_795_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc5298447121777142707_a_nat] :
      ( ! [K2: produc8208987855206359447_a_nat] :
          ( X
         != ( produc677333111835304805_a_nat @ K2 @ nil_Pr6086116330685108757_a_nat ) )
     => ~ ! [K2: produc8208987855206359447_a_nat,K4: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,Xs2: list_P4631832503587151765_a_nat] :
            ( X
           != ( produc677333111835304805_a_nat @ K2 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V ) @ Xs2 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_796_zip__with__index__from__simps_I2_J,axiom,
    ! [N2: nat,X: set_li6526943997496501093_a_nat,Xs: list_s602657755630655083_a_nat] :
      ( ( zip_wi4158353643603967848_a_nat @ N2 @ ( cons_s2546857569980105499_a_nat @ X @ Xs ) )
      = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ N2 @ X ) @ ( zip_wi4158353643603967848_a_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% zip_with_index_from_simps(2)
thf(fact_797_zip__with__index__from__simps_I2_J,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( zip_wi3407404960461264653om_nat @ N2 @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N2 @ X ) @ ( zip_wi3407404960461264653om_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% zip_with_index_from_simps(2)
thf(fact_798_n__lists__Nil,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( n_lists_nat @ N2 @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( n_lists_nat @ N2 @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_799_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_800_zip__with__index__from__def,axiom,
    ( zip_wi5053028342928268674_a_nat
    = ( ^ [N6: nat,Xs3: list_Sum_sum_a_nat] : ( zip_na2013496608136855606_a_nat @ ( upt @ N6 @ ( plus_plus_nat @ N6 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).

% zip_with_index_from_def
thf(fact_801_zip__with__index__from__def,axiom,
    ( zip_wi3407404960461264653om_nat
    = ( ^ [N6: nat,Xs3: list_nat] : ( zip_nat_nat @ ( upt @ N6 @ ( plus_plus_nat @ N6 @ ( size_size_list_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).

% zip_with_index_from_def
thf(fact_802_sunion__with_Opinduct,axiom,
    ! [A0: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P2897000184160895271_a_nat,A22: list_P2897000184160895271_a_nat,P: ( set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P2897000184160895271_a_nat > list_P2897000184160895271_a_nat > $o] :
      ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ A0 @ ( produc3731672623177972967_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_set_a @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_set_a @ K4 @ K2 )
                 => ( ( ord_less_set_a @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_set_a @ K4 @ K2 )
                   => ( ~ ( ord_less_set_a @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
              ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr236201474699349009_a_nat @ Bs ) )
         => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
                ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ As @ nil_Pr236201474699349009_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr236201474699349009_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sunion_with.pinduct
thf(fact_803_sunion__with_Opinduct,axiom,
    ! [A0: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A1: list_P4848832406072927130_a_nat,A22: list_P4848832406072927130_a_nat,P: ( nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > list_P4848832406072927130_a_nat > list_P4848832406072927130_a_nat > $o] :
      ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ A0 @ ( produc6983730236908500813_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                 => ( ( ord_less_nat @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                   => ( ~ ( ord_less_nat @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
              ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr6508300198058928388_a_nat @ Bs ) )
         => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
                ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ As @ nil_Pr6508300198058928388_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr6508300198058928388_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sunion_with.pinduct
thf(fact_804_sunion__with_Opinduct,axiom,
    ! [A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                 => ( ( ord_less_nat @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                   => ( ~ ( ord_less_nat @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr5478986624290739719at_nat @ Bs ) )
         => ( ! [F3: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sunion_with.pinduct
thf(fact_805_sinter__with_Opinduct,axiom,
    ! [A0: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P2897000184160895271_a_nat,A22: list_P2897000184160895271_a_nat,P: ( set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P2897000184160895271_a_nat > list_P2897000184160895271_a_nat > $o] :
      ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ A0 @ ( produc3731672623177972967_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_set_a @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_set_a @ K4 @ K2 )
                 => ( ( ord_less_set_a @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_set_a @ K4 @ K2 )
                   => ( ~ ( ord_less_set_a @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uu: list_P2897000184160895271_a_nat] :
              ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr236201474699349009_a_nat @ Uu ) )
         => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uv: list_P2897000184160895271_a_nat] :
                ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ Uv @ nil_Pr236201474699349009_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr236201474699349009_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sinter_with.pinduct
thf(fact_806_sinter__with_Opinduct,axiom,
    ! [A0: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A1: list_P4848832406072927130_a_nat,A22: list_P4848832406072927130_a_nat,P: ( nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > list_P4848832406072927130_a_nat > list_P4848832406072927130_a_nat > $o] :
      ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ A0 @ ( produc6983730236908500813_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                 => ( ( ord_less_nat @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                   => ( ~ ( ord_less_nat @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Uu: list_P4848832406072927130_a_nat] :
              ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr6508300198058928388_a_nat @ Uu ) )
         => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Uv: list_P4848832406072927130_a_nat] :
                ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ Uv @ nil_Pr6508300198058928388_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr6508300198058928388_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sinter_with.pinduct
thf(fact_807_sinter__with_Opinduct,axiom,
    ! [A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                 => ( ( ord_less_nat @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K4 @ K2 )
                   => ( ~ ( ord_less_nat @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Uu: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr5478986624290739719at_nat @ Uu ) )
         => ( ! [F3: nat > nat > nat > nat,Uv: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ Uv @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sinter_with.pinduct
thf(fact_808_sunion__with_Opelims,axiom,
    ! [X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_or6397690549717327911_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( ( Xa
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                  ( ( Xb
                    = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_set_a @ K4 @ K2 )
                       => ( Y
                          = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ ( rBT_or6397690549717327911_a_nat @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( ord_less_set_a @ K4 @ K2 )
                       => ( ( ( ord_less_set_a @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ ( rBT_or6397690549717327911_a_nat @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( ord_less_set_a @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or6397690549717327911_a_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ) ) ) ) ).

% sunion_with.pelims
thf(fact_809_sunion__with_Opelims,axiom,
    ! [X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_or4284765397163420882_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( ( Xa
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                  ( ( Xb
                    = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K4 @ K2 )
                       => ( Y
                          = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ ( rBT_or4284765397163420882_a_nat @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( ord_less_nat @ K4 @ K2 )
                       => ( ( ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ ( rBT_or4284765397163420882_a_nat @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or4284765397163420882_a_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ) ) ) ) ).

% sunion_with.pelims
thf(fact_810_sunion__with_Opelims,axiom,
    ! [X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or5195327883522051875at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K4 @ K2 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ ( rBT_or5195327883522051875at_nat @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( ord_less_nat @ K4 @ K2 )
                       => ( ( ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( rBT_or5195327883522051875at_nat @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or5195327883522051875at_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% sunion_with.pelims
thf(fact_811_sinter__with_Opelims,axiom,
    ! [X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_or2563069335987614682_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( ( Xa
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                  ( ( Xb
                    = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_set_a @ K4 @ K2 )
                       => ( Y
                          = ( rBT_or2563069335987614682_a_nat @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( ord_less_set_a @ K4 @ K2 )
                       => ( ( ( ord_less_set_a @ K2 @ K4 )
                           => ( Y
                              = ( rBT_or2563069335987614682_a_nat @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( ord_less_set_a @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or2563069335987614682_a_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
             => ( ( Y = nil_Pr236201474699349009_a_nat )
               => ~ ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
               => ( ( Y = nil_Pr236201474699349009_a_nat )
                 => ~ ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ) ) ) ) ).

% sinter_with.pelims
thf(fact_812_sinter__with_Opelims,axiom,
    ! [X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_or1323869261074708357_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( ( Xa
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                  ( ( Xb
                    = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K4 @ K2 )
                       => ( Y
                          = ( rBT_or1323869261074708357_a_nat @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( ord_less_nat @ K4 @ K2 )
                       => ( ( ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( rBT_or1323869261074708357_a_nat @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or1323869261074708357_a_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
             => ( ( Y = nil_Pr6508300198058928388_a_nat )
               => ~ ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
               => ( ( Y = nil_Pr6508300198058928388_a_nat )
                 => ~ ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ) ) ) ) ).

% sinter_with.pelims
thf(fact_813_sinter__with_Opelims,axiom,
    ! [X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or1698556862823036848at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K4 @ K2 )
                       => ( Y
                          = ( rBT_or1698556862823036848at_nat @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( ord_less_nat @ K4 @ K2 )
                       => ( ( ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( rBT_or1698556862823036848at_nat @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( ord_less_nat @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or1698556862823036848at_nat @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = nil_Pr5478986624290739719at_nat )
                 => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% sinter_with.pelims
thf(fact_814_ord_Osunion__with_Opinduct,axiom,
    ! [Less: set_a > set_a > $o,A0: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P2897000184160895271_a_nat,A22: list_P2897000184160895271_a_nat,P: ( set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P2897000184160895271_a_nat > list_P2897000184160895271_a_nat > $o] :
      ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ A0 @ ( produc3731672623177972967_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
              ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr236201474699349009_a_nat @ Bs ) )
         => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
                ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ As @ nil_Pr236201474699349009_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr236201474699349009_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_815_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A1: list_P4848832406072927130_a_nat,A22: list_P4848832406072927130_a_nat,P: ( nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > list_P4848832406072927130_a_nat > list_P4848832406072927130_a_nat > $o] :
      ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ A0 @ ( produc6983730236908500813_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
              ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr6508300198058928388_a_nat @ Bs ) )
         => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
                ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ As @ nil_Pr6508300198058928388_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr6508300198058928388_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_816_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr5478986624290739719at_nat @ Bs ) )
         => ( ! [F3: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_817_ord_Osunion__with_Opinduct,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr2582115297535392877at_nat @ Bs ) )
         => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_818_ord_Osunion__with_Opinduct,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,A0: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P1596508216499738619_a_nat,A22: list_P1596508216499738619_a_nat,P: ( produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P1596508216499738619_a_nat > list_P1596508216499738619_a_nat > $o] :
      ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ A0 @ ( produc8554732292819397719_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat,K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
            ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
              ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ nil_Pr8135642835551910779_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr8135642835551910779_a_nat @ Bs ) )
         => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
                ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ As @ nil_Pr8135642835551910779_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr8135642835551910779_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_819_ord_Osunion__with_Opinduct,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,A0: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,A1: list_P4631832503587151765_a_nat,A22: list_P4631832503587151765_a_nat,P: ( produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat ) > list_P4631832503587151765_a_nat > list_P4631832503587151765_a_nat > $o] :
      ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ A0 @ ( produc14175607673870551_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat,K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
            ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
              ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ nil_Pr6086116330685108757_a_nat @ Bs ) ) )
             => ( P @ F3 @ nil_Pr6086116330685108757_a_nat @ Bs ) )
         => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
                ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ As @ nil_Pr6086116330685108757_a_nat ) ) )
               => ( P @ F3 @ As @ nil_Pr6086116330685108757_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_820_sinter__with_Osimps_I1_J,axiom,
    ! [K5: set_a,K: set_a,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( ( ord_less_set_a @ K5 @ K )
       => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( ord_less_set_a @ K5 @ K )
       => ( ( ( ord_less_set_a @ K @ K5 )
           => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_or2563069335987614682_a_nat @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( ord_less_set_a @ K @ K5 )
           => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or2563069335987614682_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sinter_with.simps(1)
thf(fact_821_sinter__with_Osimps_I1_J,axiom,
    ! [K5: nat,K: nat,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( ( ord_less_nat @ K5 @ K )
       => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( ord_less_nat @ K5 @ K )
       => ( ( ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_or1323869261074708357_a_nat @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or1323869261074708357_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sinter_with.simps(1)
thf(fact_822_sinter__with_Osimps_I1_J,axiom,
    ! [K5: nat,K: nat,F: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( ord_less_nat @ K5 @ K )
       => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( ord_less_nat @ K5 @ K )
       => ( ( ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_or1698556862823036848at_nat @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or1698556862823036848at_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sinter_with.simps(1)
thf(fact_823_sunion__with_Osimps_I1_J,axiom,
    ! [K5: set_a,K: set_a,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( ( ord_less_set_a @ K5 @ K )
       => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( ord_less_set_a @ K5 @ K )
       => ( ( ( ord_less_set_a @ K @ K5 )
           => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ ( rBT_or6397690549717327911_a_nat @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( ord_less_set_a @ K @ K5 )
           => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or6397690549717327911_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sunion_with.simps(1)
thf(fact_824_sunion__with_Osimps_I1_J,axiom,
    ! [K5: nat,K: nat,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( ( ord_less_nat @ K5 @ K )
       => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( ord_less_nat @ K5 @ K )
       => ( ( ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ ( rBT_or4284765397163420882_a_nat @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or4284765397163420882_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sunion_with.simps(1)
thf(fact_825_sunion__with_Osimps_I1_J,axiom,
    ! [K5: nat,K: nat,F: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( ord_less_nat @ K5 @ K )
       => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( ord_less_nat @ K5 @ K )
       => ( ( ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ ( rBT_or5195327883522051875at_nat @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( ord_less_nat @ K @ K5 )
           => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or5195327883522051875at_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sunion_with.simps(1)
thf(fact_826_sinter__with_Oelims,axiom,
    ! [X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_or2563069335987614682_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
            ( ( Xa
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                ( ( Xb
                  = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_set_a @ K4 @ K2 )
                     => ( Y
                        = ( rBT_or2563069335987614682_a_nat @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( ord_less_set_a @ K4 @ K2 )
                     => ( ( ( ord_less_set_a @ K2 @ K4 )
                         => ( Y
                            = ( rBT_or2563069335987614682_a_nat @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( ord_less_set_a @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or2563069335987614682_a_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
           => ( Y != nil_Pr236201474699349009_a_nat ) )
         => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
             => ( Y != nil_Pr236201474699349009_a_nat ) ) ) ) ) ).

% sinter_with.elims
thf(fact_827_sinter__with_Oelims,axiom,
    ! [X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_or1323869261074708357_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
            ( ( Xa
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                ( ( Xb
                  = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K4 @ K2 )
                     => ( Y
                        = ( rBT_or1323869261074708357_a_nat @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( ord_less_nat @ K4 @ K2 )
                     => ( ( ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( rBT_or1323869261074708357_a_nat @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or1323869261074708357_a_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
           => ( Y != nil_Pr6508300198058928388_a_nat ) )
         => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
             => ( Y != nil_Pr6508300198058928388_a_nat ) ) ) ) ) ).

% sinter_with.elims
thf(fact_828_sinter__with_Oelims,axiom,
    ! [X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or1698556862823036848at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K4 @ K2 )
                     => ( Y
                        = ( rBT_or1698556862823036848at_nat @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( ord_less_nat @ K4 @ K2 )
                     => ( ( ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( rBT_or1698556862823036848at_nat @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or1698556862823036848at_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != nil_Pr5478986624290739719at_nat ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sinter_with.elims
thf(fact_829_sunion__with_Oelims,axiom,
    ! [X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_or6397690549717327911_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
            ( ( Xa
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                ( ( Xb
                  = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_set_a @ K4 @ K2 )
                     => ( Y
                        = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ ( rBT_or6397690549717327911_a_nat @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( ord_less_set_a @ K4 @ K2 )
                     => ( ( ( ord_less_set_a @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ ( rBT_or6397690549717327911_a_nat @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( ord_less_set_a @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or6397690549717327911_a_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% sunion_with.elims
thf(fact_830_sunion__with_Oelims,axiom,
    ! [X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_or4284765397163420882_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
            ( ( Xa
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                ( ( Xb
                  = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K4 @ K2 )
                     => ( Y
                        = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ ( rBT_or4284765397163420882_a_nat @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( ord_less_nat @ K4 @ K2 )
                     => ( ( ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ ( rBT_or4284765397163420882_a_nat @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or4284765397163420882_a_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% sunion_with.elims
thf(fact_831_sunion__with_Oelims,axiom,
    ! [X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or5195327883522051875at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K4 @ K2 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ ( rBT_or5195327883522051875at_nat @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( ord_less_nat @ K4 @ K2 )
                     => ( ( ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( rBT_or5195327883522051875at_nat @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( ord_less_nat @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_or5195327883522051875at_nat @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != Xa ) ) ) ) ) ).

% sunion_with.elims
thf(fact_832_sinter__with_Opsimps_I1_J,axiom,
    ! [F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: set_a,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,K5: set_a,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( accp_P344521934534433140_a_nat @ rBT_or3410422834360145725_a_nat @ ( produc4868909238200440027_a_nat @ F @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_set_a @ K5 @ K )
         => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( ord_less_set_a @ K5 @ K )
         => ( ( ( ord_less_set_a @ K @ K5 )
             => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_or2563069335987614682_a_nat @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( ord_less_set_a @ K @ K5 )
             => ( ( rBT_or2563069335987614682_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or2563069335987614682_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sinter_with.psimps(1)
thf(fact_833_sinter__with_Opsimps_I1_J,axiom,
    ! [F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K: nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,K5: nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( accp_P6999711007904097123_a_nat @ rBT_or7211739193848824296_a_nat @ ( produc3397830469334437066_a_nat @ F @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K5 @ K )
         => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( ord_less_nat @ K5 @ K )
         => ( ( ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_or1323869261074708357_a_nat @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or1323869261074708357_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or1323869261074708357_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sinter_with.psimps(1)
thf(fact_834_sinter__with_Opsimps_I1_J,axiom,
    ! [F: nat > nat > nat > nat,K: nat,V3: nat,As2: list_P6011104703257516679at_nat,K5: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K5 @ K )
         => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( ord_less_nat @ K5 @ K )
         => ( ( ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_or1698556862823036848at_nat @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or1698556862823036848at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or1698556862823036848at_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sinter_with.psimps(1)
thf(fact_835_sunion__with_Opsimps_I1_J,axiom,
    ! [F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: set_a,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,K5: set_a,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( accp_P344521934534433140_a_nat @ rBT_or6226063957824744240_a_nat @ ( produc4868909238200440027_a_nat @ F @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_set_a @ K5 @ K )
         => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( ord_less_set_a @ K5 @ K )
         => ( ( ( ord_less_set_a @ K @ K5 )
             => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ ( rBT_or6397690549717327911_a_nat @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( ord_less_set_a @ K @ K5 )
             => ( ( rBT_or6397690549717327911_a_nat @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or6397690549717327911_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sunion_with.psimps(1)
thf(fact_836_sunion__with_Opsimps_I1_J,axiom,
    ! [F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K: nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,K5: nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( accp_P6999711007904097123_a_nat @ rBT_or4679191962205916379_a_nat @ ( produc3397830469334437066_a_nat @ F @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K5 @ K )
         => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( ord_less_nat @ K5 @ K )
         => ( ( ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ ( rBT_or4284765397163420882_a_nat @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or4284765397163420882_a_nat @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or4284765397163420882_a_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sunion_with.psimps(1)
thf(fact_837_sunion__with_Opsimps_I1_J,axiom,
    ! [F: nat > nat > nat > nat,K: nat,V3: nat,As2: list_P6011104703257516679at_nat,K5: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K5 @ K )
         => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( ord_less_nat @ K5 @ K )
         => ( ( ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ ( rBT_or5195327883522051875at_nat @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( ord_less_nat @ K @ K5 )
             => ( ( rBT_or5195327883522051875at_nat @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_or5195327883522051875at_nat @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sunion_with.psimps(1)
thf(fact_838_ord_Osinter__with_Opinduct,axiom,
    ! [Less: set_a > set_a > $o,A0: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P2897000184160895271_a_nat,A22: list_P2897000184160895271_a_nat,P: ( set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P2897000184160895271_a_nat > list_P2897000184160895271_a_nat > $o] :
      ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ A0 @ ( produc3731672623177972967_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat,K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
            ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uu: list_P2897000184160895271_a_nat] :
              ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr236201474699349009_a_nat @ Uu ) )
         => ( ! [F3: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uv: list_P2897000184160895271_a_nat] :
                ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F3 @ ( produc3731672623177972967_a_nat @ Uv @ nil_Pr236201474699349009_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr236201474699349009_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_839_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,A1: list_P4848832406072927130_a_nat,A22: list_P4848832406072927130_a_nat,P: ( nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > list_P4848832406072927130_a_nat > list_P4848832406072927130_a_nat > $o] :
      ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ A0 @ ( produc6983730236908500813_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat,K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
            ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Uu: list_P4848832406072927130_a_nat] :
              ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr6508300198058928388_a_nat @ Uu ) )
         => ( ! [F3: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Uv: list_P4848832406072927130_a_nat] :
                ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F3 @ ( produc6983730236908500813_a_nat @ Uv @ nil_Pr6508300198058928388_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr6508300198058928388_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_840_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V: nat,As: list_P6011104703257516679at_nat,K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Uu: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr5478986624290739719at_nat @ Uu ) )
         => ( ! [F3: nat > nat > nat > nat,Uv: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ Uv @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_841_ord_Osinter__with_Opinduct,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uu: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr2582115297535392877at_nat @ Uu ) )
         => ( ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F3 @ ( produc1338542795132623831at_nat @ Uv @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_842_ord_Osinter__with_Opinduct,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,A0: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,A1: list_P1596508216499738619_a_nat,A22: list_P1596508216499738619_a_nat,P: ( produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat ) > list_P1596508216499738619_a_nat > list_P1596508216499738619_a_nat > $o] :
      ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ A0 @ ( produc8554732292819397719_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat,K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
            ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uu: list_P1596508216499738619_a_nat] :
              ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ nil_Pr8135642835551910779_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr8135642835551910779_a_nat @ Uu ) )
         => ( ! [F3: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Uv: list_P1596508216499738619_a_nat] :
                ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F3 @ ( produc8554732292819397719_a_nat @ Uv @ nil_Pr8135642835551910779_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr8135642835551910779_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_843_ord_Osinter__with_Opinduct,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,A0: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,A1: list_P4631832503587151765_a_nat,A22: list_P4631832503587151765_a_nat,P: ( produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat ) > list_P4631832503587151765_a_nat > list_P4631832503587151765_a_nat > $o] :
      ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ A0 @ ( produc14175607673870551_a_nat @ A1 @ A22 ) ) )
     => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat,K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
            ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K4 @ K2 )
               => ( P @ F3 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K4 @ K2 )
                 => ( ( Less @ K2 @ K4 )
                   => ( P @ F3 @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K4 @ K2 )
                   => ( ~ ( Less @ K2 @ K4 )
                     => ( P @ F3 @ As @ Bs ) ) )
                 => ( P @ F3 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Uu: list_P4631832503587151765_a_nat] :
              ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ nil_Pr6086116330685108757_a_nat @ Uu ) ) )
             => ( P @ F3 @ nil_Pr6086116330685108757_a_nat @ Uu ) )
         => ( ! [F3: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Uv: list_P4631832503587151765_a_nat] :
                ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F3 @ ( produc14175607673870551_a_nat @ Uv @ nil_Pr6086116330685108757_a_nat ) ) )
               => ( P @ F3 @ Uv @ nil_Pr6086116330685108757_a_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_844_ord_Osunion__with_Opelims,axiom,
    ! [Less: set_a > set_a > $o,X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_su8848990011291252589_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( ( Xa
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                  ( ( Xb
                    = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_845_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_su3945709818617745944_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( ( Xa
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                  ( ( Xb
                    = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_846_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_847_ord_Osunion__with_Opelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_su1546594653887917819at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) )
             => ! [K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr2582115297535392877at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ Xa @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_848_ord_Osunion__with_Opelims,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P1596508216499738619_a_nat,Xb: list_P1596508216499738619_a_nat,Y: list_P1596508216499738619_a_nat] :
      ( ( ( rBT_su2811500537751064785_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
              ( ( Xa
                = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
                  ( ( Xb
                    = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr8135642835551910779_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ nil_Pr8135642835551910779_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr8135642835551910779_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ Xa @ nil_Pr8135642835551910779_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_849_ord_Osunion__with_Opelims,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Xa: list_P4631832503587151765_a_nat,Xb: list_P4631832503587151765_a_nat,Y: list_P4631832503587151765_a_nat] :
      ( ( ( rBT_su4963355816509495275_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
              ( ( Xa
                = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
                  ( ( Xb
                    = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6086116330685108757_a_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ nil_Pr6086116330685108757_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6086116330685108757_a_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ Xa @ nil_Pr6086116330685108757_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_850_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: set_a > set_a > $o,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: set_a,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,K5: set_a,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( accp_P344521934534433140_a_nat @ ( rBT_su6954279842904320874_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_851_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K: nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,K5: nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( accp_P6999711007904097123_a_nat @ ( rBT_su849879700968488725_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_852_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > nat > nat > nat,K: nat,V3: nat,As2: list_P6011104703257516679at_nat,K5: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_853_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K5: product_prod_nat_nat,V4: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_854_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,F: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P1596508216499738619_a_nat,K5: produc5986856040925105290_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P1596508216499738619_a_nat] :
      ( ( accp_P6893916086496234062_a_nat @ ( rBT_su4889124067274007188_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ As2 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_855_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,F: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,K: produc8208987855206359447_a_nat,V3: produc8208987855206359447_a_nat,As2: list_P4631832503587151765_a_nat,K5: produc8208987855206359447_a_nat,V4: produc8208987855206359447_a_nat,Bs2: list_P4631832503587151765_a_nat] :
      ( ( accp_P8667619755534950682_a_nat @ ( rBT_su2253248761892871086_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ As2 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_856_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: set_a > set_a > $o,K5: set_a,K: set_a,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su8848990011291252589_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su8848990011291252589_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_857_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K5: nat,K: nat,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su3945709818617745944_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su3945709818617745944_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_858_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K5: nat,K: nat,F: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_859_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,K5: product_prod_nat_nat,K: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V4: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su1546594653887917819at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_860_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,K5: produc5986856040925105290_a_nat,K: produc5986856040925105290_a_nat,F: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P1596508216499738619_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P1596508216499738619_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ As2 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su2811500537751064785_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su2811500537751064785_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_861_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,K5: produc8208987855206359447_a_nat,K: produc8208987855206359447_a_nat,F: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,V3: produc8208987855206359447_a_nat,As2: list_P4631832503587151765_a_nat,V4: produc8208987855206359447_a_nat,Bs2: list_P4631832503587151765_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ As2 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_su4963355816509495275_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_su4963355816509495275_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_862_ord_Osunion__with_Oelims,axiom,
    ! [Less: set_a > set_a > $o,X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_su8848990011291252589_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
            ( ( Xa
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                ( ( Xb
                  = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su8848990011291252589_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_863_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_su3945709818617745944_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
            ( ( Xa
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                ( ( Xb
                  = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su3945709818617745944_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_864_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_865_ord_Osunion__with_Oelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_su1546594653887917819at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) )
           => ! [K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr2582115297535392877at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_866_ord_Osunion__with_Oelims,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P1596508216499738619_a_nat,Xb: list_P1596508216499738619_a_nat,Y: list_P1596508216499738619_a_nat] :
      ( ( ( rBT_su2811500537751064785_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
            ( ( Xa
              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
                ( ( Xb
                  = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su2811500537751064785_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr8135642835551910779_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr8135642835551910779_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_867_ord_Osunion__with_Oelims,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Xa: list_P4631832503587151765_a_nat,Xb: list_P4631832503587151765_a_nat,Y: list_P4631832503587151765_a_nat] :
      ( ( ( rBT_su4963355816509495275_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
            ( ( Xa
              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
                ( ( Xb
                  = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_su4963355816509495275_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6086116330685108757_a_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr6086116330685108757_a_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_868_ord_Osinter__with_Opelims,axiom,
    ! [Less: set_a > set_a > $o,X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_si5014368797561539360_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
              ( ( Xa
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                  ( ( Xb
                    = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si5014368797561539360_a_nat @ Less @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si5014368797561539360_a_nat @ Less @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si5014368797561539360_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
             => ( ( Y = nil_Pr236201474699349009_a_nat )
               => ~ ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ nil_Pr236201474699349009_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
               => ( ( Y = nil_Pr236201474699349009_a_nat )
                 => ~ ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ X @ ( produc3731672623177972967_a_nat @ Xa @ nil_Pr236201474699349009_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_869_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_si984813682529033419_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
              ( ( Xa
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                  ( ( Xb
                    = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si984813682529033419_a_nat @ Less @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si984813682529033419_a_nat @ Less @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si984813682529033419_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
             => ( ( Y = nil_Pr6508300198058928388_a_nat )
               => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ nil_Pr6508300198058928388_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
               => ( ( Y = nil_Pr6508300198058928388_a_nat )
                 => ~ ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ X @ ( produc6983730236908500813_a_nat @ Xa @ nil_Pr6508300198058928388_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_870_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
             => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si2101226886118949098at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si2101226886118949098at_nat @ Less @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = nil_Pr5478986624290739719at_nat )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_871_ord_Osinter__with_Opelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_si2205054011348398088at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) )
             => ! [K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si2205054011348398088at_nat @ Less @ X @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si2205054011348398088at_nat @ Less @ X @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr2582115297535392877at_nat )
             => ( ( Y = nil_Pr2582115297535392877at_nat )
               => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = nil_Pr2582115297535392877at_nat )
                 => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X @ ( produc1338542795132623831at_nat @ Xa @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_872_ord_Osinter__with_Opelims,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P1596508216499738619_a_nat,Xb: list_P1596508216499738619_a_nat,Y: list_P1596508216499738619_a_nat] :
      ( ( ( rBT_si4915412679432092510_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
              ( ( Xa
                = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
                  ( ( Xb
                    = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si4915412679432092510_a_nat @ Less @ X @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si4915412679432092510_a_nat @ Less @ X @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si4915412679432092510_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr8135642835551910779_a_nat )
             => ( ( Y = nil_Pr8135642835551910779_a_nat )
               => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ nil_Pr8135642835551910779_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr8135642835551910779_a_nat )
               => ( ( Y = nil_Pr8135642835551910779_a_nat )
                 => ~ ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ X @ ( produc8554732292819397719_a_nat @ Xa @ nil_Pr8135642835551910779_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_873_ord_Osinter__with_Opelims,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Xa: list_P4631832503587151765_a_nat,Xb: list_P4631832503587151765_a_nat,Y: list_P4631832503587151765_a_nat] :
      ( ( ( rBT_si7784342841749647480_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ Xa @ Xb ) ) )
       => ( ! [K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
              ( ( Xa
                = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) )
             => ! [K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
                  ( ( Xb
                    = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K4 @ K2 )
                       => ( Y
                          = ( rBT_si7784342841749647480_a_nat @ Less @ X @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K4 @ K2 )
                       => ( ( ( Less @ K2 @ K4 )
                           => ( Y
                              = ( rBT_si7784342841749647480_a_nat @ Less @ X @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K4 )
                           => ( Y
                              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si7784342841749647480_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr6086116330685108757_a_nat )
             => ( ( Y = nil_Pr6086116330685108757_a_nat )
               => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ nil_Pr6086116330685108757_a_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr6086116330685108757_a_nat )
               => ( ( Y = nil_Pr6086116330685108757_a_nat )
                 => ~ ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ X @ ( produc14175607673870551_a_nat @ Xa @ nil_Pr6086116330685108757_a_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_874_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: set_a > set_a > $o,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: set_a,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,K5: set_a,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( accp_P344521934534433140_a_nat @ ( rBT_si4138638719439722359_a_nat @ Less ) @ ( produc4868909238200440027_a_nat @ F @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si5014368797561539360_a_nat @ Less @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si5014368797561539360_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_875_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,K: nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,K5: nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( accp_P6999711007904097123_a_nat @ ( rBT_si3382426932611396642_a_nat @ Less ) @ ( produc3397830469334437066_a_nat @ F @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si984813682529033419_a_nat @ Less @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si984813682529033419_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_876_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > nat > nat > nat,K: nat,V3: nat,As2: list_P6011104703257516679at_nat,K5: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si2101226886118949098at_nat @ Less @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_877_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K5: product_prod_nat_nat,V4: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si2205054011348398088at_nat @ Less @ F @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_878_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,F: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,K: produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P1596508216499738619_a_nat,K5: produc5986856040925105290_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P1596508216499738619_a_nat] :
      ( ( accp_P6893916086496234062_a_nat @ ( rBT_si7265902633873623111_a_nat @ Less ) @ ( produc6141936104063048777_a_nat @ F @ ( produc8554732292819397719_a_nat @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si4915412679432092510_a_nat @ Less @ F @ As2 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si4915412679432092510_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_879_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,F: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,K: produc8208987855206359447_a_nat,V3: produc8208987855206359447_a_nat,As2: list_P4631832503587151765_a_nat,K5: produc8208987855206359447_a_nat,V4: produc8208987855206359447_a_nat,Bs2: list_P4631832503587151765_a_nat] :
      ( ( accp_P8667619755534950682_a_nat @ ( rBT_si149745482838135649_a_nat @ Less ) @ ( produc3315683433596007189_a_nat @ F @ ( produc14175607673870551_a_nat @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K5 @ K )
         => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
            = ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K5 @ K )
         => ( ( ( Less @ K @ K5 )
             => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( rBT_si7784342841749647480_a_nat @ Less @ F @ As2 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K5 )
             => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
                = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si7784342841749647480_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_880_splice_Opinduct,axiom,
    ! [A0: list_nat,A1: list_nat,P: list_nat > list_nat > $o] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_nat] :
            ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
           => ( P @ nil_nat @ Ys4 ) )
       => ( ! [X4: nat,Xs2: list_nat,Ys4: list_nat] :
              ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 ) )
             => ( ( P @ Ys4 @ Xs2 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 ) ) )
         => ( P @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_881_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: set_a > set_a > $o,K5: set_a,K: set_a,F: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P2897000184160895271_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P2897000184160895271_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si5014368797561539360_a_nat @ Less @ F @ As2 @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si5014368797561539360_a_nat @ Less @ F @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si5014368797561539360_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_882_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K5: nat,K: nat,F: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,V3: set_li6526943997496501093_a_nat,As2: list_P4848832406072927130_a_nat,V4: set_li6526943997496501093_a_nat,Bs2: list_P4848832406072927130_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si984813682529033419_a_nat @ Less @ F @ As2 @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si984813682529033419_a_nat @ Less @ F @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si984813682529033419_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_883_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K5: nat,K: nat,F: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si2101226886118949098at_nat @ Less @ F @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_884_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,K5: product_prod_nat_nat,K: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V4: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si2205054011348398088at_nat @ Less @ F @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si2205054011348398088at_nat @ Less @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_885_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,K5: produc5986856040925105290_a_nat,K: produc5986856040925105290_a_nat,F: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,V3: produc5986856040925105290_a_nat,As2: list_P1596508216499738619_a_nat,V4: produc5986856040925105290_a_nat,Bs2: list_P1596508216499738619_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si4915412679432092510_a_nat @ Less @ F @ As2 @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si4915412679432092510_a_nat @ Less @ F @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si4915412679432092510_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_886_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,K5: produc8208987855206359447_a_nat,K: produc8208987855206359447_a_nat,F: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,V3: produc8208987855206359447_a_nat,As2: list_P4631832503587151765_a_nat,V4: produc8208987855206359447_a_nat,Bs2: list_P4631832503587151765_a_nat] :
      ( ( ( Less @ K5 @ K )
       => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
          = ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K5 @ K )
       => ( ( ( Less @ K @ K5 )
           => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( rBT_si7784342841749647480_a_nat @ Less @ F @ As2 @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K5 )
           => ( ( rBT_si7784342841749647480_a_nat @ Less @ F @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ V3 ) @ As2 ) @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K5 @ V4 ) @ Bs2 ) )
              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K @ ( F @ K @ V3 @ V4 ) ) @ ( rBT_si7784342841749647480_a_nat @ Less @ F @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_887_ord_Osinter__with_Oelims,axiom,
    ! [Less: set_a > set_a > $o,X: set_a > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P2897000184160895271_a_nat,Xb: list_P2897000184160895271_a_nat,Y: list_P2897000184160895271_a_nat] :
      ( ( ( rBT_si5014368797561539360_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: set_a,V: produc5986856040925105290_a_nat,As: list_P2897000184160895271_a_nat] :
            ( ( Xa
              = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: set_a,V2: produc5986856040925105290_a_nat,Bs: list_P2897000184160895271_a_nat] :
                ( ( Xb
                  = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si5014368797561539360_a_nat @ Less @ X @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si5014368797561539360_a_nat @ Less @ X @ As @ ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P4489214174023894113_a_nat @ ( produc1626021218680649873_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si5014368797561539360_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr236201474699349009_a_nat )
           => ( Y != nil_Pr236201474699349009_a_nat ) )
         => ~ ( ( Xb = nil_Pr236201474699349009_a_nat )
             => ( Y != nil_Pr236201474699349009_a_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_888_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Xa: list_P4848832406072927130_a_nat,Xb: list_P4848832406072927130_a_nat,Y: list_P4848832406072927130_a_nat] :
      ( ( ( rBT_si984813682529033419_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: set_li6526943997496501093_a_nat,As: list_P4848832406072927130_a_nat] :
            ( ( Xa
              = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: set_li6526943997496501093_a_nat,Bs: list_P4848832406072927130_a_nat] :
                ( ( Xb
                  = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si984813682529033419_a_nat @ Less @ X @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si984813682529033419_a_nat @ Less @ X @ As @ ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P681071152627790164_a_nat @ ( produc4622690550020908860_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si984813682529033419_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6508300198058928388_a_nat )
           => ( Y != nil_Pr6508300198058928388_a_nat ) )
         => ~ ( ( Xb = nil_Pr6508300198058928388_a_nat )
             => ( Y != nil_Pr6508300198058928388_a_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_889_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) )
           => ! [K4: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si2101226886118949098at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si2101226886118949098at_nat @ Less @ X @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != nil_Pr5478986624290739719at_nat ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_890_ord_Osinter__with_Oelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_si2205054011348398088at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) )
           => ! [K4: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si2205054011348398088at_nat @ Less @ X @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si2205054011348398088at_nat @ Less @ X @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr2582115297535392877at_nat )
           => ( Y != nil_Pr2582115297535392877at_nat ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_891_ord_Osinter__with_Oelims,axiom,
    ! [Less: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > $o,X: produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat > produc5986856040925105290_a_nat,Xa: list_P1596508216499738619_a_nat,Xb: list_P1596508216499738619_a_nat,Y: list_P1596508216499738619_a_nat] :
      ( ( ( rBT_si4915412679432092510_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: produc5986856040925105290_a_nat,V: produc5986856040925105290_a_nat,As: list_P1596508216499738619_a_nat] :
            ( ( Xa
              = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: produc5986856040925105290_a_nat,V2: produc5986856040925105290_a_nat,Bs: list_P1596508216499738619_a_nat] :
                ( ( Xb
                  = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si4915412679432092510_a_nat @ Less @ X @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si4915412679432092510_a_nat @ Less @ X @ As @ ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P6300126698356719403_a_nat @ ( produc5595853527594814125_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si4915412679432092510_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr8135642835551910779_a_nat )
           => ( Y != nil_Pr8135642835551910779_a_nat ) )
         => ~ ( ( Xb = nil_Pr8135642835551910779_a_nat )
             => ( Y != nil_Pr8135642835551910779_a_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_892_ord_Osinter__with_Oelims,axiom,
    ! [Less: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > $o,X: produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat,Xa: list_P4631832503587151765_a_nat,Xb: list_P4631832503587151765_a_nat,Y: list_P4631832503587151765_a_nat] :
      ( ( ( rBT_si7784342841749647480_a_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: produc8208987855206359447_a_nat,V: produc8208987855206359447_a_nat,As: list_P4631832503587151765_a_nat] :
            ( ( Xa
              = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) )
           => ! [K4: produc8208987855206359447_a_nat,V2: produc8208987855206359447_a_nat,Bs: list_P4631832503587151765_a_nat] :
                ( ( Xb
                  = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K4 @ K2 )
                     => ( Y
                        = ( rBT_si7784342841749647480_a_nat @ Less @ X @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K4 @ K2 )
                     => ( ( ( Less @ K2 @ K4 )
                         => ( Y
                            = ( rBT_si7784342841749647480_a_nat @ Less @ X @ As @ ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K4 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K4 )
                         => ( Y
                            = ( cons_P8856943777243231685_a_nat @ ( produc5405793250563226055_a_nat @ K2 @ ( X @ K2 @ V @ V2 ) ) @ ( rBT_si7784342841749647480_a_nat @ Less @ X @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr6086116330685108757_a_nat )
           => ( Y != nil_Pr6086116330685108757_a_nat ) )
         => ~ ( ( Xb = nil_Pr6086116330685108757_a_nat )
             => ( Y != nil_Pr6086116330685108757_a_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_893_splice_Opelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( ( splice_nat @ X @ Xa )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ X @ Xa ) )
       => ( ( ( X = nil_nat )
           => ( ( Y = Xa )
             => ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa ) ) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( cons_nat @ X4 @ ( splice_nat @ Xa @ Xs2 ) ) )
                 => ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_894_shuffles_Opinduct,axiom,
    ! [A0: list_nat,A1: list_nat,P: list_nat > list_nat > $o] :
      ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_nat] :
            ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
           => ( P @ nil_nat @ Ys4 ) )
       => ( ! [Xs2: list_nat] :
              ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
             => ( P @ Xs2 @ nil_nat ) )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
               => ( ( P @ Xs2 @ ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( P @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
           => ( P @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_895_Cons__lenlex__iff,axiom,
    ! [M2: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N2: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M2 @ Ms ) @ ( cons_P6512896166579812791at_nat @ N2 @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
        | ( ( ( size_s5460976970255530739at_nat @ Ms )
            = ( size_s5460976970255530739at_nat @ Ns ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_896_Cons__lenlex__iff,axiom,
    ! [M2: produc5986856040925105290_a_nat,Ms: list_P4848832406072927130_a_nat,N2: produc5986856040925105290_a_nat,Ns: list_P4848832406072927130_a_nat,R2: set_Pr3551440518166979541_a_nat] :
      ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ M2 @ Ms ) @ ( cons_P681071152627790164_a_nat @ N2 @ Ns ) ) @ ( lenlex4395088088742450111_a_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s3272666785928748590_a_nat @ Ms ) @ ( size_s3272666785928748590_a_nat @ Ns ) )
        | ( ( ( size_s3272666785928748590_a_nat @ Ms )
            = ( size_s3272666785928748590_a_nat @ Ns ) )
          & ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Ms @ Ns ) @ ( lenlex4395088088742450111_a_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_897_Cons__lenlex__iff,axiom,
    ! [M2: produc8208987855206359447_a_nat,Ms: list_P2897000184160895271_a_nat,N2: produc8208987855206359447_a_nat,Ns: list_P2897000184160895271_a_nat,R2: set_Pr757995773253584751_a_nat] :
      ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ M2 @ Ms ) @ ( cons_P4489214174023894113_a_nat @ N2 @ Ns ) ) @ ( lenlex6682605546290681804_a_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s3428558496044824507_a_nat @ Ms ) @ ( size_s3428558496044824507_a_nat @ Ns ) )
        | ( ( ( size_s3428558496044824507_a_nat @ Ms )
            = ( size_s3428558496044824507_a_nat @ Ns ) )
          & ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Ms @ Ns ) @ ( lenlex6682605546290681804_a_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_898_Cons__lenlex__iff,axiom,
    ! [M2: sum_sum_a_nat,Ms: list_Sum_sum_a_nat,N2: sum_sum_a_nat,Ns: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ M2 @ Ms ) @ ( cons_Sum_sum_a_nat @ N2 @ Ns ) ) @ ( lenlex_Sum_sum_a_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ms ) @ ( size_s5283204784079214577_a_nat @ Ns ) )
        | ( ( ( size_s5283204784079214577_a_nat @ Ms )
            = ( size_s5283204784079214577_a_nat @ Ns ) )
          & ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ms @ Ns ) @ ( lenlex_Sum_sum_a_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_899_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N2: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N2 @ Ns ) ) @ ( lenlex_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_900_split__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( splice_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% split_Nil_iff
thf(fact_901_splice__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( splice_nat @ Xs @ nil_nat )
      = Xs ) ).

% splice_Nil2
thf(fact_902_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R2 ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_903_length__splice,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( splice_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% length_splice
thf(fact_904_length__splice,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_splice
thf(fact_905_splice_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( splice_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( splice_nat @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_906_splice_Osimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( splice_nat @ nil_nat @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_907_lenlex__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_908_lenlex__irreflexive,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_909_lenlex__irreflexive,axiom,
    ! [R2: set_Pr3551440518166979541_a_nat,Xs: list_P4848832406072927130_a_nat] :
      ( ! [X4: produc5986856040925105290_a_nat] :
          ~ ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Xs @ Xs ) @ ( lenlex4395088088742450111_a_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_910_lenlex__irreflexive,axiom,
    ! [R2: set_Pr757995773253584751_a_nat,Xs: list_P2897000184160895271_a_nat] :
      ( ! [X4: produc8208987855206359447_a_nat] :
          ~ ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Xs @ Xs ) @ ( lenlex6682605546290681804_a_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_911_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_912_splice_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( ( splice_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( cons_nat @ X4 @ ( splice_nat @ Xa @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_913_splice_Opsimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) )
     => ( ( splice_nat @ ( cons_nat @ X @ Xs ) @ Ys )
        = ( cons_nat @ X @ ( splice_nat @ Ys @ Xs ) ) ) ) ).

% splice.psimps(2)
thf(fact_914_splice_Opsimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ( ( splice_nat @ nil_nat @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_915_upt__rec__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_916_Cons__in__lex,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
          & ( ( size_s5460976970255530739at_nat @ Xs )
            = ( size_s5460976970255530739at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_917_Cons__in__lex,axiom,
    ! [X: produc5986856040925105290_a_nat,Xs: list_P4848832406072927130_a_nat,Y: produc5986856040925105290_a_nat,Ys: list_P4848832406072927130_a_nat,R2: set_Pr3551440518166979541_a_nat] :
      ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ ( cons_P681071152627790164_a_nat @ X @ Xs ) @ ( cons_P681071152627790164_a_nat @ Y @ Ys ) ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) )
      = ( ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X @ Y ) @ R2 )
          & ( ( size_s3272666785928748590_a_nat @ Xs )
            = ( size_s3272666785928748590_a_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Xs @ Ys ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_918_Cons__in__lex,axiom,
    ! [X: produc8208987855206359447_a_nat,Xs: list_P2897000184160895271_a_nat,Y: produc8208987855206359447_a_nat,Ys: list_P2897000184160895271_a_nat,R2: set_Pr757995773253584751_a_nat] :
      ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ ( cons_P4489214174023894113_a_nat @ X @ Xs ) @ ( cons_P4489214174023894113_a_nat @ Y @ Ys ) ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) )
      = ( ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X @ Y ) @ R2 )
          & ( ( size_s3428558496044824507_a_nat @ Xs )
            = ( size_s3428558496044824507_a_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Xs @ Ys ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_919_Cons__in__lex,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( lex_Sum_sum_a_nat @ R2 ) )
      = ( ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R2 )
          & ( ( size_s5283204784079214577_a_nat @ Xs )
            = ( size_s5283204784079214577_a_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_920_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_921_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V3: num] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V3 ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_922_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_923_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_924_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_925_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_926_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R2 ) ) ).

% Nil_notin_lex
thf(fact_927_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_928_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_929_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_930_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_931_append_Oassoc,axiom,
    ! [A3: list_nat,B3: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A3 @ B3 ) @ C2 )
      = ( append_nat @ A3 @ ( append_nat @ B3 @ C2 ) ) ) ).

% append.assoc
thf(fact_932_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_933_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_934_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_935_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_936_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_937_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_938_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_939_append_Oright__neutral,axiom,
    ! [A3: list_nat] :
      ( ( append_nat @ A3 @ nil_nat )
      = A3 ) ).

% append.right_neutral
thf(fact_940_map__append,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_Sum_sum_a_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append_Sum_sum_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) @ ( map_a_Sum_sum_a_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_941_map__append,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_942_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_943_append__eq__append__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
      ( ( ( ( size_s5283204784079214577_a_nat @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
        | ( ( size_s5283204784079214577_a_nat @ Us )
          = ( size_s5283204784079214577_a_nat @ Vs ) ) )
     => ( ( ( append_Sum_sum_a_nat @ Xs @ Us )
          = ( append_Sum_sum_a_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_944_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_945_append__in__lists__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( lists_Sum_sum_a_nat @ A5 ) )
      = ( ( member408289922725080238_a_nat @ Xs @ ( lists_Sum_sum_a_nat @ A5 ) )
        & ( member408289922725080238_a_nat @ Ys @ ( lists_Sum_sum_a_nat @ A5 ) ) ) ) ).

% append_in_lists_conv
thf(fact_946_append__in__lists__conv,axiom,
    ! [Xs: list_a,Ys: list_a,A5: set_a] :
      ( ( member_list_a @ ( append_a @ Xs @ Ys ) @ ( lists_a @ A5 ) )
      = ( ( member_list_a @ Xs @ ( lists_a @ A5 ) )
        & ( member_list_a @ Ys @ ( lists_a @ A5 ) ) ) ) ).

% append_in_lists_conv
thf(fact_947_append__in__lists__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,A5: set_nat] :
      ( ( member_list_nat @ ( append_nat @ Xs @ Ys ) @ ( lists_nat @ A5 ) )
      = ( ( member_list_nat @ Xs @ ( lists_nat @ A5 ) )
        & ( member_list_nat @ Ys @ ( lists_nat @ A5 ) ) ) ) ).

% append_in_lists_conv
thf(fact_948_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_949_length__append,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% length_append
thf(fact_950_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_951_zip__append,axiom,
    ! [Xs: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Us ) )
     => ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
        = ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Us ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_952_zip__append,axiom,
    ! [Xs: list_Sum_sum_a_nat,Us: list_nat,Ys: list_Sum_sum_a_nat,Vs: list_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Us ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_953_zip__append,axiom,
    ! [Xs: list_nat,Us: list_Sum_sum_a_nat,Ys: list_nat,Vs: list_Sum_sum_a_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Us ) )
     => ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
        = ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Us ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_954_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_955_last__appendL,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Xs ) ) ) ).

% last_appendL
thf(fact_956_last__appendR,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Ys ) ) ) ).

% last_appendR
thf(fact_957_nth__append__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_958_nth__append__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_959_nth__append__length__plus,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,N2: nat] :
      ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N2 ) )
      = ( nth_Sum_sum_a_nat @ Ys @ N2 ) ) ).

% nth_append_length_plus
thf(fact_960_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N2: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) )
      = ( nth_nat @ Ys @ N2 ) ) ).

% nth_append_length_plus
thf(fact_961_last__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_962_butlast__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_963_zip__with__index__from__append,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( zip_wi5053028342928268674_a_nat @ N2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( append338925788367110473_a_nat @ ( zip_wi5053028342928268674_a_nat @ N2 @ Xs ) @ ( zip_wi5053028342928268674_a_nat @ ( plus_plus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).

% zip_with_index_from_append
thf(fact_964_zip__with__index__from__append,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( zip_wi3407404960461264653om_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( zip_wi3407404960461264653om_nat @ N2 @ Xs ) @ ( zip_wi3407404960461264653om_nat @ ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% zip_with_index_from_append
thf(fact_965_append__butlast__last__id,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_966_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X4: nat,Xs2: list_nat] :
            ( ( P @ Xs2 )
           => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_967_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys4: list_nat,Y3: nat] :
            ( Xs
           != ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_968_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X @ Xs )
            = Zs ) )
        | ? [Ys6: list_nat] :
            ( ( ( cons_nat @ X @ Ys6 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_969_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X @ Xs ) ) )
        | ? [Ys6: list_nat] :
            ( ( Ys
              = ( cons_nat @ X @ Ys6 ) )
            & ( ( append_nat @ Ys6 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_970_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_971_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_972_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_973_append_Oleft__neutral,axiom,
    ! [A3: list_nat] :
      ( ( append_nat @ nil_nat @ A3 )
      = A3 ) ).

% append.left_neutral
thf(fact_974_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_975_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_976_map__eq__append__conv,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
      ( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
        = ( append_Sum_sum_a_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs3: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_a_Sum_sum_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_Sum_sum_a_nat @ F @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_977_map__eq__append__conv,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_a,Zs: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_978_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_979_append__eq__map__conv,axiom,
    ! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
        = ( map_a_Sum_sum_a_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs3: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_a_Sum_sum_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_Sum_sum_a_nat @ F @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_980_append__eq__map__conv,axiom,
    ! [Ys: list_a,Zs: list_a,F: nat > a,Xs: list_nat] :
      ( ( ( append_a @ Ys @ Zs )
        = ( map_nat_a @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_981_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_982_enumerate__append__eq,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( enumer3164015132978342500_a_nat @ N2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( append338925788367110473_a_nat @ ( enumer3164015132978342500_a_nat @ N2 @ Xs ) @ ( enumer3164015132978342500_a_nat @ ( plus_plus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_983_enumerate__append__eq,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N2 @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_984_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_985_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_986_last__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Ys ) ) ) ) ).

% last_append
thf(fact_987_longest__common__suffix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ss: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ( Xs
        = ( append_nat @ Xs4 @ Ss ) )
      & ( Ys
        = ( append_nat @ Ys5 @ Ss ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys5 = nil_nat )
        | ( ( last_nat @ Xs4 )
         != ( last_nat @ Ys5 ) ) ) ) ).

% longest_common_suffix
thf(fact_988_butlast__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( butlast_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_989_lex__append__leftI,axiom,
    ! [Ys: list_nat,Zs: list_nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) ) ) ).

% lex_append_leftI
thf(fact_990_same__length__different,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s5283204784079214577_a_nat @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
       => ? [Pre: list_Sum_sum_a_nat,X4: sum_sum_a_nat,Xs4: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
            ( ( X4 != Y3 )
            & ( Xs
              = ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X4 @ nil_Sum_sum_a_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y3 @ nil_Sum_sum_a_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_991_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X4: nat,Xs4: list_nat,Y3: nat,Ys5: list_nat] :
            ( ( X4 != Y3 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_992_list__induct2__rev,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
       => ( ! [X4: sum_sum_a_nat,Y3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Ys4: list_Sum_sum_a_nat] :
              ( ( P @ Xs2 @ Ys4 )
             => ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X4 @ nil_Sum_sum_a_nat ) ) @ ( append_Sum_sum_a_nat @ Ys4 @ ( cons_Sum_sum_a_nat @ Y3 @ nil_Sum_sum_a_nat ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2_rev
thf(fact_993_list__induct2__rev,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat,P: list_Sum_sum_a_nat > list_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_Sum_sum_a_nat @ nil_nat )
       => ( ! [X4: sum_sum_a_nat,Y3: nat,Xs2: list_Sum_sum_a_nat,Ys4: list_nat] :
              ( ( P @ Xs2 @ Ys4 )
             => ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X4 @ nil_Sum_sum_a_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2_rev
thf(fact_994_list__induct2__rev,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_Sum_sum_a_nat )
       => ( ! [X4: nat,Y3: sum_sum_a_nat,Xs2: list_nat,Ys4: list_Sum_sum_a_nat] :
              ( ( P @ Xs2 @ Ys4 )
             => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) @ ( append_Sum_sum_a_nat @ Ys4 @ ( cons_Sum_sum_a_nat @ Y3 @ nil_Sum_sum_a_nat ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2_rev
thf(fact_995_list__induct2__rev,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Y3: nat,Xs2: list_nat,Ys4: list_nat] :
              ( ( P @ Xs2 @ Ys4 )
             => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2_rev
thf(fact_996_lex__append__left__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).

% lex_append_left_iff
thf(fact_997_lex__append__left__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
        = ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ).

% lex_append_left_iff
thf(fact_998_lex__append__left__iff,axiom,
    ! [R2: set_Pr3551440518166979541_a_nat,Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat,Zs: list_P4848832406072927130_a_nat] :
      ( ! [X4: produc5986856040925105290_a_nat] :
          ~ ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X4 @ X4 ) @ R2 )
     => ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ ( append6733844360078942767_a_nat @ Xs @ Ys ) @ ( append6733844360078942767_a_nat @ Xs @ Zs ) ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) )
        = ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Ys @ Zs ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) ) ) ) ).

% lex_append_left_iff
thf(fact_999_lex__append__left__iff,axiom,
    ! [R2: set_Pr757995773253584751_a_nat,Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat,Zs: list_P2897000184160895271_a_nat] :
      ( ! [X4: produc8208987855206359447_a_nat] :
          ~ ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X4 @ X4 ) @ R2 )
     => ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ ( append5059809803815548476_a_nat @ Xs @ Ys ) @ ( append5059809803815548476_a_nat @ Xs @ Zs ) ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) )
        = ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Ys @ Zs ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) ) ) ) ).

% lex_append_left_iff
thf(fact_1000_lex__append__leftD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).

% lex_append_leftD
thf(fact_1001_lex__append__leftD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ).

% lex_append_leftD
thf(fact_1002_lex__append__leftD,axiom,
    ! [R2: set_Pr3551440518166979541_a_nat,Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat,Zs: list_P4848832406072927130_a_nat] :
      ( ! [X4: produc5986856040925105290_a_nat] :
          ~ ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ X4 @ X4 ) @ R2 )
     => ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ ( append6733844360078942767_a_nat @ Xs @ Ys ) @ ( append6733844360078942767_a_nat @ Xs @ Zs ) ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) )
       => ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Ys @ Zs ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) ) ) ) ).

% lex_append_leftD
thf(fact_1003_lex__append__leftD,axiom,
    ! [R2: set_Pr757995773253584751_a_nat,Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat,Zs: list_P2897000184160895271_a_nat] :
      ( ! [X4: produc8208987855206359447_a_nat] :
          ~ ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ X4 @ X4 ) @ R2 )
     => ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ ( append5059809803815548476_a_nat @ Xs @ Ys ) @ ( append5059809803815548476_a_nat @ Xs @ Zs ) ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) )
       => ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Ys @ Zs ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) ) ) ) ).

% lex_append_leftD
thf(fact_1004_lex__append__rightI,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat,Vs: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R2 ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Vs )
          = ( size_s5283204784079214577_a_nat @ Us ) )
       => ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Us ) @ ( append_Sum_sum_a_nat @ Ys @ Vs ) ) @ ( lex_Sum_sum_a_nat @ R2 ) ) ) ) ).

% lex_append_rightI
thf(fact_1005_lex__append__rightI,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R2 ) ) ) ) ).

% lex_append_rightI
thf(fact_1006_lenlex__append1,axiom,
    ! [Us: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Vs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Us @ Xs ) @ ( lenlex_Sum_sum_a_nat @ R ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Vs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
       => ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Us @ Vs ) @ ( append_Sum_sum_a_nat @ Xs @ Ys ) ) @ ( lenlex_Sum_sum_a_nat @ R ) ) ) ) ).

% lenlex_append1
thf(fact_1007_lenlex__append1,axiom,
    ! [Us: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Ys ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R ) ) ) ) ).

% lenlex_append1
thf(fact_1008_length__Suc__conv__rev,axiom,
    ! [Xs: list_Sum_sum_a_nat,N2: nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( suc @ N2 ) )
      = ( ? [Y10: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ Y10 @ nil_Sum_sum_a_nat ) ) )
            & ( ( size_s5283204784079214577_a_nat @ Ys2 )
              = N2 ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1009_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N2 ) )
      = ( ? [Y10: nat,Ys2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ Y10 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N2 ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1010_length__append__singleton,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
      = ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_1011_length__append__singleton,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_1012_nth__append,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N2 )
          = ( nth_Sum_sum_a_nat @ Xs @ N2 ) ) )
      & ( ~ ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N2 )
          = ( nth_Sum_sum_a_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1013_nth__append,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N2 )
          = ( nth_nat @ Xs @ N2 ) ) )
      & ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N2 )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1014_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = Ys )
      = ( ( Ys != nil_nat )
        & ( ( butlast_nat @ Ys )
          = Xs )
        & ( ( last_nat @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1015_take__Suc__conv__app__nth,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( take_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ nil_Sum_sum_a_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1016_take__Suc__conv__app__nth,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ ( suc @ I2 ) @ Xs )
        = ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I2 ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1017_take__Suc__Cons,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( take_nat @ ( suc @ N2 ) @ ( cons_nat @ X @ Xs ) )
      = ( cons_nat @ X @ ( take_nat @ N2 @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1018_take__eq__Nil2,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( nil_nat
        = ( take_nat @ N2 @ Xs ) )
      = ( ( N2 = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil2
thf(fact_1019_take__eq__Nil,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ( take_nat @ N2 @ Xs )
        = nil_nat )
      = ( ( N2 = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil
thf(fact_1020_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs3: list_nat] : nil_nat ) ) ).

% take0
thf(fact_1021_nth__take,axiom,
    ! [I2: nat,N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( nth_nat @ ( take_nat @ N2 @ Xs ) @ I2 )
        = ( nth_nat @ Xs @ I2 ) ) ) ).

% nth_take
thf(fact_1022_take__append,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ N2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N2 @ Xs ) @ ( take_Sum_sum_a_nat @ ( minus_minus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1023_take__append,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( take_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1024_take__Cons__numeral,axiom,
    ! [V3: num,X: nat,Xs: list_nat] :
      ( ( take_nat @ ( numeral_numeral_nat @ V3 ) @ ( cons_nat @ X @ Xs ) )
      = ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V3 ) @ one_one_nat ) @ Xs ) ) ) ).

% take_Cons_numeral
thf(fact_1025_take__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ! [I: nat] :
          ( ( take_nat @ I @ Xs )
          = ( take_nat @ I @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_1026_fo__nmlz__take,axiom,
    ! [N2: nat,AD2: set_a,Xs: list_Sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ N2 @ ( fo_nmlz_a @ AD2 @ Xs ) )
      = ( fo_nmlz_a @ AD2 @ ( take_Sum_sum_a_nat @ N2 @ Xs ) ) ) ).

% fo_nmlz_take
thf(fact_1027_take__map,axiom,
    ! [N2: nat,F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( take_Sum_sum_a_nat @ N2 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
      = ( map_a_Sum_sum_a_nat @ F @ ( take_a @ N2 @ Xs ) ) ) ).

% take_map
thf(fact_1028_take__map,axiom,
    ! [N2: nat,F: nat > a,Xs: list_nat] :
      ( ( take_a @ N2 @ ( map_nat_a @ F @ Xs ) )
      = ( map_nat_a @ F @ ( take_nat @ N2 @ Xs ) ) ) ).

% take_map
thf(fact_1029_take__map,axiom,
    ! [N2: nat,F: nat > nat,Xs: list_nat] :
      ( ( take_nat @ N2 @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( take_nat @ N2 @ Xs ) ) ) ).

% take_map
thf(fact_1030_take__Nil,axiom,
    ! [N2: nat] :
      ( ( take_nat @ N2 @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_1031_take__zip,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( take_P2173866234530122223at_nat @ N2 @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( take_nat @ N2 @ Xs ) @ ( take_nat @ N2 @ Ys ) ) ) ).

% take_zip
thf(fact_1032_take__0,axiom,
    ! [Xs: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs )
      = nil_nat ) ).

% take_0
thf(fact_1033_take__butlast,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( take_Sum_sum_a_nat @ N2 @ ( butlas5768530507476509265_a_nat @ Xs ) )
        = ( take_Sum_sum_a_nat @ N2 @ Xs ) ) ) ).

% take_butlast
thf(fact_1034_take__butlast,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ N2 @ ( butlast_nat @ Xs ) )
        = ( take_nat @ N2 @ Xs ) ) ) ).

% take_butlast
thf(fact_1035_butlast__conv__take,axiom,
    ( butlas5768530507476509265_a_nat
    = ( ^ [Xs3: list_Sum_sum_a_nat] : ( take_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).

% butlast_conv_take
thf(fact_1036_butlast__conv__take,axiom,
    ( butlast_nat
    = ( ^ [Xs3: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).

% butlast_conv_take
thf(fact_1037_take__Cons_H,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( take_nat @ N2 @ ( cons_nat @ X @ Xs ) )
          = nil_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( take_nat @ N2 @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ) ).

% take_Cons'
thf(fact_1038_lex__take__index,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( ( take_P2173866234530122223at_nat @ I @ Xs )
                  = ( take_P2173866234530122223at_nat @ I @ Ys ) )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1039_lex__take__index,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat,R2: set_Pr3551440518166979541_a_nat] :
      ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Xs @ Ys ) @ ( lex_Pr6397474375144773024_a_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3272666785928748590_a_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s3272666785928748590_a_nat @ Ys ) )
             => ( ( ( take_P3893490923519127324_a_nat @ I @ Xs )
                  = ( take_P3893490923519127324_a_nat @ I @ Ys ) )
               => ~ ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( nth_Pr4499681824854399811_a_nat @ Xs @ I ) @ ( nth_Pr4499681824854399811_a_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1040_lex__take__index,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat,R2: set_Pr757995773253584751_a_nat] :
      ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Xs @ Ys ) @ ( lex_Pr7015030108783135917_a_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3428558496044824507_a_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s3428558496044824507_a_nat @ Ys ) )
             => ( ( ( take_P8404481765139225129_a_nat @ I @ Xs )
                  = ( take_P8404481765139225129_a_nat @ I @ Ys ) )
               => ~ ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( nth_Pr7304172183617465168_a_nat @ Xs @ I ) @ ( nth_Pr7304172183617465168_a_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1041_lex__take__index,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Ys ) )
             => ( ( ( take_Sum_sum_a_nat @ I @ Xs )
                  = ( take_Sum_sum_a_nat @ I @ Ys ) )
               => ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( nth_Sum_sum_a_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1042_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I @ Xs )
                  = ( take_nat @ I @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1043_id__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( Xs
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1044_id__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( Xs
        = ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I2 ) @ ( drop_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1045_rem__nth__take__drop,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( rem_nt658808235856662061_a_nat @ I2 @ Xs )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ).

% rem_nth_take_drop
thf(fact_1046_rem__nth__take__drop,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( rem_nth_nat @ I2 @ Xs )
        = ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( drop_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ).

% rem_nth_take_drop
thf(fact_1047_drop0,axiom,
    ( ( drop_nat @ zero_zero_nat )
    = ( ^ [X5: list_nat] : X5 ) ) ).

% drop0
thf(fact_1048_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I2 @ J ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_1049_drop__drop,axiom,
    ! [N2: nat,M2: nat,Xs: list_nat] :
      ( ( drop_nat @ N2 @ ( drop_nat @ M2 @ Xs ) )
      = ( drop_nat @ ( plus_plus_nat @ N2 @ M2 ) @ Xs ) ) ).

% drop_drop
thf(fact_1050_drop__Suc__Cons,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( drop_nat @ ( suc @ N2 ) @ ( cons_nat @ X @ Xs ) )
      = ( drop_nat @ N2 @ Xs ) ) ).

% drop_Suc_Cons
thf(fact_1051_length__drop,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( drop_Sum_sum_a_nat @ N2 @ Xs ) )
      = ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N2 ) ) ).

% length_drop
thf(fact_1052_length__drop,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N2 @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).

% length_drop
thf(fact_1053_append__take__drop__id,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( drop_nat @ N2 @ Xs ) )
      = Xs ) ).

% append_take_drop_id
thf(fact_1054_drop__append,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( drop_Sum_sum_a_nat @ N2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N2 @ Xs ) @ ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_1055_drop__append,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_nat @ N2 @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( drop_nat @ N2 @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_1056_last__drop,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( last_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N2 @ Xs ) )
        = ( last_Sum_sum_a_nat @ Xs ) ) ) ).

% last_drop
thf(fact_1057_last__drop,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( last_nat @ ( drop_nat @ N2 @ Xs ) )
        = ( last_nat @ Xs ) ) ) ).

% last_drop
thf(fact_1058_drop__Cons__numeral,axiom,
    ! [V3: num,X: nat,Xs: list_nat] :
      ( ( drop_nat @ ( numeral_numeral_nat @ V3 ) @ ( cons_nat @ X @ Xs ) )
      = ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V3 ) @ one_one_nat ) @ Xs ) ) ).

% drop_Cons_numeral
thf(fact_1059_drop__take,axiom,
    ! [N2: nat,M2: nat,Xs: list_nat] :
      ( ( drop_nat @ N2 @ ( take_nat @ M2 @ Xs ) )
      = ( take_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( drop_nat @ N2 @ Xs ) ) ) ).

% drop_take
thf(fact_1060_take__drop,axiom,
    ! [N2: nat,M2: nat,Xs: list_nat] :
      ( ( take_nat @ N2 @ ( drop_nat @ M2 @ Xs ) )
      = ( drop_nat @ M2 @ ( take_nat @ ( plus_plus_nat @ N2 @ M2 ) @ Xs ) ) ) ).

% take_drop
thf(fact_1061_nth__via__drop,axiom,
    ! [N2: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( drop_nat @ N2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ( ( nth_nat @ Xs @ N2 )
        = Y ) ) ).

% nth_via_drop
thf(fact_1062_drop__zip,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N2 @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N2 @ Xs ) @ ( drop_nat @ N2 @ Ys ) ) ) ).

% drop_zip
thf(fact_1063_drop__Nil,axiom,
    ! [N2: nat] :
      ( ( drop_nat @ N2 @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_1064_drop__map,axiom,
    ! [N2: nat,F: a > sum_sum_a_nat,Xs: list_a] :
      ( ( drop_Sum_sum_a_nat @ N2 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
      = ( map_a_Sum_sum_a_nat @ F @ ( drop_a @ N2 @ Xs ) ) ) ).

% drop_map
thf(fact_1065_drop__map,axiom,
    ! [N2: nat,F: nat > a,Xs: list_nat] :
      ( ( drop_a @ N2 @ ( map_nat_a @ F @ Xs ) )
      = ( map_nat_a @ F @ ( drop_nat @ N2 @ Xs ) ) ) ).

% drop_map
thf(fact_1066_drop__map,axiom,
    ! [N2: nat,F: nat > nat,Xs: list_nat] :
      ( ( drop_nat @ N2 @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( drop_nat @ N2 @ Xs ) ) ) ).

% drop_map
thf(fact_1067_drop__0,axiom,
    ! [Xs: list_nat] :
      ( ( drop_nat @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_1068_drop__butlast,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( drop_nat @ N2 @ ( butlast_nat @ Xs ) )
      = ( butlast_nat @ ( drop_nat @ N2 @ Xs ) ) ) ).

% drop_butlast
thf(fact_1069_append__eq__conv__conj,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
        = Zs )
      = ( ( Xs
          = ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) )
        & ( Ys
          = ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_1070_append__eq__conv__conj,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Zs )
      = ( ( Xs
          = ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
        & ( Ys
          = ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_1071_take__add,axiom,
    ! [I2: nat,J: nat,Xs: list_nat] :
      ( ( take_nat @ ( plus_plus_nat @ I2 @ J ) @ Xs )
      = ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( take_nat @ J @ ( drop_nat @ I2 @ Xs ) ) ) ) ).

% take_add
thf(fact_1072_drop__Cons_H,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( drop_nat @ N2 @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ Xs ) ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( drop_nat @ N2 @ ( cons_nat @ X @ Xs ) )
          = ( drop_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ).

% drop_Cons'
thf(fact_1073_Cons__nth__drop__Suc,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) )
        = ( drop_Sum_sum_a_nat @ I2 @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1074_Cons__nth__drop__Suc,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( cons_nat @ ( nth_nat @ Xs @ I2 ) @ ( drop_nat @ ( suc @ I2 ) @ Xs ) )
        = ( drop_nat @ I2 @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1075_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
      ( ( zip_na2013496608136855606_a_nat @ Xs @ ( append_Sum_sum_a_nat @ Ys @ Zs ) )
      = ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ ( take_nat @ ( size_s5283204784079214577_a_nat @ Ys ) @ Xs ) @ Ys ) @ ( zip_na2013496608136855606_a_nat @ ( drop_nat @ ( size_s5283204784079214577_a_nat @ Ys ) @ Xs ) @ Zs ) ) ) ).

% zip_append2
thf(fact_1076_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ Xs @ ( append_nat @ Ys @ Zs ) )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs ) @ Ys ) @ ( zip_nat_nat @ ( drop_nat @ ( size_size_list_nat @ Ys ) @ Xs ) @ Zs ) ) ) ).

% zip_append2
thf(fact_1077_zip__append1,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat] :
      ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ Zs )
      = ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ ( take_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) ) @ ( zip_Su6417478541797927256at_nat @ Ys @ ( drop_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_1078_zip__append1,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) @ ( zip_nat_nat @ Ys @ ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_1079_take__hd__drop,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N2 @ Xs ) @ ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N2 @ Xs ) ) @ nil_Sum_sum_a_nat ) )
        = ( take_Sum_sum_a_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_1080_take__hd__drop,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N2 @ Xs ) ) @ nil_nat ) )
        = ( take_nat @ ( suc @ N2 ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_1081_upd__conv__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,A3: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ A3 )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ A3 @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_1082_upd__conv__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_nat,A3: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ Xs @ I2 @ A3 )
        = ( append_nat @ ( take_nat @ I2 @ Xs ) @ ( cons_nat @ A3 @ ( drop_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_1083_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt
thf(fact_1084_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_1085_length__list__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_list_update
thf(fact_1086_length__list__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I2 @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_1087_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I2 != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1088_list__update__id,axiom,
    ! [Xs: list_nat,I2: nat] :
      ( ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ I2 ) )
      = Xs ) ).

% list_update_id
thf(fact_1089_hd__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
        = ( hd_nat @ Xs ) ) ) ).

% hd_append2
thf(fact_1090_drop__update__cancel,axiom,
    ! [N2: nat,M2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( ( drop_nat @ M2 @ ( list_update_nat @ Xs @ N2 @ X ) )
        = ( drop_nat @ M2 @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_1091_list__update__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) @ Y )
      = ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1092_list__update__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
      = ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1093_hd__take,axiom,
    ! [J: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ J )
     => ( ( hd_nat @ ( take_nat @ J @ Xs ) )
        = ( hd_nat @ Xs ) ) ) ).

% hd_take
thf(fact_1094_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1095_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1096_take__update__swap,axiom,
    ! [M2: nat,Xs: list_nat,N2: nat,X: nat] :
      ( ( take_nat @ M2 @ ( list_update_nat @ Xs @ N2 @ X ) )
      = ( list_update_nat @ ( take_nat @ M2 @ Xs ) @ N2 @ X ) ) ).

% take_update_swap
thf(fact_1097_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_1098_hd__zip,axiom,
    ! [Xs: list_set_a,Ys: list_P4848832406072927130_a_nat] :
      ( ( Xs != nil_set_a )
     => ( ( Ys != nil_Pr6508300198058928388_a_nat )
       => ( ( hd_Pro4640716029831176172_a_nat @ ( zip_se7211208273872771441_a_nat @ Xs @ Ys ) )
          = ( produc1626021218680649873_a_nat @ ( hd_set_a @ Xs ) @ ( hd_Pro8871465084263545823_a_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1099_hd__zip,axiom,
    ! [Xs: list_nat,Ys: list_s602657755630655083_a_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_se550621812924229483_a_nat )
       => ( ( hd_Pro8871465084263545823_a_nat @ ( zip_na7417237922546100252_a_nat @ Xs @ Ys ) )
          = ( produc4622690550020908860_a_nat @ ( hd_nat @ Xs ) @ ( hd_set3795686063217246800_a_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1100_hd__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
          = ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1101_hd__zip,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ( ( Ys != nil_Pr5478986624290739719at_nat )
       => ( ( hd_Pro8462011474880202578at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) )
          = ( produc6161850002892822231at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( hd_Pro3460610213475200108at_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1102_hd__zip,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat] :
      ( ( Xs != nil_Pr6508300198058928388_a_nat )
     => ( ( Ys != nil_Pr6508300198058928388_a_nat )
       => ( ( hd_Pro2145305463377675232_a_nat @ ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys ) )
          = ( produc5595853527594814125_a_nat @ ( hd_Pro8871465084263545823_a_nat @ Xs ) @ ( hd_Pro8871465084263545823_a_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1103_hd__zip,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat] :
      ( ( Xs != nil_Pr236201474699349009_a_nat )
     => ( ( Ys != nil_Pr236201474699349009_a_nat )
       => ( ( hd_Pro4156817257219938426_a_nat @ ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys ) )
          = ( produc5405793250563226055_a_nat @ ( hd_Pro4640716029831176172_a_nat @ Xs ) @ ( hd_Pro4640716029831176172_a_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1104_zip__update,axiom,
    ! [Xs: list_set_a,I2: nat,X: set_a,Ys: list_P4848832406072927130_a_nat,Y: produc5986856040925105290_a_nat] :
      ( ( zip_se7211208273872771441_a_nat @ ( list_update_set_a @ Xs @ I2 @ X ) @ ( list_u6615296262896687260_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u4161903484342371241_a_nat @ ( zip_se7211208273872771441_a_nat @ Xs @ Ys ) @ I2 @ ( produc1626021218680649873_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1105_zip__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat,Ys: list_s602657755630655083_a_nat,Y: set_li6526943997496501093_a_nat] :
      ( ( zip_na7417237922546100252_a_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ ( list_u3821106994787816915_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u6615296262896687260_a_nat @ ( zip_na7417237922546100252_a_nat @ Xs @ Ys ) @ I2 @ ( produc4622690550020908860_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1106_zip__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( zip_nat_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ ( list_update_nat @ Ys @ I2 @ Y ) )
      = ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I2 @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1107_zip__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I2: nat,X: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I2 @ X ) @ ( list_u6180841689913720943at_nat @ Ys @ I2 @ Y ) )
      = ( list_u5003261594476800725at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I2 @ ( produc6161850002892822231at_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1108_zip__update,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,I2: nat,X: produc5986856040925105290_a_nat,Ys: list_P4848832406072927130_a_nat,Y: produc5986856040925105290_a_nat] :
      ( ( zip_Pr1216249236586582477_a_nat @ ( list_u6615296262896687260_a_nat @ Xs @ I2 @ X ) @ ( list_u6615296262896687260_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u5984942667395118563_a_nat @ ( zip_Pr1216249236586582477_a_nat @ Xs @ Ys ) @ I2 @ ( produc5595853527594814125_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1109_zip__update,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,I2: nat,X: produc8208987855206359447_a_nat,Ys: list_P2897000184160895271_a_nat,Y: produc8208987855206359447_a_nat] :
      ( ( zip_Pr8897620143892512999_a_nat @ ( list_u4161903484342371241_a_nat @ Xs @ I2 @ X ) @ ( list_u4161903484342371241_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u5384133507522247805_a_nat @ ( zip_Pr8897620143892512999_a_nat @ Xs @ Ys ) @ I2 @ ( produc5405793250563226055_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1110_map__update,axiom,
    ! [F: a > sum_sum_a_nat,Xs: list_a,K: nat,Y: a] :
      ( ( map_a_Sum_sum_a_nat @ F @ ( list_update_a @ Xs @ K @ Y ) )
      = ( list_u9138855634547462509_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_1111_map__update,axiom,
    ! [F: nat > a,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_nat_a @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_update_a @ ( map_nat_a @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_1112_map__update,axiom,
    ! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_1113_list_Osel_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( hd_nat @ ( cons_nat @ X21 @ X222 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_1114_list__update__code_I1_J,axiom,
    ! [I2: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_1115_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V3: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ V3 )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_1116_list_Omap__sel_I1_J,axiom,
    ! [A3: list_a,F: a > sum_sum_a_nat] :
      ( ( A3 != nil_a )
     => ( ( hd_Sum_sum_a_nat @ ( map_a_Sum_sum_a_nat @ F @ A3 ) )
        = ( F @ ( hd_a @ A3 ) ) ) ) ).

% list.map_sel(1)
thf(fact_1117_list_Omap__sel_I1_J,axiom,
    ! [A3: list_nat,F: nat > a] :
      ( ( A3 != nil_nat )
     => ( ( hd_a @ ( map_nat_a @ F @ A3 ) )
        = ( F @ ( hd_nat @ A3 ) ) ) ) ).

% list.map_sel(1)
thf(fact_1118_list_Omap__sel_I1_J,axiom,
    ! [A3: list_nat,F: nat > nat] :
      ( ( A3 != nil_nat )
     => ( ( hd_nat @ ( map_nat_nat @ F @ A3 ) )
        = ( F @ ( hd_nat @ A3 ) ) ) ) ).

% list.map_sel(1)
thf(fact_1119_hd__map,axiom,
    ! [Xs: list_a,F: a > sum_sum_a_nat] :
      ( ( Xs != nil_a )
     => ( ( hd_Sum_sum_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
        = ( F @ ( hd_a @ Xs ) ) ) ) ).

% hd_map
thf(fact_1120_hd__map,axiom,
    ! [Xs: list_nat,F: nat > a] :
      ( ( Xs != nil_nat )
     => ( ( hd_a @ ( map_nat_a @ F @ Xs ) )
        = ( F @ ( hd_nat @ Xs ) ) ) ) ).

% hd_map
thf(fact_1121_hd__map,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ( Xs != nil_nat )
     => ( ( hd_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( F @ ( hd_nat @ Xs ) ) ) ) ).

% hd_map
thf(fact_1122_hd__Nil__eq__last,axiom,
    ( ( hd_nat @ nil_nat )
    = ( last_nat @ nil_nat ) ) ).

% hd_Nil_eq_last
thf(fact_1123_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I2: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I2 ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs @ I2 @ Y ) ) ) ).

% list_update_code(3)
thf(fact_1124_hd__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( Xs = nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
          = ( hd_nat @ Ys ) ) )
      & ( ( Xs != nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
          = ( hd_nat @ Xs ) ) ) ) ).

% hd_append
thf(fact_1125_longest__common__prefix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ps: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ( Xs
        = ( append_nat @ Ps @ Xs4 ) )
      & ( Ys
        = ( append_nat @ Ps @ Ys5 ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys5 = nil_nat )
        | ( ( hd_nat @ Xs4 )
         != ( hd_nat @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_1126_hd__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( hd_nat @ Xs )
        = ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_1127_list__update__append1,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ I2 @ X )
        = ( append_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1128_list__update__append1,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I2 @ X )
        = ( append_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1129_nth__list__update,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,J: nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ( I2 = J )
         => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ J )
            = X ) )
        & ( ( I2 != J )
         => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ J )
            = ( nth_Sum_sum_a_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1130_nth__list__update,axiom,
    ! [I2: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I2 = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
            = X ) )
        & ( ( I2 != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1131_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X )
          = Xs )
        = ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1132_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I2 @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1133_hd__drop__conv__nth,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( hd_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N2 @ Xs ) )
        = ( nth_Sum_sum_a_nat @ Xs @ N2 ) ) ) ).

% hd_drop_conv_nth
thf(fact_1134_hd__drop__conv__nth,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( hd_nat @ ( drop_nat @ N2 @ Xs ) )
        = ( nth_nat @ Xs @ N2 ) ) ) ).

% hd_drop_conv_nth
thf(fact_1135_list__update__append,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N2 @ X )
          = ( append_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ N2 @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N2 @ X )
          = ( append_Sum_sum_a_nat @ Xs @ ( list_u9138855634547462509_a_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1136_list__update__append,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N2 @ X )
          = ( append_nat @ ( list_update_nat @ Xs @ N2 @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N2 @ X )
          = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1137_butlast__list__update,axiom,
    ! [K: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ( K
          = ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlas5768530507476509265_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
          = ( butlas5768530507476509265_a_nat @ Xs ) ) )
      & ( ( K
         != ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlas5768530507476509265_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
          = ( list_u9138855634547462509_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) @ K @ X ) ) ) ) ).

% butlast_list_update
thf(fact_1138_butlast__list__update,axiom,
    ! [K: nat,Xs: list_nat,X: nat] :
      ( ( ( K
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
          = ( butlast_nat @ Xs ) ) )
      & ( ( K
         != ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
          = ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X ) ) ) ) ).

% butlast_list_update
thf(fact_1139_last__list__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,K: nat,X: sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( ( K
            = ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
         => ( ( last_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
            = X ) )
        & ( ( K
           != ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
         => ( ( last_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
            = ( last_Sum_sum_a_nat @ Xs ) ) ) ) ) ).

% last_list_update
thf(fact_1140_last__list__update,axiom,
    ! [Xs: list_nat,K: nat,X: nat] :
      ( ( Xs != nil_nat )
     => ( ( ( K
            = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
            = X ) )
        & ( ( K
           != ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
            = ( last_nat @ Xs ) ) ) ) ) ).

% last_list_update
thf(fact_1141_add__nth__take__drop,axiom,
    ! [I2: nat,Zs: list_Sum_sum_a_nat,V3: sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Zs ) )
     => ( ( add_nt4212672348507122516_a_nat @ I2 @ V3 @ Zs )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Zs ) @ ( cons_Sum_sum_a_nat @ V3 @ ( drop_Sum_sum_a_nat @ I2 @ Zs ) ) ) ) ) ).

% add_nth_take_drop
thf(fact_1142_add__nth__take__drop,axiom,
    ! [I2: nat,Zs: list_nat,V3: nat] :
      ( ( ord_less_eq_nat @ I2 @ ( size_size_list_nat @ Zs ) )
     => ( ( add_nth_nat @ I2 @ V3 @ Zs )
        = ( append_nat @ ( take_nat @ I2 @ Zs ) @ ( cons_nat @ V3 @ ( drop_nat @ I2 @ Zs ) ) ) ) ) ).

% add_nth_take_drop
thf(fact_1143_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
      = ( ? [Y10: product_prod_nat_nat,N6: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N6 ) @ Y10 ) @ R2 )
            & ( ord_less_nat @ N6 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( Ys
              = ( list_u6180841689913720943at_nat @ Xs @ N6 @ Y10 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1144_listrel1__iff__update,axiom,
    ! [Xs: list_P4848832406072927130_a_nat,Ys: list_P4848832406072927130_a_nat,R2: set_Pr3551440518166979541_a_nat] :
      ( ( member2678617866712254398_a_nat @ ( produc6983730236908500813_a_nat @ Xs @ Ys ) @ ( listre5072895511082834923_a_nat @ R2 ) )
      = ( ? [Y10: produc5986856040925105290_a_nat,N6: nat] :
            ( ( member8591614759246846750_a_nat @ ( produc5595853527594814125_a_nat @ ( nth_Pr4499681824854399811_a_nat @ Xs @ N6 ) @ Y10 ) @ R2 )
            & ( ord_less_nat @ N6 @ ( size_s3272666785928748590_a_nat @ Xs ) )
            & ( Ys
              = ( list_u6615296262896687260_a_nat @ Xs @ N6 @ Y10 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1145_listrel1__iff__update,axiom,
    ! [Xs: list_P2897000184160895271_a_nat,Ys: list_P2897000184160895271_a_nat,R2: set_Pr757995773253584751_a_nat] :
      ( ( member5130819753380816600_a_nat @ ( produc3731672623177972967_a_nat @ Xs @ Ys ) @ ( listre1574470056913609720_a_nat @ R2 ) )
      = ( ? [Y10: produc8208987855206359447_a_nat,N6: nat] :
            ( ( member8751153172265411512_a_nat @ ( produc5405793250563226055_a_nat @ ( nth_Pr7304172183617465168_a_nat @ Xs @ N6 ) @ Y10 ) @ R2 )
            & ( ord_less_nat @ N6 @ ( size_s3428558496044824507_a_nat @ Xs ) )
            & ( Ys
              = ( list_u4161903484342371241_a_nat @ Xs @ N6 @ Y10 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1146_listrel1__iff__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R2 ) )
      = ( ? [Y10: sum_sum_a_nat,N6: nat] :
            ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N6 ) @ Y10 ) @ R2 )
            & ( ord_less_nat @ N6 @ ( size_s5283204784079214577_a_nat @ Xs ) )
            & ( Ys
              = ( list_u9138855634547462509_a_nat @ Xs @ N6 @ Y10 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1147_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y10: nat,N6: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N6 ) @ Y10 ) @ R2 )
            & ( ord_less_nat @ N6 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N6 @ Y10 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1148_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1149_add__le__cancel__left,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
      = ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_1150_add__le__cancel__right,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
      = ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_1151_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1152_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).

% bot_nat_0.extremum
thf(fact_1153_Suc__le__mono,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% Suc_le_mono
thf(fact_1154_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1155_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_1156_add__le__same__cancel1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1157_add__le__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1158_le__add__same__cancel1,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_1159_le__add__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_1160_le__add__diff__inverse,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1161_le__add__diff__inverse2,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1162_diff__is__0__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1163_diff__is__0__eq_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1164_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1165_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1166_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1167_take__all__iff,axiom,
    ! [N2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( take_Sum_sum_a_nat @ N2 @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N2 ) ) ).

% take_all_iff
thf(fact_1168_take__all__iff,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ( take_nat @ N2 @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).

% take_all_iff
thf(fact_1169_take__all,axiom,
    ! [Xs: list_Sum_sum_a_nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N2 )
     => ( ( take_Sum_sum_a_nat @ N2 @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1170_take__all,axiom,
    ! [Xs: list_nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 )
     => ( ( take_nat @ N2 @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1171_take__upt,axiom,
    ! [I2: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M2 ) @ N2 )
     => ( ( take_nat @ M2 @ ( upt @ I2 @ N2 ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M2 ) ) ) ) ).

% take_upt
thf(fact_1172_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( upt @ I2 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1173_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1174_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1175_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( upt @ I2 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1176_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K2 )
               => ~ ( P @ I5 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1177_le__imp__less__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_1178_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1179_less__Suc__eq__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_1180_le__less__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
        = ( N2 = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1181_Suc__le__lessD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_le_lessD
thf(fact_1182_inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ J )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ ( suc @ N ) )
                 => ( P @ N ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% inc_induct
thf(fact_1183_dec__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ I2 )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ N )
                 => ( P @ ( suc @ N ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1184_Suc__le__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_le_eq
thf(fact_1185_Suc__leI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_leI
thf(fact_1186_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_nat @ M3 @ N )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1187_Suc__diff__le,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_1188_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M2 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_1189_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C2 @ A3 )
       => ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1190_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).

% le_diff_conv
thf(fact_1191_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1192_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1193_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1194_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( minus_minus_nat @ J @ I2 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1195_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1196_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1197_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1198_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1199_Suc__leD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% Suc_leD
thf(fact_1200_le__SucE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( M2
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_1201_le__SucI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_1202_Suc__le__D,axiom,
    ! [N2: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
     => ? [M3: nat] :
          ( M7
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_1203_le__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M2 @ N2 )
        | ( M2
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_1204_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_1205_not__less__eq__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1206_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
             => ( P @ M4 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_1207_nat__induct__at__least,axiom,
    ! [M2: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( P @ M2 )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_1208_transitive__stepwise__le,axiom,
    ! [M2: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
           => ( R @ M2 @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1209_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N6: nat] :
          ( ( ord_less_eq_nat @ M5 @ N6 )
          & ( M5 != N6 ) ) ) ) ).

% nat_less_le
thf(fact_1210_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_1211_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N6: nat] :
          ( ( ord_less_nat @ M5 @ N6 )
          | ( M5 = N6 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1212_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_1213_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_1214_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J: nat] :
      ( ! [I: nat,J2: nat] :
          ( ( ord_less_nat @ I @ J2 )
         => ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1215_add__leE,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M2 @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_1216_le__add1,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).

% le_add1
thf(fact_1217_le__add2,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).

% le_add2
thf(fact_1218_add__leD1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% add_leD1
thf(fact_1219_add__leD2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_1220_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N: nat] :
          ( L
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_1221_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_1222_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_1223_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1224_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_1225_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N6: nat] :
        ? [K3: nat] :
          ( N6
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1226_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1227_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ C2 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% le_diff_iff'
thf(fact_1228_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_1229_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_1230_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1231_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1232_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1233_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y11: nat] :
                ( ( P @ Y11 )
               => ( ord_less_eq_nat @ Y11 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1234_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_1235_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_1236_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_1237_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_1238_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_1239_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1240_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M5: nat] :
        ! [X5: nat] :
          ( ( member_nat @ X5 @ N4 )
         => ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1241_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_eq_nat @ I5 @ K2 )
               => ~ ( P @ I5 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1242_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1243_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_1244_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1245_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member408289922725080238_a_nat @ vs @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ ( fo_rep_a_nat @ ( produc1626021218680649873_a_nat @ ad @ ( produc4622690550020908860_a_nat @ n @ x ) ) ) ) ) ).

%------------------------------------------------------------------------------